Controllable Quantum Switchboard
Controllable Quantum Switchboard
D. Kaszlikowski,1 L. C. Kwek,2 C. H. Lai,1 and V. Vedral3
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
Nanyang Technological University, National Institute of Education, 1, Nanyang Walk, Singapore 637616
The School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom
(Dated: September 6, 2021)
All quantum information processes inevitably requires the explicit state preparation of an en-
tangled state. Here we present the construction of a quantum switchboard which can act both as
an optimal quantum cloning machine and a quantum demultiplexer based on the preparation of a
four-qubit state.
Many quantum information processes require the ex-
plicit preparation of specially entangled quantum states.
Two-qubit maximally entangled state often called Bell
state, for instance, form an essential quantum resource
needed in quantum teleportation [1]. The preparation of
three-qubit maximally entangled state (like GHZ) could
be harnessed for secure secret sharing [2]. In one-way
quantum computing, a four-qubit entangled state called
cluster state provides an efficient implementation of a
universal quantum gate: arbitrary single-qubit unitary
operation and the CNOT gate [3].
It is interesting to note that entangled states which are
used as a common resource in quantum information pro-
cesses generally need not even be maximally entangled
at all. As long as the state is genuinely entangled, quan-
tum computation and communication will generally be
better than the classical counterparts. In particular, the
non-maximally entangled W state has been experimen-
tally implemented and proposed for controlled quantum
teleportation and secure communication [4].
An essential component of any quantum computation
is the ability to spread quantum information over vari-
ous parts of quantum computer. The parts then undergo
separate evolutions depending on the type of the quan-
tum information processing we wish to implement. Ulti-
mately we need be capable of navigating the relevant part
of the information into a designated output. In a classi-
cal computer this flow of information is achieved through
a controllable switch. Is it possible to design a quantum
analogue for such a device? An added complexity in a
quantum switch would be the requirement that the in-
formation flows down many possible channels coherently
as well as the possibility of channeling it in one selected
direction.
Ideally we would like to realize the simplest such a
device with the least number of qubits needed for this
purpose. In addition these qubits will in practice be im-
plemented in a physical system which will determine the
nature of the qubits and couplings between them. There-
fore, when designing our switch we should also take into
account realistic interaction between the qubits, which
severely limits the number of possible Hamiltonians to
execute such a quantum switch. Here we present a pos-
sible implementation of the switch that fulfills of all the
above requirements.
Let us consider an interesting four-qubit state de-
scribed by
|ψ〉 = 1√
(|(11)12〉|(11)34〉 − |(11)14〉|(11)23〉) , (1)
where |(11)ij〉 = 1√2 (|0〉i|1〉j−|1〉i|0〉j) is the singlet state.
Throughout the paper we use the following notation for
the Bell states |(ab)〉 =
(−1)kb√
|k, k + a〉 with sum-
mation modulo 2. It turns out that this state is ideally
suited for a quantum switchboard, i.e., a circuit that can
be used to direct the flow of quantum information in
a controllable manner. An interesting property of the
presented switchboard is that in the case of failure the
information is not entirely lost.
The first qubit of the state (1) belongs to Alice, the sec-
ond one to Bob, the third one to Charlene and the last
one to Dick. Suppose Alice attaches some auxiliary qubit
to the first qubit and perform a joint Bell measurement.
Immediately after getting one of the four possible out-
comes, she broadcasts two (classical) bits of information
to Bob and Charlene as it is in the usual teleportation
scheme. At this point, it is not necessary for Dick to
know these two bits of information.
Bob and Charlene can recover the state of the auxiliary
qubit with the fidelity 5
by applying appropriate unitary
transformation based on the knowledge of the broadcast
classical bits. The given state at the beginning does not
provide a universal cloning machine for three copies of
the cloned state [5]. Thus, the qubit belonging to Dick is
related to the Alice’s auxiliary qubit with the ”classical”
fidelity 1
, i.e., the fidelity that can be achieved without
prior entanglement. It is interesting to note that Bob and
Charlene possess the optimum fidelity achievable under
a symmetric cloning machine. Dick’s fidelity is allowed
since there is no limitation on the production of clones
with the fidelity below 2
Thus the presented protocol behaves like an optimal
telecloner [6]. However, there is still an unused qubit
held by Dick. Depending on Alice’s decision regarding to
whom she wishes ultimately to send her auxiliary qubit,
say Bob (Charlene) for instance, she can direct Dick to
send his qubit to Charlene (Bob). As soon as Charlene
receives Dick’s qubit, he can perform a Bell measurement
on his qubit with Dick’s qubit and send the results of his
http://arxiv.org/abs/0704.1280v1
Alice
Charlene
FIG. 1: Suppose Alice wishes to send her auxiliary qubit
to Bob. She can direct Dick to send his qubit to Charlene.
Charlene then performs a Bell measurement on his qubit with
Dick’s qubit and send the results of his measurement to Bob.
Using the information from Charlene, Bob can perfectly re-
cover the state of the Alice’s auxiliary qubit.
measurement to Bob. Using the information from Char-
lene, Bob can perfectly recover the state of the Alice’s
auxiliary qubit.
The situation is entirely symmetric, i.e., Dick can send
his qubit to Bob instead of Charlene with the result that
now Charlene can obtain Alice’s auxiliary qubit with
perfect fidelity. In short the state acts as a quantum
switchboard in which Alice can direct optimal clones to
Bob and Charlene or perform perfect quantum telepor-
tation to Bob or Charlene by utilizing Dick’s qubit as in
a quantum demultiplexer. A schematic diagram of this
quantum switchboard protocol is shown in Fig. 1. By
directing Dick’s qubit to either Bob (or Charlene), Alice
can effectively transfer the unknown auxiliary qubit to
Charlene (or Bob). Moreover, she can delay the transfer
process to a later time as long as she has effective control
over Dick’s qubit.
Incidentally, other shared states may be able to achieve
some aspect of our quantum switch but it is difficult to
find a state with all desired properties. For instance with
the GHZ state, shared among Alice, Bob and Charlene,
one could in principle provide perfect quantum telepor-
tation to both Bob and Charlene, but without the addi-
tional benefit of an optimal quantum cloner. In this case,
Alice teleclones to both Bob and Charlene with a classi-
cal fidelity of 2/3. The eventual quantum teleportation
to Bob (or Charlene) is performed with a measurement
in the basis 1/
2(|0〉 ± |1〉).
It is also interesting to note that the sheer presence of
singlets or dimer-like bonds in the four-qubit state may
make it more robust to certain types of noise. One ex-
ample would be fluctuating magnetic field or polarization
drift, depending on how we implement our qubits. This
kind of fault tolerance is absent in the GHZ state.
Let us now prove the above statements. It is conve-
nient to write the state |ψ〉 in the following way
|ψ〉 = 1
(3|(11)12〉|(11)34〉+ |(10)12〉|(10)34〉+
|(01)12〉|(01)34〉 − |(00)12〉|(00)34〉). (2)
We can immediately see that the state shared by Alice
and Bob is the Werner state with 1
of noise. Taking into
account that the fidelity of teleportation for the Werner
state with the noise fraction 1 − p is given by p+1
we see that the fidelity of Bob’s qubit is 5
. It can be
checked that the state between Alice and Dick is the
Werner state that is an equal mixture of the three Bell
states |(10)〉, |(01)〉, |(00)〉. Thus Dick’s clone of Alice’s
auxiliary qubit has the fidelity 1
, which is the fidelity
achievable classically.
The state |ψ〉 is symmetric with respect to Bob and
Charlene
|ψ〉 = 1
(3|(11)13〉|(11)24〉+ |(10)13〉|(10)24〉+
|(01)13〉|(01)24〉 − |(00)13〉|(00)24〉). (3)
therefore Charlene’s clone has the same fidelity as Bob’s
Let us now write the state |ψ〉 together with the Al-
ice’s auxiliary qubit |α〉 (particle with 0 index) in the
form suitable for further analysis. To focus our attention
we consider the scenario where Dick sends his qubit to
Charlene. We have
|α〉|ψ〉 = 1
k,l,m,n=0
λkl|(mn)01〉 ⊗
⊗ Umn,kl|α〉|(kl)34〉, (4)
where λ11 = 3, λ01 = λ10 = 1, λ00 = −1 and Umn,kl is
a usual unitary transformation that appears in the pro-
cess of teleportation with the Bell state |(kl)〉 and with
the outcome of Bell measurement (mn). For instance,
U01,11 = σx.
Suppose now that the outcome of Alice’s measurement
is (mn). The collapsed state |χmn〉 shared by Bob, Char-
lene and Dick is
|χmn〉 =
k,l=0
Umn,kl|α〉|(kl)34〉. (5)
Therefore, Bob’s state ρmn reads
ρmn =
k,l=0
|λkl|2Umn,kl|α〉〈α|U †mn,kl. (6)
After receiving two bits (mn) of classical information
from Alice, Bob can recover Alice’s state with the fi-
delity 5
as mentioned before. However, when Charlene
performs the Bell measurement on his and Dick’s qubit,
obtains the result (kl) and sends it to Bob, Bob receives
the state
|λkl|
Umn,kl|α〉, (7)
which he can transform back to the state |α〉 by applying
the inverse unitary transformation U
mn,kl
The symmetry of the state |ψ〉 allows us to repeat the
same argument for the case in which Alice decides to
send her qubit to Bob so that now Charlene can obtain
the state |α〉 with perfect fidelity. It is interesting to
note that the relative phase between the components of
the state |ψ〉 is crucial for desired functionality. Other
phase choices or, for that matter, the complete lack of
coherence, will not give us the same quantum switch.
Finally we emphasise that our quantum switch state
is a ground state, albeit degenerate, of the Majundar-
Ghosh (MG) model [10]. This spin chain belongs to a
class of many- body Hamiltonians that provide a good
qualitative account of materials like Cu2(C5H12N2)2Cl4,
CuGeO3 and YCuO2.5[11]. Therefore our quantum
switch is very realistic since it may already exist in
some solid state systems. MG is essentially a one-
dimensional quantum spin chain with nearest- and next-
nearest-neighbor exchange interactions described by the
the Hamiltonian
HMG = J
2~Si~Si+1 + α~Si ~Si+2
, (8)
where J > 0 and N is the number of sites in the one-
dimensional lattice with periodic boundary condition.
The Hamiltonian is exactly solvable for α = 1 and has
a quantum phase transition from an ordered phase to a
disordered spin-liquid-like phase as α varies from zero to
some critical value αcrit = 0.482[12].
At α = 1 and for an even N , there is a two-fold de-
generate ground state subspace spanned by two dimer
configurations
|(11)12〉|(11)34〉 . . . |(11)(N−1)N〉
|(11)23〉|(11)45〉 . . . |(11)N1〉, (9)
superposition of which, for N = 4, gives us the state |ψ〉.
In conclusion, we have provided a quantum switch-
board which could act both as an optimal quantum
cloning machine or a quantum demultiplexer. Moreover,
we also note that it is possible to extend the switchboard
to higher spins and higher dimensional spaces as long as
we have a configuration of dimer-like neighboring bonds.
We also note that apart from spin chains, it is possible
that the four-qubit state considered in this paper could
also be created from multi-photon entangled states gen-
erated with spontaneous parametric down conversion and
linear optics apparatus.
I. ACKNOWLEDGMENT
D.K. would like to thank Alastair Kay and Ravis-
hankar Ramanathan for useful discussions.
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