Braiding transformation, entanglement swapping and Berry phase in
  entanglement space
Braiding transformation, entanglement swapping and Berry phase in entanglement
space
Jing-Ling Chen,1, ∗ Kang Xue,2 and Mo-Lin Ge1, †
Liuhui Center for Applied Mathematics and Theoretical Physics Division,
Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China
Department of Physics, Northeast Normal University,
Changchun, Jilin 130024, People’s Republic of China
We show that braiding transformation is a natural approach to describe quantum entanglement, by
using the unitary braiding operators to realize entanglement swapping and generate the GHZ states
as well as the linear cluster states. A Hamiltonian is constructed from the unitary Ři,i+1(θ, ϕ)-
matrix, where ϕ = ωt is time-dependent while θ is time-independent. This in turn allows us to
investigate the Berry phase in the entanglement space.
PACS numbers: 03.67.Mn, 02.40.-k, 03.65.Vf
I. INTRODUCTION
Quantum entanglement is the most surprising non-
classical property of composite quantum systems that
Schrödinger singled out many decades ago as “the char-
acteristic trait of quantum mechanics”. Recently entan-
glement has become one of the most fascinating topics
in quantum information, because it has been shown that
entangled pairs are more powerful resources than the sep-
arable ones in a number of applications, such as quantum
cryptography [1], dense coding, teleportation [2] and in-
vestigation of quantum channels, communication proto-
cols and computation [3]. For instance, by using a maxi-
mally entangled state |Φ+〉 = 1/
2(| ↑↑〉+| ↓↓〉) (i.e., one
of Bell states and also the so-called Einstein-Podolsky-
Rosen (EPR) channel in [2]), Bennett et al. have showed
that it is faithful to transmit a one-qubit state a| ↑〉+b| ↓〉
from one location (Alice) to another (Bob) by sending
two bits of classical information.
For a two-qubit system, there has been defined a
“magic basis” consisting of four Bell states [4]:
|Φ+〉 = 1/
2(| ↑↑〉+ | ↓↓〉),
|Φ−〉 = 1/
2(| ↑↑〉 − | ↓↓〉),
|Ψ+〉 = 1/
2(| ↑↓〉+ | ↓↑〉),
|Ψ−〉 = 1/
2(| ↑↓〉 − | ↓↑〉), (1)
where spin-1/2 notation for definiteness has been used.
Any pure state of two-qubit can be expanded in this par-
ticular basis and its degree of entanglement can be ex-
pressed in a remarkably simple way [4]. It is possible to
study these Bell states from the other point of view of
transformation theory. The fact that they are all nor-
malized and mutual orthogonal naturally indicates that
the four Bell states are connected to the standard basis
∗Electronic address: chenjl@nankai.edu.cn
†Electronic address: geml@nankai.edu.cn
{| ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉} by a unitary transformation
1 0 0 1
0 1 1 0
0 −1 1 0
−1 0 0 1
. (2)
More precisely, let | ↑〉 = (1, 0)T and | ↓〉 = (0, 1)T , | ↑↑〉
is understood as | ↑〉⊗| ↑〉, one then has the matrix forms
for the standard basis as | ↑↑〉 = (1, 0, 0, 0)T , | ↑↓〉 =
(0, 1, 0, 0)T , | ↓↑〉 = (0, 0, 1, 0)T , | ↓↓〉 = (0, 0, 0, 1)T . Act-
ing the unitary matrix U on the standard basis will pro-
duce the four Bell states: U | ↑↑〉 = 1/
2(1, 0, 0,−1)T =
|Φ−〉, U | ↑↓〉 = 1/
2(0, 1,−1, 0)T = ¯|Ψ−〉, U | ↓↑〉 =
2(0, 1, 1, 0)T = |Ψ+〉, U | ↓↓〉 = 1/
2(1, 0, 0, 1)T =
|Φ+〉, in short one obtains U(| ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉) =
(|Φ−〉, |Ψ−〉, |Ψ+〉, |Φ+〉).
During the investigation of the relationships among
quantum entanglement, topological entanglement and
quantum computation, Kauffman et al. have discovered
a very significant result that the matrix U is nothing but
a braiding operator, and furthermore it can be identi-
fied to the universal quantum gate (i.e., the CNOT gate)
[5][6]. There is an earlier literature on topological quan-
tum computation and which is all about quantum com-
puting using braiding [7]. These literatures introduce
the braiding operators and Yang–Baxter equations to the
field of quantum information and quantum computation,
and also provide a novel way to study the quantum en-
tanglement.
Our aim in this work is twofold: one is to show that
braiding transformation is a natural approach describing
the quantum entanglement, the other is to investigate
the Berry phase in the entanglement space (or the Bloch
space). The paper is organized as follows. In Sec. II, we
briefly review the unitary braiding operators and apply
them to realize entanglement swapping and to generate
the Greenberger-Horne-Zeilinger (GHZ) states as well as
the linear cluster states. In Sec. III, after briefly review-
ing the Yang–Baxterization approach, we construct a
Hamiltonian from the unitary Ři,i+1(θ, ϕ)-matrix, where
ϕ is time-dependent while θ is time-independent. This in
http://arxiv.org/abs/0704.0709v3
mailto:chenjl@nankai.edu.cn
mailto:geml@nankai.edu.cn
turn allows us to investigate the Berry phase in the en-
tanglement space. Conclusion and discussion are made
in the last section.
II. BRAIDING TRANSFORMATION AND ITS
APPLICATIONS
Hereafter for convenience, we shall denote the spin up
| ↑〉 and down | ↓〉 as |0〉 and |1〉, respectively. Braiding
operators are the generalizations of the usual permuta-
tion operators. ForN spin-1/2 particles, the permutation
operator for the particles i and i+ 1 reads
Pi,i+1 =
(1 + ~σi · ~σi+1) =
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
, (3)
Here Pi,i+1 is understood as 11 ⊗ 12 ⊗ · · · ⊗ 1i−1 ⊗ (1 +
~σi · ~σi+1)/2⊗ 1i+2 ⊗ · · · ⊗ 1N , where 1 is the 2× 2 unit
matrix. The permutation operator Pi,i+1 exchanges the
spin state |k〉i ⊗ |l〉i+1 to be |l〉i ⊗ |k〉i+1.
The braiding operators satisfy the following braid re-
lations:
bi,i+1bi+1,i+2bi,i+1 = bi+1,i+2bi,i+1bi+1,i+2, i ≤ N − 2,
bi,i+1bj,j+1 = bj,j+1bi,i+1, |i− j| ≥ 2. (4)
The usual permutation operator Pi,i+1 is a solution of
Eq. (4) with the constraint P 2i,i+1 = 1. Physics prefers
to the unitary transformations. One may observe that
both U and Pi,i+1 are unitary. Two more general unitary
braiding transformations satisfying the braiding relations
Bi,i+1 =
1 0 0 e−iϕ
0 1 1 0
0 −1 1 0
−eiϕ 0 0 1
, (5)
Pi,i+1 =
eiξ00 0 0 0
0 0 eiξ10 0
0 eiξ01 0 0
0 0 0 eiξ11
, (6)
which allow additional phase factors. Braiding opera-
tors Bi,i+1 and Pi,i+1 transform the direct-product states
|kl〉 ≡ |k〉i ⊗ |l〉i+1 in the following way
Bi,i+1
|00〉 − eiϕ|11〉
|01〉 − |10〉
|01〉+ |10〉
e−iϕ|00〉+ |11〉
, (7)
Pi,i+1
eiξ00 |00〉
eiξ10 |10〉
eiξ01 |01〉
eiξ11 |11〉
. (8)
B12 B34
B12 B34
|ψ〉ABCD = |Φ
−〉AB ⊗ |Φ
|ψ′〉ABCD = −|Φ
−〉AD ⊗ |Φ
|0〉A |0〉B |0〉D|0〉C
FIG. 1: Realizing ES by braiding transformations. After act-
ing B34B12 on a separable state |0000〉ABCD , one prepares
a state |ψ〉ABCD = |Φ−〉AB ⊗ |Φ−〉CD needed for quantum
entanglement swapping. After performing successive braid-
ing transformations B23B34B12B23 on |ψ〉ABCD , the entan-
glement involved in the state |ψ〉ABCD is swapped to the state
|ψ′〉ABCD = −|Φ−〉AD ⊗ |Φ+〉BC .
They may generate entangled states from disentangled
ones: (i) The braiding matrix Bi,i+1 yields directly the
four Bell states |Φ±〉 and |Ψ±〉 with the relative phase
factor e−iϕ. The phase factor e−iϕ originates from the
q-deformation of the braiding operator U with q = e−iϕ
[8][9], and ϕ may have a physical significance of mag-
netic flux [10]. In the next section, we shall vary adia-
batically the parameter ϕ to obtain the Berry phase in
the entanglement space. (ii) When Pi,i+1 acts on an ini-
tial separable state 1/
2(|0〉+ |1〉)i⊗ 1/
2(|0〉+ |1〉)i+1,
it produces an entangled state (eiξ00 |00〉 + eiξ01 |01〉 +
eiξ10 |10〉 + eiξ11 |11〉)/2 whose degree of entanglement
equals to |ei(ξ00+ξ11) − ei(ξ01+ξ10)|/2. Thus it is indeed
a very natural way for the braiding operators to describe
and to generate quantum entanglement. To strengthen
such a viewpoint, we would like to provide two explicit
examples as applications of braiding transformations as
follows.
Example 1: Entanglement swapping. Entanglement
swapping (ES) is a very interesting quantum mechanical
phenomenon, which was originally proposed by Żukowski
et al. [11], generalized to multipartite quantum systems
by Zeilinger et al. [12] and Bose et al. [13] independently,
and experimentally realized by Pan et al. [14]. The origi-
nal ES is based on quantum measurement: Suppose Alice
and Bob share an entangled state, similarly Claire and
Danny also share some entangled states, if Bob and Claire
come together and make a measurement in a suitable
basis and communicate their measurement results clas-
sically, then Alice’s and Danny’s particles may become
entangled. Now we come to use the braiding transforma-
tions to realize the ES. Starting from a separable state
|0000〉ABCD ≡ |0000〉1234, we prepare a state |ψ〉ABCD
needed for quantum entanglement swapping due to the
braiding transformations B12 and B34 as follows:
|ψ〉ABCD = B34B12|0000〉ABCD
= |Φ−〉AB ⊗ |Φ−〉CD, (9)
(|00〉 − |11〉)AB ⊗
(|00〉 − |11〉)CD,
here for simplicity we have set ϕ = 0, and |Φ±〉 are the
usual Bell states. One may verify that
|ψ′〉ABCD = B23B34B12B23|ψ〉ABCD
= −|Φ−〉AD ⊗ |Φ+〉BC , (10)
(−|00〉+ |11〉)AD ⊗
(|00〉+ |11〉)BC ,
in other words, after making the successive braiding
transformations B23B34B12B23, the entanglement in-
volved in the state |ψ〉ABCD is swapped to |ψ′〉ABCD,
therefore we have realized the ES (see Fig. 1). The dif-
ference between the original ES scenario and ours is that
the former based on quantum measurement, while the
latter based on unitary braiding transformations with-
out quantum measurement. It is worthy to mention that
the approach of realizing ES by braiding transformations
is not unique. For instance, ES can be done even simpler
by using only two permutations P34P23 that acting on
the state |ψ〉ABCD.
Example 2: Generating the GHZ states and the linear
cluster states. These are some kinds of important en-
tangled states in quantum information, such as the well-
known GHZ state and the linear cluster state. (i) It is
easy to check that, after acting B12B23 on the initially
separable three-qubit state |111〉123, one obtains a state
|ψ′〉GHZ = B12B23|111〉123 (11)
(|100〉123 + |010〉123 + |001〉123 + |111〉123),
which is equivalent to the standard three-qubit GHZ
state |ψ〉GHZ = 1/
2(|000〉123 + |111〉123) up to a local
unitary transformation:
|ψ′〉GHZ = Ua ⊗ Ub ⊗ Uc|ψ〉GHZ , (12)
where Ua = Ub = Uc = V , and
, (13)
i.e., the unitary transformation V is decomposed as a
product of the Hadamard gate and the phase gate of σz.
In general, one may obtain the N -qubit GHZ states by
acting B12B23 · · ·BN−1,N on the initially separable N -
qubit state |11 · · · 1〉12···N . (ii) The linear cluster state
is the highly entangled multiparticle state on which one-
way quantum computation is based [15][16]. The linear
cluster state is locally equivalent to the N -qubits ring
cluster state. The random quantum measurement er-
ror can be overcome by applying a feed-forward tech-
nique, such that the future measurement basis depends
on earlier measurement results. This technique is crucial
for achieving deterministic quantum computation once
a cluster state is prepared. For four qubits, the linear
cluster state reads
|ψ〉cluster =
(|0〉1|0〉2|0〉3|0〉4 + |0〉1|0〉2|1〉3|1〉4 +
|1〉1|1〉2|0〉3|0〉4 − |1〉1|1〉2|1〉3|1〉4). (14)
However, it is not easy to generate |ψ〉cluster by us-
ing only one kind of unitary braiding transformations
Bi,i+1. In the following, starting from the initial separa-
ble four-qubit state |0000〉1234, we would like to mathe-
matically generate the four-qubit linear cluster state by
combined using two kinds of unitary braiding transfor-
mations Bi,i+1 and Pi,i+1, namely
|ψ〉cluster = P23P23B34B12|0000〉1234, (15)
where the phases in P23 are chosen as ξ00 = 0, ξ01 =
ξ10 = ξ11 = π, and P23 is the usual permutation opera-
tor in Eq. (3). Moreover, one can mathematically gener-
ate 16 orthogonal four-qubit linear cluster states by act-
ing P23P23B34B12 on the initial states |ijkl〉1234, where
i, j, k, l run from 0 to 1.
Significantly such realizations of entanglement swap-
ping as well as the GHZ states are purely based on one
kind of braiding transformations Bi,i+1. Eqs. (9)-(13)
are hopeful to provide an alternative approach for the
experimenter to realize the ES and also generate the
GHZ states through a network of quantum logic gates in
the future. Recent realization of the linear cluster states
is based on quantum measurements [16]. By using two
kinds of braiding transformations, Eq. (15) has mathe-
matically produced the state |ψ〉cluster . Since Bi,i+1 and
Pi,i+1 do not have the same eigenvalues and they can-
not be the matrices representing exchanges within the
same braid group representation, there is still a distance
between the mathematical realization Eq. (15) and the
actual physical realization.
III. R-MATRIX, HAMILTONIAN AND BERRY
PHASE IN ENTANGLEMENT SPACE
In Ref. [6], the unitary matrix Ři,i+1(θ, ϕ) has been in-
troduced from the Yang–Baxterization approach [8] in or-
der to include the general discussion of the nonmaximally
entangled states. To make the paper be self-contained,
we briefly review it in the following.
The Yang-Baxterization of the unitary braiding oper-
ator Bi,i+1 is
Ři,i+1(x) =
1 + x2
(Bi,i+1 + xB
i,i+1), (16)
namely, Ři,i+1(x)-matrix is a linear superposition of ma-
trices Bi,i+1 and B
i,i+1, where B
−1 = B† is the inverse
matrix of B. The unitary Ř-matrix is a generalization
of the unitary braiding matrix Bi,i+1, which satisfies the
Yang–Baxter equation:
Ři(x) Ři+1(xy) Ři(y) = Ři+1(y) Ři(xy) Ři+1(x), (17)
where x and y are called the spectral parameters. The
braid relations (4) can be viewed as an asymptotic be-
havior of the Yang–Baxter equation. By introducing the
new variables of angles θ as cos θ = (1− x)/
2(1 + x2),
sin θ = (1+x)/
2(1 + x2), the matrix Ři,i+1(x) may be
recast to Ři,i+1(θ, ϕ) = sin θ 1i ⊗ 1i+1 + cos θ Mi,i+1.
where Mi,i+1 = e
−iϕS+i ⊗ S
i+1 − eiϕS
i ⊗ S
i+1 + S
S−i+1 − S
i ⊗ S
i+1, and S
± = Sx ± iSy are the matrices
for spin-1/2 angular momentum operators.
Similar to Eq. (7), when the unitary matrix
Ři,i+1(θ, ϕ) acts on the direct-product states |kl〉, it is
expected to produce the nonmaximally entangled states
Ři,i+1(θ, ϕ)
sin θ|00〉 − eiϕ cos θ|11〉
sin θ|01〉 − cos θ|10〉
cos θ|01〉+ sin θ|10〉)
e−iϕ cos θ|00〉+ sin θ|11〉
Remarkably, the four states in the right-hand side of Eq.
(18) possess the same degree of entanglement (or the con-
currence [17]) equals to | sin(2θ)|. When θ = π/4, they
reduce to the four Bell basis and correspondingly the ma-
trix Ři,i+1(θ, ϕ) reduces to the braiding operator Bi,i+1.
There are two parameters θ, ϕ in the unitary matrix
Ři,i+1(θ, ϕ). If let θ be time-dependent while ϕ be time-
independent, one can construct a Hamiltonian as in Ref.
[6]. However, the eigenstates of such a Hamiltonian are
separable states, which do not allow us to study the Berry
phases for entangled states. To reach this purpose, in this
paper we will let ϕ = ωt be time-dependent while θ be
time-independent.
Equation (18) can be abbreviated as
Ři,i+1(θ, ϕ)|ψ(π/2, 0)〉 = |ψ(θ, ϕ)〉. Taking
the Schrödinger equation ih̄∂|ψ(θ, ϕ)〉/∂t =
H(θ, ϕ)|ψ(θ, ϕ)〉 into account, one obtains
ih̄∂/∂t[Ři,i+1(θ, ϕ)|ψ(π/2, 0)〉] = ih̄∂/∂t[|ψ(θ, ϕ)〉] =
H(θ, ϕ)|ψ(θ, ϕ)〉 = H(θ, ϕ)Ři,i+1(θ, ϕ)|ψ(π/2, 0)〉. Now
let the parameters θ be time-independent and ϕ(t) = ωt,
one may arrive at a Hamiltonian through the unitary
transformation Ři,i+1(θ, ϕ) as
H(θ, ϕ) = ih̄
∂Ři,i+1(θ, ϕ)
i,i+1(θ, ϕ). (19)
More precisely, the Hamiltonian reads
H(θ, ϕ) = h̄ϕ̇ cos θ
cos θ 0 0 e−iϕ sin θ
0 0 0 0
0 0 0 0
eiϕ sin θ 0 0 − cos θ
,(20)
or, H(θ, ϕ) = h̄ϕ̇ cos θ[cos θ(Szi ⊗ 1i+1 + 1i ⊗ Szi+1) +
sin θ(e−iϕS+i ⊗ S
i+1 + e
iϕS−i ⊗ S
i+1]. In the standard
basis {|00〉, |01〉, |10〉, |11〉}, one observes that H(θ, ϕ)
has contributions merely on {|00〉, |11〉}, i.e., it makes
four-dimensions “collapse” to two-dimensions since θ
is assumed to be time-independent. In the basis of
{|01〉, |10〉}, the two eigenstates |χ01〉 = |01〉, |χ10〉 = |10〉
FIG. 2: Berry phases in Bloch space (or the entanglement
space). The parameter θ comes from the Yang–Baxterization
of the unitary braiding operators, while parameters ϕ origi-
nates from the q-deformation of the braiding operators. They
define a point on the unit three-dimensional sphere named
the Bloch sphere, and have definite geometric meanings as
angles of longitude and latitude respectively. Let θ be time-
independent, when the parameter ϕ(t) evolves adiabatically
from 0 to 2π, one obtains the Berry phases for χ±(θ, ϕ)
as shown in Eq. (22). The relation between Berry phases
and concurrence of the entangled states χ±(θ, ϕ) is γ± =
∓π(1−
1− C2), where C = | sin θ| is the concurrence.
are degenerate with zero eigenvalues E01 = E10 = 0,
they will not give rise to Berry phases so we would not
like to discuss them here. In the basis of {|00〉, |11〉}, the
two eigenvalues E± = ±h̄ϕ̇ cos θ with two corresponding
eigenstates read
|χ+(θ, ϕ)〉 = cos
|00〉+ eiϕ sin
|11〉,
|χ−(θ, ϕ)〉 = −e−iϕ sin
|00〉+ cos θ
|11〉. (21)
Interestingly, the interval between E+ and E− depends
on θ that related to the degree of entanglement of the
states. According to Berry’s theory [18], when ϕ(t)
evolves adiabatically from 0 to 2π, the corresponding
Berry phases for the entangled states are
γ± = i
dt 〈χ±(θ, ϕ)|
|χ±(θ, ϕ)〉 = ∓
, (22)
where Ω = 2π(1 − cos θ) is the familiar solid angle en-
closed by the loop on the Bloch sphere (see Fig. 2).
Actually, the eigenstates |χ±(θ, ϕ)〉 are the SU(2) spin
coherent states. If we express the Hamiltonian in terms
of SU(2) generators as [19]
H(θ, ϕ) = X1J1 +X2J2 +X3J3, (23)
where X1 = 2h̄ϕ̇ cos θ sin θ cosϕ, X2 =
2h̄ϕ̇ cos θ sin θ sinϕ, X3 = 2h̄ϕ̇ cos θ cos θ, and the
SU(2) generators are
J1 = (S
i ⊗ S
i+1 + S
i ⊗ S
i+1)/2,
J2 = (S
i ⊗ S
i+1 − S
i ⊗ S
i+1)/2i,
J3 = (S
i ⊗ 1i+1 + 1i ⊗ Szi+1)/2, (24)
based on which one can verify directly that
|χ+(θ, ϕ)〉 = exp[ζJ+ − ζ∗J−] |00〉,
|χ−(θ, ϕ)〉 = exp[ζJ+ − ζ∗J−] |11〉, (25)
where exp[ζJ+ − ζ∗J−] is the spin coherence operators
(and also the usual D
2 (θ, ϕ)-matrix in the angular mo-
mentum theory), J± = J1 ± iJ2 and ζ = e−iϕθ/2. Berry
phase for spin coherence states has been discussed in [19],
where the corresponding result coincides with Eq. (22).
IV. CONCLUSION AND DISCUSSION
In summary, we have shown that braiding transfor-
mation is a natural approach to describe quantum en-
tanglement, by applying the unitary braiding operators
to realize entanglement swapping and to generate the
GHZ states as well as the linear cluster states. The uni-
tary braiding matrix Bi,i+1 describes the Bell states and
the Yang–Baxter matrix Ři,i+1(θ, ϕ) describes generally
entangled states with arbitrary degree of entanglement.
Varying the parameter θ continuously from 0 to 2π, one
may obtain an “oscillating entanglement” phenomenon
for the entangled states. A Hamiltonian is constructed
from the unitary Ři,i+1(θ, ϕ)-matrix, where ϕ = ωt is
time-dependent while θ is time-independent. This in turn
allows us to investigate the Berry phases for the entan-
gled states in the entanglement space.
Let us make two discussions to end this paper.
(i) Very recently, geometric phases for mixed states
[20] have been observed in experiments by using NMR
interferometry [21] as well as single photon interferome-
try [22]. Under a certain noisy environment, the states
|χ±(θ, ϕ)〉 may become mixed states as
ρ±(r, θ, ϕ) = r |χ±〉〈χ±|+ (1− r)ρnoise, (26)
where 0 ≤ r ≤ 1. Usually, ρnoise is chosen as 1i ⊗
1i+1/4 = (|00〉〈00| + |01〉〈01| + |10〉〈10| + |11〉〈11|)/4
and ρ±(r, θ, ϕ) become the generalized Werner states
[3]. Following Ref. [23], one may calculate the geo-
metric phases for the mixed states ρ±(r, θ, ϕ), however,
the computation becomes complicated since ρ±(r, θ, ϕ)
have two nonzero degenerate eigenvalues in the subspace
spanned by {|01〉, |10〉}. Geometric phases for degen-
erate mixed states are complicated and we will discuss
them elsewhere. In the following, we would like to dis-
cuss a more simpler case for geometric phases of mixed
states, by restricting the noise in the subspace spanned
by {|00〉, |11〉}. The analysis on such a restriction to
the noisy environment is limited, for it assumes that the
states |01〉 and |10〉 are decoupled, and the environment
only affects the |00〉 and |11〉 subspace.
For simplicity, let us denote |0〉 ≡ |00〉, |1〉 ≡ |11〉,
then the Hamiltonian can be rewritten in a very sim-
ple form as H(θ, ϕ) = h̄ϕ̇ cos θ r̂ · σ, where r̂ =
(sin θ cosϕ, sin θ sinϕ, cos θ) is a unit vector on the Bloch
sphere, and σ = (σ1, σ2, σ3) is the Pauli matrix vector
in the basis of {|0〉, |1〉}, namely, σ1 = |0〉〈1| + |1〉〈0|,
σ2 = −i|0〉〈1| + i|1〉〈0|, σ3 = |0〉〈0| − |1〉〈1|. Based
on which, the pure states |χ±(θ, ϕ)〉 can be rewritten
in a density matrix form as |χ±〉〈χ±| = (11 ± r̂ · σ)/2,
where 11 = |0〉〈0| + |1〉〈1|. In other words, in the basis
of {|0〉, |1〉}, |χ±〉 may be viewed as states of a single
“qubit”, which allows us to introduce mixed states and
discuss their geometric phases in a particular noisy en-
vironment as follows. By choosing ρnoise = 11/2, one has
from Eq. (26) that
ρ±(r, θ, ϕ) =
(11± r · σ), (27)
where r = rr̂. The state |χ+〉 corresponds to a point r̂
on the Bloch sphere; ρnoise is located on the center of
the Bloch sphere; the unit vector r̂ shrinks to r when the
particular noise is presented and then |χ±〉〈χ±| turn to
be mixed states ρ±(r, θ, ϕ). Follow the same calculations
in [23], let r and θ be time-independent, when parameter
ϕ(t) evolves adiabatically from 0 to 2π, one obtains the
geometric phase for the mixed states (27) as
γmixed± = ∓ arctan(r tan
), (28)
which reduces to Eq. (22) for r = 1.
(ii) The Berry phases in Eq. (22) can be expressed
in terms of the concurrence of the states |χ±(θ, ϕ)〉 as
γ± = ∓π(1 −
1− C2), with C = | sin θ| being the con-
currence. It is well-known that C is an invariant of entan-
glement for the entangled states |χ±(θ, ϕ)〉, while Berry
phase is related to some certain topological structures.
This might bridge a connection between quantum entan-
glement and topological quantum computation. Even-
tually, when one restricts the discussion to the basis of
{|0〉, |1〉}, by taking θ = π/4, φ = −π/2 (or q = i), the
matrix Ři,i+1 may reduce to the two-dimensional repre-
sentation of braiding operators as in Eq. (140) of [9],
which has physical applications in non-Abelian quantum
Hall systems and topological quantum field theory.
ACKNOWLEDGMENTS The authors thank Prof.
L. D. Faddeev and Prof. K. Fijikawa for their encourage-
ment and useful discussions. This work was supported
in part by NSF of China (Grant No. 10575053 and No.
10605013) and Program for New Century Excellent Tal-
ents in University.
[1] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). [2] C. H. Bennett, and S. J. Wiesner, Phys. Rev. Lett. 69,
2881 (1992); C. H. Bennett, G. Brassard, C. Crépeau, R.
Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett.
70, 1895 (1993).
[3] M. A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information (Cambridge University Press,
2000).
[4] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W.
K. Wootters, Phys. Rev. A 54, 3824 (1996).
[5] L. H. Kauffman and S. J. Lomonaco Jr., New J. Phys. 6,
134 (2004); J. M. Franko, E. C. Rowell, and Z. Wang, J.
Knot Theory Ramifications 15, 413 (2006).
[6] Y. Zhang, L. H. Kauffman and M. L. Ge, Int. J. Quant.
Inform. 3, 669 (2005).
[7] A. Y. Kitaev, Annals Phys. 303, 2 (2003); e-print
quant-ph/9707021.
[8] Yang–Baxter Equations in Integrable Systems, edited by
M. Jimbo (World Scientific, Singapore, 1990).
[9] J. K. Slingerland, and F. A. Bais, Nucl. Phys. B 612, 229
(2001).
[10] G. Badurek, H. Rauch, A.Zeilinger, W. Bauspiess, and
U. Bonse, Phys.Rev. D 14, 1177 (1976); A. Zeilinger,
Physica B 137, 235 (1986).
[11] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert,
Phys. Rev. Lett. 71, 4287 (1993).
[12] A. Zeilinger, M. A. Horne, H. Weinfurter, and M.
Żukowski, Phys. Rev. Lett. 78, 3031 (1997).
[13] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A 57,
822 (1998).
[14] J. W. Pan, D. Bouwmeester, H. Weinfurter, A. Zeilinger,
Phys. Rev. Lett. 80, 3891 (1998).
[15] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,
5188 (2001).
[16] R. Prevedel, P. Walther, F. Tiefenbacher, P. Böhi, R.
Kaltenbaek, T. Jennewein, and A. Zeilinger, Nature 445,
65 (2007).
[17] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[18] Geometric Phases in Physics, edited by A. Shapere and
F. Wilczek (World Scientific, Singapore, 1989).
[19] S. Chaturvedi, M. S. Sriram, and V. Srinivasan, J. Phys.
A 20, L1091 (1987).
[20] E. Sjöqvist, A. K. Pati, A. Ekert, J. S. Anandan, M.
Ericsson, D. K. L. Oi, and V. Vedral, Phys. Rev. Lett.
85, 2845 (2000).
[21] J. Du, P. Zou, M. Shi, L. C. Kwek, J. W. Pan, C. H. Oh,
A. Ekert, D. K. L. Oi, and M. Ericsson, Phys. Rev. Lett.
91, 100403 (2003).
[22] M. Ericsson, D. Achilles, J. T. Barreiro, D. Branning, N.
A. Peters, and P. G. Kwiat, Phys. Rev. Lett. 94, 050401
(2005).
[23] K. Singh, D. M. Tong, K. Basu, J. L. Chen, and J. F.
Du, Phys. Rev. A 67, 032106 (2003).
http://arxiv.org/abs/quant-ph/9707021
