Quantum State Transfer with Spin Chains
Quantum State Transfer with Spin Chains
Daniel Klaus Burgarth
A thesis submitted to the University of London
for the degree of Do
tor of Philosophy
Department of Physi
s and Astronomy
University College London
De
ember 2006
http://arxiv.org/abs/0704.1309v1
De
laration
I, Daniel Klaus Burgarth, 
on�rm that the work presented in this thesis is my own.
Where information has been derived from other sour
es, I 
on�rm that this has been
indi
ated in the thesis.
Abstra
t
In the last few de
ades the idea 
ame up that by making use of the superposition
prin
iple from Quantum Me
hani
s, one 
an pro
ess information in a new and mu
h
faster way. Hen
e a new �eld of information te
hnology, QIT (Quantum Information
Te
hnology), has emerged. From a physi
s point of view it is important to �nd ways of
implementing these new methods in real systems. One of the most basi
 tasks required
for QIT is the ability to 
onne
t di�erent 
omponents of a Quantum Computer by
quantum wires that obey the superposition prin
iple. Sin
e superpositions 
an be
very sensitive to noise this turns out to be already quite di�
ult. Re
ently, it was
suggested to use 
hains of permanently 
oupled spin-1/2 parti
les (quantum 
hains)
for this purpose. They have the advantage that no external 
ontrol along the wire is
required during the transport of information, whi
h makes it possible to isolate the
wire from sour
es of noise. The purpose of this thesis is to develop and investigate
advan
ed s
hemes for using quantum 
hains as wires. We �rst give an introdu
tion to
basi
 quantum state transfer and review existing advan
ed s
hemes by other authors.
We then introdu
e two new methods whi
h were 
reated as a part of this thesis. First,
we show how the �delity of transfer 
an be made perfe
t by performing measurements
at the re
eiving end of the 
hain. Then we introdu
e a s
heme whi
h is based on
performing unitary operations at the end of the 
hain. We generalise both methods
and dis
uss them from the more fundamental point of view of mixing properties of a
quantum 
hannel. Finally, we study the e�e
ts of a non-Markovian environment on
quantum state transfer.
A
knowledgements
Most of all, I would like to thank my supervisor Sougato Bose for mu
h inspira-
tion and advi
e. I am very grateful for many inspiring and fruitful dis
ussions and

ollaborations with Vittorio Giovannetti, and with Floor Paauw, Christoph Bruder,
Jason Twamley, Andreas Bu
hleitner and Vladimir Korepin. Furthermore I would
like to thank all my tea
hers and those who have guided and motivated me along my
journey through physi
s, in
luding Heinz-Peter Breuer, Fran
es
o Petru

ione, Lewis
Ryder, John Strange, Werner Riegler, Carsten S
huldt and Rolf Bussmann. I a
knowl-
edge �nan
ial support by the UK Engineering and Physi
al S
ien
es Resear
h Coun
il
through the grant GR/S62796/01. Finally I would like to thank my parents for their
loving support.
Notation
X,Y,Z Pauli matri
es
Xn, Yn, Zn Pauli matri
es a
ting on the Hilbert-spa
e of qubit n
|0〉, |1〉 Single qubit state in the 
anoni
al basis
|0〉 Quantum 
hain in the produ
t state |0〉 ⊗ · · · ⊗ |0〉
|n〉 �Single ex
itation� state Xn|0〉
TrX Partial tra
e over subsystem X
|| . . . || Eu
lidean ve
tor norm
|| . . . ||1 Tra
e norm
|| . . . ||2 Eu
lidean matrix norm
We also use the following graphi
al representation:
|n〉 ≡
|0〉 ≡
|0〉 ≡
|1〉 ≡
nth qubit
|ψ〉 = α|0〉 + β|1〉 ≡
controlled region:
quantum gates
and measurements
uncontrolled region
coupling
receiver
("Bob")
sender
("Alice")
Contents
1 Introdu
tion 9
1.1 Quantum Computation and Quantum Information . . . . . . . . . . . 10
1.2 Quantum state transfer along short distan
es . . . . . . . . . . . . . . 11
1.3 Implementations and experiments . . . . . . . . . . . . . . . . . . . . . 15
1.4 Basi
 
ommuni
ation proto
ol . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Initialisation and end-gates . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Transfer fun
tions . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 Heisenberg Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 20
1.4.5 Dynami
 and Dispersion . . . . . . . . . . . . . . . . . . . . . . 21
1.4.6 How high should p(t) be? . . . . . . . . . . . . . . . . . . . . . 27
1.5 Advan
ed 
ommuni
ation proto
ols . . . . . . . . . . . . . . . . . . . . 29
1.5.1 Engineered Hamiltonians . . . . . . . . . . . . . . . . . . . . . 29
1.5.2 Weakly 
oupled sender and re
eiver . . . . . . . . . . . . . . . . 29
1.5.3 En
oding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.4 Time-dependent 
ontrol . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Motivation and outline of this work . . . . . . . . . . . . . . . . . . . . 31
2 Dual Rail en
oding 34
2.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 S
heme for 
on
lusive transfer . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Arbitrarily perfe
t state transfer . . . . . . . . . . . . . . . . . . . . . 38
2.4 Estimation of the time-s
ale the transfer . . . . . . . . . . . . . . . . . 40
2.5 De
oheren
e and imperfe
tions . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Disordered 
hains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.7 Con
lusive transfer in the presen
e of disorder . . . . . . . . . . . . . . 45
2.8 Arbitrarily perfe
t transfer in the presen
e of disorder . . . . . . . . . 48
2.9 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.10 Numeri
al Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Contents
2.11 Coupled 
hains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.12 Con
lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Multi Rail en
oding 58
3.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 E�
ient en
oding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Perfe
t transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Convergen
e theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Quantum 
hains with nearest-neighbour intera
tions . . . . . . . . . . 69
3.7 Comparison with Dual Rail . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8 Con
lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Ergodi
ity and mixing 72
4.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Topologi
al ba
kground . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Generalised Lyapunov Theorem . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Topologi
al spa
es . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Metri
 spa
es . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.1 Mixing 
riteria for Quantum Channels . . . . . . . . . . . . . . 84
4.4.2 Beyond the density matrix operator spa
e: spe
tral properties . 86
4.4.3 Ergodi
 
hannels with pure �xed points . . . . . . . . . . . . . 88
4.5 Con
lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Read and write a

ess by lo
al 
ontrol 93
5.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Proto
ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 De
omposition equations . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Coding transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Fidelities for reading and writing . . . . . . . . . . . . . . . . . . . . . 99
5.6 Appli
ation to spin 
hain 
ommuni
ation . . . . . . . . . . . . . . . . . 101
5.7 Con
lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 A valve for probability amplitude 104
6.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 Arbitrarily Perfe
t State Transfer . . . . . . . . . . . . . . . . . . . . . 104
6.3 Pra
ti
al Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Contents
6.4 Con
lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 External noise 110
7.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4 Con
lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8 Con
lusion and outlook 120
List of Figures 121
List of Tables 126
Bibliography 127
Index 141
1 Introdu
tion
The Hilbert spa
e that 
ontains the states of quantum me
hani
al obje
ts is huge,
s
aling exponentially with the number of parti
les des
ribed. In 1982, Ri
hard Feyn-
man suggested to make use of this as a resour
e for simulating quantum me
hani
s in a
quantum 
omputer, i.e. a devi
e where the physi
al intera
tion 
ould be �programmed�
to yield a spe
i�
 Hamiltonian. This has led to the new �elds of Quantum Computa-
tion and Quantum Information. A quantum 
omputer 
an solve questions one 
ould
never imagine to solve using an ordinary 
omputer. For example, it 
an fa
torise
large numbers into primes e�
iently, a task of greatest importan
e for 
ryptography.
It may thus be a surprise that more than twenty years after the initial ideas, these
devi
es still haven't been built or only in ridi
ulously small size. The largest quantum

omputer so far 
an only solve problems that any 
hild 
ould solve within se
onds.
A 
loser look reveals that the main problem in the realisation of quantum 
omput-
ers is the �programming�, i.e. the design of a spe
i�
 (time-dependent) Hamiltonian,
usually des
ribed as a set of dis
rete unitary gates. This turns out to be extremely
di�
ult be
ause we need to 
onne
t mi
ros
opi
 obje
ts (those behaving quantum
me
hani
ally) with ma
ros
opi
 devi
es that 
ontrol the mi
ros
opi
 behaviour. Even
if one manages to �nd a link between the mi
ro- and the ma
ros
opi
 world, su
h as
laser pulses and ele
tri
 or magneti
 �elds, then the 
onne
tion introdu
es not only

ontrol but also noise (dissipation and de
oheren
e) to the mi
ros
opi
 system, and
its quantum behaviour is diminished.
The vision of this thesis is to develop theoreti
al methods narrowing the gap between
what is imagined theoreti
ally and what 
an be done experimentally. As a method
we 
onsider 
hains (or more general graphs) of permanently 
oupled quantum systems.
This idea has been originally put forward by S. Bose for the spe
i�
 task of quantum

ommuni
ation [1℄. Due to the permanent 
oupling, these devi
es 
an in prin
iple be
built in su
h a way that they don't require external 
ontrol to perform their tasks,
just like a me
hani
al 
lo
kwork. This also over
omes the problem of de
oheren
e as
they 
an be separated from any sour
e of noise. Unfortunately, most s
hemes that
have been developed so far still require external 
ontrol, though mu
h less than an
�ordinary� quantum 
omputer. Furthermore, internal dispersion in these devi
es is
1 Introdu
tion
leading to a de
rease of their �delity. A third problem is, that for building these
devi
es the permanent 
ouplings still need to be realised, although only on
e, and
experimental 
onstraints su
h as resolution and errors need to be 
onsidered. We are
thus left with the following questions: whi
h is the best way to perform quantum
state transfer using a permanently 
oupled graph? How mu
h 
ontrol do we need,
and how di�
ult will it be to implement the 
ouplings? How do errors and noise
a�e
t the s
heme? All these points are highly related and it 
annot be expe
ted to
�nd an absolute, i.e. system independent answer. The purpose of this resear
h is to
develop advan
ed s
hemes for the transfer of quantum information, to improve and
generalise existing ideas, to relate them to ea
h other and to investigate their stability
and e�
ien
y.
1.1 Quantum Computation and Quantum Information
In this Se
tion we review some of the basi
 
on
epts of Quantum Computation. We
will be very brief and only fo
us on those aspe
ts that we require later on in the thesis.
A more detailed introdu
tion 
an be found in [2℄.
In information s
ien
e, an algorithm is a list of instru
tions that a 
omputer performs
on a given input to a
hieve a spe
i�
 task. For instan
e, a fa
toring algorithm has an
arbitrary integer as its input, and gives its prime fa
tors as an output. A quantum
fa
toring algorithm 
an be thought of in a similar way, i.e. it has an integer as input,
and its prime fa
tors as an output. In-between however it en
odes information in a
quantum me
hani
al system. Due to the superposition prin
iple, the information of a
quantum system 
annot be represented as bits. The valid generalisation of the bit to
the quantum 
ase is 
alled qubit. The possible states of a qubit are written as
α|0〉 + β|1〉, (1.1)
where α, β are normalised 
omplex 
oe�
ients, and |0〉 and |1〉 are ve
tors of a two-
dimensional 
omplex ve
tor spa
e. Peter W. Shor has shown in a famous paper [3℄
that the detour of representing the intermediate part of a fa
toring algorithm in a
quantum system (as well as using quantum gates, see below) 
an be very bene�
ial:
it runs mu
h faster. This is important, be
ause many 
ryptographi
 methods rely
on fa
toring algorithms being slow. Shor's algorithm is de�nitely not the only reason
why it would be very ni
e to have a quantum 
omputer, i.e. a ma
hine that represents
information in a quantum way and 
an perform instru
tions on it, and many more
details 
an be found in the textbook mentioned above.
1 Introdu
tion
Algorithms on a 
omputer 
an be represented as list of logi
al operations on bits.
Likewise, a (standard) quantum algorithm 
an be represented as a list of quantum
logi
al operations, or quantum gates, a
ting on qubits. The most general quantum
algorithm is given by an arbitrary unitary operator. A universal set of gates is a
set su
h that any quantum algorithm (i.e. unitary operator) 
an be de
omposed
into a sequen
e of gates belonging to this set. In the standard model of quantum

omputation, one assumes that su
h a set is available on the ma
hine [4℄. Also the
ability to perform measurements is assumed. We refer to this as the full 
ontrol 
ase.
From a information theoreti
 point of view, qubits are not only useful obje
ts to
perform algorithms with, but also very interesting from a fundamental point of view.
To give a (too simple) analogy 
onsider the following. If you read the word �
ho
olate�,
you 
an asso
iate a positive/negative or neutral feeling of whether you would like to eat
some 
ho
olate now. However, what was the state of your mind 
on
erning 
ho
olate
before you read the word? Unless you were already 
raving for 
ho
olate beforehand,
or you have just eaten a lot, your mind was probably unde
ided. Moreover, it would
have been very di�
ult - if not impossible - to des
ribe to someone in plain language
whi
h opinion you had about the 
ho
olate before you read the word.
In a similar manner, the quantum information 
ontained in a single arbitrary and
unknown qubit 
annot be des
ribed by 
lassi
al information. When it is measured, it
behaves like a normal bit in the sense that the out
ome is only 0 or 1, but when it
is not measured, it behaves in some way as if it was unde
ided between 0 and 1. Of

ourse one has to be very 
areful with these analogies. But for the purpose of this
thesis it is important to stress that quantum information 
annot be transported by
any 
lassi
al methods [5℄. This is why it is so important and also so di�
ult to develop
new wires, dubbed quantum wires, that are 
apable of doing this.
1.2 Quantum state transfer along short distan
es
In theory, additional devi
es for the transfer of unknown quantum states are not
required for building a quantum 
omputer, unless it is being used for typi
al quantum

ommuni
ation purposes, su
h as se
ret key distribution [4℄. This is be
ause the
universal set of gates on the quantum 
omputer 
an be used to transfer quantum
states by applying sequen
es of two-qubit swap gates (Fig. 1.1).
However in pra
ti
e it is 
ru
ial to minimise the required number of quantum gates,
as ea
h gate typi
ally introdu
es errors. In this light it appears 
ostly to perform N−1
swap gates between nearest neighbours to just move a qubit state over a distan
e of
N sites. For example, Shor's algorithm on N qubits 
an be implemented by only
1 Introdu
tion
SN−1,N
Figure 1.1: In areas of universal 
ontrol, quantum states 
an easily be transferred by
sequen
es of unitary swap gates Sj,k between nearest neighbours.
logN quantum gating operations [6℄ if long distant qubit gates are available. These
long distant gates 
ould 
onsist of lo
al gates followed by a quantum state transfer.
If however the quantum state transfer is implemented as a sequen
e of lo
al gates,
then the number of operations blows up to the order of N gates. The quantum state
transfer 
an even be thought of as the sour
e of the power of quantum 
omputation, as
any quantum 
ir
uit with logN gates and lo
al gates only 
an be e�
iently simulated
on a 
lassi
al 
omputer [7, 8℄.
A se
ond reason to 
onsider devi
es for quantum state transfer is related to s
ala-
bility . While small quantum 
omputers have already been built [9℄, it is very di�
ult
to build large arrays of fully 
ontrollable qubits. A bla
k box that transports unknown
quantum states 
ould be used to build larger quantum 
omputers out of small 
ompo-
nents by 
onne
ting them. Likewise, quantum state transfer 
an be used to 
onne
t
di�erent 
omponents of a quantum 
omputer, su
h as the pro
essor and the memory
(see also Fig. 1.2). On larger distan
es, �ying qubits su
h as photons, ballisti
 ele
trons
and guided atoms/ions are 
onsidered for this purpose [10, 11℄. However, 
onverting
ba
k and forth between stationary qubits and mobile 
arriers of quantum information
and interfa
ing between di�erent physi
al implementations of qubits is very di�
ult
and worthwhile only for short 
ommuni
ation distan
es. This is the typi
al situation
one has to fa
e in solid state systems, where quantum information is usually 
ontained
in the states of �xed obje
ts su
h as quantum dots or Josephson jun
tions. In this 
ase
permanently 
oupled quantum 
hains have re
ently been proposed as prototypes of re-
liable quantum 
ommuni
ation lines [1,12℄. A quantum 
hain (also referred to as spin

hain) is a one-dimensional array of qubits whi
h are 
oupled by some Hamiltonian
(
f. Fig. 1.3). These 
ouplings 
an transfer states without external 
lassi
al 
ontrol.
1 Introdu
tion
In many 
ases, su
h permanent 
ouplings are easy to build in solid state devi
es (in
fa
t a lot of e�ort usually goes into suppressing them). The qubits 
an be of the same
type as the other qubits in the devi
e, so no interfa
ing is required.
Quantum
processor
Quantum
memory
Controller
Input
Output
Figure 1.2: S
hemati
 layout of a quantum 
omputer. The solid arrows represent the
�ow of quantum information, and the dashed arrows the �ow of 
lassi
al information.
Figure 1.3: Permanently 
oupled quantum 
hains 
an transfer quantum states without

ontrol along the line. Note that the ends still need to be 
ontrollable to initialise and
read out quantum states.
Another related motivation to 
onsider quantum 
hains is that they 
an simplify the
layout of quantum devi
es on wafers. A typi
al 
hip 
an 
ontain millions of qubits, and
the fabri
ation of many qubits is in prin
iple no more di�
ult than the fabri
ation of
a single one. In the last 
ouple of years, remarkable progress was made in experiments
with quantum dots [13, 14℄ and super-
ondu
ting qubits [15, 16℄. It should however
be emphasised that for initialisation, 
ontrol and readout, those qubits have to be

onne
ted to the ma
ros
opi
 world (see Fig. 1.2). For example, in a typi
al �ux qubit
gate, mi
rowave pulses are applied onto spe
i�
 qubits of the sample. This requires
many (
lassi
al) wires on the 
hip, whi
h is thus a 
ompound of quantum and 
lassi
al

omponents. The ma
ros
opi
 size of the 
lassi
al 
ontrol is likely to be the bottlene
k
of the s
alability as a whole. In this situation, quantum 
hains are useful in order to
keep some distan
e between the 
ontrolled quantum parts. A possible layout for su
h
1 Introdu
tion
a quantum 
omputer is shown in Fig. 1.4. It is built out of blo
ks of qubits, some
of whi
h are dedi
ated to 
ommuni
ation and therefore 
onne
ted to another blo
k
through a quantum 
hain. Within ea
h blo
k, arbitrary unitary operations 
an be
performed in a fast and reliable way (they may be de
omposed into single and two-
qubit operations). Su
h blo
ks do not 
urrently exist, but they are the fo
us of mu
h
work in solid state quantum 
omputer ar
hite
ture. The distan
e between the blo
ks
is determined by the length of the quantum 
hains between them. It should be large
enough to allow for 
lassi
al 
ontrol wiring of ea
h blo
k, but short enough so that
the time-s
ale of the quantum 
hain 
ommuni
ation is well below the time-s
ale of
de
oheren
e in the system.
Figure 1.4: Small blo
ks (grey) of qubits (white 
ir
les) 
onne
ted by quantum 
hains.
Ea
h blo
k 
onsists of (say) 13 qubits, 4 of whi
h are 
onne
ted to outgoing quantum

hains (the thi
k bla
k lines denote their nearest-neighbour 
ouplings). The blo
ks are

onne
ted to the ma
ros
opi
 world through 
lassi
al wires (thin bla
k lines with bla
k

ir
les at their ends) through whi
h arbitrary unitary operations 
an be triggered on
the blo
k qubits. The quantum 
hains require no external 
ontrol.
Finally, an important reason to study quantum state transfer in quantum 
hains
stems from a more fundamental point of view. Su
h systems in prin
iple allow tests of
Bell-inequalities and non-lo
ality in solid-state experiments well before the realisation
of a quantum 
omputer. Although quantum transport is quite an established �eld, the
quantum information point of view o�ers many new perspe
tives. Here, one looks at
the transport of information rather than ex
itations, and at entanglement [17,18,19,20℄
rather than 
orrelation fun
tions. It has re
ently been shown that this sheds new
light on well-known physi
al phenomena su
h as quantum phase transitions [21, 22,
1 Introdu
tion
23,24℄, quantum 
haos [25, 26, 27, 28℄ and lo
alisation [29, 30℄. Furthermore, quantum
information takes on a more a
tive attitude. The 
orrelations of the system are not
just 
al
ulated, but one also looks at how they may be 
hanged.
1.3 Implementations and experiments
As we have seen above, the main advantage of state transfer with quantum 
hains is
that the qubits 
an be of the same type as those used for the quantum 
omputation.
Therefore, most systems that are thought of as possible realisations of a quantum

omputer 
an also be used to build quantum 
hains. Of 
ourse there has to be some

oupling between the qubits. This is typi
ally easy to a
hieve in solid state sys-
tems, su
h as Josephson jun
tions with 
harge qubits [31, 32℄, �ux qubits [33, 34℄ (see
also Fig. 1.5) or quantum dots dots using the ele
trons [35, 36℄ or ex
itons [37, 38℄.
Other systems where quantum 
hain Hamiltonians 
an at least be simulated are NMR
qubits [39,40,41℄ and opti
al latti
es [42℄. Su
h a simulation is parti
ularly useful in the
latter 
ase, where lo
al 
ontrol is extremely di�
ult. Finally, qubits in 
avities [43,44℄
and 
oupled arrays of 
avities were 
onsidered [45, 46℄.
Figure 1.5: A quantum 
hain 
onsisting of N = 20 �ux qubits [34℄ (pi
ture and exper-
iment by Floor Paauw, TU Delft). The 
hain is 
onne
ted to four larger SQUIDS for
readout and gating.
For the more fundamental questions, su
h as studies of entanglement transfer, non-
lo
ality and 
oherent transport, the quantum 
hains 
ould also be realised by systems
whi
h are not typi
ally thought of as qubits, but whi
h are natural spin 
hains. These

an be mole
ular systems [47℄ or quasi-1D solid state materials [48, 49℄.
1.4 Basi
 
ommuni
ation proto
ol
We now review the most basi
 transport proto
ol for quantum state transfer, initially
suggested in [1℄. For the sake of simpli
ity, we 
on
entrate on the linear 
hain setting,
though more general graphs of qubits 
an be 
onsidered in the same way. The proto
ol

onsists of the following steps:
1 Introdu
tion
1. Initialise the quantum 
hain in the ground state
|G〉. (1.2)
2. Put an arbitrary and unknown qubit with (possibly mixed) state ρ at the sending
end of the 
hain
ρ⊗ Tr1 {|G〉〈G|} . (1.3)
3. Let the system evolve under its Hamiltonian H for a time t
exp {−iHt} ρ⊗ Tr1 {|G〉〈G|} exp {iHt} . (1.4)
4. Pi
k up the quantum state at the end of the 
hain
σ ≡ Tr1,...,N−1 [exp {−iHt} ρ⊗ Tr1 {|G〉〈G|} exp {iHt}] . (1.5)
Some pra
ti
al aspe
ts how to realise these steps are dis
ussed in the next se
tion.
For the moment, we will 
on
entrate on the quality of quantum state transfer given
that the above steps 
an be performed. From a quantum information perspe
tive, the
above equations des
ribe a quantum 
hannel [5℄ τ that maps input states ρ at one end
of the 
hain to output states τ(ρ) = σ on the other end. A very simple measure of the
quality of su
h a quantum 
hannel is the �delity [50, 51, 2℄
F (ρ, σ) ≡
ρ1/2σρ1/2
. (1.6)
More advan
ed measures of the quality of transfer will be dis
ussed in Chapter 3. Note
also that some authors de�ne the �delity without taking the square of the tra
e. It is
a real-valued, symmetri
 fun
tion with range between 0 and 1, assuming unity if and
only if ρ = σ. Sin
e the transported state that is an unknown result of some quantum

omputation, we are interested in the minimal �delity
F0 ≡ minρF (ρ, τ(ρ)). (1.7)
We remark that some authors also assume an equal distribution of input states and

ompute the average �delity [1℄. Using the strong 
on
avity of the �delity [2℄ and the
linearity of τ we �nd that the minimum must be assumed on pure input states,
F0 = minψ〈ψ|τ(ψ)|ψ〉. (1.8)
1 Introdu
tion
In the present 
ontext, F0 = F0(H, t) is a fun
tion of of the Hamiltonian H of the
quantum 
hain (through the spe
i�
 role of the ground state in the proto
ol and
through the time evolution), and of the time interval t that the system is evolving in
the third step of the proto
ol.
1.4.1 Initialisation and end-gates
There are two strong assumptions in the proto
ol from the last se
tion. The �rst one
is that the 
hain 
an be initialised in the ground state |G〉. How 
an that be a
hieved
if there is no lo
al 
ontrol along the 
hain? The answer appears to be quite easy: one
just applies a strong global magneti
 �eld and strong 
ooling (su
h as laser 
ooling or
dilution refrigeration) and lets the system rea
h its ground state by relaxation. The

ooling needs to be done for the remaining parts of the quantum 
omputer anyway,
so no extra devi
es are required. However there is a problem with the time-s
ale of
the relaxation. If the system is brought to the ground state by 
ooling, it must be

oupled to some environment. But during the quantum 
omputation, one 
learly does
not want su
h an environment. This is usually solved by having the time-s
ale of the

omputation mu
h smaller (say mi
rose
onds) than the time-s
ale of the 
ooling (say
se
onds or minutes). But if the quantum 
hain should be used multiple times during
one 
omputation, then how is it reset between ea
h usage? This is important to avoid
memory e�e
ts [52℄, and there are two solutions to this problem. Either the proto
ol
is su
h that at the end the 
hain is automati
ally in the ground state. Su
h a proto
ol
usually 
orresponds to perfe
t state transfer. The other way is to use the 
ontrol at
the ends of the 
hain to bring it ba
k to the ground state. A simple 
ooling proto
ol
is given by the following: one measures the state of the last qubit of the 
hain. If it
is in |0〉, then one just lets the 
hain evolve again and repeats. If however it is found
to be in |1〉, one applies the Pauli operator X to �ip it before evolving and repeating.
It will be
ome 
lear later on in the thesis that su
h a proto
ol typi
ally 
onverges
exponentially fast to the ground state of the 
hain.
The se
ond assumption in the last se
tion is that the sender and re
eiver are 
apable
of swapping in and out the state mu
h qui
ker than the time-s
ale of the intera
tion of
the 
hain. Alternatively, it is assumed that they 
an swit
h on and o� the intera
tion
between the 
hain and their memory in su
h time-s
ale. It has re
ently been shown [33℄
that this is not a fundamental problem, and that �nite swit
hing times 
an even slightly
improve the �delity if they are 
arefully in
luded in the proto
ol. But this requires to
solve the full time-dependent S
hrödinger equation, and introdu
es further parameters
to the model (i.e. the raise and fall time of the 
ouplings). For the sake of simpli
ity,
1 Introdu
tion
we will therefore assume that the end gates are mu
h faster then the time evolution
of the 
hain (see also Se
tion 6.3).
1.4.2 Symmetries
The dimensionality of the Hilbert spa
e H of a quantum 
hain of N qubits is 2N .
This makes it quite hopeless in general to determine the minimal �delity Eq. (1.8) for
long quantum 
hains. Most investigations on quantum state transfer with quantum

hains up to date are therefore 
on
entrating on Hamiltonians with additional sym-
metries. With few ex
eptions [34,21,22,53℄ Hamiltonians that 
onserve the number of
ex
itations are 
onsidered. In this 
ase the Hilbert spa
e is a dire
t sum of subspa
es
invariant under the time evolution,
Hℓ, (1.9)
with dimHℓ =
, and where ℓ is the number of ex
itations. These Hamiltonians are
mu
h easier to handle both analyti
ally and numeri
ally, and it is also easier to get
an intuition of the dynami
s. Furthermore, they o

ur quite naturally as a 
oupling
between qubits in the relevant systems. We stress though that there is no fundamental
reason to restri
t quantum 
hain 
ommuni
ation to this 
ase.
1.4.3 Transfer fun
tions
The spa
e H0 only 
ontains the state |0〉 whi
h is thus always an eigenstate of H. We
will assume here that it is also the ground state,
|G〉 = |0〉. (1.10)
This 
an be a
hieved by applying a strong global magneti
 �eld (or equivalent) to the
system. The spa
e H1 is spanned by the ve
tors {|k〉, k = 1, . . . , N} having exa
tly
one ex
itation. The above proto
ol be
omes:
1. Initialise the quantum 
hain in the ground state
|0〉 (1.11)
2. Put an arbitrary and unknown qubit in the pure state |ψ〉 = α|0〉 + β|1〉 at the
1 Introdu
tion
sending end of the 
hain
α|0〉+ β|1〉 (1.12)
3. Let the system evolve for a time t
α|0〉+ β exp {−iHt} |1〉 (1.13)
4. Pi
k up the quantum state at the end of the 
hain (see [1℄)
τ(ψ) = (1− p(t))|0〉〈0| + p(t)|ψ〉〈ψ|, (1.14)
with the minimal �delity given by
F0 = minψ〈ψ|τ(ψ)|ψ〉 (1.15)
= p(t) + (1− p(t))minψ |〈0|ψ〉|2 = p(t). (1.16)
The fun
tion p(t) is the transition probability from the state |1〉 to |N 〉 given by
p(t) = |〈N | exp {−iHt} |1〉|2 . (1.17)
We see that in the 
ontext of quantum state transfer, a single parameter su�
es to

hara
terise the properties of an ex
itation 
onserving 
hain. The averaged �delity [1℄
is also easily 
omputed as
. (1.18)
Even more 
omplex measures of transfer su
h as the quantum 
apa
ity only depend on
p(t) [54℄. It is also a physi
ally intuitive quantity, namely a parti
ular matrix element
of the time evolution operator,
fn,m(t) ≡ 〈n| exp {−iHt} |m〉 (1.19)
e−iEkt〈n|Ek〉〈Ek|m〉, (1.20)
where |Ek〉 and Ek are the eigenstates and energy levels of the Hamiltonian in H1.
1 Introdu
tion
1.4.4 Heisenberg Hamiltonian
The Hamiltonian 
hosen in [1℄ is a Heisenberg Hamiltonian
H = −J
(XnXn+1 + YnYn+1 + ZnZn+1)−B
Zn + c, (1.21)
with a 
onstant term
J(N − 1)
+NB (1.22)
added to set the ground state energy to 0. For J > 0 it ful�ls all the assumptions
dis
ussed above, namely its ground state is given by |0〉 and it 
onserves the number
of ex
itations in the 
hain. The Heisenberg intera
tion is very 
ommon and serves
here as a typi
al and analyti
ally solvable model for quantum state transfer.
In the �rst ex
itation subspa
e H1, the Heisenberg Hamiltonian Eq. (1.21) is ex-
pressed in the basis {|n〉} as
−1 2 −1
−1 2 . . .
−1 2 −1
. (1.23)
A more general study of su
h tridiagonal matri
es 
an be found in a series of arti
les
on 
oherent dynami
s [55, 56, 57, 58℄. Some interesting analyti
ally solvable models
have also been identi�ed [59, 56, 57℄ (we shall 
ome ba
k to that point later).
For the present 
ase, the eigenstates of Eq (1.23) are [1℄
|Ek〉 =
1 + δk0
(2n− 1)
|n〉 (k = 0, . . . , N − 1), (1.24)
with the 
orresponding energies given by
Ek = 2B + 2J
1− cos πk
. (1.25)
The parameter B has no relevan
e for the �delity but determines the stability of the
ground state (the energy of the �rst ex
ited state is given by 2B). The minimal �delity
1 Introdu
tion
for a Heisenberg 
hain is given by
p(t) = N−2
−2iJt(1− cos πk
(−1)k
1 + cos
. (1.26)
As an example, Fig 1.6 shows p(t) for N = 50.
 0.02
 0.04
 0.06
 0.08
 0.12
 0.14
 0  10  20  30  40  50  60  70  80  90  100
Time [1/J]
Figure 1.6: Minimal �delity p(t) for a Heisenberg 
hain of length N = 50.
1.4.5 Dynami
 and Dispersion
Already in [1℄ has been realised that the �delity for quantum state transfer along spin

hains will in general not be perfe
t. The reason for the imperfe
t transfer is the
dispersion [60℄ of the information along the 
hain. Initially the quantum information
is lo
alised at the sender, but as it travels through the 
hain it also spreads (see Fig. 1.7
and Fig. 1.8). This is not limited to the Heisenberg 
oupling 
onsidered here, but a
very 
ommon quantum e�e
t. Due to the dispersion, the probability amplitude peak
that rea
hes Bob is typi
ally small, and be
omes even smaller as the 
hains get longer.
The �delity given Eq. (1.26) is shown in Fig. 1.6. We 
an see that a wave of
quantum information is travelling a
ross the 
hain. It rea
hes the other end at a time
of approximately
(1.27)
1 Introdu
tion
 0  10  20  30  40  50
Position along chain
Figure 1.7: Snapshots of the time evolution of a Heisenberg 
hain with N = 50. Shown
is the distribution |fn,1(t)|2 of the wave-fun
tion in spa
e at di�erent times if initially
lo
alised at the �rst qubit.
 0  20  40  60  80  100
Time [1/J]
Relative mean
Relative variance
Figure 1.8: Mean and varian
e of the state |1〉 as a fun
tion of time. Shown is the 
ase
N = 50 with the y-axis giving the value relative to the mean N/2 + 1 and varian
e
(N2 − 1)/12 of an equal distribution 1√
1 Introdu
tion
As a rough estimate of the s
aling of the �delity with respe
t to the 
hain length
around this peak we 
an use [1, 61℄ (see also Fig. 1.9)
|fN,1(t)|2 ≈ |2JN (
)|2 ≈ |
(N − 2t
)]|2, (1.28)
where JN (x) is a Bessel fun
tion of �rst kind and ai(x) is the Airy fun
tion. The airy
fun
tion ai(x) has a maximum of 0.54 at x = −1.02. Hen
e we have
) = |fN,1(
)|2 ≈ 1.82N−2/3. (1.29)
It is however possible to �nd times where the �delity of the 
hain is mu
h higher.
The reason for this is that the wave-pa
ket is re�e
ted at the ends of the 
hain and
starts interfering with itself (Fig 1.6). As the time goes on, the probability distribution
be
omes more and more random. Sometimes high peaks at the re
eiving end o

ur.
From a theoreti
al point of view, it is interesting to determine the maximal peak
o

urring, i.e.
pM (T ) ≡ max
0<t<T
p(t). (1.30)
As we 
an see in Fig. 1.10 there is quite a potential to improve from the estimate
Eq. (1.29).
 0.02
 0.04
 0.06
 0.08
 0.12
 0.14
 20  22  24  26  28  30
Time [1/J]
|fN,1(t)|
|2 JN(2t)|
|(16/N)1/3 ai[(2/N)1/3(N-2t)]|2
Figure 1.9: Approximation of the transfer amplitude for N = 50 around the �rst
maximum by Bessel and Airy fun
tions [1, 61℄.
1 Introdu
tion
 0.1  1  10  100  1000  10000  100000  1e+06  1e+07
Time [1/J]
Figure 1.10: pM (T ) as a fun
tion of T for di�erent 
hain lengths. The solid 
urve is
given by 1.82(2T )
and 
orresponds to the �rst peak of the probability amplitude
(Eq. 1.29)
We will now show a perhaps surprising 
onne
tion of the fun
tion pM (T ) to number
theory. Some spe
ulations on the dependen
e of the �delity on the 
hain length being
divisible by 3 were already made in [1℄, but not rigorously studied. As it turns out, for

hains with prime number length the maximum of the �delity is a
tually 
onverging
to unity (see Fig. 1.10). To show this, we �rst prove the following
Lemma 1.1 Let N be an odd prime. Then the set
(k = 0, 1, . . . , ,
N − 1
(1.31)
is linear independent over the rationals Q.
Proof Assume that
λk cos
= 0 (1.32)
with λk ∈ Q. It follows that
+ exp
= 0 (1.33)
1 Introdu
tion
and hen
e
λk exp
λk exp
i(N − k)π
= 0. (1.34)
Changing indexes on the se
ond sum we get
λk exp
k=N+1
λN−k exp
= 0. (1.35)
and �nally
λ̃k exp
= 0, (1.36)
where
λ̃0 = 2λ0 (1.37)
λ̃k = λk (k = 1, . . . ,
N − 1
) (1.38)
λ̃k = −λN−k (k =
N + 1
, . . . , N − 1). (1.39)
Sin
e N is prime, the roots of unity in Eq. (1.36) are all primitive and therefore linearly
independent over Q [62, Theorem 3.1, p. 313℄. Hen
e λk = 0 for all k. �
Theorem 1.1 (Half re
urren
e) Let N be an odd prime. For a Heisenberg

hain of length N we have
pM (T ) = lim
0<t<T
= 1. (1.40)
Proof The eigenfrequen
ies of the Hamiltonian in the �rst ex
itation se
tor H1 are
given by
Ek = 2B + 2J
1− cos πk
(k = 0, 1, . . . , , N − 1). (1.41)
Using Krone
ker's theorem [63℄ and Lemma 1.1, the equalities
exp {itEk} = (−1)k e2(B+J)t (k = 0, 1, . . . , ,
N − 1
) (1.42)
1 Introdu
tion

an be ful�lled arbitrarily well by 
hoosing an appropriate t. Sin
e
= − cos (N − k)π
, (1.43)
the equalities (1.42) are then also ful�lled arbitrarily well for k = 0, . . . , N − 1. This
is known as as su�
ient 
ondition for perfe
t state transfer in mirror symmetri

hains [64℄, where the eigenstates 
an be 
hosen su
h that they are alternately sym-
metri
 and antisymmetri
. Roughly speaking, Eq. (1.42) introdu
es the 
orre
t phases
(a sign 
hange for the antisymmetri
 eigenstates) to move the state |1〉 to |N〉 and
hen
e the theorem. �
Remark 1.1 The time-s
ale for �nding high valued peaks is however exponential in
the 
hain length [63℄. Therefore the above theorem has little pra
ti
al use. For non-
prime 
hain lengths, the eigenfrequen
ies are not su�
iently independent to guarantee
a perfe
t state transfer, with the algebrai
 dimensionality of the roots of unity for non-
prime N given by the Euler totient fun
tion φ(N) [62, Theorem 3.1, p. 313℄. We also
remark that due to its asymptoti
 
hara
ter, the above result is not 
ontradi
ting [65℄,
where it was shown that 
hains longer than N ≥ 4 never have perfe
t �delity.
Having proved that there are many 
hains that 
an in prin
iple perform arbitrarily well,
it is important to �nd a 
ut-o� time for the optimisation Eq. (1.30). Faster transfer
than linear in N using lo
al Hamiltonians is impossible due to the Lieb-Robinson
bound [66, 67℄, whi
h is a �speed limit� in non-relativisti
 quantum me
hani
s giving
rise to a well de�ned group velo
ity. Transport faster than this group velo
ity is
exponentially suppressed. Going ba
k to the motivation of quantum state transfer, a
natural 
omparison [37℄ for the time-s
ale of quantum state transfer is given by the
time it would take to perform a sequen
e of swap gates (
f. Fig 1.1) that are realised
by a pairwise swit
hable 
oupling Hamiltonian
(XnXn+1 + YnYn+1). (1.44)
This time is linear in the 
hain length:
(N − 1)π
. (1.45)
Ideally one 
ould say that the time for quantum state transfer should not take mu
h
longer than this. However one may argue that there is a trade-o� between qui
k
transfer on one hand, and minimising 
ontrol on the other hand. A se
ond 
ut-o�
1 Introdu
tion
time may be given by the de
oheren
e time of the spe
i�
 implementation. But short
de
oheren
e times 
ould always be 
ountera
ted by in
reasing the 
hain 
oupling J.
A more general and implementation independent limit is given by the requirement
that the peak width ∆t
should not be too small with respe
t to the total time.
Otherwise it is di�
ult to pi
k up the state at the 
orre
t time. For the �rst peak,
we 
an estimate the width by using the full width at half height of the airy fun
tion.
From Eq. (1.28) we get an absolute peak width of ∆t
≈ 0.72N1/3/J and a relative
width of
≈ 1.44N−2/3. (1.46)
This is already quite demanding from an experimental perspe
tive and we 
on
lude
that the transfer time should not be 
hosen mu
h longer than those of the �rst peak.
1.4.6 How high should p(t) be?
We have not dis
ussed yet what the a
tual value of p(t) should be to make su
h a
spin 
hain useful as a devi
e for quantum state transfer. p(t) = 0 
orresponds to no
state transfer, p(t) = 1 to a perfe
t state transfer. But what are the relevant s
ales
for intermediate p(t)? In pra
ti
e, the quantum transfer will su�er from additional
external noise (Chapter 7) and also the quantum 
omputer itself is likely to be very
noisy. From this point of view, requiring p(t) = 1 seems a bit too demanding.
From a theoreti
al perspe
tive, it is interesting that for any p(t) > 0, one 
an al-
ready do things whi
h are impossible using 
lassi
al 
hannels, namely entanglement
transfer and distillation [2℄. The entanglement of formation between the sender (Al-
i
e) and the re
eiver (Bob) is simply given by
p(t) [1℄. This entanglement 
an be
partially distilled [68℄ into singlets, whi
h 
ould be used for state transfer using tele-
portation [2℄. It is however not known how mu
h, i.e. at whi
h rate, entanglement

an be distilled (we will develop lower bounds for the entanglement of distillation in
Se
tion 2.2 and Se
tion 3.4). Also, entanglement distillation is a quite 
omplex pro-

edure that requires lo
al unitary operations and measurements, additional 
lassi
al

ommuni
ation, and multiple 
hain usages; and few expli
it proto
ols are known. This
is likely to preponderate the bene�ts of using a quantum 
hain.
When the 
hain is used without en
oding and further operation, the averaged �delity
Eq. (1.18) be
omes better than the 
lassi
al
averaged �delity [1℄ when p(t) > 3 −
By �
lassi
al �delity�, we mean the �delity that 
an be a
hieved by optimising the following pro-
to
ol: Ali
e performs measurements on her state and sends Bob the out
ome through a 
lassi
al

ommuni
ation line. Bob then tries to rebuild the state that Ali
e had before the measurement
based on the information she sent. For qubits, the 
lassi
al �delity is given by 2/3 [69℄.
1 Introdu
tion
2. Following the 
on
lusion from the last subse
tion that the �rst peak is the most
relevant one, this would mean that only 
hains with length until N = 33 perform
better than the 
lassi
al �delity.
Finally, the quantum 
apa
ity [54, 70℄ of the 
hannel be
omes non-zero only when
p(t) > 1/2, 
orresponding to 
hain lengths up to N = 6. Roughly speaking, it is a
measure of the number of perfe
tly transmitted qubits per 
hain usage that 
an be
a
hieved asymptoti
ally using en
oding and de
oding operations on multiple 
hannel
usages. The quantum 
apa
ity 
onsidered here is not assumed to be assisted by a

lassi
al 
ommuni
ation, and the threshold of p(t) > 0.5 to have a non-zero quantum

apa
ity is a result of the non-
loning theorem [2℄. This is not 
ontradi
ting the
fa
t that entanglement distillation is possible for any p(t) > 0, as the entanglement
distillation proto
ols require additional 
lassi
al 
ommuni
ation.
All the above points are summarised in Fig. 1.11. We 
an see that only very short

hains rea
h reasonable values (say > 0.6) for the minimal �delity.
 0  0.2  0.4  0.6  0.8  1
235610153380300
Corresponding chain length (first peak)
Quantum capacity
EOD (lower bound)
Averaged fidelity
Classical threshold
Figure 1.11: Quantum 
apa
ity, entanglement of formation (EOF), a lower bound for
the entanglement of distillation (EOD) and the averaged �delity as a fun
tion of p(t).
We also show the 
orresponding 
hain length whi
h rea
hes this value as a �rst peak
and the 
lassi
al threshold 3− 2
2. The expli
it expression for the quantum 
apa
ity
plotted here is given in [54℄, and the lower bound of the entanglement of distillation
will be derived in Se
tion 3.4.
1 Introdu
tion
1.5 Advan
ed 
ommuni
ation proto
ols
We have seen in the last se
tion that without mu
h further e�ort, i.e. entanglement
distillation, unmodulated Heisenberg 
hains are useful only when they are very short.
Shortly after the initial proposal [1℄ it has been shown that there are ways to a
hieve
even perfe
t state transfer with arbitrarily long 
hains. These advan
ed proposals 
an
roughly be grouped into four 
ategories, whi
h we will now brie�y des
ribe.
1.5.1 Engineered Hamiltonians
The Heisenberg model 
hosen by Bose features many typi
al aspe
ts of 
oherent trans-
port, i.e. the wave-like behaviour, the dispersion, and the almost-periodi
ity of the
�delity. These features do not depend so mu
h on the spe
i�
 
hoi
es of the parame-
ters of the 
hain, su
h as the 
ouplings strengths. There are however spe
i�
 
ouplings
for quantum 
hains that show a quite di�erent time evolution, and it was suggested
in [71℄ and independently in [72℄ to use these to a
hieve a perfe
t state transfer:
H = −J
n(N − n) (XnXn+1 + YnYn+1) (1.47)
These values for engineered 
ouplings also appear in a di�erent 
ontext in [57,73℄. The
time evolution under the Hamiltonian (1.47) features an additional mirror symmetry:
the wave-pa
ket disperses initially, but the dispersion is reversed after its 
entre has
passed the middle of the 
hain (Fig. 1.12). This approa
h has been extended by
various authors [64, 74, 75, 76, 77, 78, 79, 80, 53, 81, 82, 83, 84, 65, 59, 19℄, and many other

hoi
es of parameters for perfe
t or near perfe
t state transfer in various settings were
found [59, 83, 81℄.
1.5.2 Weakly 
oupled sender and re
eiver
A di�erent approa
h of tuning the Hamiltonian was suggested in [85℄. There, only
the �rst and the last 
ouplings j of the 
hain are engineered to be mu
h weaker than
the remaining 
ouplings J of the 
hain, whi
h 
an be quite arbitrary. The �delity

an be made arbitrarily high by making the edge 
oupling strengths smaller. It was
shown [86, 87℄ that to a
hieve a �delity of 1 − δ in a 
hain of odd length, it takes
approximately a time of
δ (1.48)
1 Introdu
tion
 0  10  20  30  40  50
Position along chain
t=pi/8
t=pi/4
t=3pi/8
t=pi/2
Figure 1.12: Snapshots of the time evolution of a quantum 
hain with engineered

ouplings (1.47) for N = 50. Shown is the distribution of the wave-fun
tion in spa
e
at di�erent times if initially lo
alised at the �rst qubit (
ompare Fig. 1.7).
1 Introdu
tion
and the 
oupling ratio has to be approximately j/J ≈
δ/N. Some spe
i�
 types of
quantum 
hains whi
h show high �delity for similar reasons were also investigated [88,
89, 90, 91℄.
1.5.3 En
oding
We have seen in Subse
. 1.4.6 that if p(t) < 1/2, the �delity 
annot be improved
by using any en
oding/de
oding strategy (be
ause the quantum 
apa
ity is zero).
However it is possible to 
hange the proto
ol des
ribed in Se
. 1.4 slightly su
h that
the �delity is mu
h higher. This 
an be thought of as a �hardware en
oding�, and
was suggested �rst in [60℄. There, it was assumed that the 
hain 
onsists of three
se
tions: one part of length ≈ 2N1/3 
ontrolled by the sending party, one �free� part
of length N and one part of length ≈ 2.8N1/3 
ontrolled by the re
eiving party. The
sender en
odes the qubit not only in a single qubit of the 
hain, but in a Gaussian-
modulated superposition of his qubits. These Gaussian pa
kets are known to have
minimal dispersion. Likewise, the re
eiver performs a de
oding operation on all qubits
he 
ontrols. Near-perfe
t �delity 
an be rea
hed.
1.5.4 Time-dependent 
ontrol
Finally, a number of authors found ways of improving the �delity by time-dependent

ontrol of some parameters of the Hamiltonian. In [92℄ it is shown that if the end

ouplings 
an be 
ontrolled as arbitrary (in general 
omplex valued) smooth fun
tions
of time the en
oding s
heme [60℄ 
ould be simulated without the requirement of ad-
ditional operations and qubits. Another possibility to a
hieve perfe
t state transfer
is to have an Ising intera
tion with additionally pulsed global rotations [93, 40, 94℄.
Further related methods of manipulating the transfer by global �elds were reported
in [95, 96, 25, 97, 98, 28℄.
1.6 Motivation and outline of this work
While the advan
ed transfer proto
ols have shown that in prin
iple high �delity 
an
be a
hieved with arbitrarily long 
hains, they have all 
ome at a 
ost. Engineering
ea
h 
oupling of the Hamiltonian puts extra demands on the experimental realisation,
whi
h is often already at its very limits just to ensure the 
oheren
e of the system.
Furthermore, the more a s
heme relies on parti
ular properties of the Hamiltonian, the
more it will be a�e
ted by imperfe
tions in its implementation [99, 84℄. For example,
simulating an engineered 
hain of length N = 50 with a (relative) disorder of 5%, we
1 Introdu
tion
get a �delity peak of 0.95±0.02. For a disorder of 10% we get 0.85±0.05. The weakly

oupled system is very stable for o�-site disorder [85℄, but su�ers strongly from on-site
disorder (i.e. magneti
 �elds in z−dire
tion) at the ends of the 
hain. For example,
for a 
hain of N = 50 with edge 
ouplings j = 0.01 and the remaining 
ouplings
being J = 1, we �nd that already a magneti
 �eld of the order of 0.00001 lowers the
�delity to 0.87 ± 0.12. For �elds of the order of 0.00005 we �nd 0.45 ± 0.32. This is
be
ause these �u
tuations must be small with respe
t to the small 
oupling, so there
is a double s
aling. Also, the time-s
ale of the transfer is longer than in other s
hemes
(note though that this may sometimes even be useful for having enough time to pi
k up
the re
eived state). On the other hand, en
oding and time-dependent 
ontrol require
additional resour
es and gating operations. It is not possible to judge independently of
the realisation whi
h of the above s
hemes is the �most pra
ti
al� one. We summarise
the di�erent aspe
ts that are important in the following �ve 
riteria for quantum state
transfer:
1. High e�
ien
y: How does the �delity depend on the length of the 
hain? Whi
h
rate [100, 81, 74℄ 
an be a
hieved?
2. Minimal 
ontrol : How many operations are required to a
hieve a 
ertain �delity?
Where
is 
ontrol required?
3. Minimal resour
es: What additional resour
es are required?
4. Minimal design: How general is the 
oupling type
? What values of the 
oupling
strengths are allowed?
5. Robustness: How is the �delity a�e
ted by stati
 disorder, by time-dependent
disorder, by gate and timing errors, and by external noise su
h as de
oheren
e
and dissipation?
At the start of this resear
h, only the engineering and en
oding s
hemes were available.
The engineering s
hemes are strong in the points 2 and 3, but quite weak in the points
4 and 5. The en
oding s
heme on the other hand has its weakness in points 2 and 3.
It was hen
e desirable to develop more balan
ed s
hemes. Sin
e most experiments
in Quantum Information are extremely sensitive and at the 
utting edge of their
parameters (i.e. requiring extremely low temperatures, well tuned lasers, and so forth,
For example, gates at the ends of the 
hain are always needed for write-in and read-out, and thus
�
heaper� than gates along the 
hain. Global 
ontrol along the whole 
hain is often easier than
lo
al 
ontrol.
Often the 
oupling type is already �xed by the experiment
1 Introdu
tion
to maintain their quantum behaviour), we parti
ularly wanted to �nd s
hemes whi
h
are strong in the points 4 and 5. Also, from a more fundamental point of view, we
were interested in seeing how mu
h information on the state of a quantum 
hain 
ould
be obtained by the re
eiver in prin
iple, and how the re
eiver might even be able to
prepare states on the whole 
hain.
The main a
hievements of this thesis are two s
hemes for the transfer of quantum
information using measurements (Chapter 2 and 3) or unitary operations (Chapter 5
and 6) at the re
eiving end of the 
hain. Sin
e both s
hemes use 
onvergen
e properties
of quantum operations, it seemed natural to investigate these properties in a more
abstra
t way (Chapter 4). There, we found a new way of 
hara
terising mixing maps,
whi
h has appli
ations beyond quantum state transfer, and may well be relevant for
other �elds su
h as 
haos theory or statisti
al physi
s. Finally, in Chapter 7 we
dis
uss problems quantum state transfer in the presen
e of external noise. The results
in Chapters 3-6 were developed in 
ollaboration with Vittorio Giovannetti from S
uola
Normale, Pisa. Mu
h of the material dis
ussed in this thesis has been published or
submitted for publi
ation [101, 102, 103, 104, 105, 106, 107, 108, 109, 110℄.
2 Dual Rail en
oding
2.1 Introdu
tion
The role of measurement in quantum information theory has be
ome more a
tive
re
ently. Measurements are not only useful to obtain information about some state
or for preparation, but also, instead of gates, for quantum 
omputation [111℄. In the

ontext of quantum state transfer, it seems �rst that measurements would spoil the

oheren
e and destroy the state. The �rst indi
ation that measurements 
an a
tually
be used to transfer quantum information along anti-ferromagneti
 
hains was given
in [24℄. However there the measurements had to be performed along the whole 
hain.
This may in some 
ases be easier than to perform swap gates, but still requires high
lo
al a

essibility. We take a �hybrid� approa
h here: along the 
hain, we let the
system evolve 
oherently, but at the re
eiving end, we try to help the transfer by
measuring. The main disadvantage of the en
oding used in the proto
ols above is
that on
e the information dispersed, there is no way of �nding out where it is without
destroying it. A dual rail en
oding [112℄ as used in quantum opti
s on the other
hand allows us to perform parity type measurements that do not spoil the 
oheren
e
of the state that is sent. The out
ome of the measurement tells us if the state has
arrived at the end (
orresponding to a perfe
t state transfer) or not. We 
all this

on
lusively perfe
t state transfer. Moreover, by performing repetitive measurements,
the probability of su

ess 
an be made arbitrarily 
lose to unity. As an example of
su
h an amplitude delaying 
hannel, we show how two parallel Heisenberg spin 
hains

an be used as quantum wires. Perfe
t state transfer with a probability of failure lower
than P in a Heisenberg 
hain of N qubits 
an be a
hieved in a time-s
ale of the order
of 0.33J−1N1.7| lnP |. We demonstrate that our s
heme is more robust to de
oheren
e
and non-optimal timing than any s
heme using single spin 
hains.
We then generalise the dual rail en
oding to disordered quantum 
hains. The s
heme
performs well for both spatially 
orrelated and un
orrelated �u
tuations if they are
relatively weak (say 5%). Furthermore, we show that given a quite arbitrary pair of
quantum 
hains, one 
an 
he
k whether it is 
apable of perfe
t transfer by only lo
al
operations at the ends of the 
hains, and the system in the middle being a bla
k box.
2 Dual Rail en
oding
We argue that unless some spe
i�
 symmetries are present in the system, it will be

apable of perfe
t transfer when used with dual rail en
oding. Therefore our s
heme
puts minimal demand not only on the 
ontrol of the 
hains when using them, but also
on the design when building them.
This Chapter is organised as follows. In Se
tion 2.2, we suggest a s
heme for quan-
tum 
ommuni
ation using two parallel spin 
hains of the most natural type (namely
those with 
onstant 
ouplings). We require modest en
odings (or gates) and measure-
ments only at the ends of the 
hains. The state transfer is 
on
lusive, whi
h means
that it is possible to tell by the out
ome of a quantum measurement, without destroy-
ing the state, if the transfer took pla
e or not. If it did, then the transfer was perfe
t.
The transmission time for 
on
lusive transfer is not longer than for single spin 
hains.
In Se
tion 2.3, we demonstrate that our s
heme o�ers even more: if the transfer was
not su

essful, then we 
an wait for some time and just repeat the measurement,
without having to resend the state. By performing su�
iently many measurements,
the probability for perfe
t transfer approa
hes unity. Hen
e the transfer is arbitrarily
perfe
t. We will show in Se
tion 2.4 that the time needed to transfer a state with a
given probability s
ales in a reasonable way with the length of the 
hain. In Se
tion
2.5 we show that en
oding to parallel 
hains and the 
on
lusiveness also makes our
proto
ol more robust to de
oheren
e (a hitherto unaddressed issue in the �eld of quan-
tum 
ommuni
ation through spin 
hains). In the last part of this 
hapter, we show
how this s
heme 
an be generalised to disordered 
hains (Se
tions 2.6-2.10) and even

oupled 
hains (Se
tion 2.11).
2.2 S
heme for 
on
lusive transfer
We intend to propose our s
heme in a system-independent way with o

asional refer-
en
es to systems where 
onditions required by our s
heme are a
hieved. We assume
that our system 
onsists of two identi
al un
oupled spin-1/2-
hains (1) and (2) of
length N , des
ribed by the Hamiltonian
H = H(1) ⊗ I(2) + I(1) ⊗H(2) − EgI(1) ⊗ I(2). (2.1)
The term identi
al states that H(1) and H(2) are the same apart from the label of the
Hilbert spa
e they a
t on. The requirement of parallel 
hains instead of just one is
not a real problem, sin
e in many experimental realisations of spin 
hains, it is mu
h
easier to produ
e a whole bun
h of parallel un
oupled [48,49℄ 
hains than just a single
2 Dual Rail en
oding
Figure 2.1: Two quantum 
hains inter
onne
ting A and B. Control of the systems is
only possible at the two qubits of either end.
We assume that the ground state of ea
h 
hain is |0〉i, i.e. a ferromagneti
 ground
state, with H(i) |0〉i = Eg |0〉i , and that the subspa
e 
onsisting of the single spin
ex
itations |n〉i is invariant under H(i). Let us assume that the state that Ali
e wants
to send is at the �rst qubit of the �rst 
hain, i.e.
|ψA〉1 ≡ α |0〉1 + β |1〉1 , (2.2)
and that the se
ond 
hain is in the ground state |0〉2. The aim of our proto
ol is to
transfer quantum information from the 1st (�Ali
e�) to the Nth (�Bob�) qubit of the
�rst 
hain:
|ψA〉1 → |ψB〉1 ≡ α |0〉1 + β |N 〉1 . (2.3)
The �rst step (see also Fig. 2.2) is to en
ode the input qubit in a dual rail [112℄ by
applying a NOT gate on the �rst qubit of system (2) 
ontrolled by the �rst qubit of
system (1) being zero, resulting in a superposition of ex
itations in both systems,
|s(0)〉 = α |0, 1〉+ β |1, 0〉 , (2.4)
where we have introdu
ed the short notation |n,m〉 ≡ |n〉1 ⊗ |m〉2. This is assumed
to take pla
e in a mu
h shorter time-s
ale than the system dynami
s. Even though a
2-qubit gate in solid state systems is di�
ult, su
h a gate for 
harge qubits has been
reported [15℄. For the same qubits, Josephson arrays have been proposed as single spin

hains for quantum 
ommuni
ation [31℄. For this system, both requisites of our s
heme
are thus available. In fa
t, the demand that Ali
e and Bob 
an do measurements and
apply gates to their lo
al qubits (i.e. the ends of the 
hains) will be naturally ful�lled
in pra
ti
e sin
e we are suggesting a s
heme to transfer information between quantum

omputers (as des
ribed in Se
tion 1.2).
2 Dual Rail en
oding
(1) ◦ spin chain (1) tℓ • |ψB〉
(2) ⊕ spin chain (2) tℓ ⊕
✙ ❴❴❴❴❴❴❴❴
✤✤✤✤✤✤✤
❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
wait again if 0
success
 if 1
Alice Bob
Figure 2.2: Quantum 
ir
uit representation of 
on
lusive and arbitrarily perfe
t state
transfer. The �rst gate at Ali
e's qubits represents a NOT gate applied to the se
ond
qubit 
ontrolled by the �rst qubit being zero. The qubit |ψA〉1 on the left hand side
represents an arbitrary input state at Ali
e's site, and the qubit |ψB〉1 represents the
same state, su

essfully transferred to Bob's site. The tℓ-gate represents the unitary
evolution of the spin 
hains for a time interval of tℓ.
Under the system Hamiltonian, the ex
itation in Eq. (2.4) will travel along the two
systems. The state after the time t1 
an be written as
|φ(t1)〉 =
fn,1(t1) |s(n)〉 , (2.5)
where |s(n)〉 = α |0, n〉 + β |n, 0〉 and the 
omplex amplitudes fn,1(t1) are given by
Eq. (1.19). We 
an de
ode the qubit by applying a CNOT gate at Bob's site. Assuming
that this happens on a time-s
ale mu
h shorter than the evolution of the 
hain, the
resulting state is given by
fn,1(t1) |s(n)〉+ fN,1(t1) |ψB〉1 ⊗ |N〉2 . (2.6)
Bob 
an now perform a measurement on his qubit of system (2). If the out
ome of
this measurement is 1, he 
an 
on
lude that the state |ψ〉(1)1 has been su

essfully
transferred to him. This happens with the probability |fN,1(t1)|2 . If the out
ome is
0, the system is in the state
P (1)
fn,1(t1) |s(n)〉 , (2.7)
where P (1) = 1− |fN,1(t1)|2 is the probability of failure for the �rst measurement. If
the proto
ol stopped here, and Bob would just assume his state as the transferred one,
2 Dual Rail en
oding
the 
hannel 
ould be des
ribed as an amplitude damping 
hannel [54℄, with exa
tly the
same �delity as the single 
hain s
heme dis
ussed in [1℄. Note that here the en
oding is
symmetri
 with respe
t to α and β, so the minimal �delity is the same as the averaged
But su

ess probability is more valuable than �delity: Bob has gained knowledge
about his state, and may reje
t it and ask Ali
e to retransmit (this is known as a
quantum erasure 
hannel [113℄). Of 
ourse in general the state that Ali
e sends is the
unknown result of some quantum 
omputation and 
annot be sent again easily. This

an be over
ome in the following way: Ali
e sends one e-bit on the dual rail �rst. If
Bob measures a su

ess, he tells Ali
e, and they both start to teleport the unknown
state. If he measures a failure, they reset the 
hains and start again. Sin
e the joint
probability of failure 
onverges exponentially fast to zero this is quite e�
ient. In fa
t
the 
on
lusive transfer of entanglement is possible even on a single 
hain by using the
same 
hain again instead of a se
ond one [114℄. This 
an be seen as a very simple
entanglement distillation pro
edure, a
hieving a rate of |fN,1(t)|2/2. However the 
hain
needs to be reset between ea
h transmission (see Se
tion 1.4.1 for problems related
to this), and Ali
e and Bob require 
lassi
al 
ommuni
ation. We will show in the
next se
tion, that the reuse of the 
hain(s) is not ne
essary, as arbitrarily perfe
t state
transfer 
an already a
hieved in the �rst transmission.
2.3 Arbitrarily perfe
t state transfer
Be
ause Bob's measurement has not revealed anything about the input state (the
su

ess probability is independent of the input state), the information is still residing
in the 
hain. By letting the state (2.7) evolve for another time t2 and applying the
CNOT gate again, Bob has another 
han
e of re
eiving the input state. The state
before performing the se
ond measurement is easily seen to be
P (1)
{fn,1(t2 + t1)− fn,N(t2)fN,1(t1)} |s(n)〉 . (2.8)
Hen
e the probability to re
eive the qubit at Bobs site at the se
ond measurement is
P (1)
|fN,1(t2 + t1)− fN,N (t2)fN,1(t1)|2 . (2.9)
If the transfer was still unsu

essful, this strategy 
an be repeated over and over.
Ea
h time Bob has a probability of failed state transfer that 
an be obtained from the
2 Dual Rail en
oding
generalisation of Eq. (2.8) to an arbitrary number of iterations. The joint probability
that Bob fails to re
eive the state all the time is just the produ
t of these probabilities.
We denote the joint probability of failure for having done l unsu

essful measurements
as P (ℓ). This probability depends on the time intervals tℓ between the (ℓ− 1)th and
ℓth measurement, and we are interested in the 
ase where the tℓ are 
hosen su
h that
the transfer is fast. It is possible to write a simple algorithm that 
omputes P (ℓ)
for any transition amplitude fr,s(t). Figure 2.3 shows some results for the Heisenberg
Hamiltonian given by Eq. (1.21).
 1e-06
 1e-05
 1e-04
 0.001
 0.01
 0  5  10  15  20  25
Number of measurements
N=150
N=100
N=  50
N=  20
N=  10
N=    5
Figure 2.3: Semilogarithmi
 plot of the joint probability of failure P (ℓ) as a fun
tion of
the number of measurements ℓ. Shown are Heisenberg spin-1/2-
hains with di�erent
lengths N . The times between measurements tℓ have been optimised numeri
ally.
An interesting question is whether the joint probability of failure 
an be made
arbitrarily small with a large number of measurements. In fa
t, the times tℓ 
an be

hosen su
h that the transfer be
omes arbitrarily perfe
t. We will prove this in the
next Chapter, where a generalisation of the dual rail s
heme and a mu
h wider 
lass
of Hamiltonians is 
onsidered. In the limit of large number of measurements, the spin

hannel will not damp the initial amplitude, but only delay it.
2 Dual Rail en
oding
2.4 Estimation of the time-s
ale the transfer
The a
hievable �delity is an important, but not the only 
riterion of a state transfer
proto
ol. In this Se
tion, we give an heuristi
 approa
h to estimate the time that it
needs to a
hieve a 
ertain �delity in a Heisenberg spin 
hain. The 
omparison with
numeri
 examples is 
on�rming this approa
h.
Let us �rst des
ribe the dynami
 of the 
hain in a very qualitative way. On
e Ali
e
has initialised the system, an ex
itation wave pa
ket will travel along the 
hain. As
shown in Subse
tion 1.4.5, it will rea
h Bob at a time of the order of
, (2.10)
with an amplitude of
∣fN,1(t
≈ 1.82N−2/3. (2.11)
It is then re�e
ted and travels ba
k and forth along the 
hain. Sin
e the wave pa
ket
is also dispersing, it starts interfering with its tail, and after a 
ouple of re�e
tions
the dynami
 is be
oming quite randomly. This e�e
t be
omes even stronger due to
Bobs measurements, whi
h 
hange the dynami
s by proje
ting away parts of the wave
pa
ket. We now assume that 2t
(the time it takes for a wave pa
ket to travel
twi
e along the 
hain) remains a good estimate of the time-s
ale in whi
h signi�
ant
probability amplitude peaks at Bobs site o

ur, and that Eq. (2.11) remains a good
estimate of the amplitude of these peaks
. Therefore, the joint probability of failure
is expe
ted to s
ale as
P (ℓ) ≈
1− 1.82N−2/3
(2.12)
in a time of the order of
t(ℓ) ≈ 2tmaxℓ = J−1Nℓ. (2.13)
If we 
ombine Eq. (2.12) and (2.13) and solve for the time t(P ) needed to rea
h a

ertain probability of failure P , we get for N ≫ 1
t(P ) ≈ 0.55J−1N5/3 |lnP | . (2.14)
We 
ompare this rough estimate with exa
t numeri
al results in Fig. 2.4. The best �t
This is not a strong assumption. If the ex
itation was fully randomly distributed, the probability
would s
ale as N−1. By sear
hing for good arrival times, this 
an be slightly in
reased to N−2/3.
2 Dual Rail en
oding
for the range shown in the �gure is given by
t(P ) = 0.33J−1N5/3 |lnP | . (2.15)
We 
an 
on
lude that the transmission time for arbitrarily perfe
t transfer is s
aling
not mu
h worse with the length N of the 
hains than the single spin 
hain s
hemes.
Despite of the logarithmi
 dependen
e on P, the time it takes to a
hieve high �delity
is still reasonable. For example, a system with N = 100 and J = 20K ∗ kB will take
approximately 1.3ns to a
hieve a �delity of 99%. In many systems, de
oheren
e is

ompletely negligible within this time-s
ale. For example, some Josephson jun
tion
systems [115℄ have a de
oheren
e time of Tφ ≈ 500ns, while trapped ions have even
larger de
oheren
e times.
 0.1 0.2 0.3
 0.4 0.5 0.6
 0.7 0.8 0.9
 1000
 1500
 2000
 2500
Transfer Time [1/J]
Numerical data
Chain length
Joint probability of failure
Transfer Time [1/J]
Figure 2.4: Time t needed to transfer a state with a given joint probability of failure
P a
ross a 
hain of length N . The points denote exa
t numeri
al data, and the �t is
given by Eq. (2.15).
2 Dual Rail en
oding
2.5 De
oheren
e and imperfe
tions
If the 
oupling between the spins J is very small, or the 
hains are very long, the
transmission time may no longer be negligible with respe
t to the de
oheren
e time.
It is interesting to note that the dual rail en
oding then o�ers some signi�
ant general
advantages over single 
hain s
hemes. Sin
e we are suggesting a system-independent
s
heme, we will not study the e�e
ts of spe
i�
 environments on our proto
ol, but just
qualitatively point out its general advantages.
At least theoreti
ally, it is always possible to 
ool the system down or to apply a
strong magneti
 �eld so that the environment is not 
ausing further ex
itations. For
example in �ux qubit systems, the system is 
ooled to ≈ 25mK to ensure that the
energy splitting∆ ≫ kBT [116℄. Then, there are two remaining types of quantum
noise that will o

ur: phase noise and amplitude damping. Phase noise is a serious
problem and arises here only when an environment 
an distinguish between spin �ips
on the �rst 
hain and spin �ips on the se
ond 
hain. It is therefore important that the
environment 
annot resolve their di�eren
e. In this 
ase, the environment will only

ouple with the total z-
omponent
Z(1)n + Z
n (2.16)
of the spins of both 
hains at ea
h position n. This has been dis
ussed for spin-boson
models in [117,118℄ but also holds for spin environments as long as the 
hains are 
lose
enough. The qubit is en
oded in a de
oheren
e-free subspa
e [119℄ and the s
heme is
fully robust to phase noise. Even though this may not be true for all implementations
of dual rail en
oding, it is worthwhile noti
ing it be
ause su
h an opportunity does
not exist at all for single 
hain s
hemes, where the 
oheren
e between two states with
di�erent total z-
omponent of the spin has to be preserved. Having shown one way of
avoiding phase noise, at least in some systems, we now pro
eed to amplitude damping.
The evolution of the system in presen
e of amplitude damping of a rate Γ 
an be
easily derived using a quantum-jump approa
h [120℄. This is based on a quantum
master equation approa
h, whi
h is valid in the Born-Markov approximation [121℄
(i.e. it holds for weakly 
oupled environments without memory e�e
ts). Similarly to
phase noise, it is ne
essary that the environment a
ts symmetri
ally on the 
hains.
The dynami
s is then given by an e�e
tive non-Hermitian Hamiltonian
Heff = H + iΓ
Z(1)n + Z
n + 2
/2 (2.17)
2 Dual Rail en
oding
if no jump o

urs. If a jump o

urs, the system is ba
k in the ground state |0〉. The
state of the system before the �rst measurement 
onditioned on no jump is given by
fn,1(t) |s(n)〉 , (2.18)
and this happens with the probability of e−2Γt (the norm of the above state). If a
jump o

urs, the system will be in the ground state
1− e−2Γt |0, 0〉 . (2.19)
The density matrix at the time t is given by a mixture of (2.18) and (2.19). In 
ase
of (2.19), the quantum information is 
ompletely lost and Bob will always measure
an unsu

essful state transfer. If Bob however measures a su

ess, it is 
lear that no
jump has o

urred and he has the perfe
tly transferred state. Therefore the proto
ol
remains 
on
lusive, but the su

ess probability is lowered by e−2Γt. This result is
still valid for multiple measurements, whi
h leave the state (2.19) unaltered. The
probability of a su

essful transfer at ea
h parti
ular measurement ℓ will de
rease
by e−2Γt(ℓ), where t(ℓ) is the time at whi
h the measurement takes pla
e. After a

ertain number of measurements, the joint probability of failure will no longer de
rease.
Thus the transfer will no longer be arbitrarily perfe
t, but 
an still rea
h a very high
�delity. Some numeri
al examples of the minimal joint probability of failure that 
an
be a
hieved,
P (ℓ) ≈
1− 1.35N−2/3e−
(2.20)
are given in Fig. 2.5. For J/Γ = 50K ns nearly perfe
t transfer is still possible for

hains up to a length of N ≈ 40.
Even if the amplitude damping is not symmetri
, its e�e
t is weaker than in single
spin s
hemes. This is be
ause it 
an be split in a symmetri
 and asymmetri
 part.
The symmetri
 part 
an be over
ome with the above strategies. For example, if the
amplitude damping on the 
hains is Γ1 and Γ2 with Γ1 > Γ2, the state (2.18) will be
fn,1(t)
αe−Γ2t |0, n〉+ βe−Γ1t |n, 0〉
(2.21)
≈ e−Γ2t
fn,1(t) |s(n)〉 (2.22)
2 Dual Rail en
oding
provided that t ≪ (Γ1 − Γ2)−1 . Using a 
hain of length N = 20 with J = 20K ∗ kB
and Γ−11 = 4ns, Γ
2 = 4.2ns we would have to ful�l t ≪ 164ns. We 
ould perform
approximately 10 measurements (
f. Eq. (2.13)) without deviating too mu
h from the
state (2.22). In this time, we 
an use our proto
ol in the normal way. The resulting
su

ess probability given by the �nite version of Eq. (2.20) would be 75%. A similar
reasoning is valid for phase noise, where the environment 
an be split into 
ommon
and separate parts. If the 
hains are 
lose, the 
ommon part will dominate and the
separate parts 
an be negle
ted for short times.
 0  20  40  60  80  100  120  140  160  180  200
Chain length
1 K ns
10 K ns
25 K ns
50 K ns
100 K ns
200 K ns
Figure 2.5: The minimal joint probability of failure P (ℓ) for 
hains with length N in
the presen
e of amplitude damping. The parameter J/Γ of the 
urves is the 
oupling
of the 
hain (in Kelvin) divided by the de
ay rate (ns−1).
2.6 Disordered 
hains
The main requirement for perfe
t transfer with dual rail en
oding in the above is
that two identi
al quantum 
hains have to be designed. While this is not so mu
h
a theoreti
al problem, for possible experimental realizations of the s
heme [31℄ the
question arises naturally how to 
ope with slight asymmetries of the 
hannels. We
are now going to demonstrate that in many 
ases, perfe
t state transfer with dual rail
en
oding is possible for quantum 
hains with di�ering Hamiltonians.
By doing so, we also o�er a solution to another and perhaps more general problem:
2 Dual Rail en
oding
if one implements any of the s
hemes for quantum state transfer, the Hamiltonians will
always be di�erent from the theoreti
al ones by some random perturbation. This will
lead to a de
rease of �delity in parti
ular where spe
i�
 energy levels were assumed
(see [99,84℄ for an analysis of �u
tuations a�e
ting the engineered 
hains des
ribed in
Subse
tion 1.5.1). This problem 
an be avoided using the s
heme des
ribed below. In
general, disorder 
an lead to a Anderson lo
alisation [122,29,30℄ of the eigenstates (and
therefore to low �delity transport of quantum information). In this se
tion however
this is not relevant, as we 
onsider only short 
hains (N < 100) and small disorder
(≈ 10% of the 
oupling strength), and the lo
alisation length is mu
h longer then
the length of the 
hain. We will show numeri
ally that the dual rail s
heme 
an still
a
hieve arbitrarily perfe
t transfer for a uniformly 
oupled Heisenberg Hamiltonian
with disordered 
oupling strengths (both for the 
ase of spatially 
orrelated and un-

orrelated disorder). Moreover, for any two quantum 
hains, we show that Bob and
Ali
e 
an 
he
k whether their system is 
apable of dual rail transfer without dire
tly
measuring their Hamiltonians or lo
al properties of the system along the 
hains but
by only measuring their part of the system.
2.7 Con
lusive transfer in the presen
e of disorder
We 
onsider two un
oupled quantum 
hains (1) and (2), as shown in Fig. 2.6. The

hains are des
ribed by the two Hamiltonians H(1) and H(2) with total Hamiltonian
given by
H = H(1) ⊗ I(2) + I(1) ⊗H(2), (2.23)
and the time evolution operator fa
torising as
U(t) = exp
−iH(1)t
⊗ exp
−iH(2)t
. (2.24)
For the moment, we assume that both 
hains have equal length N , but it will be
ome

lear in Se
tion 2.9 that this is not a requirement of our s
heme. All other assumptions
remain as in the �rst part of the 
hapter.
Initially, Ali
e en
odes the state as
α |0, 1〉+ β |1, 0〉 . (2.25)
This is a superposition of an ex
itation in the �rst qubit of the �rst 
hain and an
2 Dual Rail en
oding
Figure 2.6: Two disordered quantum 
hains inter
onne
ting A and B. Control of the
systems is only possible at the two qubits of either end.
ex
itation in the �rst qubit of the se
ond 
hain. The state will evolve into
{αgn,1(t) |0, n〉+ βfn,1(t) |n, 0〉} , (2.26)
fn,1(t) ≡ 〈n, 0 |U(t)| 1, 0〉 (2.27)
gn,1(t) ≡ 〈0, n |U(t)| 0, 1〉 . (2.28)
In Se
tion 2.2, these fun
tions were identi
al. For di�ering 
hains this is no longer the

ase. We may, however, �nd a time t1 su
h that the modulus of their amplitudes at
the last spins are the same (see Fig. 2.7),
gN,1(t1) = e
iφ1fN,1(t1). (2.29)
At this time, the state (2.26) 
an be written as
{αgn,1(t1) |0, n〉+ βfn,1(t1) |n, 0〉}+
fN,1(t1)
eiφ1α |0,N 〉+ β |N, 0〉
. (2.30)
Bob de
odes the state by applying a CNOT gate on his two qubits, with the �rst qubit
2 Dual Rail en
oding
 0  5  10  15  20  25
Time [1/J]
|fN,1(t)|
|gN,1(t)|
Figure 2.7: The absolute values of the transition amplitudes fN,1(t) and gN,1(t) for
two Heisenberg 
hains of length N = 10. The 
ouplings strengths of both 
hains were

hosen randomly from the interval [0.8J, 1.2J ] . The 
ir
les show times where Bob 
an
perform measurements without gaining information on α and β.
as the 
ontrol bit. The state thereafter is
{αgn,1(t1) |0, n〉+ βfn,1(t1) |n, 0〉}+
fN,1(t1)
eiφ1α |0〉(1) + β |N 〉(1)
⊗ |N 〉(2) . (2.31)
Bob then measures his se
ond qubit. Depending on the out
ome of this measurement,
the systems will either be in the state
{αgn,1(t1) |0, n〉+ βfn,1(t1) |n, 0〉} (2.32)
or in
eiφ1α |0〉(1) + β |N〉(1)
⊗ |N 〉(2) , (2.33)
where p1 = 1−|fN,1(t1)|2 = 1−|gN,1(t1)|2 is the probability that Bob has not re
eived
the state. The state (2.33) 
orresponds to the 
orre
tly transferred state with a known
phase error (whi
h 
an be 
orre
ted by Bob using a simple phase gate). If Bob �nds the
system in the state (2.32), the transfer has been unsu

essful, but the information is
2 Dual Rail en
oding
still in the 
hain. We thus see that 
on
lusive transfer is still possible with randomly

oupled 
hains as long as the requirement (2.29) is met. This requirement will be
further dis
ussed and generalised in the next se
tion.
2.8 Arbitrarily perfe
t transfer in the presen
e of disorder
If the transfer was unsu

essful, the state (2.32) will evolve further, o�ering Bob
further opportunities to re
eive Ali
e's message. For identi
al quantum 
hains, leads
to a su

ess for any reasonable Hamiltonian (Se
tion 3.6). For di�ering 
hains, this is
not ne
essarily the 
ase, be
ause measurements are only allowed at times where the
probability amplitude at the end of the 
hains is equal, and there may be systems where
this is never the 
ase. In this se
tion, we will develop a 
riterion that generalises Eq.
(2.29) and allows to 
he
k numeri
ally whether a given system is 
apable of arbitrarily
perfe
t state transfer.
The quantity of interest for 
on
lusive state transfer is the joint probability P (ℓ)
that after having 
he
ked l times, Bob still has not re
eived the proper state at his end
of the 
hains. Optimally, this should approa
h zero if ℓ tends to in�nity. In order to
derive an expression for P (ℓ), let us assume that the transfer has been unsu

essful for
ℓ− 1 times with time intervals tℓ between the the ℓth and the (ℓ− 1)th measurement,
and 
al
ulate the probability of failure at the ℓth measurement. In a similar manner,
we assume that all the ℓ − 1 measurements have met the requirement of 
on
lusive
transfer (that is, Bob's measurements are unbiased with respe
t to α and β) and derive
the requirement for the ℓth measurement.
To 
al
ulate the probability of failure for the ℓth measurement, we need to take
into a

ount that Bob's measurements disturb the unitary dynami
s of the 
hain. If
the state before a measurement with the out
ome �failure� is |ψ〉 , the state after the
measurement will be
Q |ψ〉 , (2.34)
where Q is the proje
tor
Q = I − |0, N 〉 〈0, N | − |N, 0〉 〈N, 0| , (2.35)
and pℓ is the probability of failure at the lth measurement. The dynami
s of the

hain is alternating between unitary and proje
tive, su
h that the state before the ℓth
2 Dual Rail en
oding
measurement is given by
P (ℓ− 1)
{U(tk)Q} {α |1, 0〉+ β |0, 1〉} , (2.36)
where
P (ℓ− 1) =
pk. (2.37)
Note that the operators in (2.36) do not 
ommute and that the time ordering of the
produ
t (the index k in
reases from right to left) is important. The probability that
there is an ex
itation at the Nth site of either 
hain is given by
P (ℓ− 1)
|α|2 |F (ℓ)|2 + |β|2 |G(ℓ)|2
, (2.38)
F (ℓ) ≡ 〈N, 0|
{U(tk)Q} |1, 0〉 , (2.39)
G(ℓ) ≡ 〈0, N |
{U(tk)Q} |0, 1〉 . (2.40)
Bob's measurements are therefore unbiased with respe
t to α and β if and only if
|F (ℓ)| = |G(ℓ)| ∀ℓ. (2.41)
In this 
ase, the state 
an still be transferred 
on
lusively (up to a known phase). The
probability of failure at the ℓth measurement is given by
pℓ = 1−
|F (ℓ)|2
P (ℓ− 1) . (2.42)
It is easy (but not very enlightening) to show [103℄ that the 
ondition (2.41) is equiv-
alent to ∥
{U(tk)Q} |1, 0〉
{U(tk)Q} |0, 1〉
∀ℓ, (2.43)
and that the joint probability of failure - if at ea
h measurement the above 
ondition
2 Dual Rail en
oding
is ful�lled - is simply given by
P (ℓ) =
{U(tk)Q} |1, 0〉
. (2.44)
It may look as if Eq. (2.43) was a 
ompli
ated multi-time 
ondition for the measuring
times tℓ, that be
omes in
reasingly di�
ult to ful�l with a growing number of mea-
surements. This is not the 
ase. If proper measuring times have been found for the
�rst ℓ − 1 measurements, a trivial time tℓ that ful�ls Eq. (2.43) is tℓ = 0. In this

ase, Bob measures immediately after the (ℓ− 1)th measurement and the probability
amplitudes on his ends of the 
hains will be equal - and zero (a useless measurement).
But sin
e the left and right hand side of Eq. (2.43) when seen as fun
tions of tℓ are
both almost-periodi
 fun
tions with initial value zero, it is likely that they interse
t
many times, unless the system has some spe
i�
 symmetry or the systems are 
om-
pletely di�erent. Note that we do not 
laim at this point that any pair of 
hains will
be 
apable of arbitrary perfe
t transfer. We will dis
uss in the next system how one

an 
he
k this for a given system by performing some simple experimental tests.
2.9 Tomography
Suppose someone gives you two di�erent experimentally designed spin 
hains. It may
seem from the above that knowledge of the full Hamiltonian of both 
hains is ne
essary
to 
he
k how well the system 
an be used for state transfer. This would be a very
di�
ult task, be
ause we would need a

ess to all the spins along the 
hannel to
measure all the parameters of the Hamiltonian. In fa
t by expanding the proje
tors in
Eq. (2.43) one 
an easily see that the only matrix elements of the evolution operator
whi
h are relevant for 
on
lusive transfer are
fN,1(t) = 〈N, 0|U(t) |1, 0〉 (2.45)
fN,N(t) = 〈N, 0|U(t) |N, 0〉 (2.46)
gN,1(t) = 〈0, N |U(t) |0, 1〉 (2.47)
gN,N (t) = 〈0, N |U(t) |0, N〉 . (2.48)
Physi
ally, this means that the only relevant properties of the system are the transition
amplitudes to arrive at Bob's ends and to stay there. The modulus of fN,1(t) and
fN,N (t) 
an be measured by initialising the system in the states |1, 0〉 and |N, 0〉 and
then performing a redu
ed density matrix tomography at Bob's site at di�erent times
2 Dual Rail en
oding
t, and the 
omplex phase of these fun
tions is obtained by initialising the system
in (|0, 0〉+ |1, 0〉) /
2 and (|0, 0〉+ |N, 0〉) /
2 instead. In the same way, gN,1(t)
and gN,N (t) are obtained. All this 
an be done in the spirit of minimal 
ontrol at
the sending and re
eiving ends of the 
hain only, and needs to be done only on
e.
It is interesting to note that the dynami
s in the middle part of the 
hain is not
relevant at all. It is a bla
k box (see Fig. 2.8) that may involve even 
ompletely
di�erent intera
tions, number of spins, et
., as long as the total number of ex
itations
is 
onserved. On
e the transition amplitudes [Equations (2.45)-(2.48)℄ are known, one
Figure 2.8: The relevant properties for 
on
lusive transfer 
an be determined by mea-
suring the response of the two systems at their ends only.

an sear
h numeri
ally for optimised measurement times tℓ using Eq. (2.44) and the

ondition from Eq. (2.43).
One weakness of the s
heme des
ribed here is that the times at whi
h Bob measures
have to be very pre
ise, be
ause otherwise the measurements will not be unbiased
with respe
t to α and β. This demand 
an be relaxed by measuring at times where
not only the probability amplitudes are similar, but also their slope (see Fig. 2.7).
The 
omputation of these optimal timings for a given system may be 
ompli
ated, but
they only need to be done on
e.
2.10 Numeri
al Examples
In this se
tion, we show some numeri
al examples for two 
hains with Heisenberg

ouplings J whi
h are �u
tuating. The Hamiltonians of the 
hains i = 1, 2 are given
2 Dual Rail en
oding
H(i) =
J(1 + δ(i)n )
X(i)n X
n+1 + Y
n+1 + Z
, (2.49)
where δ
n are uniformly distributed random numbers from the interval [−∆,∆] . We
have 
onsidered two di�erent 
ases: in the �rst 
ase, the δ
n are 
ompletely un
orre-
lated (i.e. independent for both 
hains and all sites along the 
hain). In the se
ond

ase, we have taken into a

ount a spa
ial 
orrelation of the signs of the δ
n along ea
h
of the 
hains, while still keeping the two 
hains un
orrelated. For both 
ases, we �nd
that arbitrarily perfe
t transfer remains possible ex
ept for some very rare realisations
of the δ
Be
ause measurements must only be taken at times whi
h ful�l the 
ondition (2.43),
and these times usually do not 
oin
ide with the optimal probability of �nding an
ex
itation at the ends of the 
hains, it is 
lear that the probability of failure at ea
h
measurement will in average be higher than for 
hains without �u
tuations. Therefore,
more measurements have to be performed in order to a
hieve the same probability of
su

ess. The pri
e for noisy 
ouplings is thus a longer transmission time and a higher
number of gating operations at the re
eiving end of the 
hains. Some averaged values
are given in Table 2.1 for the Heisenberg 
hain with un
orrelated 
oupling �u
tuations.
∆ = 0 ∆ = 0.01 ∆ = 0.03 ∆ = 0.05 ∆ = 0.1
377 524 ± 27 694 ± 32 775± 40 1106 ± 248
M 28 43± 3 58± 3 65± 4 110 ± 25
Table 2.1: The total time t and the number of measurements M needed to a
hieve a
probability of su

ess of 99% for di�erent �u
tuation strengths ∆ (un
orrelated 
ase).
Given is the statisti
al mean and the standard deviation. The length of the 
hain is
N = 20 and the number of random samples is 10. For strong �u
tuations ∆ = 0.1,
we also found parti
ular samples where the su

ess probability 
ould not be a
hieved
within the time range sear
hed by the algorithm.
For the 
ase where the signs of the δ
n are 
orrelated, we have used the same model
as in [99℄, introdu
ing the parameter c su
h that
δ(i)n δ
n−1 > 0 with propability c, (2.50)
δ(i)n δ
n−1 < 0 with propability 1− c. (2.51)
2 Dual Rail en
oding
For c = 1 (c = 0) this 
orresponds to the 
ase where the signs of the 
ouplings
are 
ompletely 
orrelated (anti-
orrelated). For c = 0.5 one re
overs the 
ase of
un
orrelated 
ouplings. We 
an see from the numeri
al results in Table 2.2 that
arbitrarily perfe
t transfer is possible for the whole range of c.
c = 0 c = 0.1 c = 0.3 c = 0.7 c = 0.9 c = 1
666± 20 725± 32 755± 41 797± 35 882± 83 714± 41
M 256± 2 62± 3 65± 4 67± 4 77± 7 60± 4
Table 2.2: The total time t and the number of measurements M needed to a
hieve a
probability of su

ess of 99% for di�erent 
orrelations c between the 
ouplings [see Eq.
(2.50) and Eq. (2.51)℄. Given is the statisti
al mean and the standard deviation for a
�u
tuation strength of ∆ = 0.05. The length of the 
hain is N = 20 and the number
of random samples is 20.
For ∆ = 0, we know from Se
tion 2.4 that the time to transfer a state with proba-
bility of failure P s
ales as
t(P ) = 0.33J−1N1.6 |lnP | . (2.52)
If we want to obtain a similar formula in the presen
e of noise, we 
an perform a �t
to the exa
t numeri
al data. For un
orrelated �u
tuations of ∆ = 0.05, this is shown
in Fig. 2.9. The best �t is given by
t(P ) = 0.2J−1N1.9 |lnP | . (2.53)
We 
on
lude that weak �u
tuations (say up to 5%) in the 
oupling strengths do not
deteriorate the performan
e of our s
heme mu
h for the 
hain lengths 
onsidered. Both
the transmission time and the number of measurements raise, but still in a reasonable
way [
f. Table 2.1 and Fig. 2.9℄. For larger �u
tuations, the s
heme is still appli
able in
prin
iple, but the amount of junk (i.e. 
hains not 
apable of arbitrary perfe
t transfer)
may get too large.
Note that we have 
onsidered the 
ase where the �u
tuations δin are 
onstant in
time. This is a reasonable assumption if the dynami
 �u
tuations (e.g. those arising
from thermal noise) 
an be negle
ted with respe
t to the 
onstant �u
tuations (e.g.
those arising from manufa
turing errors). If the �u
tuations were varying with time,
the tomography measurements in Se
. 2.9 would involve a time-average, and Bob
would not measure exa
tly at the 
orre
t times. The transferred state (2.33) would
then be a�e
ted by both phase and amplitude noise.
2 Dual Rail en
oding
 0.1 0.2 0.3
 0.4 0.5 0.6
 0.7 0.8 0.9
 1000
 1500
 2000
 2500
 3000
 3500
 4000
 4500
 5000
Transfer Time [1/J]
Numerical data
Chain length
Joint probability of failure
Transfer Time [1/J]
Figure 2.9: Time t needed to transfer a state with a given joint probability of failure
P a
ross a 
hain of length N with un
orrelated �u
tuations of ∆ = 0.05. The points
denote numeri
al data averaged over 100 realisations, and the �t is given by Eq. (2.53).
This �gure should be 
ompared with Fig. 2.4 where ∆ = 0.
2.11 Coupled 
hains
Let us look at the 
ondition for 
on
lusive transfer in the more general s
enario indi-

ated by Fig. 2.10: Ali
e and Bob have a bla
k box a
ting as an amplitude damping

hannel in the following way. It has two inputs and two outputs. If Ali
e puts in state
in the dual rail,
|ψ〉 = α|01〉 + β|10〉, (2.54)
where α and β are arbitrary and unknown normalised amplitudes, then the output at
Bob is given by
p|φ〉〈φ| + (1− p) |00〉〈00|, (2.55)
with a normalised �su

ess� state
|φ〉 = 1√
αf |01〉 + βg|10〉 + αf̃ |10〉 + βg̃|01〉
. (2.56)
2 Dual Rail en
oding
This bla
k box des
ribes the behaviour of an arbitrarily 
oupled qubit system that

onserves the number of ex
itations and that is initialised in the all zero state, in
luding
parallel un
oupled 
hains, and 
oupled 
hains.
Figure 2.10: Most general setting for 
on
lusive transfer: A bla
k box with two inputs
and two outputs, a
ting as an amplitude damping 
hannel de�ned by Eqs. (2.54) and
(2.55)
From the normalisation of |φ〉 it follows that
p = p(α, β) = |αf + βg̃|2 +
∣βg + αf̃
. (2.57)
We are interested in 
on
lusive transfer: by measuring the observable |00〉〈00| the
Bob 
an proje
t the output onto either the failure state |00〉 or |φ〉. This is 
learly
possible, but the question is if the output |φ〉 and the input |ψ〉 are related by a
unitary operation.
If Bob is able to re
over the full information that Ali
e sent, then p(α, β) must be
independent of α and β (otherwise, some information on these amplitudes 
ould be
obtained by the measurement already, whi
h 
ontradi
ts the non-
loning theorem [2℄).
This implies that p(1, 0) = p(0, 1), i.e.
|f |2 +
= |g̃|2 + |g|2 . (2.58)
Be
ause
|f + g̃|2 + 1
∣g + f̃
(2.59)
= p(1, 0) + Re
f∗g̃ + gf̃∗
(2.60)
it also implies that
f∗g̃ + gf̃∗
= 0. (2.61)
2 Dual Rail en
oding
Using the same tri
k for p
we get that Im
f∗g̃ + gf̃∗
= 0 and therefore
f∗g̃ + gf̃∗ = 0. (2.62)
If we write |ψ〉 = U |φ〉 we get
, (2.63)
whi
h is a unitary operator if Eq. (2.58) and (2.62) hold. We thus 
ome to the

on
lusion that 
on
lusive transfer with the bla
k box de�ned above is possible if and
only if the probability p is independent of α and β. It is interesting to note that a
verti
al mirror symmetry of the system does not guarantee this. A 
ounterexample
is sket
hed in Fig. 2.11: 
learly the initial (�dark�) state |01〉 − |10〉 does not evolve,
whereas |01〉 + |10〉 does. Hen
e the probability must depend on α and β. A trivial

ase where 
on
lusive transfer works is given by two un
oupled 
hains, at times where
|f |2 = |g|2. This was dis
ussed in Se
t. 2.8. A non-trivial example is given by the

oupled system sket
hed in Fig. 2.12. This 
an be seen by splitting the Hamiltonian
in a horizontal and verti
al 
omponent,
H = Hv +Hz. (2.64)
By applying HvHz and HzHv on single-ex
itation states it is easily 
he
ked that they

ommute in the �rst ex
itation se
tor (this is not longer true in higher se
tors). Sin
e
the probability is independent of α and β in the un
oupled 
ase it must also be true
in the 
oupled 
ase (a rotation in the subspa
e {|01〉, |10〉} does not harm).
Figure 2.11: A simple 
ounterexample for a verti
ally symmetri
 system where dual
rail en
oding is not possible. The bla
k lines represent ex
hange 
ouplings.
A �nal remark - as Ali
e and Bob alway only deal with the states {|00〉, |10〉, |01〉}
it is obvious that the en
oding used in this 
hapter is really living on qutrits. In some
sense it would be more natural to 
onsider permanently 
oupled systems of qutrits,
2 Dual Rail en
oding
Figure 2.12: An example for a verti
ally symmetri
 system where dual rail en
oding
is possible. The bla
k lines represent ex
hange 
ouplings of equal strength.
su
h as SU(3) 
hains [123,102,124,125℄. The �rst level of the qutrit |0〉 is then used as
a marker for �no information here�, whereas the information is en
oded in the states
|1〉 and |2〉. One would have to ensure that there is no transition between |0〉 and
|1〉, |2〉, and that the system is initialised in the all zero state.
2.12 Con
lusion
In 
on
lusion, we have presented a simple s
heme for 
on
lusive and arbitrarily per-
fe
t quantum state transfer. To a
hieve this, two parallel spin 
hains (individually
amplitude damping 
hannels) have been used as one amplitude delaying 
hannel. We
have shown that our s
heme is more robust to de
oheren
e and imperfe
t timing than
the single 
hain s
hemes. We have also shown that the s
heme is appli
able to dis-
ordered and 
oupled 
hains. The s
heme 
an be used as a way of improving any of
the other s
hemes from the introdu
tion. For instan
e, one may try to engineer the

ouplings to have a very high probability of su

ess already at the �rst measurement,
and use further measurements to 
ompensate the errors of implementing the 
orre
t
values for the 
ouplings. We remark that the dual rail proto
ol is unrelated to error
�ltration [126℄ where parallel 
hannels are used for �ltering out environmental e�e
ts
on �ying qubits, whereas the purpose of the dual rail proto
ol is to ensure the ar-
rival of the qubit. Indeed one 
ould 
ombine both proto
ols to send a qubit on say
four rails to ensure the arrival and �lter errors. Finally, we note that in some re
ent
work [80℄ it was shown that our en
oding 
an be used to perform quantum gates while
the state is transferred, and that it 
an in
rease the 
onvergen
e speed if one performs
measurements at intermediate positions [110, 127℄.
3 Multi Rail en
oding
3.1 Introdu
tion
In quantum information theory the rate R of transferred qubits per 
hannel is an
important e�
ien
y parameter [70℄. Therefore one question that naturally arises is
whether or not there is any spe
ial meaning in the 1/2 value of R a
hieved in the
dual rail proto
ol of the last 
hapter. We will show now that this is not the 
ase,
be
ause there is a way of bringing R arbitrarily 
lose to 1 by 
onsidering multi rail
en
odings. Furthermore, in Se
tion 2.3 it was still left open for whi
h Hamiltonians
the probability of su

ess 
an be made arbitrarily 
lose to 1. Here, we give a su�
ient
and easily attainable 
ondition for a
hieving this goal.
This 
hapter is organised as follows: the model and the notation are introdu
ed in
Se
. 3.2. The e�
ien
y and the �delity of the proto
ol are dis
ussed in Se
. 3.3 and
in Se
. 3.4, respe
tively. Finally in Se
. 3.5 we prove a theorem whi
h provides us
with a su�
ient 
ondition for a
hieving e�
ient and perfe
t state transfer in quantum

hains.
3.2 The model
Assume that the two 
ommuni
ating parties operate on M independent (i.e. non
intera
ting) 
opies of the 
hain. This is quite a 
ommon attitude in quantum informa-
tion theory [70℄ where su

essive uses of a memoryless 
hannel are formally des
ribed
by introdu
ing many parallel 
opies of the 
hannel (see [54℄ for a dis
ussion on the
possibility of applying this formal des
ription to quantum 
hain models). Moreover for
the 
ase at hand the assumption of Ali
e and Bob dealing with �real� parallel 
hains
seems reasonable also from a pra
ti
al point of view [48, 49℄. The idea is to use these

opies to improve the overall �delity of the 
ommuni
ation. As usual, we assume Ali
e
and Bob to 
ontrol respe
tively the �rst and last qubit of ea
h 
hain (see Fig. 3.1). By
preparing any superposition of her spins Ali
e 
an in prin
iple transfer up toM logi
al
qubits. However, in order to improve the 
ommuni
ation �delity the two parties will
�nd it more 
onvenient to redundantly en
ode only a small number (say Q(M) 6M)
3 Multi Rail en
oding
Length N
Figure 3.1: S
hemati
 of the system: Ali
e and Bob operateM 
hains, ea
h 
ontaining
N spins. The spins belonging to the same 
hain intera
t through the Hamiltonian H
whi
h a

ounts for the transmission of the signal in the system. Spins of di�erent

hains do not intera
t. Ali
e en
odes the information in the �rst spins of the 
hains
by applying unitary transformations to her qubits. Bob re
overs the message in the
last spins of the 
hains by performing joint measurements.
of logi
al qubits in the M spins. By adopting these strategies Ali
e and Bob are ef-
fe
tively sa
ri�
ing the e�
ien
y R(M) = Q(M)/M of their 
ommuni
ation line in
order to in
rease its �delity. This is typi
al of any 
ommuni
ation s
heme and it is
analogous to what happens in quantum error 
orre
tion theory, where a single logi
al
qubit is stored in many physi
al qubits. In the last 
hapter we have seen that for
M = 2 it is possible to a
hieve perfe
t state transfer of a single logi
al qubit with an
e�
ien
y equal to 1/2. Here we will generalise su
h result by proving that there exist
an optimal en
oding-de
oding strategy whi
h asymptoti
ally allows to a
hieve perfe
t
state transfer and optimal e�
ien
y, i.e.
R(M) = 1 . (3.1)
Our strategy requires Ali
e to prepare superpositions of the M 
hains where ∼ M/2
of them have a single ex
itation in the �rst lo
ation while the remaining are in |0〉.
Sin
e in the limit M >> 1 the number of qubit transmitted is log
≈ M , this
ar
hite
ture guarantees optimal e�
ien
y (3.1). On the other hand, our proto
ol
requires Bob to perform 
olle
tive measurements on his spins to determine if all the
∼ M/2 ex
itations Ali
e is transmitting arrived at his lo
ation. We will prove that
3 Multi Rail en
oding
by repeating these dete
tions many times, Bob is able to re
over the messages with
asymptoti
ally perfe
t �delity.
Before beginning the analysis let us introdu
e some notation. The following de�-
nitions look more 
ompli
ated than they really are; unfortunately we need them to

arefully de�ne the states that Ali
e uses for en
oding the information. In order to
distinguish the M di�erent 
hains we introdu
e the label m = 1, · · · ,M : in this for-
malism |n〉m represents the state of m-th 
hain with a single ex
itation in the n-th
spin. In the following we will be interested in those 
on�gurations of the whole system
where K 
hains have a single ex
itation while the remaining M −K are in |0〉, as in
the 
ase
|1〉1 ⊗ |1〉2 · · · ⊗ |1〉K ⊗ |0〉K+1 · · · ⊗ |0〉M (3.2)
where for instan
e the �rst K 
hains have an ex
itation in the �rst 
hain lo
ation.
Another more general example is given in Fig. 3.2. The 
omplete 
hara
terisation of
these ve
tors is obtained by spe
ifying i) whi
h 
hains possess a single ex
itation and
ii) where these ex
itations are lo
ated horizontally along the 
hains. In answering to
the point i) we introdu
e the K-element subsets Sℓ, 
omposed by the labels of those

hains that 
ontain an ex
itation. Ea
h of these subsets Sℓ 
orresponds to a subspa
e
of the Hilbert spa
e H(Sℓ) with a dimension NK . The total number of su
h subsets
is equal to the binomial 
oe�
ient
, whi
h 
ounts the number of possibilities in
whi
h K obje
ts (ex
itations) 
an be distributed among M parties (parallel 
hains).
In parti
ular for any ℓ = 1, · · · ,
the ℓ-th subset Sℓ will be spe
i�ed by assigning
its K elements, i.e. Sℓ ≡ {m(ℓ)1 , · · · ,m
K } with m
j ∈ {1, · · · ,M} and m
j < m
for all j = 1, · · · ,K. To 
hara
terise the lo
ation of the ex
itations, point ii), we will
introdu
e instead theK-dimensional ve
tors ~n ≡ (n1, · · · , nK) where nj ∈ {1, · · · , N}.
We 
an then de�ne
|~n; ℓ〉〉 ≡
|nj〉m(ℓ)j
m′∈Sℓ
|0〉m′ , (3.3)
where Sℓ is the 
omplementary of Sℓ to the whole set of 
hains.
The state (3.3) represents a 
on�guration where the j-th 
hain of the subset Sℓ is in
|nj〉 while the 
hains that do not belong to Sℓ are in |0〉 (see Fig. 3.2 for an expli
it
example). The kets |~n; ℓ〉〉 are a natural generalisation of the states |n〉1 ⊗ |0〉2 and
|0〉1⊗|n〉2 whi
h were used for the dual rail en
oding. They are useful for our purposes
be
ause they are mutually orthogonal, i.e.
〈〈~n; ℓ|~n′; ℓ′〉〉 = δℓℓ′ δ~n~n′ , (3.4)
3 Multi Rail en
oding
Length 6
Figure 3.2: Example of our notation for M = 5 
hains of length N = 6 with K = 2
ex
itations. The state above, given by |0〉1 ⊗ |3〉2 ⊗ |0〉3 ⊗ |1〉4 ⊗ |0〉5, has ex
itations
in the 
hains m1 = 2 and m2 = 4 at the horizontal position n1 = 3 and n2 = 1. It
is in the Hilbert spa
e H(S6) 
orresponding to the subset S6 = {2, 4} (assuming that
the sets Sℓ are ordered in a 
anoni
al way, i.e. S1 = {1, 2}, S2 = {1, 3} and so on) and
will be written as |(3, 1); 6〉〉. There are
= 10 di�erent sets Sℓ and the number of
qubits one 
an transfer using these states is log2 10 ≈ 3. The e�
ien
y is thus given
by R ≈ 3/5 whi
h is already bigger than in the dual rail s
heme.
and their time evolution under the Hamiltonian does not depend on ℓ. Among the
ve
tors (3.3) those where all the K ex
itations are lo
ated at the beginning of the Sℓ

hains play an important role in our analysis. Here ~n = ~1 ≡ (1, · · · , 1) and we 
an
write
|~1; ℓ〉〉 ≡
m′∈Sℓ
|0〉m′ . (3.5)
A

ording to Eq. (3.4), for ℓ = 1, · · · ,
these states form orthonormal set of
elements. Analogously by 
hoosing ~n = ~N ≡ (N, · · · , N) we obtain the orthonormal
set of
ve
tors
| ~N ; ℓ〉〉 ≡
|N 〉m
m′∈Sℓ
|0〉m′ , (3.6)
where all the K ex
itations are lo
ated at the end of the 
hains.
3 Multi Rail en
oding
3.3 E�
ient en
oding
If all theM 
hains of the system are originally in |0〉, the ve
tors (3.5) 
an be prepared
by Ali
e by lo
ally operating on her spins. Moreover sin
e these ve
tors span a
dimensional subspa
e, Ali
e 
an en
ode in the 
hain Q(M,K) = log2
qubits of
logi
al information by preparing the superpositions,
|Φ〉〉 =
Aℓ |~1; ℓ〉〉 , (3.7)
with Aℓ 
omplex 
oe�
ients. The e�
ien
y of su
h en
oding is hen
e R(M,K) =
log2 (
whi
h maximised with respe
t to K gives,
R(M) =
for M even
(M−1)/2
for M odd .
(3.8)
The Stirling approximation 
an then be used to prove that this en
oding is asymptot-
i
ally e�
ient (3.1) in the limit of large M , e.g.
≈ log2
(M/2)M
=M. (3.9)
Note that already for M = 5 the en
oding is more e�
ient (
f. Fig. 3.2) than in the
dual rail en
oding. In the remaining of the 
hapter we show that the en
oding (3.7)
provides perfe
t state transfer by allowing Bob to perform joint measurements at his
end of the 
hains.
3.4 Perfe
t transfer
Sin
e the M 
hains do not intera
t with ea
h other and possess the same free Hamil-
tonian H, the unitary evolution of the whole system is des
ribed by U(t) ≡ ⊗mum(t),
with um(t) being the operator a
ting on the m-th 
hain. The time evolved of the input
|~1; ℓ〉〉 of Eq. (3.5) is thus equal to
U(t)|~1; ℓ〉〉 =
F [~n,~1; t] |~n; ℓ〉〉 , (3.10)
where the sum is performed for all nj = 1, · · · , N and
F [~n, ~n′; t] ≡ fn1,n′1(t) · · · fnK ,n′K (t) , (3.11)
3 Multi Rail en
oding
is a quantity whi
h does not depend on ℓ. In Eq. (3.10) the term ~n = ~N 
orresponds
to having all the K ex
itations in the last lo
ations of the 
hains. We 
an thus write
U(t)|~1; ℓ〉〉 = γ1(t)| ~N ; ℓ〉〉+
1− |γ1(t)|2 |ξ(t); ℓ〉〉 , (3.12)
where
γ1(t) ≡ 〈〈 ~N ; ℓ|U(t)|~1; ℓ〉〉 = F [ ~N,~1; t] (3.13)
is the probability amplitude that all the K ex
itation of |~1; ℓ〉〉 arrive at the end of the

hains, and
|ξ(t); ℓ〉〉 ≡
~n 6= ~N
F1[~n,~1; t] |~n; ℓ〉〉 , (3.14)
F1[~n,~1; t] ≡
F [~n,~1; t]
1− |γ1(t)|2
, (3.15)
is a superposition of terms where the number of ex
itations arrived to the end of the

ommuni
ation line is stri
tly less then K. It is worth noti
ing that Eq. (3.4) yields
the following relations,
〈〈 ~N ; ℓ|ξ(t); ℓ′〉〉 = 0, 〈〈ξ(t); ℓ|ξ(t); ℓ′〉〉 = δℓℓ′ , (3.16)
whi
h shows that {||ξ(t); ℓ〉〉} is an orthonormal set of ve
tors whi
h spans a subspa
e
orthogonal to the states | ~N ; ℓ〉〉. The time evolution of the input state (3.7) follows by
linearity from Eq. (3.12), i.e.
|Φ(t)〉〉 = γ1(t) |Ψ〉〉+
1− |γ1(t)|2 |Ψ(t)〉〉 , (3.17)
|Ψ(t)〉〉 ≡
Aℓ |ξ(t); ℓ〉〉 ,
|Ψ〉〉 ≡
Aℓ | ~N ; ℓ〉〉 . (3.18)
The ve
tors |Ψ〉〉 and |Ψ(t)〉〉 are unitary transformations of the input message (3.7)
where the orthonormal set {|~1; ℓ〉〉} has been rotated into {| ~N ; ℓ〉〉} and {|ξ(t); ℓ〉〉}
respe
tively. Moreover |Ψ〉〉 is the 
on�guration we need to have for perfe
t state
3 Multi Rail en
oding
transfer at the end of the 
hain. In fa
t it is obtained from the input message (3.7)
by repla
ing the 
omponents |1〉 (ex
itation in the �rst spin) with |N 〉 (ex
itation in
the last spin). From Eq. (3.16) we know that |Ψ〉〉 and |Ψ(t)〉〉 are orthogonal. This
property helps Bob to re
over the message |Ψ〉〉 from |Φ(t)〉〉: he only needs to perform
a 
olle
tive measurement on the M spins he is 
ontrolling to establish if there are K
or less ex
itations in those lo
ations. The above is 
learly a proje
tive measurement
that 
an be performed without destroying the quantum 
oheren
e asso
iated with the

oe�
ients Aℓ. Formally this 
an des
ribed by introdu
ing the observable
Θ ≡ 1−
| ~N ; ℓ〉〉〈〈 ~N ; ℓ| . (3.19)
A single measurement of Θ on |Φ(t1)〉〉 yields the out
ome 0 with probability p1 ≡
|γ1(t1)|2, and the out
ome +1 with probability 1 − p1. In the �rst 
ase the system
will be proje
ted in |Ψ〉〉 and Bob will get the message. In the se
ond 
ase instead the
state of the system will be
ome |Ψ(t1)〉〉. Already at this stage the two 
ommuni
ating
parties have a su

ess probability equal to p1. Moreover, as in the dual rail proto
ol,
the 
hannels have been transformed into a quantum erasure 
hannel [113℄ where the
re
eiver knows if the transfer was su

essful. Just like the dual rail en
oding, this
en
oding 
an be used as a simple entanglement puri�
ation method in quantum 
hain
transfer (see end of Se
tion 2.2). The rate of entanglement that 
an be distilled is
given by
∣F [ ~N,~1; t]
= R(M)p(t)⌊M/2⌋, (3.20)
where we used Eq. (3.11) and p(t) ≡ |fN,1(t)|2 . As we 
an see, in
reasing M on
one hand in
reases R(M), but on the other hand de
reases the fa
tor p(t)⌊M/2⌋. Its
maximum with respe
t toM gives us a lower bound of the entanglement of distillation
for a single spin 
hain, as shown in Fig. 1.11. We 
an also see that it be
omes worth
en
oding on more than three 
hains for 
on
lusive transfer only when p(t) > 0.8.
Consider now what happens when Bob fails to get the right answer from the mea-
surement. The state on whi
h the 
hains is proje
ted is expli
itly given by
|Ψ(t1)〉〉 =
~n 6= ~N
F1[~n,~1; t1]
Aℓ|~n; ℓ〉〉 . (3.21)
Let us now 
onsider the evolution of this state for another time interval t2. By repeat-
ing the same analysis given above we obtain an expression similar to (3.17), i.e.
|Φ(t2, t1)〉〉 = γ2 |Ψ〉〉+
1− |γ2|2 |Ψ(t2, t1)〉〉 , (3.22)
3 Multi Rail en
oding
where now the probability amplitude of getting all ex
itation in the N -th lo
ations is
des
ribed by
~n 6= ~N
F [ ~N,~n; t2] F1[~n,~1; t1]. (3.23)
In this 
ase |Ψ(t)〉〉 is repla
ed by
|Ψ(t2, t1)〉〉 =
Aℓ |ξ(t2, t1); ℓ〉〉 , (3.24)
|ξ(t2, t1); ℓ〉〉 =
~n 6= ~N
F2[~n,~1; t2, t1]|~n; ℓ〉〉, (3.25)
and F2 de�ned as in Eq. (3.27) (see below). In other words, the state |Φ(t2, t1)〉〉 
an
be obtained from Eq. (3.17) by repla
ing γ1 and F1 with γ2 and F2. Bob 
an hen
e
try to use the same strategy he used at time t1: i.e. he will 
he
k whether or not
his M qubits 
ontain K ex
itations. With (
onditional) probability p2 ≡ |γ2|2 he will
get a positive answer and his quantum register will be proje
ted in the state |Ψ〉〉 of
Eq. (3.18). Otherwise he will let the system evolve for another time interval t3 and
repeat the proto
ol. By reiterating the above analysis it is possible to give a re
ursive
expression for the 
onditional probability of su

ess pq ≡ |γq|2 after q − 1 su

essive
unsu

essful steps. The quantity γq is the analogue of γ2 and γ1 of Eqs. (3.13) and
(3.22). It is given by
~n 6= ~N
F [ ~N,~n; tq] Fq−1[~n,~1, tq−1, · · · , t1] , (3.26)
where
Fq−1[~n,~1; tq−1, · · · , t1] (3.27)
~n′ 6= ~N
F [ ~N,~n′; tq−1]
1− |γq−1|2
Fq−2[~n
′,~1; tq−2, · · · , t1]
and F1[~n,~1, t] is given by Eq. (3.15). In these equations tq, · · · , t1 are the time-intervals
that o

urred between the various proto
ol steps. Analogously the 
onditional proba-
bility of failure at the step q is equal to 1−pq. The probability of having j−1 failures
and a su

ess at the step j-th 
an thus be expressed as
π(j) = pj(1− pj−1)(1 − pj−2) · · · (1− p1) , (3.28)
3 Multi Rail en
oding
while the total probability of su

ess after q steps is obtained by the sum of π(j) for
all j = 1, · · · , q, i.e.
π(j) . (3.29)
Sin
e pj > 0, Eq. (3.29) is a monotoni
 fun
tion of q. As a matter of fa
t in the next
se
tion we prove that under a very general hypothesis on the system Hamiltonian, the
probability of su

ess Pq 
onverges to 1 in the limit of q → ∞. This means that by
repeating many times the 
olle
tive measure des
ribed by Θ Bob is guaranteed to get,
sooner or later, the answer 0 and hen
e the message Ali
e sent to him. In other words
our proto
ol allows perfe
t state transfer in the limit of repetitive 
olle
tive measures.
Noti
e that the above analysis applies for all 
lasses of subsets Sℓ. The only di�eren
e
between di�erent 
hoi
es of K is in the velo
ity of the 
onvergen
e of Pq → 1. In any

ase, by 
hoosing K ∼ M/2 Ali
e and Bob 
an a
hieve perfe
t �delity and optimal
e�
ien
y.
3.5 Convergen
e theorem
Theorem 3.1 (Arbitrarly perfe
t transfer) If there is no eigenve
tor |em〉 of
the quantum 
hain Hamiltonian H whi
h is orthogonal to |N 〉, then there is a

hoi
e of the times intervals tq, tq−1, · · · , t1 su
h that the �delity 
onverges to 1 as
q → ∞.
Before proving this Theorem, let us give an intuitive reasoning for the 
onvergen
e.
The unitary evolution 
an be thought of of a rotation in some abstra
t spa
e, while
the measurement 
orresponds to a proje
tion. The dynami
s of the system is then
represented by alternating rotations and proje
tions. In general this will de
rease the
norm of ea
h ve
tor to null, unless the rotation axis is the same as the proje
tion axis.
Proof The state of the system at a time interval of tq after the (q− 1)-th failure 
an
be expressed in 
ompa
t form as follows
|Φ(tq, · · · , t1)〉〉 =
U(tq)ΘU(tq−1)Θ · · ·U(t1)Θ|Φ〉〉
(1− pq−1) · · · (1− p1)
(3.30)
with U(t) the unitary time evolution generated by the system Hamiltonian, and with
3 Multi Rail en
oding
Θ the proje
tion de�ned in Eq. (3.19). One 
an verify for instan
e that for q = 2, the
above equation 
oin
ides with Eq. (3.22). [For q = 1 this is just (3.17) evaluated at
time t1℄. By de�nition the 
onditional probability of su

ess at step q-th is equal to
pq ≡ |〈〈Ψ|Φ(tq, · · · , t1)〉〉|2. (3.31)
Therefore, Eq. (3.28) yields
π(q) = |〈〈Ψ|U(tq)ΘU(tq−1)Θ · · ·U(t1)Θ|Φ〉〉|2 (3.32)
= |〈〈 ~N ; ℓ|U(tq)ΘU(tq−1)Θ · · ·U(t1)Θ|~1; ℓ〉〉|2 ,
where the se
ond identity stems from the fa
t that, a

ording to Eq. (3.4), U(t)Θ
preserves the orthogonality relation among states |~n; ℓ〉〉 with distin
t values of ℓ. In
analogy to the 
ases of Eqs. (3.11) and (3.13), the se
ond identity of (3.32) establishes
that π(q) 
an be 
omputed by 
onsidering the transfer of the input |~1; ℓ〉〉 for arbitrary
ℓ. The expression (3.32) 
an be further simpli�ed by noti
ing that for a given ℓ
the 
hains of the subset Sℓ 
ontribute with a unitary fa
tor to π(q) and 
an be thus
negle
ted (a

ording to (3.5) they are prepared in |0〉 and do not evolve under U(t)Θ).
Identify |~1〉〉ℓ and | ~N 〉〉ℓ with the 
omponents of |~1; ℓ〉〉 and | ~N ; ℓ〉〉 relative to the 
hains
belonging to the subset Sℓ. In this notation we 
an rewrite Eq. (3.32) as
π(q) = |ℓ〈〈 ~N |Uℓ(tq)Θℓ · · ·Uℓ(t1)Θℓ|~1〉〉ℓ|2 , (3.33)
where Θℓ = 1 − | ~N 〉〉ℓ〈〈 ~N | and Uℓ(t) is the unitary operator ⊗m∈Sℓum(t) whi
h de-
s
ribes the time evolution of the 
hains of Sℓ. To prove that there exist suitable

hoi
es of tℓ su
h that the series (3.29) 
onverges to 1 it is su�
ient to 
onsider the

ase tℓ = t > 0 for all j = 1, · · · , q: this is equivalent to sele
ting de
oding proto-

ols with 
onstant measuring intervals. By introdu
ing the operator Tℓ ≡ Uℓ(t)Θℓ,
Eq. (3.33) be
omes thus
π(q) = |ℓ〈〈 ~N | (Tℓ)q|~1〉〉ℓ|2 (3.34)
=ℓ〈〈~1|(T †ℓ )
q| ~N 〉〉ℓ〈〈 ~N | (Tℓ)q|~1〉〉ℓ = w(q)− w(q + 1) ,
where
w(j) ≡ℓ 〈〈~1|(T †ℓ )
j (Tℓ)
j |~1〉〉ℓ = ‖(Tℓ)j |~1〉〉ℓ‖2 , (3.35)
3 Multi Rail en
oding
is the norm of the ve
tor (Tℓ)
j |~1〉〉ℓ. Substituting Eq. (3.34) in Eq. (3.29) yields
[w(j) − w(j + 1)] = 1− w(q + 1) (3.36)
where the property w(1) = ℓ〈〈~1|Θℓ|~1〉〉ℓ = 1 was employed. Proving the thesis is hen
e
equivalent to prove that for q → ∞ the su

ession w(q) nulli�es. This last relation 
an
be studied using properties of power bounded matri
es [128℄. In fa
t, by introdu
ing
the norm of the operator (Tℓ)
we have,
w(q) = ‖(Tℓ)q|~1〉〉ℓ‖2 6 ‖(Tℓ)q‖2 6 c
1 + ρ(Tℓ)
(3.37)
where c is a positive 
onstant whi
h does not depend on q (if S is the similarity
transformation that puts Tℓ into the Jordan 
anoni
al form, i.e. J = S
−1TℓS, then
c is given expli
itly by c = ‖S‖ ‖S−1‖) and where ρ(Tℓ) is the spe
tral radius of Tℓ,
i.e. the eigenvalue of Tℓ with maximum absolute value (N.B. even when Tℓ is not
diagonalisable this is a well de�ned quantity). Equation (3.37) shows that ρ(Tℓ) < 1
is a su�
ient 
ondition for w(q) → 0. In our 
ase we note that, given any normalised
eigenve
tor |λ〉〉ℓ of Tℓ with eigenvalue λ we have
|λ| = ‖Tℓ|λ〉〉ℓ‖ = ‖Θℓ|λ〉〉ℓ‖ 6 1 , (3.38)
where the inequality follows from the fa
t that Θℓ is a proje
tor. Noti
e that in
Eq. (3.38) the identity holds only if |λ〉〉 is also an eigenve
tor of Θℓ with eigenvalue +1,
i.e. only if |λ〉〉ℓ is orthogonal to | ~N 〉〉ℓ. By de�nition |λ〉〉ℓ is eigenve
tor Tℓ = Uℓ(t)Θℓ:
therefore the only possibility to have the equality in Eq. (3.38) is that i) |λ〉〉ℓ is
an eigenve
tor of Uℓ(t) (i.e. an eigenve
tor of the Hamiltonian
1 Htotℓ of the 
hain
subset Sℓ) and ii) it is orthogonal to | ~N 〉〉ℓ. By negating the above statement we
get a su�
ient 
ondition for the thesis. Namely, if all the eigenve
tors | ~E〉〉ℓ of Htotℓ
are not orthogonal to | ~N 〉〉ℓ than the absolute values of the eigenvalues λ of Tℓ are
stri
tly smaller than 1 whi
h implies ρ(Tℓ) < 1 and hen
e the thesis. Sin
e the Sℓ

hannels are identi
al and do not intera
t, the eigenve
tors | ~E〉〉ℓ ≡
m∈Sℓ |em〉m are
tensor produ
t of eigenve
tors |em〉 of the single 
hain Hamiltonians H. Therefore the
Noti
e that stri
tly speaking the eigenve
tors of the Hamiltonian are not the same as those of the
time evolution operators. The latter still 
an have evolution times at whi
h additional degenera
y

an in
rease the set of eigenstates. A trivial example is given for t = 0 where all states be
ome
eigenstates. But it is always possible to �nd times t at whi
h the eigenstates of U(t) 
oin
ide with
those of H .
3 Multi Rail en
oding
su�
ient 
ondition be
omes
ℓ〈〈 ~E| ~N 〉〉ℓ =
m〈N |em〉m 6= 0 , (3.39)
whi
h 
an be satis�ed only if 〈N |em〉 6= 0 for all eigenve
tors |em〉 of the single 
hain
Hamiltonian H. �
Remark 3.1 While we have proven here that for equal time intervals the probability
of su

ess is 
onverging to unity, in pra
ti
e one may use optimal measuring time
intervals ti for a faster transfer (see also Se
tion 2.4). We also point out that timing
errors may delay the transfer, but will not de
rease its �delity.
3.6 Quantum 
hains with nearest-neighbour intera
tions
It is worth noti
ing that Eq. (3.39) is a very weak 
ondition, be
ause eigenstates
of Hamiltonians are typi
ally entangled. For instan
e, it holds for open 
hains with
nearest neighbour-intera
tions:
Theorem 3.2 (Multi rail proto
ol) Let H be the Hamiltonian of an open
nearest-neighbour quantum 
hain that 
onserves the number of ex
itations. If there
is a time t su
h that f1,N (t) 6= 0 (i.e. the Hamiltonian is 
apable of transport be-
tween Ali
e and Bob) then the state transfer 
an be made arbitrarily perfe
t by
using the multi rail proto
ol.
Proof We show by 
ontradi
tion that the 
riterion of Theorem 3.1 is ful�lled. As-
sume there exists a normalised eigenve
tor |e〉 of the single 
hain Hamiltonian H su
h
〈N |e〉 = 0. (3.40)
Be
ause |e〉 is an eigenstate, we 
an 
on
lude that also
〈e |H|N 〉 = 0. (3.41)
If we a
t with the Hamiltonian on the ket in Eq. (3.41) we may get some term propor-
tional to 〈e|N 〉 (
orresponding to an Ising-like intera
tion) and some part proportional
to 〈e|N − 1〉 (
orresponding to a hopping term; if this term did not exist, then 
learly
3 Multi Rail en
oding
f1,N (t) = 0 for all times). We 
an thus 
on
lude that
〈e|N − 1〉 = 0. (3.42)
Note that for a 
losed 
hain, e.g. a ring, this need not be the 
ase, be
ause then also
a term proportional to 〈e|N + 1〉 = 〈e|1〉 would o

ur. If we insert the Hamiltonian
into Eq. (3.42) again, we 
an use the same reasoning to see that
〈e|N − 2〉 = · · · = 〈e|1〉 = 0 (3.43)
and hen
e |e〉 = 0, whi
h is a 
ontradi
tion to |e〉 being normalised. �
3.7 Comparison with Dual Rail
As we have seen above, the Multi Rail proto
ol allows us in prin
iple to rea
h in
prin
iple a rate arbitrarily 
lose to one. However for a fair 
omparison with the Dual
Rail proto
ol, we should also take into a

ount the time-s
ale of the transfer. For the

on
lusive transfer of entanglement, we have seen in Se
tion 3.4 that only for 
hains
whi
h have a su

ess probability higher than p(t) = 0.8 it is worth en
oding on more
than three rails. The reason is that if the probability of su

ess for a single ex
itation
is p, then the probability of su

ess for ⌊M/2⌋ ex
itations on on M parallel 
hains is
lowered to p⌊M/2⌋. The proto
ol for three rails is always more e�
ient than on two, as
still only one ex
itation is being used, but three 
omplex amplitudes 
an be transferred
per usage.
For arbitrarily perfe
t transfer, the situation is slightly more 
ompli
ated as the
optimal 
hoi
e of M also depends on the joint probability of failure that one plans to
a
hieve. Let us assume that at ea
h step of the proto
ol, the su

ess probability on a
single 
hain is p. Then the number of steps to a
hieve a given probability of failure P
using M 
hains is given by
ℓ(P,M) = max
ln(1− p⌊M/2⌋)
. (3.44)
If we assume that the total time-s
ale of the transfer is proportional to the number of
steps, then the number of qubits that 
an be transferred per time interval is given by
v(P,M) ∝ R(M)/ℓ(P,M). (3.45)
Optimising this rate with respe
t to M we �nd three di�erent regimes of the joint
3 Multi Rail en
oding
probability of failure (see Fig. 3.3). If one is happy with a large P, then the Multi Rail
proto
ol be
omes superior to the Dual Rail for medium p. For intermediate P, the
threshold is 
omparable to the threshold of p = 0.8 for 
on
lusive transfer of entangle-
ment. Finally for very low P the Multi Rail only be
omes useful for p very 
lose to one.
In all three 
ases the threshold is higher than the p(t) that 
an usually a
hieved with
unmodulated Heisenberg 
hains. We 
an thus 
on
lude that the Multi Rail proto
ol
only be
omes useful for 
hains whi
h already have a very good performan
e.
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
Success probability on a single chain
P=0.50
P=0.90
P=0.99
Figure 3.3: Optimal rates (maximisation of Eq. (3.45 with respe
t toM) for the Multi
Rail proto
ol. Shown are three 
urves 
orresponding to di�erent values of the joint
probability of failure P one plans to a
hieve.
3.8 Con
lusion
We thus 
on
lude that any nearest-neighbour Hamiltonian that 
an transfer quantum
information with nonzero �delity (in
luding the Heisenberg 
hains analysed above) is

apable of e�
ient and perfe
t transfer when used in the 
ontext of parallel 
hains.
Hamiltonians with non-nearest neighbour intera
tions [89,81℄ 
an also be used as long
as the 
riterion of Theorem 3.1 is ful�lled.
4 Ergodi
ity and mixing
4.1 Introdu
tion
We have seen above that by applying measurements at the end of parallel 
hains, the
state of the 
hain is 
onverging to the ground state, and the quantum information is
transferred to the re
eiver. Indeed, repetitive appli
ation of the same transformation
is the key ingredient of many 
ontrols te
hniques. Beside quantum state transfer, they
have been exploited to inhibit the de
oheren
e of a system by frequently perturbing
its dynami
al evolution [129,130,131,132,133℄ (Bang-Bang 
ontrol) or to improve the
�delity of quantum gates [134℄ by means of frequent measurements (quantum Zeno-
e�e
t [135℄). Re
ently analogous strategies have also been proposed in the 
ontext
of state preparation [136, 137, 138, 139, 140, 141, 142℄. In Refs. [138, 139℄ for instan
e,
a homogenisation proto
ol was presented whi
h allows one to transform any input
state of a qubit into a some pre-�xed target state by repetitively 
oupling it with an
external bath. A similar thermalisation proto
ol was dis
ussed in Ref. [140℄ to study
the e�
ien
y of simulating 
lassi
al equilibration pro
esses on a quantum 
omputer.
In Refs. [141, 142℄ repetitive intera
tions with an externally monitored environment
were exploited instead to implement puri�
ation s
hemes whi
h would allow one to
extra
t pure state 
omponents from arbitrary mixed inputs.
ergodic mixing
Figure 4.1: S
hemati
 examples of the orbits of a ergodi
 and a mixing map.
4 Ergodi
ity and mixing
The 
ommon trait of the proposals [136,137,138,139,140,141,142℄ and the dual and
multi rail proto
ols is the requirement that repeated appli
ations of a properly 
hosen
quantum operation τ 
onverges to a �xed density matrix x∗ independently from the
input state x of the system, i.e.
τn(x) ≡ τ ◦ τ ◦ · · · ◦ τ
︸ ︷︷ ︸
−→ x∗ , (4.1)
with �◦� representing the 
omposition of maps. Following the notation of Refs. [143,
144℄ we 
all Eq. (4.1) the mixing property of τ . It is related with another important
property of maps, namely ergodi
ity (see Fig. 4.1). The latter requires the existen
e of
a unique input state x0 whi
h is left invariant under a single appli
ation of the map
τ(x) = x ⇐⇒ x = x0 . (4.2)
Ergodi
ity and the mixing property are of high interest not only in the 
ontext of the
above quantum information s
hemes. They also o

ur on a more fundamental level
in statisti
al me
hani
s [147℄ and open quantum systems [121, 148℄, where one would
like to study irreversibility and relaxation to thermal equilibrium.
In the 
ase of quantum transformations one 
an show that mixing maps with 
on-
vergen
e point x∗ are also ergodi
 with �xed point x0 = x∗. The opposite impli
ation
however is not generally true sin
e there are examples of ergodi
 quantum maps whi
h
are not mixing (see the following). Su�
ient 
onditions for mixing have been dis
ussed
both in the spe
i�
 
ase of quantum 
hannel [140, 143, 146℄ and in the more abstra
t

ase of maps operating on topologi
al spa
es [147℄. In parti
ular the Lyapunov dire
t
method [147℄ allows one to prove that an ergodi
 map τ is mixing if there exists a 
on-
tinuous fun
tional S whi
h, for all points but the �xed one, is stri
tly in
reasing under
τ . Here we strengthen this 
riterion by weakening the requirement on S: our gener-
alised Lyapunov fun
tions are requested only to have limiting values S(τn(x))|n→∞
whi
h di�er from S(x) for all x 6= x0. It turns out that the existen
e of su
h S is
not just a su�
ient 
ondition but also a ne
essary 
ondition for mixing. Exploiting
this fa
t one 
an easily generalise a previous result on stri
tly 
ontra
tive maps [143℄
De�nition (4.2) may sound unusual for readers who are familiar with a de�nition of ergodi
ity from
statisti
al me
hani
s, where a map is ergodi
 if its invariant sets have measure 0 or 1. The notion
of ergodi
ity used here is 
ompletely di�erent, and was introdu
ed in [143, 145, 146℄. The set X
one should have in mind here is not a measurable spa
e, but the 
ompa
t 
onvex set of quantum
states. A perhaps more intuitive de�nition of ergodi
ity based on the time average of observables
is given by Lemma 4.5).
4 Ergodi
ity and mixing
by showing that maps whi
h are asymptoti
 deformations (see De�nition 4.14) are
mixing. This has, unlike 
ontra
tivity, the advantage of being a property independent
of the 
hoi
e of metri
 (see however [144℄ for methods of �nding �tight� norms). In
some 
ases, the generalised Lyapunov method permits also to derive an optimal mix-
ing 
ondition for quantum 
hannels based on the quantum relative entropy. Finally
a slightly modi�ed version of our approa
h whi
h employs multi-
entral Lyapunov
fun
tions yields a 
hara
terisation of (not ne
essarily mixing) maps whi
h in the limit
of in�nitely many appli
ations move all points toward a proper subset (rather than a
single point) of the input spa
e.
The introdu
tion of a generalised Lyapunov method seems to be sound not only
from a mathemati
al point of view, but also from a physi
al point of view. In e�e
t,
it often happens that the informations available on the dynami
s of a system are only
those related on its asymptoti
 behaviour (e.g. its thermalisation pro
ess), its �nite
time evolution being instead di�
ult to 
hara
terise. Sin
e our method is expli
itly

onstru
ted to exploit asymptoti
 features of the mapping, it provides a more e�e
tive
way to probe the mixing property of the pro
ess.
Presenting our results we will not restri
t ourself to the 
ase of quantum operations.
Instead, following [147℄ we will derive them in the more general 
ontext of 
ontinuous
maps operating on topologi
al spa
es [149℄. This approa
h makes our results stronger
by allowing us to invoke only those hypotheses whi
h, to our knowledge, are stri
tly
ne
essary for the derivation. It is important to stress however that, as a parti
ular
instan
e, all the Theorems and Lemmas presented in this 
hapter hold for any linear,

ompletely positive, tra
e preserving map (i.e. quantum 
hannels) operating on a

ompa
t subset of normed ve
tors (i.e. the spa
e of the density matri
es of a �nite
dimensional quantum system). Therefore readers who are not familiar with topologi
al
spa
es 
an simply interpret our derivations as if they were just obtained for quantum

hannels a
ting on a �nite dimensional quantum system.
This 
hapter is organised as follows. In Se
. 4.3 the generalised Lyapunov method
along with some minor results is presented in the 
ontext of topologi
al spa
es. Then
quantum 
hannels are analysed in Se
. 4.4 providing a 
omprehensive summary of the
ne
essary and su�
ient 
onditions for the mixing property of these maps. Con
lusions
and remarks form the end of the 
hapter in Se
. 4.5.
4.2 Topologi
al ba
kground
Let us �rst introdu
e some basi
 topologi
al ba
kground required for this 
hapter. A
more detailed introdu
tion is given in [149℄. Topologi
al spa
es are a very elegant way
4 Ergodi
ity and mixing
of de�ning 
ompa
tness, 
onvergen
e and 
ontinuity without requiring more than the
following stru
ture:
De�nition 4.1 A topologi
al spa
e is a pair (X ,O) of a set X and a set O of subsets
of X (
alled open sets) su
h that
1. X and ∅ are open
2. Arbitrary unions of open sets are open
3. Interse
tions of two open sets are open
Example 4.1 If X is an arbitrary set, and O = {X , ∅}, then (X ,O) is a topologi
al
spa
e. O is 
alled the trivial topology.
De�nition 4.2 A topologi
al spa
e X is 
ompa
t if any open 
over (i.e. a set of open
sets su
h that X is 
ontained in their union) 
ontains a �nite sub-
over.
De�nition 4.3 A sequen
e xn ∈ X is 
onvergent with limit x∗ if ea
h open neigh-
bourhood O(x∗) (i.e. a set su
h that x∗ ∈ O(x∗) ∈ O 
ontains all but �nitely many
points of the sequen
e.
De�nition 4.4 A map on a topologi
al spa
e is 
ontinous if the preimage of any open
set is open.
This is already all we require to make useful statements about ergodi
ity and mixing.
However, there are some subtleties whi
h we need to take 
are of:
De�nition 4.5 A topologi
al spa
e is sequentially 
ompa
t if every sequen
e has a

onvergent subsequen
e.
Sequentially 
ompa
tness is in general not related to 
ompa
tness! Another subtlety is
that with the above de�nition, a sequen
e 
an 
onverge to many di�erent points. For
example, in the trivial topology, any sequen
e 
onverges to any point. This motivates
De�nition 4.6 A topologi
al spa
e is Hausdor� if any two distin
t points 
an by
separated by open neighbourhoods.
A limit of a sequen
e in a Hausdor� spa
e is unique. All these problems disappear in
metri
al spa
es:
4 Ergodi
ity and mixing
De�nition 4.7 Ametri
 spa
e is a pair (X , d) of a set X and a fun
tion d : X×X → R
su
h that
1. d(x, y) ≥ 0 and d(x, y) = 0 ⇔ x = y
2. d(x, y) = d(y, x)
3. d(x, z) ≤ d(x, y) + d(y, z)
A metri
 spa
e be
omes a topologi
al spa
e with the 
anoni
al topology
De�nition 4.8 A subset O of a metri
 spa
e X is open if ∀x ∈ O there is an ǫ > 0
su
h that {y ∈ X |d(x, y) ≤ ǫ} ⊂ O.
In a metri
 spa
e with the 
anoni
al topology, 
ompa
tness and sequentially 
ompa
t-
ness be
ome equivalent. Furthermore, it is automati
ally Hausdor� (see Fig. 4.2).
Compact
spaces
spaces
Compact
Hausdorff
Topological spaces
metric spaces
spaces
Sequentially
compact
Figure 4.2: Relations between topologi
al spa
es [149℄. The spa
e of density matri
es
on whi
h quantum 
hannels are de�ned, is a 
ompa
t and 
onvex subset of a normed
ve
tors spa
e (the spa
e of linear operators of the system) whi
h, in the above graphi
al
representation �ts within the set of 
ompa
t metri
 spa
es.
4.3 Generalised Lyapunov Theorem
4.3.1 Topologi
al spa
es
In this se
tion we introdu
e the notation and derive our main result (the Generalised
Lyapunov Theorem).
4 Ergodi
ity and mixing
De�nition 4.9 Let X be a topologi
al spa
e and let τ : X → X be a map. The
sequen
e xn ≡ τn(x), where τn is a short-hand notation for the n−fold 
omposition
of τ, is 
alled the orbit of x. An element x∗ ∈ X is 
alled a �xed point of τ if and only
τ(x∗) = x∗ . (4.3)
τ is 
alled ergodi
 if and only if it has exa
tly one �xed point. τ is 
alled mixing if
and only if there exists a 
onvergen
e point x∗ ∈ X su
h that any orbit 
onverges to
it, i.e.
xn = x∗ ∀x ∈ X . (4.4)
A dire
t 
onne
tion between ergodi
ity and mixing 
an be established as follows.
Lemma 4.1 Let τ : X → X be a 
ontinuous mixing map on a topologi
al Hausdor�
spa
e X . Then τ is ergodi
.
Proof Let x∗ be the 
onvergen
e point of τ and let x ∈ X arbitrary. Sin
e τ is

ontinuous we 
an perform the limit in the argument of τ, i.e.
τ(x∗) = τ
τn(x)
= lim
τn+1(x) = x∗, (4.5)
whi
h shows that x∗ is a �xed point of τ . To prove that it is unique assume by

ontradi
tion that τ possesses a se
ond �xed point y∗ 6= x∗. Then limn→∞ τn(y∗) =
y∗ 6= x∗, so τ 
ould not be mixing (sin
e the limit is unique in a Hausdor� spa
e � see
Fig. 4.2). Hen
e τ is ergodi
. �
Remark 4.1 The 
onverse is not true in general, i.e. not every ergodi
 map is mixing
(not even in Hausdor� topologi
al spa
es). A simple 
ounterexample is given by
τ : [−1, 1] → [−1, 1] with τ(x) ≡ −x and the usual topology of R, whi
h is ergodi
 with
�xed point 0, but not mixing sin
e for x 6= 0, τn(x) = (−1)nx is alternating between
two points. A similar 
ounterexample will be dis
ussed in the quantum 
hannel se
tion
(see Example 4.2).
A well known 
riterion for mixing is the existen
e of a Lyapunov fun
tion [147℄.
De�nition 4.10 Let τ : X → X be a map on a topologi
al spa
e X . A 
ontinuous
map S : X → R is 
alled a (stri
t) Lyapunov fun
tion for τ around x∗ ∈ X if and only
S (τ(x)) > S(x) ∀x 6= x∗. (4.6)
4 Ergodi
ity and mixing
Remark 4.2 At this point is is neither assumed that x∗ is a �xed point, nor that τ is
ergodi
. Both follows from the theorem below.
Theorem 4.1 (Lyapunov fun
tion) Let τ : X → X be a 
ontinuous map on a
sequentially 
ompa
t topologi
al spa
e X . Let S : X → R be a Lyapunov fun
tion for
τ around x∗. Then τ is mixing with the �xed point x∗.
The proof of this theorem is given in [147℄. We will not reprodu
e it here, be
ause
we will provide a general theorem that in
ludes this as a spe
ial 
ase. In fa
t, we will
show that the requirement of the stri
t monotoni
ity 
an be mu
h weakened, whi
h
motivates the following de�nition.
De�nition 4.11 Let τ : X → X be a map on a topologi
al spa
e X . A 
ontinuous
map S : X → R is 
alled a generalised Lyapunov fun
tion for τ around x∗ ∈ X if and
only if the sequen
e S (τn(x)) is point-wise 
onvergent2 for any x ∈ X and S ful�ls
S∗(x) ≡ lim
S (τn(x)) 6= S(x) ∀x 6= x∗. (4.7)
In general it may be di�
ult to prove the point-wise 
onvergen
e. However if S is
monotoni
 under the a
tion of τ and the spa
e is 
ompa
t, the situation be
omes

onsiderably simpler. This is summarised in the following Lemma.
Lemma 4.2 Let τ : X → X be map on a 
ompa
t topologi
al spa
e. A 
ontinuous
map S : X → R whi
h ful�ls
S (τ(x)) > S(x) ∀x ∈ X , (4.8)
S∗(x) ≡ lim
S (τn(x)) > S(x) ∀x 6= x∗. (4.9)
for some �xed x∗ ∈ X is a generalised Lyapunov fun
tion for τ around x∗.
Proof It only remains to show the (point-wise) 
onvergen
e of S (τn(x)). Sin
e S is
a 
ontinuous fun
tion on a 
ompa
t spa
e, it is bounded. By Eq. (4.8) the sequen
e
is monotoni
. Any bounded monotoni
 sequen
e 
onverges. �
Corollary 4.1 Let τ : X → X be a map on a 
ompa
t topologi
al spa
e. A 
ontinuous
map S : X → R whi
h ful�ls
S (τ(x)) > S(x) ∀x ∈ X , (4.10)
Point-wise 
onvergen
e in this 
ontext means that for any �xed x the sequen
e Sn ≡ S (τ
n(x)) is

onvergent.
4 Ergodi
ity and mixing
τN (x)
> S(x) ∀x 6= x∗, (4.11)
for some �xed N ∈ N and for some x∗ ∈ X is a generalised Lyapunov fun
tion for τ
around x∗.
Remark 4.3 This implies that a stri
t Lyapunov fun
tion is a generalised Lyapunov
fun
tion (with N = 1).
We 
an now state the main result of this se
tion:
Theorem 4.2 (Generalized Lyapunov fun
tion) Let τ : X → X be a 
on-
tinuous map on a sequentially 
ompa
t topologi
al spa
e X . Let S : X → R be a
generalised Lyapunov fun
tion for τ around x∗. Then τ is mixing with �xed point
Proof Consider the orbit xn ≡ τn(x) of a given x ∈ X . Be
ause X is sequen-
tially 
ompa
t, the sequen
e xn has a 
onvergent subsequen
e (see Fig. 4.2), i.e.
limk→∞ xnk ≡ x̃. Let us assume that x̃ 6= x∗ and show that this leads to a 
on-
tradi
tion. By Eq. (4.7) we know that there exists a �nite N ∈ N su
h that
τN (x̃)
6= S(x̃). (4.12)
Sin
e τN is 
ontinuous we 
an perform the limit in the argument, i.e.
τN (xnk) = τ
N (x̃). (4.13)
Likewise, by 
ontinuity of S we have
S (xnk) = S(x̃), (4.14)
and on the other hand
S (xN+nk) = lim
τN (xnk)
= S(τN x̃), (4.15)
where the se
ond equality stems from the 
ontinuity of the map S and τN . Be
ause
S is a generalised Lyapunov fun
tion, the sequen
e S (xn) is 
onvergent. Therefore
the subsequen
es (4.14) and (4.15) must have the same limit. We 
on
lude that
4 Ergodi
ity and mixing
S(τN x̃) = S(x̃) whi
h 
ontradi
ts Eq. (4.12). Hen
e x̃ = x∗. Sin
e we have shown
that any 
onvergent subsequen
e of τn(x) 
onverges to the same limit x∗, it follows by
Lemma 4.3 that τn(x) is 
onverging to x∗. Sin
e that holds for arbitrary x, it follows
that τ is mixing. �
Lemma 4.3 Let xn be a sequen
e in a sequentially 
ompa
t topologi
al spa
e X su
h
that any 
onvergent subsequen
e 
onverges to x∗. Then the sequen
e 
onverges to x∗.
Proof We prove by 
ontradi
tion: assume that the sequen
e does not 
onverge to x∗.
Then there exists an open neighbourhood O(x∗) of x∗ su
h that for all k ∈ N, there is
a nk su
h that xnk /∈ O(x∗). Thus the subsequen
e xnk is in the 
losed spa
e X\O(x∗),
whi
h is again sequentially 
ompa
t. xnk has a 
onvergent subsequen
e with a limit
in X\O(x∗), in parti
ular this limit is not equal to x∗. �
There is an even more general way of de�ning Lyapunov fun
tions whi
h we state here
for 
ompleteness. It requires the 
on
ept of the quotient topology [149℄.
De�nition 4.12 Let τ : X → X be a map on a topologi
al spa
e X . A 
ontinuous
map S : X → R is 
alled a multi-
entral Lyapunov fun
tion for τ around F ⊆ X if
and only if the sequen
e S (τn(x)) is point-wise 
onvergent for any x ∈ X and if S
and τ ful�l the following three 
onditions: S is 
onstant on F , τ(F) ⊆ F , and
S∗(x) ≡ lim
S (τn(x)) 6= S(x) ∀x /∈ F . (4.16)
For these fun
tions we 
annot hope that the orbit is mixing. We 
an however show
that the orbit is �
onverging� to the set F in the following sense:
Theorem 4.3 (Multi-
entral Lyapunov fun
tion) Let τ : X → X be a 
ontinu-
ous map on a sequentially 
ompa
t topologi
al spa
e X . Let S : X → R be a multi-

entral Lyapunov fun
tion for τ around F . Let ϕ : X → X/F be the 
ontinuous
mapping into the quotient spa
e (i.e. ϕ(x) = [x] for x ∈ X\F and ϕ(x) = [F ] for
x ∈ F). Then τ̃ : X/F → X/F given by τ̃([x]) = ϕ
ϕ−1([x])
is mixing with �xed
point [F ].
Proof First note that τ̃ is well de�ned be
ause ϕ is invertible on X/F\[F ] and
τ(F) ⊆ F , so that τ̃([F ]) = [F ]. Sin
e X is sequentially 
ompa
t, the quotient spa
e
X/F is also sequentially 
ompa
t. Note that for O open, τ̃−1(O) = ϕ
ϕ−1 (O)
is the image of ϕ of an open set in X and therefore (by de�nition of the quotient topol-
ogy) open in X/F . Hen
e τ̃ is 
ontinuous. The fun
tion S̃([x]) : X/F → X/F given
by S̃([x]) = S(ϕ−1([x])) is 
ontinuous and easily seen to be a generalised Lyapunov
fun
tion around [F ]. By Theorem 4.2 it follows that τ̃ is mixing. �
4 Ergodi
ity and mixing
4.3.2 Metri
 spa
es
We now show that for the parti
ular 
lass of 
ompa
t topologi
al sets whi
h posses a
metri
, the existen
e of a generalised Lyapunov fun
tion is also a ne
essary 
ondition
for mixing.
Theorem 4.4 (Lyapunov 
riterion) Let τ : X → X be a 
ontinuous map on a

ompa
t metri
 spa
e X . Then τ is mixing with �xed point x∗ if and only if a generalised
Lyapunov fun
tion around x∗ exists.
Proof Firstly, in metri
 spa
es 
ompa
tness and sequential 
ompa
tness are equiva-
lent, so the requirements of Theorem 4.2 are met. Se
ondly, for any mixing map τ with
�xed point x∗, a generalised Lyapunov fun
tion around x∗ is given by S(x) ≡ d(x∗, x).
In fa
t, it is 
ontinuous be
ause of the 
ontinuity of the metri
 and satis�es
S (τn(x)) = d(x∗, x∗) = 0 6 d(x∗, x) = S(x), (4.17)
where the equality holds if and only x = x∗. We 
all d(x∗, x) the trivial generalised
Lyapunov fun
tion. �
Remark 4.1 In the above Theorem we have not used all the properties of the metri
.
In fa
t a 
ontinuous semi-metri
 (i.e. without the triangle inequality) would su�
e.
The trivial Lyapunov fun
tion requires knowledge of the �xed point of the map. There
is another way of 
hara
terising mixing maps as those whi
h bring elements 
loser to
ea
h other (rather than 
loser to the �xed point).
De�nition 4.13 A map τ : X → X is on a metri
 spa
e is 
alled a non-expansive
map if and only if
d(τ(x), τ(y)) 6 d(x, y) ∀x, y ∈ X , (4.18)
a weak 
ontra
tion if and only if
d(τ(x), τ(y)) < d(x, y) ∀x, y ∈ X , x 6= y, (4.19)
and a stri
t 
ontra
tion if and only if there exists a k < 1 su
h that
d(τ(x), τ(y)) 6 k d(x, y) ∀x, y ∈ X . (4.20)
Remark 4.2 The notation adopted here is slightly di�erent from the de�nitions used by
other Authors [143, 150, 5℄ who use 
ontra
tion to indi
ate our non-expansive maps.
4 Ergodi
ity and mixing
Our 
hoi
e is motivated by the need to 
learly distinguish between non-expansive
transformation and weak 
ontra
tions.
We 
an generalise the above de�nition in the following way:
De�nition 4.14 A map τ : X → X on a metri
 spa
e is 
alled an asymptoti
 defor-
mation if and only if the sequen
e d(τn(x), τn(y)) 
onverges point-wise for all x, y ∈ X
d(τn(x), τn(y)) 6= d(x, y) ∀x, y ∈ X , x 6= y. (4.21)
Lemma 4.4 Let τ : X → X be a non-expansive map on a metri
 spa
e X , and let
d(τN (x), τN (y)) < d(x, y) ∀x, y ∈ X , x 6= y (4.22)
for some �xed N ∈ N. Then τ is an asymptoti
 deformation. Then τ is an asymptoti
deformation.
Proof The existen
e of the limit limn→∞ d(τ
n(x), τn(y)) follows from the monotoni
-
ity and the fa
t the any metri
 is lower bounded. �
Remark 4.4 Any weak 
ontra
tion is an asymptoti
 deformation (with N = 1).
Theorem 4.5 (Asymptoti
 deformations) Let τ : X → X be a 
ontinuous
map on a 
ompa
t metri
 spa
e X with at least one �xed point. Then τ is mixing
if and only if τ is an asymptoti
 deformation.
Proof Firstly assume that τ is an asymptoti
 deformation. Let x∗ be a �xed point
and de�ne S(x) = d(x∗, x).
S(τn(x)) = lim
d(x∗, τ
n(x))
= lim
d(τn(x∗), τ
n(x)) 6= d(x∗, x) = S(x) ∀x 6= x∗, (4.23)
hen
e S(x) is a generalised Lyapunov fun
tion. By Theorem 4.2 it follows that τ is
mixing. Se
ondly, if τ is mixing, then
d(τn(x), τn(y)) = d(x∗, x∗) = 0 6= d(x, y) ∀x, y ∈ X , x 6= y, (4.24)
so τ is an asymptoti
 deformation. �
4 Ergodi
ity and mixing
Remark 4.5 Note that the existen
e of a �xed point is assured if τ is a weak 
ontra
tion
on a 
ompa
t spa
e [151℄, or if the metri
 spa
e is 
onvex 
ompa
t [152℄.
As a spe
ial 
ase, we get the following result:
Corollary 4.2 Any weak 
ontra
tion τ on a 
ompa
t metri
 spa
e is mixing.
Proof Sin
e the spa
e is 
ompa
t τ has at least one �xed point. Moreover from
Lemma 4.4 we know that τ is an asymptoti
 deformation. Then Theorem 4.5 applies.�
Remark 4.6 This result 
an be seen as an instan
e of Bana
h 
ontra
tion prin
iple on

ompa
t spa
es. In the se
ond part of the 
hapter we will present a 
ounterexample
whi
h shows that weak 
ontra
tivity is only a su�
ient 
riterion for mixing (see Ex-
ample 4.3). In the 
ontext of quantum 
hannels an analogous 
riterion was suggested
in [146, 143℄ whi
h applied to stri
t 
ontra
tions. We also note that for weak and
stri
t 
ontra
tions, the trivial generalised Lyapunov fun
tion (Theorem 4.4) is a stri
t
Lyapunov fun
tion.
Lemma 4.5 states the ergodi
 theorem by Birkho� [153℄ whi
h, in the 
ontext of
normed ve
tor spa
es, shows the equivalen
e between the de�nition of ergodi
ity of
Eq. (4.4) and the standard time average de�nition.
Lemma 4.5 Let X be a 
onvex and 
ompa
t subset of a normed ve
tor spa
e, and let
τ : X → X be a 
ontinuous map. If τ is ergodi
 with �xed point x∗, then
τ ℓ(x) = x∗ . (4.25)
Proof De�ne the sequen
e An ≡ 1n+1
ℓ=0 τ
ℓ(x). Let then M be the upper bound
for the norm of ve
tors in X , i.e. M ≡ supx∈X ‖x‖ < ∞. whi
h exists be
ause X is

ompa
t. The sequen
e An has a 
onvergent subsequen
e Ank with limit Ã. Sin
e τ
is 
ontinuous one has limk→∞ τ(Ank) = τ(Ã). On the other hand, we have
‖τ(Ank)−Ank‖ =
nk + 1
‖τnk+1(x)− x‖ 6 ‖τ
nk+1(x)‖+ ‖x‖
nk + 1
nk + 1
, (4.26)
so the two sequen
es must have the same limit, i.e. τ(Ã) = Ã. Sin
e τ is ergodi
, we
have Ã = x∗ and limn→∞An = x∗ by Lemma 4.3. �
Remark 4.7 Note that if τ has a se
ond �xed point y∗ 6= x∗, then for all n one has
ℓ=0 τ
ℓ(y∗) = y∗, so Eq. (4.25) would not apply.
4 Ergodi
ity and mixing
4.4 Quantum Channels
In this Se
tion we dis
uss the mixing properties of quantum 
hannels [2℄ whi
h a

ount
for the most general evolution a quantum system 
an undergo in
luding measure-
ments and 
oupling with external environments. In this 
ontext solving the mixing
problem (4.1) is equivalent to determine if repetitive appli
ation of a 
ertain physi
al
transformation will drive any input state of the system (i.e. its density matri
es) into
a unique output 
on�guration. The relationship between the di�erent mixing 
riteria
one 
an obtain in this 
ase is summarised in Fig. 4.3.
At a mathemati
al level quantum 
hannels 
orrespond to linear maps a
ting on the
density operators ρ of the system and satisfying the requirement of being 
ompletely
positive and tra
e preserving (CPT). For a formal de�nition of these properties we
refer the reader to [154, 5, 155℄: here we note only that a ne
essary and su�
ient

ondition to being CPT is to allow Kraus de
omposition [154℄ or, equivalently, Stine-
spring dilation [156℄. Our results are appli
able if the underlying Hilbert spa
e is
�nite-dimensional. In su
h regime there is no ambiguity in de�ning the 
onvergen
e
of a sequen
e sin
e all operator norms are equivalent (i.e. given two norms one 
an

onstru
t an upper and a lower bound for the �rst one by properly s
aling the se
ond
one). Also the set of bounded operators and the set of operators of Hilbert-S
hmidt

lass 
oin
ide. For the sake of de�niteness, however, we will adopt the tra
e-norm
whi
h, given the linear operator Θ : H → H, is de�ned as ‖Θ‖1 = Tr[
Θ†Θ] with
Tr[· · · ] being the tra
e over H and Θ† being the adjoint of Θ. This 
hoi
e is in part
motivated by the fa
t [150℄ that any quantum 
hannel is non-expansive with respe
t
to the metri
 indu
ed
by ‖ ·‖1 (the same property does not ne
essarily apply to other
operator norms, e.g. the Hilbert-S
hmidt norm, also when these are equivalent to
‖ · ‖1).
We start by showing that the mixing 
riteria dis
ussed in the �rst half of the 
hapter
do apply to the 
ase of quantum 
hannel. Then we will analyse these maps by studying
their linear extensions in the whole ve
tor spa
e formed by the linear operators of H.
4.4.1 Mixing 
riteria for Quantum Channels
Let H be a �nite dimensional Hilbert spa
e and let S(H) be the set of its density
matri
es ρ. The latter is a 
onvex and 
ompa
t subset of the larger normed ve
tor
spa
e L(H) 
omposed by the linear operators Θ : H → H of H. From this and from
the fa
t that CPT maps are 
ontinuous (indeed they are linear) it follows that for
This is just the tra
e distan
e d(ρ, σ) = ‖ρ− σ‖1.
4 Ergodi
ity and mixing
asymptotic
deformation mixing
ergodic
spectral gap
generalized Lyapunov
function exists
ergodic with
pure fixpoint
strict Lyapunov
function exists
contraction
Figure 4.3: Relations between the di�erent properties of a quantum 
hannel.
a quantum 
hannel there always exists at least one density operator whi
h is a �xed
point [140℄. It also follows that all the results of the previous se
tion apply to quantum

hannels. In parti
ular Lemma 4.1 holds, implying that any mixing quantum 
hannel
must be ergodi
. The following example shows, however, that it is possible to have
ergodi
 quantum 
hannels whi
h are not mixing.
Example 4.2 Consider the qubit quantum 
hannel τ obtained by 
as
ading a 
om-
pletely de
oherent 
hannel with a NOT gate. Expli
itly τ is de�ned by the transfor-
mations τ(|0〉〈0|) = |1〉〈1|, τ(|1〉〈1|) = |0〉〈0|, and τ(|0〉〈1|) = τ(|1〉〈0|) = 0 with |0〉, |1〉
being the 
omputational basis of the qubit. This map is ergodi
 with �xed point given
by the 
ompletely mixed state (|0〉〈0| + |1〉〈1|)/2. However it is trivially not mixing
sin
e, for instan
e, repetitive appli
ation of τ on |0〉〈0| will os
illate between |0〉〈0|
and |1〉〈1|.
Theorems 4.5 implies that a quantum 
hannel τ : S(H) → S(H) is mixing if and
only if it is an asymptoti
 deformation. As already pointed out in the introdu
tion,
this property is metri
 independent (as opposed to 
ontra
tivity). Alternatively, if the
�xed point of a quantum 
hannel is known, then one may use the trivial generalised
Lyapunov fun
tion (Theorem 4.4) to 
he
k if it is mixing. However both 
riteria
depend on the metri
 distan
e, whi
h usually has no easy physi
al interpretation. A
more useful 
hoi
e is the quantum relative entropy, whi
h is de�ned as
H(ρ, σ) ≡ Trρ(log ρ− log σ). (4.27)
The quantum relative entropy is 
ontinuous in �nite dimension [157℄ and 
an be used
as a measure of distan
e (though it is not a metri
). It is �nite if the support of ρ is

ontained in the support of σ. To ensure that it is a 
ontinuous fun
tion on a 
ompa
t
spa
e, we 
hoose σ to be faithful:
Theorem 4.6 (Relative entropy 
riterion) A quantum 
hannel with faithful �xed
point ρ∗ is mixing if and only if the quantum relative entropy with respe
t to ρ∗ is a
4 Ergodi
ity and mixing
generalised Lyapunov fun
tion.
Proof Be
ause of Theorem 4.2 we only need to prove the se
ond part of the thesis,
i.e. that mixing 
hannels admit the quantum relative entropy with respe
t to the �xed
point, S(ρ) ≡ H(ρ, ρ∗), as a generalised Lyapunov fun
tion. Firstly noti
e that the
quantum relative entropy is monotoni
 under quantum 
hannels [158,159℄. Therefore
the limit S∗(ρ) ≡ limn→∞ S (τn(ρ)) does exist and satis�es the 
ondition S∗(ρ) > S(ρ).
Suppose now there exists a ρ su
h that S∗(ρ) = S(ρ). Be
ause τ is mixing and S is

ontinuous we have
S(ρ) = S∗(ρ) = lim
S (τn(ρ)) = S(ρ∗) = 0, (4.28)
and hen
e H(ρ, ρ∗) = 0. Sin
e H(ρ, σ) = 0 if and only if ρ = σ it follows that S is a
Lyapunov fun
tion around ρ∗. �
Another important investigation tool is Corollary 4.2: weak 
ontra
tivity of a quantum

hannel is a su�
ient 
ondition for mixing. As already mentioned in the previous
se
tion, unfortunately this not a ne
essary 
ondition. Here we present an expli
it

ounterexample based on a quantum 
hannel introdu
ed in Ref. [140℄.
Example 4.3 Consider a three-level quantum system 
hara
terised by the orthogo-
nal ve
tors |0〉, |1〉, |2〉 and the quantum 
hannel τ de�ned by the transformations
τ(|2〉〈2|) = |1〉〈1|, τ(|1〉〈1|) = τ(|0〉〈0|) = |0〉〈0|, and τ(|i〉〈j|) = 0 for all i 6= j. Its
easy to verify that after just two iterations any input state ρ will be transformed into
the ve
tor |0〉〈0|. Therefore the map is mixing. On the other hand it is expli
itly not
a weak 
ontra
tion with respe
t to the tra
e norm sin
e, for instan
e, one has
‖ τ(|2〉〈2|) − τ(|0〉〈0|) ‖1 = ‖ |1〉〈1| − |0〉〈0| ‖1 = ‖ |2〉〈2| − |0〉〈0| ‖1 , (4.29)
where in the last identity we used the invarian
e of ‖ · ‖1 with respe
t to unitary
transformations.
4.4.2 Beyond the density matrix operator spa
e: spe
tral properties
Exploiting linearity quantum 
hannels 
an be extended beyond the spa
e S(H) of
density operators to be
ome maps de�ned on the full ve
tor spa
e L(H) of the linear
operators of the system, in whi
h basi
 linear algebra results hold. This allows one
to simplify the analysis even though the mixing property (4.1) is still de�ned with
respe
t to the density operators of the system.
4 Ergodi
ity and mixing
Mixing 
onditions for quantum 
hannels 
an be obtained by 
onsidering the stru
-
ture of their eigenve
tors in the extended spa
e L(H). For example, it is easily shown
that the spe
tral radius [160℄ of any quantum 
hannel is equal to unity [140℄, so its
eigenvalues are 
ontained in the unit 
ir
le. The eigenvalues λ on the unit 
ir
le (i.e.
|λ| = 1) are referred to as peripheral eigenvalues. Also, as already mentioned, sin
e
S(H) is 
ompa
t and 
onvex, CPT maps have always at least one �xed point whi
h
is a density matrix [140℄.
Theorem 4.7 (Spe
tral gap 
riterion) Let τ be a quantum 
hannel. τ is mixing
if and only if its only peripheral eigenvalue is 1 and this eigenvalue is simple.
Proof The �if� dire
tion of the proof is a well known result from linear algebra (see
for example [160, Lemma 8.2.7℄). Now let us assume τ is mixing towards ρ∗. Let Θ be a
generi
 operator in L(H). Then Θ 
an be de
omposed in a �nite set of non-orthogonal
density operators
, i.e. Θ =
ℓ cℓρℓ, with ρℓ ∈ S(H) and cℓ 
omplex. Sin
e Tr [ρℓ] = 1,
we have have Tr [Θ] =
ℓ cℓ. Moreover sin
e τ is mixing we have limn→∞ τ
n (ρℓ) = ρ∗
for all ℓ, with 
onvergen
e with respe
t to the tra
e-norm. Be
ause of linearity this
implies
τn (Θ) =
cℓ ρ∗ = Tr [Θ] ρ∗ . (4.30)
If there existed any other eigenve
tor Θ∗ of τ with eigenvalue on the unit 
ir
le, then
limn→∞ τ
n(Θ∗) would not satisfy Eq. (4.30). �
The speed of 
onvergen
e 
an also be estimated by [140℄
‖τn (ρ)− ρ∗‖1 6 CN nN κn , (4.31)
where N is the dimensionality of the underlying Hilbert spa
e, κ is the se
ond largest
eigenvalue of τ , and CN is some 
onstant depending only on N and on the 
hosen
norm. Hen
e, for n ≫ N the 
onvergen
e be
omes exponentially fast. As mentioned
in [143℄, the 
riterion of Theorem 4.7 is in general di�
ult to 
he
k. This is be
ause
one has to �nd all eigenvalues of the quantum 
hannel, whi
h is hard espe
ially in
the high dimensional 
ase. Also, if one only wants to 
he
k if a parti
ular 
hannel
To show that this is possible, 
onsider an arbitrary operator basis of L(H). If N is the �nite
dimension of H the basis will 
ontain N2 elements. Ea
h element of the basis 
an then be
de
omposed into two Hermitian operators, whi
h themselves 
an be written as linear 
ombinations
of at most N proje
tors. Therefore there exists a generating set of at most 2N3 positive operators,
whi
h 
an be normalised su
h that they are quantum states. There even exists a basis (i.e. a
minimal generating set) 
onsisting of density operators, but in general it 
annot be orthogonalised.
4 Ergodi
ity and mixing
is mixing or not, then the amount of information obtained is mu
h higher than the
required amount.
Example 4.4 As an appli
ation 
onsider the non mixing CPT map of Example 4.2.
One 
an verify that apart from the eigenvalue 1 asso
iated with its �xed point (i.e.
the 
ompletely mixed state), it possess another peripheral eigenvalue. This is λ = −1
whi
h is asso
iated with the Pauli operator |0〉〈0| − |1〉〈1|.
Corollary 4.3 The 
onvergen
e speed of any mixing quantum 
hannel is exponentially
fast for su�
iently high values of n.
Proof From Theorem 4.7 mixing 
hannels have exa
tly one peripheral eigenvalue,
whi
h is also simple. Therefore the derivation of Ref. [140℄ applies and Eq. (4.31)
holds. �
This result should be 
ompared with the 
ase of stri
tly 
ontra
tive quantum 
hannels
whose 
onvergen
e was shown to be exponentially fast along to whole traje
tory [143,
146℄.
4.4.3 Ergodi
 
hannels with pure �xed points
An interesting 
lass of ergodi
 quantum 
hannel is formed by those CPT maps whose
�xed point is a pure density matrix. Among them we �nd for instan
e the maps
employed in the 
ommuni
ation proto
ols dis
ussed in this thesis or those of the pu-
ri�
ation s
hemes of Refs. [142, 141℄. We will now show that within this parti
ular

lass, ergodi
ity and mixing are indeed equivalent properties.
We �rst need the following Lemma, whi
h dis
usses a useful property of quantum

hannels (see also [161℄).
Lemma 4.6 Let τ be a quantum 
hannel and Θ be an eigenve
tor of τ with peripheral
eigenvalue λ = eiϕ. Then, given g = Tr
> 0, the density matri
es ρ =
ΘΘ†/g and σ =
Θ†Θ/g are �xed points of τ .
Proof Use the left polar de
omposition to write Θ = g ρU where U is a unitary
operator. The operator ρU is 
learly an eigenve
tor of τ with eigenvalue eiϕ, i.e.
τ(ρU) = λ ρU . (4.32)
4 Ergodi
ity and mixing
Hen
e introdu
ing a Kraus set {Kn}n of τ [154℄ and the spe
tral de
omposition of the
density matrix ρ =
j pj|ψj〉〈ψj | with pj > 0 being its positive eigenvalues, one gets
λ = Tr[τ(ρU)U †] =
j,ℓ,n
pj〈φℓ|Kn|ψj〉〈ψj |UK†nU †|φℓ〉 , (4.33)
where the tra
e has been performed with respe
t to an orthonormal basis {|φℓ〉}ℓ of
H. Taking the absolute values of both terms gives
|λ| = |
j,ℓ,n
pj〈φℓ|Kn|ψj〉〈ψj |UK†nU †|φℓ〉|
j,ℓ,n
pj〈φℓ|Kn|ψj〉〈ψj |K†n|φℓ〉
j,ℓ,n
pj〈φℓ|UKnU †|ψj〉〈ψj |UK†nU †|φℓ〉
Tr[τ(ρ)]
Tr[τ̃(ρ)] = 1, (4.34)
where the inequality follows from the Cau
hy-S
hwartz inequality. The last identity
instead is a 
onsequen
e of the fa
t that the transformation τ̃(ρ) = Uτ(U †ρU)U † is
CPT and thus tra
e preserving. Sin
e |λ| = 1 it follows that the inequality must be
repla
ed by an identity. This happens if and only if there exist eiϑ su
h that
pj{〈φℓ|Kn|ψj〉}∗ =
pj〈ψj |K†n|φℓ〉 = eiϑ
pj〈ψj |UK†nU †|φℓ〉 , (4.35)
for all j, ℓ and n. Sin
e the |φℓ〉 form a basis of H, and pj > 0 this implies
〈ψj |K†n = eiϑ 〈ψj |UK†nU † ⇒ 〈ψj |UK†n = e−iϑ 〈ψj |K†nU , (4.36)
for all n and for all the not null eigenve
tors |ψj〉 of ρ. This yields
τ(ρU) =
Kn|ψj〉〈ψj |UK†n = e−iϑ
Kn|ψj〉〈ψj |K†nU
= e−iϑ τ(ρ)U (4.37)
whi
h, repla
ed in (4.32) gives e−iϑ τ(ρ) = eiϕ ρ, whose only solution is e−iϑ = eiϕ.
Therefore τ(ρ) = ρ and ρ is a �xed point of τ . The proof for σ goes along similar
lines: simply 
onsider the right polar de
omposition of Θ instead of the left polar
de
omposition. �
Corollary 4.4 Let τ be an ergodi
 quantum 
hannel. It follows that its eigenve
tors
asso
iated with peripheral eigenvalues are normal operators.
4 Ergodi
ity and mixing
Proof Let Θ be an eigenoperator with peripheral eigenvalue eiϕ su
h that τ (Θ) =
eiϕ Θ. By Lemma 4.6 we know that, given g = Tr
the density matri
es
ΘΘ†/g and σ =
Θ†Θ/g must be �xed points of τ . Sin
e the map is ergodi
we must have ρ = σ, i.e. ΘΘ† = Θ†Θ. �
Theorem 4.8 (Purely ergodi
 maps) Let |ψ1〉〈ψ1| be the pure �xed point of
an ergodi
 quantum 
hannel τ . It follows that τ is mixing.
Proof We will use the spe
tral gap 
riterion showing that |ψ1〉〈ψ1| is the only pe-
ripheral eigenve
tor of τ . Assume in fa
t that Θ ∈ L(H) is a eigenve
tor of τ with
peripheral eigenvalue, i.e.
τ (Θ) = eiϕΘ . (4.38)
From Lemma 4.6 we know that the density matrix
ΘΘ†/g, (4.39)
with g = Tr
> 0, must be a �xed point of τ . Sin
e this is an ergodi
 map we
must have ρ = |ψ1〉〈ψ1|. This implies Θ = g|ψ1〉〈ψ2|, with |ψ2〉 some normalised ve
tor
of H. Repla
ing it into Eq. (4.38) and dividing both terms by g yields τ (|ψ1〉〈ψ2|) =
eiϕ|ψ1〉〈ψ2| and
|〈ψ1|τ(|ψ1〉〈ψ2|)|ψ2〉| = 1 . (4.40)
Introdu
ing a Kraus set {Kn}n of τ and employing Cau
hy-S
hwartz inequality one

an then write
1 = |〈ψ1|τ(|ψ1〉〈ψ2|)|ψ2〉| = |
〈ψ1|Kn|ψ1〉〈ψ2|K†n|ψ2〉| (4.41)
〈ψ1|Kn|ψ1〉〈ψ1|K†n|ψ1〉
〈ψ2|Kn|ψ2〉〈ψ2|K†n|ψ2〉
〈ψ1|τ(|ψ1〉〈ψ1|)|ψ1〉
〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 =
〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 ,
where we used the fa
t that |ψ1〉 is the �xed point of τ . Sin
e τ is CPT the quantity
〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 is upper bounded by 1. Therefore in the above expression the
4 Ergodi
ity and mixing
inequality must be repla
ed by an identity, i.e.
〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 = 1 ⇐⇒ τ(|ψ2〉〈ψ2|) = |ψ2〉〈ψ2| . (4.42)
Sin
e τ is ergodi
, we must have |ψ2〉〈ψ2| = |ψ1〉〈ψ1|. Therefore Θ ∝ |ψ1〉〈ψ1| whi
h
shows that |ψ1〉〈ψ1| is the only eigenve
tor of τ with peripheral eigenvalue of. �
An appli
ation of the previous Theorem is obtained as follows.
Lemma 4.7 LetMAB =MA⊗1B+1A⊗MB be an observable of the 
omposite system
HA ⊗HB and τ the CPT linear map on HA of Stinespring form [156℄
τ(ρ) = TrB
U (ρ⊗ |φ〉B〈φ|)U †
, (4.43)
(here TrX [· · · ] is the partial tra
e over the system X, and U is a unitary operator of
HA⊗HB). Assume that [MAB , U ] = 0 and that |φ〉B is the eigenve
tor 
orresponding
to a non-degenerate maximal or minimal eigenvalue of MB. Then τ is mixing if and
only if U has one and only one eigenstate that fa
torises as |ν〉A ⊗ |φ〉B .
Proof Let ρ be an arbitrary �xed point of τ (sin
e τ is CPT it has always at least
one), i.e. TrB
U (ρ⊗ |φ〉B〈φ|)U †
= ρ. Sin
e MAB is 
onserved and TrA [MAρ] =
TrA [MAτ(ρ)], the system B must remain in the maximal state, whi
h we have assumed
to be unique and pure, i.e.
U (ρ⊗ |φ〉B〈φ|)U † = ρ⊗ |φ〉B〈φ| =⇒ [U, ρ⊗ |φ〉B〈φ|] = 0 . (4.44)
Thus there exists a orthonormal basis {|uk〉}k ofHA⊗HB diagonalising simultaneously
both U and ρ ⊗ |φ〉B〈φ|. We express the latter in this basis, i.e. ρ ⊗ |φ〉B〈φ| =
k pk|uk〉〈uk| with pk > 0, and 
ompute the von Neumann entropy of subsystem B.
This yields
0 = H(|φ〉B〈φ|) = H
pk|uk〉〈uk|
pk H (TrA [|uk〉〈uk|]) .(4.45)
From the 
onvexity of the von Neumann entropy the above inequality leads to a

ontradi
tion unless TrA [|uk〉〈uk|] = |φ〉B〈φ| for all k. The |uk〉 must therefore be
fa
torising,
|uk〉 = |νk〉A ⊗ |φ〉B . (4.46)
If the fa
torising eigenstate of U is unique, it must follow that ρ = |ν〉〈ν| for some
|ν〉 and that τ is ergodi
. By Theorem 4.8 it then follows that τ is also mixing. If on
4 Ergodi
ity and mixing
the other hand there exists more than one fa
torising eigenstate, than all states of the
form of Eq. (4.46) 
orrespond to a �xed point ρk = |νk〉〈νk| and τ is neither ergodi
nor mixing. �
Remark 4.3 An appli
ation of this Lemma is the proto
ol for read and write a

ess
by lo
al 
ontrol dis
ussed in the next 
hapter.
4.5 Con
lusion
In reviewing some known results on the mixing property of 
ontinuous maps, we
obtained a stronger version of the dire
t Lyapunov method. For 
ompa
t metri
 spa
es
(in
luding quantum 
hannels operating over density matri
es) it provides a ne
essary
and su�
ient 
ondition for mixing. Moreover it allows us to prove that asymptoti
deformations with at least one �xed point must be mixing.
In the spe
i�
 
ontext of quantum 
hannels we employed the generalised Lyapunov
method to analyse the mixing properties. Here we also analysed di�erent mixing

riteria. In parti
ular we have shown that an ergodi
 quantum 
hannel with a pure
�xed point is also mixing.
5 Read and write a

ess by lo
al

ontrol
5.1 Introdu
tion
The unitarity of Quantum Me
hani
s implies that information is 
onserved. Whatever
happens to a quantum system - as long as it is unitary, the original state 
an in
prin
iple be re
overed by applying the inverse unitary transformation. However it is
well known that in open quantum systems [121℄ the redu
ed dynami
s is no longer
unitary. The redu
ed dynami
s is des
ribed by a 
ompletely positive, tra
e preserving
maps, and we have seen in the last 
hapter that there are extreme examples, namely
mixing maps, where all information about the initial state is eventually lost. Where
has it gone? If the whole system evolves unitary, then this information must have been
transferred in the 
orrelations between redu
ed system and environment [162℄, and/or
in the environment. We 
an see that this may be useful for quantum state transfer, in
parti
ular the 
ase where all information is transferred into the �environment�, whi
h

ould be another quantum system (the re
eiver). A parti
ularly useful 
ase is given by
mixing maps with pure 
onvergen
e points, be
ause a pure state 
annot be 
orrelated,
and be
ause we have a simple 
onvergen
e 
riterion in this 
ase (Subse
tion 4.4.3).
This is an example of homogenisation [138, 139℄. Furthermore, if the mixing property
arises from some operations, we 
an expe
t that by applying the inverse operations,
information 
an also be transferred ba
k to the system. This property was used
in [137, 163℄ to generate arbitrary states of a 
avity �eld by sending atoms through
the 
avity. The 
ru
ial di�eren
e is that in our system 
ontrol is only assumed to be
available on a subsystem (su
h as, for example, the ends of a quantum 
hain). Hen
e
we will show in this 
hapter how arbitrary quantum states 
an be written to (i.e.
prepared on) a large system, and read from it, by lo
al 
ontrol only. This is similar
in spirit to universal quantum interfa
es [164℄, but our di�erent approa
h allows us to
spe
ify expli
it proto
ols and to give lower bounds for �delities. We also demonstrate
how this 
an be used to signi�
antly improve the quantum 
ommuni
ation between two
parties if the re
eiver is allowed to store the re
eived signals in a quantum memory
5 Read and write a

ess by lo
al 
ontrol
before de
oding them. In the limit of an in�nite memory, the transfer is perfe
t.
We prove that this s
heme allows the transfer of arbitrary multi-partite states along
Heisenberg 
hains of spin-1/2 parti
les with random 
oupling strengths.
Even though the 
onvergen
e of a mixing map is essentially exponentially fast
(Corollary 4.3), we still have to deal with in�nite limits. Looking at the environment
this in turn would require to study states on an in�nite dimensional Hilbert spa
e,
and unfortunately this 
an introdu
e many mathemati
al di�
ulties. We are mainly
interested in bounds for the �nite 
ase: if the proto
ol stops after �nitely many steps,
what is the �delity of the reading/writing? Whi
h en
oding and de
oding operations
must be applied? By stressing on these questions, we 
an a
tually avoid the in�nite
dimensional 
ase, but the pri
e we have to pay is that our 
onsiderations be
ome a
bit te
hni
ally involved.
5.2 Proto
ol
We 
onsider a tripartite �nite dimensional Hilbert spa
e given by H = HC⊗HC̄⊗HM .
We assume that full 
ontrol (the ability to prepare states and apply unitary transfor-
mations) is possible on system C and M, but no 
ontrol is available on system C̄.
However, we assume that C and C̄ are 
oupled by some time-independent Hamilto-
nian H. We show here that under 
ertain assumptions, if the system CC̄ is initialised
in some arbitrary state we 
an transfer (�read�) this state into the systemM by apply-
ing some operations between M and C. Likewise, by initialising the system M in the

orre
t state, we 
an prepare (�write�) arbitrary states on the system CC̄. The system
M fun
tions as a quantum memory and must be at least as large as the system CC̄.
As sket
hed in Fig. 5.1 we 
an imagine it to be split into se
tors Mℓ, I.e..
HMℓ (5.1)
dimHMℓ = dimHC . (5.2)
For the reading 
ase, we assume that the memory is initialised in the state
|0〉M ≡
|0〉Mℓ (5.3)
5 Read and write a

ess by lo
al 
ontrol
where |0〉 
an stand for some generi
 state1. Like in the multi rail proto
ols 
onsidered
in Chapter 3, we let the system evolve for a while, perform an operation, let it evolve
again and so forth, only that now the operation is not a measurement, but a unitary
gate. More spe
i�
ally, at step ℓ of the proto
ol we perform a unitary swap Sℓ between
system C and systems Mℓ. After the Lth swap operation the proto
ol stops. The
proto
ol for reading is thus represented by the unitary operator
W ≡ SLUSL−1U · · ·SℓU · · ·S1U, (5.4)
where U ∈ L(HCC̄) is the time-evolution operator U = exp {−iHt} for some �xed
time interval t. As we will see in the next se
tion, the redu
ed evolution of the system
C̄ under the proto
ol 
an be expressed in terms of the CPT map
τ(ρC̄) ≡ trC
U (ρC̄ ⊗ |0〉C〈0|)U †
, (5.5)
where |0〉C is the state that is swapped in from the memory. Our main assumption now
is that τ is ergodi
 with a pure �xed point (whi
h we denote as |0〉C̄). By Theorem 4.8
this implies that τ is mixing, and therefore asymptoti
ally all information is transferred
into the memory.
For writing states on the system, we just make use of the unitarity of W. Roughly
speaking, we initialise the memory in the state that it would have ended up in after
applying W if system CC̄ had started in the state we want to initialise. Then we
apply the inverse of W given by
W † = U †S1 · · ·U †Sℓ · · ·U †SL−1U †SL. (5.6)
We will see in Se
tion 5.4 how this gives rise to a unitary 
oding transformation on
the memory system, su
h that arbitrary and unknown states 
an be initialised on the
system. The reader has probably noti
ed that the inverse ofW is generally unphysi
al
in the sense that it requires ba
kward time evolution, i.e. one has to wait negative
time steps between the swaps. But we will see later how this 
an be �xed by a simple
transformation. For the moment, we just assume that W † is physi
al.
Later on we will give an example where |0〉 represents a multi-qubit state with all qubits aligned,
but here we don't need to assume this.
5 Read and write a

ess by lo
al 
ontrol
Mℓ+1 Mℓ+2MℓMℓ−2 Mℓ−1
Writing
Reading
Figure 5.1: The system CC̄ 
an only be 
ontrolled by a
ting on a (small) subsystem
C. However system C is 
oupled to system C̄ by a unitary operator U = exp {−iHt} .
This 
oupling 
an - in some 
ases - mediate the lo
al 
ontrol on C to the full system
CC̄. In our 
ase, system C is 
ontrolled by performing regular swap operations Sℓ
between it and a quantum memory Mℓ.
5.3 De
omposition equations
In this se
tion we give a de
omposition of the state after applying the proto
ol whi
h
will allow us to estimate the �delities for state transfer in terms of the mixing properties
of the map τ. Let |ψ〉CC̄ ∈ HCC̄ be an arbitrary state. We noti
e that the C 
omponent
of W |ψ〉CC̄ |0〉M is always |0〉C . Therefore we 
an de
ompose it as follows
W |ψ〉CC̄ |0〉M = |0〉C ⊗
η|0〉C̄ |φ〉M +
1− η|∆〉C̄M
(5.7)
with |∆〉C̄M being a normalised ve
tor of C̄ and M whi
h satis�es the identity
C̄〈0|∆〉C̄M = 0 . (5.8)
5 Read and write a

ess by lo
al 
ontrol
It is worth stressing that in the above expression η, |φ〉M and |∆〉C̄M are depending
on |ψ〉CC̄ . We de
ompose W † a
ting on the �rst term of Eq. (5.7) as
W †|0〉CC̄ |φ〉M =
η̃ |ψ〉CC̄ |0〉M +
1− η̃ |∆̃〉CC̄M , (5.9)
where |∆̃〉CC̄M is the orthogonal 
omplement of |ψ〉CC̄ |0〉M , i.e.
C̄C〈ψ|M 〈0|∆̃〉CC̄M = 0 . (5.10)
Multiplying Eq. (5.9) from the left with CC̄〈ψ|M 〈0| and using the 
onjugate of Eq.
(5.7) we �nd that η = η̃. An expression of η in terms of τ 
an be obtained by noti
ing
that for any ve
tor |ψ〉C̄C the following identity applies
τ(ρC̄) = trC
U (ρC̄ ⊗ |0〉C〈0|)U †
= trCM
USℓ (|ψ〉C̄C〈ψ| ⊗ |0〉M 〈0|) SℓU †
(5.11)
with ρC̄ being the redu
ed density matrix trC [|ψ〉C̄C〈ψ|]. Reiterating this expression
one gets
W (|ψ〉CC̄〈ψ| ⊗ |0〉M 〈0|)W †
= τL−1
(5.12)
with ρ′
= trC
U (|ψ〉C̄C〈ψ|)U †
. Therefore from Eq. (5.7) and the orthogonality
relation (5.8) it follows that
η = C̄〈0|τL−1
|0〉C̄ , (5.13)
whi
h, sin
e τ is mixing, shows that η → 1 for L→ ∞. Moreover we 
an use Eq. (4.31)
to 
laim that
|η − 1| = |C̄〈0|τL−1
|0〉C̄ − 1|
≤ ‖τL−1
− |0〉C̄〈0|‖1 ≤ R (L− 1)dC̄ κL−1, (5.14)
where R is a 
onstant whi
h depends upon dC̄ ≡ dimHC̄ and where κ ∈]0, 1[ is the
se
ond largest eigenvalue of τ.
5.4 Coding transformation
Here we derive the de
oding/en
oding transformation that relates states on the mem-
ory M to the states that are on the system CC̄. We �rst apply the above de
omposi-
5 Read and write a

ess by lo
al 
ontrol
tions Eqs. (5.7) and (5.9) to a �xed orthonormal basis {|ψk〉CC̄} of HCC̄ , i.e.
W |ψk〉CC̄ |0〉M = |0〉C ⊗
ηk|0〉C̄ |φk〉M +
1− ηk|∆k〉C̄M
W †|0〉CC̄ |φk〉M =
ηk |ψk〉CC̄ |0〉M +
1− ηk |∆̃k〉CC̄M . (5.15)
De�ne a linear operator D on HM whi
h performs the following transformation
D|ψk〉M = |φk〉M . (5.16)
Here |ψk〉M are orthonormal ve
tors ofM whi
h represent the states {|ψk〉CC̄} of HCC̄
(formally they are obtained by a partial isometry from C̄C to M). The ve
tors |φk〉M
are de�ned through Eq. (5.15) - typi
ally they will not be orthogonal. We �rst show
that for large L they be
ome approximately orthogonal.
From the unitarity ofW † and from Eq. (5.15) we 
an establish the following identity
M 〈φk|φk′〉M =
ηk ηk′ δkk′ +
ηk (1− ηk′) C̄CM 〈ψk0|∆̃k′〉C̄CM (5.17)
ηk′ (1− ηk) C̄CM 〈∆̃k|ψk′0〉C̄CM +
(1− η̃k)(1 − η̃k′) CC̄M 〈∆̃k|∆̃k′〉CC̄M .
De�ning η0 ≡ mink ηk it follows for k 6= k′ that
|M 〈φk|φk′〉M | ≤
ηk (1− ηk′) |C̄CM 〈ψk0|∆̃k′〉C̄CM | (5.18)
ηk′ (1− ηk) |C̄CM 〈∆̃k|ψk′0〉C̄CM |
(1− η̃k)(1− η̃k′) |CC̄M 〈∆̃k|∆̃k′〉CC̄M |
1− η0 + (1− η0) ≤ 3
1− η0. (5.19)
Therefore for all k, k′ the inequality
|M 〈φk|φk′〉M − δk,k′ | ≤ 3
1− η0 (5.20)
holds. It is worth noti
ing that, sin
e Eq. (5.14) applies for all input states |ψ〉C̄C , we
|η0 − 1| ≤ C (L− 1)dC̄ κL−1 . (5.21)
Eq. (5.20) allows us to make an estimation of the eigenvalues λk of D
†D as
|λk − 1| ≤ 3 dCC̄
1− η0, (5.22)
5 Read and write a

ess by lo
al 
ontrol
with dCC̄ ≡ dimHCC̄ . We now take a polar de
omposition D = PV of D. V is the
best unitary approximation to D [160, p 432℄ and we have
||D − V ||22 =
λk − 1
|λk − 1|
≤ 3 d2
1− η0. (5.23)
Therefore
||D − V ||2 ≤
3 dCC̄ (1− η0)1/4, (5.24)
whi
h, thanks to Eq. (5.21), shows that D 
an be approximated arbitrary well by a
unitary operator V for L→ ∞.
5.5 Fidelities for reading and writing
In what follows we will use V † and V as our reading and writing transformation,
respe
tively. In parti
ular, V † will be used to re
over the input state |ψ〉CC̄ of the

hain after we have (partially) transferred it into M through the unitary W (i.e. we
�rst a
t on |ψ〉CC̄ ⊗ |0〉M with W , and then we apply V † on M). Vi
e-versa, in order
to prepare a state |ψ〉CC̄ on CC̄ we �rst prepare M into |ψ〉M , then we apply to it
the unitary transformation V and �nally we apply W †. We now give bounds on the
�delities for both pro
edures.
The �delity for reading the state |ψ〉M is given by
Fr(ψ) ≡ M 〈ψ|V † RM V |ψ〉M (5.25)
where RM is the state of the memory after W , i.e.
RM ≡ trCC̄
W (|ψ〉CC̄〈ψ| ⊗ |0〉M 〈0|)W †
= η |φ〉M 〈φ|+ (1− η) σM . (5.26)
In the above expression we used Eqs. (5.7) and (5.8) and de�ned σM = trC̄ [|∆〉C̄M 〈∆|].
Therefore by linearity we get
Fr(ψ) = η |M 〈φ|V |ψ〉M |2 + (1− η) M 〈ψ|V † σM V |ψ〉M ≥ η |M 〈φ|V |ψ〉M |2 . (5.27)
5 Read and write a

ess by lo
al 
ontrol
Noti
e that
|M 〈φ|V |ψ〉M | = |M 〈φ|V −D +D|ψ〉M | ≥ |M 〈φ|D|ψ〉M | − |M 〈φ|D − V |ψ〉M | .(5.28)
Now we use the inequality (5.24) to write
|M 〈φ|D − V |ψ〉M | ≤ ||D − V ||2 ≤
3 dCC̄ (1− η0)1/4 . (5.29)
If |ψ〉M was a basis state |ψk〉M , then |M 〈φ|D|ψ〉M | = 1 by the de�nition Eq. (5.16)
of D. For generi
 |ψ〉M we 
an use the linearity to �nd after some algebra that
η |M 〈φ|D|ψ〉M | ≥
η0 − 3 dCC̄
1− η0 . (5.30)
Therefore Eq. (5.28) gives
η |M 〈φ|V |ψ〉M | >
η0 − 5 dCC̄ (1− η0)1/4 . (5.31)
By Eq. (5.27) it follows that
Fr ≥ η0 − 10 dCC̄ (1− η0)1/4 . (5.32)
The �delity for writing a state |ψ〉C̄C into C̄C is given by
Fw(ψ) ≡ CC̄〈ψ|trM
W †V (|ψ〉M 〈ψ| ⊗ |0〉C̄C〈0|) V †W
|ψ〉CC̄ . (5.33)
A lower bound for this quantity is obtained by repla
ing the tra
e over M with the
expe
tation value on |0〉M , i.e.
Fw(ψ) ≥ CC̄〈ψ|M 〈0|W †V (|ψ〉M 〈ψ| ⊗ |0〉C̄C〈0|) V †W |0〉M |ψ〉CC̄
∣CC̄〈0|M 〈ψ|V †W |0〉M |ψ〉CC̄
∣M 〈ψ|V †|φ〉M
= η |M 〈φ|V |ψ〉M |2 (5.34)
where Eqs. (5.7) and the orthogonality relation (5.8) have been employed to derive the
se
ond identity. Noti
e that the last term of the inequality (5.34) 
oin
ides with the
lower bound (5.27) of the reading �delity. Therefore, by applying the same derivation
of the previous se
tion we 
an write
F ≥ η0 − 10 dCC̄ (1− η0)1/4, (5.35)
5 Read and write a

ess by lo
al 
ontrol
NR NB S
Figure 5.2: Ali
e and Bob 
ontrol the spins NA and NB inter
onne
ted by the spins
NR. At time jt Bob performs a swap Sj between his spins and the memory Mj .
whi
h shows that the reading and writing �delities 
onverge to 1 in the limit of large
L. Note that this lower bound 
an probably be largely improved.
5.6 Appli
ation to spin 
hain 
ommuni
ation
We now show how the above proto
ol 
an be used to improve quantum state transfer
on a spin 
hain. The main advantage of using su
h a memory proto
ol is that - opposed
to all other s
hemes - Ali
e 
an send arbitrary multi-qubit states with a single usage
of the 
hannel. She needs no en
oding, all the work is done by Bob. The proto
ol
proposed here 
an be used to improve the performan
es of any s
heme mentioned in
Se
tion 1.5, and it works for a large 
lass of Hamiltonians, in
luding Heisenberg and
XY models with arbitrary (also randomly distributed) 
oupling strengths.
Consider a 
hain of spin-1/2 parti
les des
ribed by a Hamiltonian H whi
h 
onserves
the number of ex
itations. The 
hain is assumed to be divided in three portions
A (Ali
e), B (Bob) and R (the remainder of the 
hain, 
onne
ting Ali
e and Bob)

ontaining respe
tively the �rst NA spins of the 
hain, the last NB spins and the
intermediate NR spins, and the total length of the 
hain is N = NA+NR+NB (see Fig
5.2). Bob has a

ess also to a 
olle
tion of quantum memories M1, · · · ,Mj · · · ,ML
isomorphi
 with B, i.e. ea
h having dimension equal to the dimension 2NB of B.
We assume that Bob's memory is initialised in the zero ex
itation state |0〉M . Ali
e
prepares an arbitrary and unknown state |ψ〉A on her NA qubits. By de�ning the
(from Bob's perspe
tive) 
ontrolled part of system C = B and the un
ontrolled part
C̄ = AR, we 
an apply the results of the last se
tions and get the following
Theorem 5.1 (Memory swapping) Let H be the Hamiltonian of an open
nearest-neighbour quantum 
hain that 
onserves the number of ex
itations. If there
is a time t su
h that f1,N (t) 6= 0 (i.e. the Hamiltonian is 
apable of transport be-
tween Ali
e and Bob) then the state transfer 
an be made arbitrarily perfe
t by
using the memory swapping proto
ol.
5 Read and write a

ess by lo
al 
ontrol
Proof We only have to show that the redu
ed dynami
s on the 
hain is mixing with
a pure �xed point. Using the number of ex
itations as a 
onserved additive observable,
we 
an use the 
riterion of Lemma 4.7: If there exists exa
tly one eigenstate |E〉 of
fa
torising form with |0〉B , i.e.
∃1 |λ〉AR : H|λ〉AR ⊗ |0〉B = E|λ〉AR ⊗ |0〉B , (5.36)
then the redu
ed dynami
s is mixing toward |0〉AR. Assume by 
ontradi
tion that has
an eigenve
tor |E〉AR 6= |0〉AR whi
h falsi�es Eq. (5.36). Su
h an eigenstate 
an be
written as
|E〉AR ⊗ |0〉B = a|µ〉AR ⊗ |0〉B + b|µ̄〉AR ⊗ |0〉B , (5.37)
where a and b are 
omplex 
oe�
ients and where the spin just before the se
tion B
(with position NA + NR) is in the state |0〉 for |µ〉AR and in the state |1〉 for |µ̄〉AR.
Sin
e the intera
tion between this spin and the �rst spin of se
tion B in
ludes an
ex
hange term (otherwise f1,N (t)=0 for all t), then the a
tion of H on the se
ond
term of (5.37) yields exa
tly one state whi
h 
ontains an ex
itation in the se
tor
B. It 
annot be 
ompensated by the a
tion of H on the �rst term of (5.37). But
by assumption |E〉AR ⊗ |0〉B is an eigenstate of H, so we 
on
lude that b = 0. This
argument 
an be repeated for the se
ond last spin of se
tion R, the third last spin, and
so on, to �nally yield |E〉AR = |0〉AR, as long as all the nearest neighbour intera
tions

ontain ex
hange parts. �
Remark 5.1 Theorem 5.1 should be 
ompared to Theorem 3.2 for the multi rail pro-
to
ol. They are indeed very similar. However the 
urrent theorem is mu
h stronger,
sin
e it allows to send arbitrary multi-ex
itation states, and also to write states ba
k
onto the 
hain. It is interesting to note that Lemma 4.7 and Theorem 5.1 indi
ate a

onne
tion between the dynami
al 
ontrollability of a system and its stati
 entangle-
ment properties. It may be interesting to obtain a quantitative relation between the
amount of entanglement and the 
onvergen
e speed.
Let us now 
ome ba
k to the question raised in Se
tion 5.2 about the operation W †
being unphysi
al. As mentioned before, this 
an be �xed using a simple transfor-
mation: if the Hamiltonian H ful�ls the requirements of Lemma 4.7, then also the
Hamiltonian −H ful�ls them. Now derive the 
oding transformation Ṽ as given in
Se
tion 5.4 for the Hamiltonian H̃ = −H. In this pi
ture, the reading proto
ol W is
unphysi
al, whereas the writing proto
ol be
omes physi
al. In the more general 
ase
5 Read and write a

ess by lo
al 
ontrol
where the 
ondition of Lemma 4.7 is not valid, but the map
τ(ρC̄) ≡ trC
U (ρC̄ ⊗ |0〉C〈0|)U †
(5.38)
is still ergodi
 with a pure �xed point, we then require the map
τ̃(ρC̄) ≡ trC
U † (ρC̄ ⊗ |0〉C〈0|)U
(5.39)
to be also ergodi
 with pure �xed point to be able to use this tri
k.
5.7 Con
lusion
We have given an expli
it proto
ol for 
ontrolling a large permanently 
oupled system
by a

essing a small subsystem only. In the 
ontext of quantum 
hain 
ommuni
ation
this allows us to make use of the quantum memory of the re
eiving party to improve
the �delity to a value limited only by the size of the memory. We have shown that
this s
heme 
an be applied to a Heisenberg spin 
hain. The main advantage of this
method is that arbitrary multi-ex
itation states 
an be transferred. Also, our method

an be applied to 
hains that do not 
onserve the number of ex
itations in the system,
as long as the redu
ed dynami
 is ergodi
 with a pure �xed point.
It remains an open question how mu
h of our results remain valid if the 
hannel is
mixing toward a mixed state. In this 
ase, a part of the quantum information will in
general remain in the 
orrelations between the system and the memory, and it 
annot
be expe
ted that the �delity 
onverges to one. However, by 
on
entrating only on the
eigenstate of the �xed point density operator with the largest eigenvalue, it should be
possible to derive some bounds of the amount of information that 
an be extra
ted.
6 A valve for probability amplitude
6.1 Introdu
tion
We have mainly dis
ussed two methods for quantum state transfer so far. In the
�rst one, multiple 
hains where used, and in the se
ond one, a single 
hain was used
in 
ombination with a large quantum memory. Can we 
ombine the best of the two
s
hemes, i.e. is it possible to use only a single 
hain and a single memory qubit? In this

hapter we will show that this is indeed the 
ase and that the �delity 
an be improved
easily by applying in 
ertain time-intervals two-qubit gates at the re
eiving end of the

hain. These gates a
t as a valve whi
h takes probability amplitude out of the system
without ever putting it ba
k. The required sequen
e is determined a priori by the
Hamiltonian of the system. Su
h a proto
ol is optimal in terms of resour
es, be
ause
two-qubit gates at the sending and re
eiving end are required in order to 
onne
t the

hain to the blo
ks in all above proto
ols (though often not mentioned expli
itly). At
the same time, the engineering demands are not higher then for the memory swapping
proto
ol. Our s
heme has some similarities with [92℄, but the gates used here are
mu
h simpler, and arbitrarily high �delity is guaranteed by a 
onvergen
e theorem
for arbitrary 
oupling strengths and all non-Ising 
oupling types that 
onserve the
number of ex
itations. Furthermore, we show numeri
ally that our proto
ol 
ould
also be realised by a simple swit
hable intera
tion.
6.2 Arbitrarily Perfe
t State Transfer
We now show how the re
eiver 
an improve the �delity to an arbitrarily high value
by applying two-qubit gates between the end of the 
hain and a �target qubit� of the
blo
k. We label the qubits of the 
hain by 1, 2, · · · , N and the target qubit by N + 1
(see Fig. 6.1). The 
oupling of the 
hain is des
ribed by a Hamiltonian H. We assume
that the Hamiltonian H 
onserves the number of ex
itations and that the target qubit
N + 1 is un
oupled,
H|N + 1〉 = 0 (6.1)
6 A valve for probability amplitude
and set the energy of the ground state |0〉 to zero. For what follows we restri
t all
operators to the N + 2 dimensional Hilbert spa
e
H = span {|n〉; n = 0, 1, 2, . . . , N + 1} . (6.2)
Our �nal assumption about the Hamiltonian of the system is that there exists a time
t su
h that
fN,t(t) ≡ 〈N | exp {−itH} |1〉 6= 0. (6.3)
Physi
ally this means that the Hamiltonian has the 
apability of transporting from
the �rst to the last qubit of the 
hain. As mentioned in the introdu
tion, the �delity
of this transport may be very bad in pra
ti
e.
1 2 N N+1
Figure 6.1: A quantum 
hain (qubits 1, 2, · · · , N) and a target qubit (N + 1). By
applying a sequen
e of two-qubit unitary gates Vk on the last qubit of the 
hain and
the target qubit, arbitrarily high �delity 
an be a
hieved.
We denote the unitary evolution operator for a given time tk as Uk ≡ exp {−itkH}
and introdu
e the proje
tor
P = 1− |0〉〈0| − |N 〉〈N | − |N + 1〉〈N + 1|. (6.4)
A 
ru
ial ingredient to our proto
ol is the operator
V (c, d) ≡ P + |0〉〈0|+ d|N 〉〈N |+ d∗|N + 1〉〈N + 1|
+c∗|N + 1〉〈N | − c|N 〉〈N + 1|, (6.5)
where c and d are 
omplex normalised amplitudes. It is easy to 
he
k that
V V † = V †V = 1, (6.6)
so V is a unitary operator on H. V a
ts as the identity on all but the last two
qubits, and 
an hen
e be realised by a lo
al two-qubit gate on the qubits N and N +1.
6 A valve for probability amplitude
Furthermore we have V P = P and
V (c, d) [{c|N 〉+ d|N + 1〉}] = |N + 1〉. (6.7)
The operator V (c, d) has the role of moving probability amplitude c from the Nth
qubit to target qubit, without moving amplitude ba
k into the system, and 
an be
thought of as a valve. Of 
ourse as V (c, d) is unitary, there are also states su
h that
V (c, d) a
ting on them would move ba
k probability amplitude into the system, but
these do not o

ur in the proto
ol dis
ussed here.
Using the time-evolution operator and two-qubit unitary gates on the qubits N and
N +1 we will now develop a proto
ol that transforms the state |1〉 into |N + 1〉. Let
us �rst look at the a
tion of U1 on |1〉. Using the proje
tor P we 
an de
ompose this
time-evolved state as
U1|1〉 = PU1|1〉+ |N 〉〈N |U1|1〉
≡ PU1|1〉+
p1 {c1|N 〉+ d1|N + 1〉} , (6.8)
where p1 = |〈N |U1|1〉|2 , c1 = 〈N |U1|1〉/
p1 and d1 = 0. Let us now 
onsider the
a
tion of V1 ≡ V (c1, d1) on the time-evolved state. By Eq. (6.7) it follows that
V1U1|1〉 = PU1|1〉+
p1|N + 1〉. (6.9)
Hen
e with a probability of p1, the ex
itation is now in the position N +1, where it is
�frozen� (sin
e that qubit is not 
oupled to the 
hain. We will now show that at the
next step, this probability is in
reased. Applying U2 to Eq. (6.9) we get
U2V1U1|1〉
= PU2PU1|1〉+ 〈N |U2PU1|1〉|N 〉+
p1|N + 1〉
= PU2PU1|1〉+
p2 {c2|N 〉+ d2|N + 1〉} (6.10)
with c2 = 〈N |U2PU1|1〉/
p2, d2 =
p2 and
p2 = p1 + |〈N |U2PU1|1〉|2 ≥ p1. (6.11)
Applying V2 ≡ V (c2, d2) we get
V2U2V1U1|1〉 = PU2PU1|1〉+
p2|N + 1〉. (6.12)
6 A valve for probability amplitude
Repeating this strategy ℓ times we get
|1〉 =
|1〉+√pℓ|N + 1〉, (6.13)
where the produ
ts are arranged in the time-ordered way. Using the normalisation of
the r.h.s. of Eq. (6.13) we get
pℓ = 1−
. (6.14)
From Se
tion 3.5 we know that there exists a t > 0 su
h that for equal time intervals
t1 = t2 = . . . = tk = t we have limℓ→∞ pℓ = 1. Therefore the limit of in�nite gate
operations for Eq. (6.13) is given by
|1〉 = |N + 1〉. (6.15)
It is also easy to see that limk→∞ dℓ = 1, limk→∞ cℓ = 0 and hen
e the gates Vk

onverge to the identity operator. Furthermore, sin
e VkUk|0〉 = |0〉 it also follows
that arbitrary superpositions 
an be transferred. As dis
ussed in Theorem 4.31, this

onvergen
e is asymptoti
ally exponentially fast in the number of gate applied (a
detailed analysis of the relevant s
aling 
an be found in Chapter 2). Equation (6.15)
is a surprising result, whi
h shows that any non-perfe
t transfer 
an be made arbitrarily
perfe
t by only applying two-qubit gates on one end of the quantum 
hain. It avoids
restri
ting the gate times to spe
i�
 times (as opposed to the dual rail s
heme) while
requiring no additional memory qubit (as opposed to the memory swapping s
heme).
The sequen
e Vk that needs to be applied to the end of the 
hain to perform the
state transfer only depends on the Hamiltonian of the quantum 
hain. The relevant
properties 
an in prin
iple be determined a priori by pre
eding measurements and
tomography on the quantum 
hain (as dis
ussed in Se
t. 2.9).
6.3 Pra
ti
al Considerations
Motivated by the above result we now investigate how the above proto
ol may be
implemented in pra
ti
e, well before the realisation of the quantum 
omputing blo
ks
from Fig. 1.4. The two-qubit gates Vk are essentially rotations in the {|01〉, |10〉} spa
e
of the qubits N and N +1. It is therefore to be expe
ted that they 
an be realised (up
6 A valve for probability amplitude
to a irrelevant phase) by a swit
hable Heisenberg or XY type 
oupling between the
Nth and the target qubit. However in the above, we have assumed that the gates Vk

an be applied instantaneously, i.e. in a time-s
ale mu
h smaller than the time-s
ale
of the dynami
s of the 
hain. This 
orresponds to a swit
hable 
oupling that is mu
h
stronger than the 
oupling strength of the 
hain.
 0  100  200  300  400  500  600  700  800
Time [1/J]
switched magnetic field
switched interaction
Figure 6.2: Numeri
al example for the 
onvergen
e of the su

ess probability. Sim-
ulated is a quantum 
hain of length N = 20 with the Hamiltonian from Eq. (6.16)
(dashed line) and Eq. (6.17) with B/J = 20 (solid line). Using the original proto-

ol [1℄, the same 
hain would only rea
h a su

ess probability of 0.63 in the above time
interval.
Here, we numeri
ally investigate if a 
onvergen
e similar to the above results is still
possible when this assumption is not valid. We do however assume that the swit
hing
of the intera
tion is still des
ribable by an instantaneous swit
hing (i.e. the sudden
approximation is valid). This assumption is mainly made to keep the numeri
s simple.
We do not expe
t qualitative di�eren
es when the swit
hing times be
ome �nite as
long as the time-dependent Hamiltonian is still 
onserving the number of ex
itations
in the 
hain. In fa
t it has re
ently been shown that the �nite swit
hing time 
an even
improve the �delity [33℄. Intuitively, this happens be
ause by gradually de
reasing the

oupling, he not only re
eives the probability amplitude of the last qubit of the 
hain,
but 
an also �swallow� a bit of the dispersed wave-pa
ked (similar to the situation
dis
ussed in [92℄).
We have investigated two types of swit
hing. For the �rst type, the 
oupling itself
6 A valve for probability amplitude
is swit
hable, i.e.
H(t) = J
σ−n σ
n+1 +∆(t)σ
N+1 + h.
., (6.16)
where ∆(t) 
an be 0 or 1. For the se
ond type, the target qubit is permanently 
oupled
to the remainder of the 
hain, but a strong magneti
 �eld on the last qubit 
an be
swit
hed,
H(t) = J
σ−n σ
n+1 + h.
.+B∆(t)σ
N+1, (6.17)
where again ∆(t) 
an be 0 or 1 and B ≫ 1. This suppresses the 
oupling between the
Nth and N + 1th qubit due to an energy mismat
h.
In both 
ases, we �rst numeri
ally optimise the times for unitary evolution tk over a
�xed time interval su
h that the probability amplitude at the Nth qubit is maximal.
The algorithm then �nds the optimal time interval during whi
h ∆(t) = 1 su
h that
the probability amplitude at the target qubit is in
reased. In some 
ases the phases
are not 
orre
t, and swit
hing on the intera
tion would result in probability amplitude
�oating ba
k into the 
hain. In this situation, the target qubit is left de
oupled and the

hain is evolved to the next amplitude maximum at the Nth qubit. Surprisingly, even
when the time-s
ale of the gates is 
omparable to the dynami
s, near-perfe
t transfer
remains possible (Fig 6.2). In the 
ase of the swit
hed magneti
 �eld, the a
hievable
�delity depends on the strength of the applied �eld. This is be
ause the magneti
 �eld
does not fully suppress the 
oupling between the two last qubits. A small amount of
probability amplitude is lost during ea
h time evolution Uk, and when the gain by the
gate is 
ompensated by this loss, the total su

ess probability no longer in
reases.
6.4 Con
lusion
We have seen that by having a simple swit
hable intera
tion a
ting as a valve for
probability amplitude, arbitrarily perfe
t state transfer is possible on a single spin

hain. In fa
t, by using the inverse proto
ol, arbitrary
states in the �rst ex
itation
se
tor 
an also be prepared on the 
hain. Furthermore, this proto
ol 
an easily be
adopted to arbitrary graphs 
onne
ting multiple senders and re
eivers (as dis
ussed
for weakly 
oupled systems in [86℄).
Opposed to the method for state preparation developed in the last 
hapter this allows the 
reation
of known states only (as the valve operations Vk depend expli
itly on the state that one wants to
prepare).
7 External noise
7.1 Introdu
tion
An important question that was left open so far is what happens to quantum state
transfer in the presen
e of external noise. It is well known from the theory of open
quantum systems [121℄ that this 
an lead to dissipation and de
oheren
e, whi
h also
means that quantum information is lost. The evolution of a 
losed quantum system
is des
ribed by the S
hrödinger equation
∂t|ψ〉 = −iH|ψ〉. (7.1)
If a system is very strongly 
oupled to a environment, the dynami
 is 
ompletely in
o-
herent and des
ribed by some simple rate equations for the o

upation probabilities,
∂tPn =
kn→mPn −
km→nPm. (7.2)
In the more general 
ase where the dynami
 
onsists of 
oherent and in
oherent parts,
the evolution 
an sometimes be expressed as a Lindblad equation [121℄
∂tρ = Lρ (7.3)
for the redu
ed density matrix. These three regimes are shown in Fig. 7.1. For
quantum information theory, 
oheren
e is essential [2℄, and one has to try to isolate
the quantum 
hain as mu
h as possible from the environment. In the partially 
oherent
regime, typi
ally the quantum behaviour de
ays exponentially with a rate depending
on the temperature of the environment. Not surprisingly, this has also been found
in the 
ontext of quantum state transfer [165, 166, 167℄. From a theoreti
al point
of view it is perhaps more interesting to look at the low temperature and strong

oupling regime, where the dynami
s is often non-Markovian [121℄ and 
an no longer
expressed as a simple Lindblad equation. This is also interesting from a pra
ti
al
perspe
tive, 
orresponding to e�e
ts of the environment whi
h 
annot be avoided by

ooling. Here we 
onsider a model where the system is 
oupled to a spin environment
7 External noise
coherent
partially
coherent
incoherent
System coupling
Figure 7.1: Dominant regimes of dynami
s depending on the relative strength of the
system Hamiltonian and the environmental 
oupling [47℄.
through an ex
hange intera
tion. This 
oupling o�ers the unique opportunity of an
analyti
 solution of our problem without any approximations regarding the strength
of system-environment 
oupling (in most treatments of the e�e
t of an environment on
the evolution of a quantum system, the system-environment 
oupling is assumed to be
weak) and allows us to in
lude inhomogeneous intera
tions of the bath spins with the
system. For su
h 
oupling, de
oheren
e is possible for mixed (thermal) initial bath
states [168, 169℄. However if the system and bath are both initially 
ooled to their
ground states, is there still a non-trivial e�e
t of the environment on the �delity? In
this 
hapter we �nd that there are two important e�e
ts: the spin transfer fun
tions
(Eq. 1.19) are slowed down by a fa
tor of two, and destabilised by a modulation of
|cosGt| , where G is the mean square 
oupling to the environment. This has both
positive and negative impli
ations for the use of strongly 
oupled spin systems as
quantum 
ommuni
ation 
hannels. The spin transfer fun
tions also o

ur in the 
harge
and energy transfer dynami
s in mole
ular systems [47℄ and in 
ontinuous time random
walks [170℄ to whi
h our results equally apply.
7 External noise
7.2 Model
We 
hoose to start with a spe
i�
 spin system, i.e. an open spin 
hain of arbitrary
length N, with a Hamiltonian given by
HS = −
Jℓ (XℓXℓ+1 + YℓYℓ+1) , (7.4)
where Jℓ are some arbitrary 
ouplings and Xℓ and Yℓ are the Pauli-X and Y matri
es
for the ℓth spin. Toward the end of the se
tion we will however show that our results
hold for any system where the number of ex
itations is 
onserved during dynami
al
evolution. In addition to the 
hain Hamiltonian, ea
h spin ℓ of the 
hain intera
ts with
an independent bath of Mℓ environmental spins (see Fig 7.2) via an inhomogeneous
Hamiltonian,
I = −
k + YℓY
. (7.5)
Figure 7.2: A spin 
hain of length N = 5 
oupled to independent baths of spins.
In the above expression, the Pauli matri
es Xℓ and Yℓ a
t on the ℓth spin of the

hain, whereas X
k and Y
k a
t on the kth environmental spin atta
hed to the ℓth
spin of the 
hain. We denote the total intera
tion Hamiltonian by
I . (7.6)
The total Hamiltonian is given by H = HS +HI , where it is important to note that
[HS,HI ] 6= 0.We assume that a homogeneous magneti
 �eld along the z-axis is applied.
The ground state of the system is then given by the fully polarised state |0, 0〉, with
all 
hain and bath spins aligned along the z-axis. The above Hamiltonian des
ribes an
extremely 
omplex and disordered system with a Hilbert spa
e of dimension 2N+NM .
In the 
ontext of state transfer however, only the dynami
s of the �rst ex
itation se
tor
is relevant. We pro
eed by mapping this se
tor to a mu
h simpler system [171, 172,
7 External noise
173,174, 175℄. For ℓ = 1, 2, . . . , N we de�ne the states
|ℓ, 0〉 ≡ Xℓ|0, 0〉 (7.7)
|0, ℓ〉 ≡ 1
k |0, 0〉 (7.8)
. (7.9)
It is easily veri�ed that (setting J0 = JN = 0)
HS|ℓ, 0〉 = −Jℓ−1|ℓ− 1, 0〉 − Jℓ|ℓ+ 1, 0〉
HS|0, ℓ〉 = 0, (7.10)
HI |ℓ, 0〉 = −Gℓ|0, ℓ〉 (7.11)
HI |0, ℓ〉 = −Gℓ|ℓ, 0〉. (7.12)
Hen
e these states de�ne a 2N−dimensional subspa
e that is invariant under the
a
tion of H. This subspa
e is equivalent to the �rst ex
itation se
tor of a system of
2N spin 1/2 parti
les, 
oupled as it is shown in Fig 7.3.
Figure 7.3: In the �rst ex
itation se
tor, the system 
an be mapped into an e�e
tive
spin model where the bath spins are repla
ed by a single e�e
tive spin, as indi
ated
here for N = 5.
Our main assumption is that the bath 
ouplings are in e�e
t the same, i.e. Gℓ = G
for all ℓ. Note however that the individual number of bath spinsMℓ and bath 
ouplings
may still depend on ℓ and k as long as their means square average is the same.
Also, our analyti
 solution given in the next paragraph relies on this assumption, but
numeri
s show that our main result [Equation (7.28)℄ remains a good approximation
if the Gℓ slightly vary and we take G ≡ 〈Gℓ〉 . Disorder in the verti
al 
ouplings is
treated exa
tly in the sense that our results hold for any 
hoi
e of 
ouplings Jℓ.
7 External noise
7.3 Results
In this paragraph, we solve the S
hrödinger equation for the model outlined above and
dis
uss the spin transfer fun
tions. Firstly, let us denote the orthonormal eigenstates
of HS alone by
HS|ψk〉 = ǫk|ψk〉 (k = 1, 2 . . . , N) (7.13)
|ψk〉 =
akℓ|ℓ, 0〉. (7.14)
For what follows, it is not important whether analyti
 expressions for the eigensystem
ofHS 
an be found. Our result holds even for models that are not analyti
ally solvable,
su
h as the randomly 
oupled 
hains 
onsidered in Se
tion 2.6. We now make an ansatz
for the eigenstates of the full Hamiltonian, motivated by the fa
t that the states
|φnℓ 〉 ≡
(|ℓ, 0〉 + (−1)n |0, ℓ〉) (n = 1, 2) (7.15)
are eigenstates of H
I with the 
orresponding eigenvalues ±G [this follows dire
tly
from Eqs. (7.11) and (7.12)℄. De�ne the ve
tors
|Ψnk〉 ≡
akℓ|φnℓ 〉 (7.16)
with k = 1, 2, . . . , N and n = 0, 1. The |Ψnk〉 form an orthonormal basis in whi
h we
express the matrix elements of the Hamiltonian. We 
an easily see that
HI |Ψnk〉 = − (−1)
G|Ψnk〉 (7.17)
HS|Ψnk〉 =
akℓ|ℓ, 0〉 =
|Ψ0k〉+ |Ψ1k〉
. (7.18)
Therefore the matrix elements of the full Hamiltonian H = HS +HI are given by
〈Ψn′k′ |H|Ψnk〉 = δkk′
− (−1)nGδnn′ +
. (7.19)
The Hamiltonian is not diagonal in the states of Eq. (7.16). But H is now blo
k
diagonal 
onsisting of N blo
ks of size 2, whi
h 
an be easily diagonalised analyti
ally.
7 External noise
The orthonormal eigenstates of the Hamiltonian are given by
|Enk 〉 = c−1kn
((−1)n∆k − 2G) |Ψ0k〉+ ǫk|Ψ1k〉
(7.20)
with the eigenvalues
Enk =
(ǫk + (−1)n∆k) (7.21)
and the normalisation
ckn ≡
((−1)n∆k − 2G)2 + ǫ2k, (7.22)
where
4G2 + ǫ2k. (7.23)
Note that the ansatz of Eq. (7.16) that put H in blo
k diagonal form did not depend
on the details of HS and H
I . The methods presented here 
an be applied to a mu
h
larger 
lass of systems, in
luding the generalised spin star systems (whi
h in
lude an
intera
tion within the bath) dis
ussed in [175℄.
After solving the S
hrödinger equation, let us now turn to quantum state transfer.
The relevant quantity [1, 92℄ is given by the transfer fun
tion
fN,1(t) ≡ 〈N, 0| exp {−iHt} |1, 0〉
exp {−iEnk t} 〈Enk |1, 0〉〈N, 0|Enk 〉.
The modulus of fN,1(t) is between 0 (no transfer) and 1 (perfe
t transfer) and fully
determines the �delity of state transfer. Sin
e
〈ℓ, 0|Enk 〉 = c−1kn
((−1)n∆k − 2G) 〈ℓ, 0|Ψ0k〉+ ǫk〈ℓ, 0|Ψ1k〉
c−1kn√
((−1)n∆k − 2G+ ǫk) akℓ
we get
fN,1(t) = (7.24)
(ǫk+(−1)n∆k) ((−1)
∆k − 2G+ ǫk)2
((−1)n∆k − 2G)2 + ǫ2k
Eq. (7.24) is the main result of this se
tion, fully determining the transfer of quantum
information and entanglement in the presen
e of the environments. In the limit G→ 0,
7 External noise
we have ∆k ≈ ǫk and fN,1(t) approa
hes the usual result without an environment,
f0N,1(t) ≡
exp {−itǫk} ak1a∗kN . (7.25)
In fa
t, a series expansion of Eq. (7.24) yields that the �rst modi�
ation of the transfer
fun
tion is of the order of G2,
exp {−itǫk}
. (7.26)
Hen
e the e�e
t is small for very weakly 
oupled baths. However, as the 
hains get
longer, the lowest lying energy ǫ1 usually approa
hes zero, so the 
hanges be
ome more
signi�
ant (s
aling as 1/ǫk). For intermediate G, we evaluated Eq. (7.24) numeri
ally
and found that the �rst peak of the transfer fun
tion generally be
omes slightly lower,
and gets shifted to higher times (Figures 7.4 and 7.5). A numeri
 sear
h in the

oupling spa
e {Jℓ, ℓ = 1, . . . , N − 1} however also revealed some rare examples where
an environment 
an also slightly improve the peak of the transfer fun
tion (Fig 7.6).
 0  2  4  6  8  10  12  14  16  18  20
Time [1/J]
Figure 7.4: The absolute value of the transport fun
tion fN,1(t) of an uniform spin

hain (i.e. Jℓ = 1) with length N = 10 for three di�erent values of the bath 
oupling
G. The �lled grey 
urve is the envelope of the limiting fun
tion for G≫ ǫk/2 given by
|f0( t
)|. We 
an see that Eq. (7.28) be
omes a good approximation already at G = 4.
In the strong 
oupling regimeG≫ ǫk/2, we 
an approximate Eq. (7.23) by∆k ≈ 2G.
7 External noise
 0  2  4  6  8  10  12  14  16  18  20
Time [1/J]
Figure 7.5: The same as Fig. 7.4, but now for an engineered spin 
hain [i.e. Jℓ =√
ℓ(N − ℓ)℄ as in Subse
tion 1.5.1. For 
omparison, we have res
aled the 
ouplings
su
h that
ℓ Jℓ is the same as in the uniform 
oupling 
ase.
Inserting it in Eq. (7.24) then be
omes
fN,1(t) ≈
e−iGt
−itǫk
−itǫk
= cos(Gt)f0N,1(
). (7.27)
This surprisingly simple result 
onsists of the normal transfer fun
tion, slowed down
by a fa
tor of 1/2, and modulated by a qui
kly os
illating term (Figures 7.4 and 7.5).
We 
all this e�e
t destabilisation. Our derivation a
tually did not depend on the
indexes of f(t) and we get for the transfer from the nth to the mth spin of the 
hain
fn,m(t) ≈ cos(Gt)f0n,m(
). (7.28)
It may look surprising that the matrix fn,m is no longer unitary. This is be
ause
we are 
onsidering the dynami
s of the 
hain only, whi
h is an open quantum sys-
tem [121℄. A heuristi
 interpretation of Eq. (7.28) is that the ex
itation os
illates ba
k
and forth between the 
hain and the bath (hen
e the modulation), and spends half
7 External noise
of the time trapped in the bath (hen
e the slowing). If the time of the maximum of
the transfer fun
tion |f0n,m(t)| for G = 0 is a multiple of π/2G then this maximum is
also rea
hed in the presen
e of the bath. We remark that this behaviour is strongly
non-Markovian [121℄.
Finally, we want to stress that Eq. (7.28) is universal for any spin Hamiltonian that

onserves the number of ex
itations, i.e. with [HS,
ℓ Zℓ] = 0. Thus our restri
tion
to 
hain-like topology and ex
hange 
ouplings for HS is not ne
essary. In fa
t the only
di�eren
e in the whole derivation of Eq. (7.28) for a more general Hamiltonian is that
Eq. (7.10) is repla
ed by
HS |ℓ, 0〉 =
hℓ′ |ℓ′, 0〉. (7.29)
The Hamiltonian 
an still be formally diagonalised in the �rst ex
itation se
tor as in
Eq. (7.14), and the states of Eq. (7.20) will still diagonalise the total Hamiltonian
HS +HI . Also, rather than 
onsidering an ex
hange Hamiltonian for the intera
tion
with the bath, we 
ould have 
onsidered a Heisenberg intera
tion [176℄, but only for
the spe
ial 
ase where all bath 
ouplings g
k are all the same [177℄. Up to some
irrelevant phases, this leads to the same results as for the ex
hange intera
tion.
 0  5  10  15  20  25  30  35  40  45  50
Time [1/J]
G=0.066
Figure 7.6: A weakly 
oupled bath may even improve the transfer fun
tion for some
spe
i�
 
hoi
es of the Jℓ. This plot shows the transfer fun
tion |fN,1(t)| for N = 10.
The 
ouplings Jℓ were found numeri
ally.
7 External noise
7.4 Con
lusion
We found a surprisingly simple and universal s
aling law for the spin transfer fun
tions
in the presen
e of spin environments. In the 
ontext of quantum state transfer this
result is double-edged: on one hand, it shows that even for very strongly 
oupled
baths quantum state transfer is possible, with the same �delity and only reasonable
slowing. On the other hand, it also shows that the �delity as a fun
tion of time
be
omes destabilised with a qui
kly os
illating modulation fa
tor. In pra
ti
e, this
fa
tor will restri
t the time-s
ale in whi
h one has to be able to read the state from
the system. The results here are very spe
i�
 to the simple bath model and do not hold
in more general models (su
h as these dis
ussed in [165, 167℄, where true de
oheren
e
and dissipation takes pla
e). What we intended to demonstrate is that even though
a bath 
oupling need not introdu
e de
oheren
e or dissipation to the system, it 
an

ause other dynami
al pro
esses that 
an be problemati
 for quantum information
pro
essing. Be
ause the e�e
ts observed here 
annot be avoided by 
ooling the bath,
they may be
ome relevant in some systems as a low temperature limit.
8 Con
lusion and outlook
Our resear
h on quantum state transfer with spin 
hains has taken us on a journey
from a very pra
ti
al motivation to quite fundamental issues and ba
k again. On one
hand, our results are quite abstra
t and fundamental, and have related state transfer
to number theory, topology and quantum 
onvergen
e. On the other hand, we have
developed s
hemes whi
h are simple and pra
ti
al, taking into a

ount experimental
hurdles su
h as disorder and restri
ted 
ontrol. While the multi rail s
heme and
the memory swapping s
heme will probably be
ome useful only after mu
h further
progress in experimental QIT, the dual rail s
heme and in parti
ular the valve s
heme
have some good 
han
es to be realised in the near future.
State transfer with quantum 
hains has be
ome an area of large interest, with more
than seventy arti
les on the subje
t over the last three years. The most important
goal now is an experiment that demonstrates 
oherent transfer on a short 
hain (say
of length N ≥ 5). Su
h an experiment is not only useful building a quantum 
om-
puter, but also from a fundamental perspe
tive. For instan
e, the violation of a Bell-
inequality between distant entangled solid state qubits would be a milestone in the
�eld. Sin
e this requires a very high transfer �delity, the design of su
h an experiment
would probably require system dependent theoreti
al resear
h on how to over
ome
spe
i�
 types of noise and how to improve the �delity for spe
i�
 Hamiltonians.
List of Figures
1.1 In areas of universal 
ontrol, quantum states 
an easily be transferred
by sequen
es of unitary swap gates Sj,k between nearest neighbours. . 12
1.2 S
hemati
 layout of a quantum 
omputer. The solid arrows represent
the �ow of quantum information, and the dashed arrows the �ow of

lassi
al information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Permanently 
oupled quantum 
hains 
an transfer quantum states with-
out 
ontrol along the line. Note that the ends still need to be 
ontrol-
lable to initialise and read out quantum states. . . . . . . . . . . . . . 13
1.4 Small blo
ks (grey) of qubits (white 
ir
les) 
onne
ted by quantum

hains. Ea
h blo
k 
onsists of (say) 13 qubits, 4 of whi
h are 
onne
ted
to outgoing quantum 
hains (the thi
k bla
k lines denote their nearest-
neighbour 
ouplings). The blo
ks are 
onne
ted to the ma
ros
opi
world through 
lassi
al wires (thin bla
k lines with bla
k 
ir
les at their
ends) through whi
h arbitrary unitary operations 
an be triggered on
the blo
k qubits. The quantum 
hains require no external 
ontrol. . . 14
1.5 A quantum 
hain 
onsisting of N = 20 �ux qubits [34℄ (pi
ture and
experiment by Floor Paauw, TU Delft). The 
hain is 
onne
ted to four
larger SQUIDS for readout and gating. . . . . . . . . . . . . . . . . . . 15
1.6 Minimal �delity p(t) for a Heisenberg 
hain of length N = 50. . . . . . 21
1.7 Snapshots of the time evolution of a Heisenberg 
hain with N = 50.
Shown is the distribution |fn,1(t)|2 of the wave-fun
tion in spa
e at
di�erent times if initially lo
alised at the �rst qubit. . . . . . . . . . . 22
1.8 Mean and varian
e of the state |1〉 as a fun
tion of time. Shown is
the 
ase N = 50 with the y-axis giving the value relative to the mean
N/2 + 1 and varian
e (N2 − 1)/12 of an equal distribution 1√
|n〉. 22
1.9 Approximation of the transfer amplitude for N = 50 around the �rst
maximum by Bessel and Airy fun
tions [1, 61℄. . . . . . . . . . . . . . 23
List of Figures
1.10 pM (T ) as a fun
tion of T for di�erent 
hain lengths. The solid 
urve is
given by 1.82(2T )
and 
orresponds to the �rst peak of the proba-
bility amplitude (Eq. 1.29) . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.11 Quantum 
apa
ity, entanglement of formation (EOF), a lower bound
for the entanglement of distillation (EOD) and the averaged �delity as
a fun
tion of p(t). We also show the 
orresponding 
hain length whi
h
rea
hes this value as a �rst peak and the 
lassi
al threshold 3 − 2
The expli
it expression for the quantum 
apa
ity plotted here is given
in [54℄, and the lower bound of the entanglement of distillation will be
derived in Se
tion 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.12 Snapshots of the time evolution of a quantum 
hain with engineered

ouplings (1.47) for N = 50. Shown is the distribution of the wave-
fun
tion in spa
e at di�erent times if initially lo
alised at the �rst qubit
(
ompare Fig. 1.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1 Two quantum 
hains inter
onne
ting A and B. Control of the systems
is only possible at the two qubits of either end. . . . . . . . . . . . . . 36
2.2 Quantum 
ir
uit representation of 
on
lusive and arbitrarily perfe
t
state transfer. The �rst gate at Ali
e's qubits represents a NOT gate
applied to the se
ond qubit 
ontrolled by the �rst qubit being zero. The
qubit |ψA〉1 on the left hand side represents an arbitrary input state at
Ali
e's site, and the qubit |ψB〉1 represents the same state, su

essfully
transferred to Bob's site. The tℓ-gate represents the unitary evolution
of the spin 
hains for a time interval of tℓ. . . . . . . . . . . . . . . . . 37
2.3 Semilogarithmi
 plot of the joint probability of failure P (ℓ) as a fun
tion
of the number of measurements ℓ. Shown are Heisenberg spin-1/2-

hains with di�erent lengths N . The times between measurements tℓ
have been optimised numeri
ally. . . . . . . . . . . . . . . . . . . . . . 39
2.4 Time t needed to transfer a state with a given joint probability of failure
P a
ross a 
hain of length N . The points denote exa
t numeri
al data,
and the �t is given by Eq. (2.15). . . . . . . . . . . . . . . . . . . . . . 41
2.5 The minimal joint probability of failure P (ℓ) for 
hains with length N
in the presen
e of amplitude damping. The parameter J/Γ of the 
urves
is the 
oupling of the 
hain (in Kelvin) divided by the de
ay rate (ns−1). 44
2.6 Two disordered quantum 
hains inter
onne
ting A and B. Control of
the systems is only possible at the two qubits of either end. . . . . . . 46
List of Figures
2.7 The absolute values of the transition amplitudes fN,1(t) and gN,1(t)
for two Heisenberg 
hains of length N = 10. The 
ouplings strengths
of both 
hains were 
hosen randomly from the interval [0.8J, 1.2J ] .
The 
ir
les show times where Bob 
an perform measurements without
gaining information on α and β. . . . . . . . . . . . . . . . . . . . . . . 47
2.8 The relevant properties for 
on
lusive transfer 
an be determined by
measuring the response of the two systems at their ends only. . . . . . 51
2.9 Time t needed to transfer a state with a given joint probability of failure
P a
ross a 
hain of length N with un
orrelated �u
tuations of∆ = 0.05.
The points denote numeri
al data averaged over 100 realisations, and
the �t is given by Eq. (2.53). This �gure should be 
ompared with Fig.
2.4 where ∆ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.10 Most general setting for 
on
lusive transfer: A bla
k box with two inputs
and two outputs, a
ting as an amplitude damping 
hannel de�ned by
Eqs. (2.54) and (2.55) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.11 A simple 
ounterexample for a verti
ally symmetri
 system where dual
rail en
oding is not possible. The bla
k lines represent ex
hange 
ouplings. 56
2.12 An example for a verti
ally symmetri
 system where dual rail en
oding
is possible. The bla
k lines represent ex
hange 
ouplings of equal strength. 57
3.1 S
hemati
 of the system: Ali
e and Bob operate M 
hains, ea
h 
on-
taining N spins. The spins belonging to the same 
hain intera
t through
the Hamiltonian H whi
h a

ounts for the transmission of the signal in
the system. Spins of di�erent 
hains do not intera
t. Ali
e en
odes the
information in the �rst spins of the 
hains by applying unitary trans-
formations to her qubits. Bob re
overs the message in the last spins of
the 
hains by performing joint measurements. . . . . . . . . . . . . . . 59
3.2 Example of our notation for M = 5 
hains of length N = 6 with K = 2
ex
itations. The state above, given by |0〉1⊗|3〉2⊗|0〉3⊗|1〉4⊗|0〉5, has
ex
itations in the 
hains m1 = 2 and m2 = 4 at the horizontal position
n1 = 3 and n2 = 1. It is in the Hilbert spa
e H(S6) 
orresponding
to the subset S6 = {2, 4} (assuming that the sets Sℓ are ordered in
a 
anoni
al way, i.e. S1 = {1, 2}, S2 = {1, 3} and so on) and will
be written as |(3, 1); 6〉〉. There are
= 10 di�erent sets Sℓ and the
number of qubits one 
an transfer using these states is log2 10 ≈ 3. The
e�
ien
y is thus given by R ≈ 3/5 whi
h is already bigger than in the
dual rail s
heme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
List of Figures
3.3 Optimal rates (maximisation of Eq. (3.45 with respe
t to M) for the
Multi Rail proto
ol. Shown are three 
urves 
orresponding to di�erent
values of the joint probability of failure P one plans to a
hieve. . . . . 71
4.1 S
hemati
 examples of the orbits of a ergodi
 and a mixing map. . . . 72
4.2 Relations between topologi
al spa
es [149℄. The spa
e of density matri-

es on whi
h quantum 
hannels are de�ned, is a 
ompa
t and 
onvex
subset of a normed ve
tors spa
e (the spa
e of linear operators of the
system) whi
h, in the above graphi
al representation �ts within the set
of 
ompa
t metri
 spa
es. . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Relations between the di�erent properties of a quantum 
hannel. . . . 85
5.1 The system CC̄ 
an only be 
ontrolled by a
ting on a (small) subsystem
C. However system C is 
oupled to system C̄ by a unitary operator
U = exp {−iHt} . This 
oupling 
an - in some 
ases - mediate the lo
al

ontrol on C to the full system CC̄. In our 
ase, system C is 
ontrolled
by performing regular swap operations Sℓ between it and a quantum
memory Mℓ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Ali
e and Bob 
ontrol the spins NA and NB inter
onne
ted by the
spins NR. At time jt Bob performs a swap Sj between his spins and
the memory Mj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1 A quantum 
hain (qubits 1, 2, · · · , N) and a target qubit (N + 1). By
applying a sequen
e of two-qubit unitary gates Vk on the last qubit of
the 
hain and the target qubit, arbitrarily high �delity 
an be a
hieved. 105
6.2 Numeri
al example for the 
onvergen
e of the su

ess probability. Sim-
ulated is a quantum 
hain of length N = 20 with the Hamiltonian from
Eq. (6.16) (dashed line) and Eq. (6.17) with B/J = 20 (solid line). Us-
ing the original proto
ol [1℄, the same 
hain would only rea
h a su

ess
probability of 0.63 in the above time interval. . . . . . . . . . . . . . . 108
7.1 Dominant regimes of dynami
s depending on the relative strength of
the system Hamiltonian and the environmental 
oupling [47℄. . . . . . 111
7.2 A spin 
hain of length N = 5 
oupled to independent baths of spins. . 112
7.3 In the �rst ex
itation se
tor, the system 
an be mapped into an e�e
tive
spin model where the bath spins are repla
ed by a single e�e
tive spin,
as indi
ated here for N = 5. . . . . . . . . . . . . . . . . . . . . . . . . 113
List of Figures
7.4 The absolute value of the transport fun
tion fN,1(t) of an uniform spin

hain (i.e. Jℓ = 1) with length N = 10 for three di�erent values of the
bath 
oupling G. The �lled grey 
urve is the envelope of the limiting
fun
tion for G ≫ ǫk/2 given by |f0( t2 )|. We 
an see that Eq. (7.28)
be
omes a good approximation already at G = 4. . . . . . . . . . . . . 116
7.5 The same as Fig. 7.4, but now for an engineered spin 
hain [i.e. Jℓ =
ℓ(N − ℓ)℄ as in Subse
tion 1.5.1. For 
omparison, we have res
aled
the 
ouplings su
h that
ℓ Jℓ is the same as in the uniform 
oupling

ase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.6 A weakly 
oupled bath may even improve the transfer fun
tion for some
spe
i�
 
hoi
es of the Jℓ. This plot shows the transfer fun
tion |fN,1(t)|
for N = 10. The 
ouplings Jℓ were found numeri
ally. . . . . . . . . . 118
List of Tables
2.1 The total time t and the number of measurements M needed to a
hieve
a probability of su

ess of 99% for di�erent �u
tuation strengths ∆
(un
orrelated 
ase). Given is the statisti
al mean and the standard
deviation. The length of the 
hain is N = 20 and the number of random
samples is 10. For strong �u
tuations ∆ = 0.1, we also found parti
ular
samples where the su

ess probability 
ould not be a
hieved within the
time range sear
hed by the algorithm. . . . . . . . . . . . . . . . . . . 52
2.2 The total time t and the number of measurements M needed to a
hieve
a probability of su

ess of 99% for di�erent 
orrelations c between the

ouplings [see Eq. (2.50) and Eq. (2.51)℄. Given is the statisti
al mean
and the standard deviation for a �u
tuation strength of ∆ = 0.05. The
length of the 
hain is N = 20 and the number of random samples is 20. 53
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Index
amplitude damping, 41
amplitude delaying 
hannel, 33
Anderson lo
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arbitrary and unknown qubit, 11
asymptoti
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bla
k box, 12, 49, 53

ho
olate, 11

lassi
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on
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CPT, 83

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dual rail, 35
e�
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entanglement of formation, 27
entanglement transfer, 27
ergodi
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�delity, 16
�x-point, 76
�ux qubits, 15
generalised Lyapunov fun
tion, 77
Heisenberg Hamiltonian, 20
homogenisation, 92
Lindblad equation, 109
maximal peak, 23
minimal �delity, 16, 20
mixing, 76
multi rail, 68
non-expansive map, 80
non-Markovian, 117
peak width, 27
peripheral eigenvalues, 86
phase noise, 41
pure �x-points, 87
quantum 
apa
ity, 28
quantum 
hain, 12
quantum 
hannel, 16, 83
quantum 
omputer, 10
quantum erasure 
hannel, 37
quantum gates, 11
quantum memory, 93
quantum relative entropy, 84
Index
quantum-jump approa
h, 41
qutrits, 55
reading and writing �delities, 100
s
alability, 12
S
hrödinger equation, 109
Shor's algorithm, 10
spe
tral radius, 67
spin 
hain, 12
stri
t 
ontra
tion, 80
swap gates, 26
time-s
ale, 38
tomography, 49
topologi
al spa
e, 74
transfer fun
tions, 18
valve, 103
weak 
ontra
tion, 80
	Introduction
	Quantum Computation and Quantum Information
	Quantum state transfer along short distances
	Implementations and experiments
	Basic communication protocol
	Initialisation and end-gates
	Symmetries
	Transfer functions
	Heisenberg Hamiltonian
	Dynamic and Dispersion
	How high should p(t) be?
	Advanced communication protocols
	Engineered Hamiltonians
	Weakly coupled sender and receiver
	Encoding
	Time-dependent control
	Motivation and outline of this work
	Dual Rail encoding
	Introduction
	Scheme for conclusive transfer
	Arbitrarily perfect state transfer
	Estimation of the time-scale the transfer
	Decoherence and imperfections
	Disordered chains
	Conclusive transfer in the presence of disorder
	Arbitrarily perfect transfer in the presence of disorder
	Tomography
	Numerical Examples
	Coupled chains
	Conclusion
	Multi Rail encoding
	Introduction
	The model
	Efficient encoding
	Perfect transfer
	Convergence theorem
	Quantum chains with nearest-neighbour interactions
	Comparison with Dual Rail
	Conclusion
	Ergodicity and mixing
	Introduction
	Topological background
	Generalised Lyapunov Theorem
	Topological spaces
	Metric spaces
	Quantum Channels
	Mixing criteria for Quantum Channels
	Beyond the density matrix operator space: spectral properties
	Ergodic channels with pure fixed points
	Conclusion
	Read and write access by local control
	Introduction
	Protocol
	Decomposition equations
	Coding transformation
	Fidelities for reading and writing
	Application to spin chain communication
	Conclusion
	A valve for probability amplitude
	Introduction
	Arbitrarily Perfect State Transfer
	Practical Considerations
	Conclusion
	External noise
	Introduction
	Model
	Results
	Conclusion
	Conclusion and outlook
	List of Figures
	List of Tables
	Bibliography
	Index
