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Physics Letters A www.elsevier.com/locate/pla

Entanglement of Bell diagonal mixed states Hui Zhao College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

a r t i c l e

i n f o

Article history: Received 20 January 2009 Received in revised form 4 May 2009 Accepted 21 August 2009 Available online 25 August 2009 Communicated by P.R. Holland

a b s t r a c t A suﬃcient and necessary condition for separability of Bell diagonal mixed states for bipartite systems in higher dimensions is presented. Moreover, we present a necessary condition for genuine entanglement of Bell diagonal mixed states in higher dimensions for multipartite systems. © 2009 Published by Elsevier B.V.

PACS: 03.67.Mn 03.65.Ud Keywords: Bell diagonal states Entanglement

1. Introduction Quantum entanglement has played very important roles in quantum information processing such as quantum teleportation [1], quantum cryptography [2], quantum dense coding [3] and parallel computing [4]. Historically, Einstein, Podolsky and Rosen (EPR) [5] and Schrödinger [6] ﬁrst recognized entanglement. Bell considered Bell states to test the EPR problem and proposed the famous inequality [7]. Since then there have been many important generalizations [8–14] and interesting applications [15] for Bell states. Peres [16] discovered a very strong necessary condition for separability, namely, the separable states remain positive if subjected to partial transposition (PPT criterion). This was soon shown by Horodecki et al. to be suﬃcient for 2 × 2 and 2 × 3 bipartite systems [17,18]. The reduction criterion for separability was proposed independently in Ref. [19] and Ref. [20]. This criterion is equivalent to the PPT criterion for 2 × N composite systems, but it also is not suﬃcient for separability in general. Nielsen et al. presented another necessary criterion called majorization criterion [21]. Still more efforts have been done to analyze entanglement of quantum states [22–29]. In the meantime, considerable interest has been devoted to multipartite entanglement. In Ref. [30] a necessary condition for genuine entanglement of Bell diagonal states in 2 × 2 × · · · × 2 systems was presented. In Ref. [31] the authors showed that ﬁnding generic Bell states diagonal entanglement witnesses for d1 × d2 × · · · × dn systems reduces to a convex optimization problem. But they only presented Bell states diagonal entanglement witnesses for multi-qubits, 2 × N and 3 × 3 systems. Multipartite entanglement is more complicated than bipartite cases, an operational criterion to determine whether an arbitrary multipartite mixed state is entangled has not yet been obtained till now. The goal of this Letter is twofold. First, we present a suﬃcient and necessary condition for separability of Bell diagonal mixed states for bipartite systems in higher dimensions. Then we study entanglement of Bell diagonal mixed states in higher-dimensional for multipartite systems. The result of 2 × 2 × · · · × 2 systems in Ref. [30] is generalized to the case of 2 × 2 × · · · × N systems. 2. Separability condition for Bell diagonal mixed states in bipartite systems Bell diagonal mixed states for 2 × N systems are deﬁned as [31]

ρ2×N =

1 N −1

qi 1 i 2 |ψi 1 i 2 ψi 1 i 2 |,

i 1 =0 i 2 =0

E-mail address: [email protected] 0375-9601/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.physleta.2009.08.048

H. Zhao / Physics Letters A 373 (2009) 3924–3930

1

where 0 q i 1 i 2 1 and

i 1 =0

N −1

i 2 =0 q i 1 i 2

3925

= 1. |ψi 1 i 2 is a generalized Bell basis

|ψi 1 i 2 = I 2×2 ⊗ ( S )i 2 (Ω)i 1 |ψ00 , 1

where |ψ00 = √1

k=0 |k|k,

2

⎛

0

1

I , S and Ω denote identical operator, shift operator and phase module, respectively,

···

0

⎞

0

⎛

⎜ 1 0 0 ··· 0 ⎟ ⎟ ⎜ ⎟ ⎜ S =⎜ 0 0 1 ··· 0 ⎟ ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠ 0

0

···

0

1

1

0

···

0

0

⎞

⎜ 0 −1 0 · · · 0 ⎟ ⎟ ⎜ ⎟ ⎜ Ω =⎜ 0 0 1 ··· 0 ⎟ ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠

,

0

N ×N

0

···

0

1

. N ×N

Theorem 1. Bell diagonal mixed states ρ2× N , where N is even, is separable if and only if it is a PPT state.

ρ2×N is a PPT state and N = 2d, we have

Proof. Suppose that

|ψ0i 2 = I 2×2 ⊗ I 2d×2d |ψ00 , |ψ0i 2 = I 2×2 ⊗ S |ψ00 , The density matrix

⎛

|ψ1i 2 = I 2×2 ⊗ Ω|ψ00

|ψ1i 2 = I 2×2 ⊗ S Ω|ψ00

(i 2 = 2k, k = 0, . . . , d − 1), (i 2 = 2k + 1, k = 0, . . . , d − 1).

ρ2×2d can be written as

⎞

⎜ 0 c 0(2k+1) ⎜ ⎜ ⎜ ··· ··· ⎜ ⎜ 0 0 ⎜ ρ2×2d = ⎜ ⎜ 0 ˜c 0(2k+1) ⎜ ⎜ a˜ 0 ⎜ 0(2k) ⎜ ⎝ ··· ···

···

0

0

a˜ 0(2k)

···

···

0

c˜ 0(2k+1)

0

···

0 ⎟ ⎟

···

··· 0

0

···

···

0

a0(2k)

0

··· ··· ···

0

0

···

0

c 0(2k+1)

0

···

0

0

a0(2k)

···

···

0

0

··· ···

0

0

0

⎟ ··· ···⎟ ⎟ ··· 0 ⎟ ⎟ ⎟, ··· 0 ⎟ ⎟ ··· 0 ⎟ ⎟ ⎟ ··· ···⎠

where d −1

a0(2k) =

d −1

1 2

a˜ 0(2k) =

(q0(2k) + q1(2k) ),

k =0

1 2

(q0(2k) − q1(2k) ),

k =0

d −1

d −1

1

c 0(2k+1) =

2

c˜ 0(2k+1) =

(q0(2k+1) + q1(2k+1) ),

k =0

1 2

(q0(2k+1) − q1(2k+1) ).

k =0

After a partial transposition for the second system, we get

⎛

⎞

⎜ 0 c 0(2k+1) ⎜ ⎜ ⎜ ··· ··· ⎜ ⎜ 0 0 ⎜ ρ2T×2 2d = ⎜ ⎜ 0 a˜ 0(2k) ⎜ ⎜ c˜ 0 ⎜ 0(2k+1) ⎜ ⎝ ··· ···

···

0

0

c˜ 0(2k+1)

···

···

0

a˜ 0(2k)

0

···

0 ⎟ ⎟

···

··· 0

0

···

···

0

a0(2k)

0

0

The nonzero eigenvalues of

ρ11 =

···

0

0

···

0

c 0(2k+1)

0

···

0

0

a0(2k)

···

···

0

0

··· ··· 0

⎟ ··· ···⎟ ⎟ ··· 0 ⎟ ⎟ ⎟. ··· 0 ⎟ ⎟ ··· 0 ⎟ ⎟ ⎟ ··· ···⎠

ρ2T×2 2d are

λ1 = a0(2k) + c˜ 0(2k+1) , Because of 0 q i 1 i 2 1 and

··· ···

λ2 = a0(2k) − c˜ 0(2k+1) ,

1

a0(2k)

0

0

c 0(2k+1)

i 1 =0

2d−1 i 2 =0

ρ11 ρ12

qi 1 i 2 = 1, the matrix ρ˜ = ρ † ρ 12 22

,

λ3 = a˜ 0(2k) + c 0(2k+1) ,

ρ12 =

0

a˜ 0(2k)

c˜ 0(2k+1)

0

,

λ4 = c 0(2k+1) − a˜ 0(2k) .

is also PPT, where

ρ˜ is separable if and only if it is PPT, therefore ρ˜ can be written as ρ˜ = p i ρ i (1 ) ⊗ ρ i (2 ) , i

0

ρ22 =

c 0(2k+1)

0

0

a0(2k)

.

3926

where

H. Zhao / Physics Letters A 373 (2009) 3924–3930

ρi(1) and ρi(2) are density matrices for the ﬁrst and the second systems, respectively. Let

I 2×2 02×(d−2) ⊗ ρ i (2 ) , ρ˜i (2) = 0(d−2)×2

we have

ρ2×2d =

0(d−2)×(d−2)

p i ρi (1) ⊗ ρ˜i

(2 )

.

i

So

ρ2×2d is separable. T2 Conversely, if ρ2×2d is separable, ρ2× 0 applying the PPT criterion [16]. 2 2d

Remark 1. In Ref. [31] Bell states diagonal entanglement witnesses for 2 × N systems was derived. Such witnesses are in general only suﬃcient for the presence of entanglement. But our result is a suﬃcient and necessary condition for entanglement. Remark 2. In fact, if we choose appropriate shift operator and phase module for arbitrary dimensions Bell diagonal mixed states, we can construct a class of Bell diagonal mixed states which are similar to ρ2× N . For example, consider Bell diagonal mixed states for 2d1 × 2d2 systems 2d1 −1 2d2 −1

ρ2d1 ×2d2 =

i 1 =0

qi 1 i 2 |ψi 1 i 2 ψi 1 i 2 |,

i 2 =0

its Bell basis |ψi 1 i 2 can be constructed as follows:

|ψi 1 i 2 = (Ω)i 1 ⊗ ( S )i 2 |ψ00 , 1 1

|ψ00 = √

2 k =0

|k|k,

(1)

where Ω and S denote phase module and shift operator, respectively,

⎛

1

0

0

...

⎞

0

⎛

⎜ 0 −1 0 · · · 0 ⎟ ⎟ ⎜ ⎟ ⎜ Ω =⎜ 0 0 1 ··· 0 ⎟ ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠ 0

0

···

0

1

0

1

···

0

0

⎞

⎜ 1 0 0 ··· 0 ⎟ ⎟ ⎜ ⎟ ⎜ S =⎜ 0 0 1 ··· 0 ⎟ ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠

,

0

2d1 ×2d1

0

···

0

1

.

2d2 ×2d2

Eq. (1) can be rewritten equivalently as [32] 1 1

|ψi 1 i 2 = √

2 j =0

In this case,

exp 2π i

ji 1

2

| j |( j + i 2 ) mod 2 (i 1 = 0, 1, . . . , 2d1 − 1; i 2 = 0, 1, . . . , 2d2 − 1).

ρ2d1 ×2d2 can be written as ⎛ a(2i )(2 j )

ρ2d1 ×2d2

0

⎜ 0 c (2i )(2 j +1) ⎜ ⎜ ⎜ ··· ··· ⎜ ⎜ 0 0 ⎜ =⎜ ⎜ 0 ˜c (2i )(2 j +1) ⎜ ⎜ a˜ 0 ⎜ (2i )(2 j ) ⎜ ⎝ ··· ··· 0

···

0

0

a˜ (2i )(2 j )

···

···

0

c˜ (2i )(2 j +1)

0

···

···

···

··· ··· ···

0

0

0

···

0

c (2i )(2 j +1)

0

···

0

0

a(2i )(2 j )

···

···

0

0

··· ··· ···

0

0

0

(2)

⎞

0 ⎟ ⎟

⎟ ··· ···⎟ ⎟ ··· 0 ⎟ ⎟ ⎟, ··· 0 ⎟ ⎟ ··· 0 ⎟ ⎟ ⎟ ··· ···⎠ ···

(3)

0

where d 1 −1 d 2 −1

a(2i )(2 j ) =

q(2i )(2 j ) + q(2i +1)(2 j ) i =0

j =0

2

d 1 −1 d 2 −1

,

a˜ (2i )(2 j ) =

q(2i )(2 j ) − q(2i +1)(2 j ) i =0

d 1 −1 d 2 −1

c (2i )(2 j +1) =

q(2i )(2 j +1) + q(2i +1)(2 j +1) i =0

j =0

2

2

j =0

,

d 1 −1 d 2 −1

,

c˜ (2i )(2 j +1) =

q(2i )(2 j +1) − q(2i +1)(2 j +1) i =0

Using the similar method as above, we can obtain the following theorem. Theorem 2. ρ2d1 ×2d2 of the form Eq. (3) is separable if and only if it is a PPT state.

j =0

2

.

H. Zhao / Physics Letters A 373 (2009) 3924–3930

3927

3. Entanglement of Bell diagonal mixed states in multipartite systems Let the dimensions of systems A 1 , . . . , A n , A n+1 be 2, . . . , 2, N, respectively. A multipartite quantum state if it can be written as

ρ2×···×2×N =

A1

p i ρi

A n +1

⊗ ρiA 2 ⊗ · · · ⊗ ρi

ρ2×···×2×N is called separable

,

i A

A

A

where ρi 1 , ρi 2 . . . , ρi n+1 are density matrices for systems A 1 , A 2 , . . . , A n+1 respectively, p i > 0 and i p i = 1. An entangled state which is not biseparable, triseparable, etc., in arbitrary decomposition, is said to be genuinely entangled. Bell diagonal mixed states for 2 ×· · ·× 2 × N systems can be written as [31]

ρ2×···×2×N =

qi 1 ···in+1 |ψi 1 ···in+1 ψi 1 ···in+1 |.

|ψi 1 ···in+1 is a generalized Bell basis

|ψi 1 ···in+1 = (σz )i 1 ⊗ (σx )i 2 ⊗ · · · ⊗ (σx )in ⊗ ( S )in+1 |ψ0···0 , where |ψ0···0 = √1

1

k=0 |k|k,

2

⎛

0

1

σz and σx denote Pauli operators, S denotes shift operator,

···

0

0

⎞

⎜ 1 0 0 ··· 0 ⎟ ⎟ ⎜ ⎟ ⎜ S =⎜ 0 0 1 ··· 0 ⎟ ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠ 0

0

···

0

(4)

1

. N ×N

Eq. (4) can be rewritten equivalently as 1 1

|ψi 1 ···in+1 = √

2 j =0

exp 2π i

ji 1 2

Suppose that N = 2d, the density matrix

n +1 ( j + il ) mod 2 (i 1 = 0, 1; i 2 = 0, 1; . . . ; in = 0, 1; in+1 = 0, 1, . . . , N − 1). | j 1 l

ρ2×···×2×N can be written as

ρ2×···×2×N = ( b1 b2 · · · b(2n−1 −1)N +1 b(2n−1 −1)N +2 · · · b2n N )T ,

(6)

where

b1 = (a00···0(2k) , 0, . . . , 0, a˜ 00···0(2k) , . . . , 0), b2 = (0, c 00···0(2k+1) , . . . , c˜ 00···0(2k+1) , 0, . . . , 0),

...,

b(2n−1 −1) N +1 = (0, . . . , a01···1(2k) , 0, . . . , 0, a˜ 01···1(2k) , . . . , 0), b(2n−1 −1) N +2 = (0, . . . , 0, c 01···1(2k+1) , . . . , c˜ 01···1(2k+1) , 0, . . . , 0),

...,

b2n N = (0, . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , 0), and d −1

a00···0(2k) =

d −1

1 2

c 00···0(2k+1) =

(q00···0(2k) + q10···0(2k) ),

k =0 d −1

1 2

···

(q00···0(2k+1) + q10···0(2k+1) ),

k =0

d −1 1 a01···1(2k) = (q01···1(2k) + q11···1(2k) ),

2

c 01···1(2k+1) =

k =0 d −1

1 2

(5)

l =2

(q01···1(2k+1) + q11···1(2k+1) ),

a˜ 00···0(2k) =

1 2

c˜ 00···0(2k+1) =

··· a˜ 01···1(2k) =

(q00···0(2k) − q10···0(2k) ),

k =0 d −1

1 2

(q00···0(2k+1) − q10···0(2k+1) ),

k =0

d −1

1 2

c˜ 01···1(2k+1) =

k =0

For example, the density matrix for 2 × 2 × 2d systems can be written as

(q01···1(2k) − q11···1(2k) ),

k =0 d −1

1 2

(q01···1(2k+1) − q11···1(2k+1) ).

k =0

3928

H. Zhao / Physics Letters A 373 (2009) 3924–3930

⎛

a00(2k)

0

⎜ 0 c 00(2k+1) ⎜ ⎜ ⎜ ··· ··· ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ 0 0 ⎜ ⎜ ··· ··· ⎜ ⎜ ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎜ ⎜ ··· · ·· ⎜ ⎜ 0 c˜ 00(2k+1) ⎜ ⎜ ⎝ a˜ 00(2k) 0 ···

···

a˜ 00(2k)

···

···

···

···

···

···

···

···

···

···

···

···

···

· · · c˜ 00(2k+1)

···

···

···

···

···

···

···

···

···

··· 0 a˜ 01(2k) · · · · · · c˜ 01(2k+1) 0 ···

···

···

···

···

···

· · · a01(2k)

0

···

0

···

···

···

···

0

c˜ 01(2k+1)

c 01(2k+1)

· · · a˜ 01(2k)

···

· · · c 01(2k+1)

0

0

···

···

···

···

0

···

···

···

0

···

0

a01(2k)

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

· · · c 00(2k+1)

0

···

···

···

···

···

···

···

0

a00(2k)

···

···

···

···

···

···

···

···

···

···

⎞

···⎟ ⎟ ⎟ ···⎟ ⎟ ···⎟ ⎟ ⎟ ···⎟ ⎟ ···⎟ ⎟ ⎟, ···⎟ ⎟ ···⎟ ⎟ ⎟ ···⎟ ⎟ ···⎟ ⎟ ⎟ ···⎠ ···

where d −1

a00(2k) =

2

c 00(2k+1) = a01(2k) =

d −1

1

1 2

c 01(2k+1) =

a˜ 00(2k) =

(q00(2k) + q10(2k) ),

k =0 d −1

1

(q00(2k+1) + q10(2k+1) ),

2

k =0 d −1

k =0 d −1

1 2

(q01(2k+1) + q11(2k+1) ),

2

c˜ 00(2k+1) = a˜ 01(2k) =

(q01(2k) + q11(2k) ),

1

1 2

c˜ 01(2k+1) =

k =0

(q00(2k) − q10(2k) ),

k =0 d −1

1

(q00(2k+1) − q10(2k+1) ),

2

k =0 d −1

(q01(2k) − q11(2k) ),

k =0 d −1

1 2

(q01(2k+1) − q11(2k+1) ).

k =0

Now using the similar method to Ref. [30], we can get a necessary condition for genuine entanglement of Bell diagonal mixed states. Theorem 3. If the density matrix (n + 1)-th qudit. Proof. TA

ρ2×···×2×N of the form Eq. (6) is a genuinely entangled state, it is negative under partial transposition for the

TA

n+1 ρ2×···× denotes the density matrix ρ2×···×2×2d under partial transposition for the (n + 1)-th qudit. The non-zero eigenvalues of 2×2d

n+1 ρ2×···× are 2×2d

d −1

1 2

(q0i 2 ···in (2k) + q1i 2 ···in (2k) + q0i 2 ···in (2k+1) − q1i 2 ···in (2k+1) ),

k =0 d −1

1 2

(q0i 2 ···in (2k) + q1i 2 ···in (2k) − q0i 2 ···in (2k+1) + q1i 2 ···in (2k+1) ),

k =0 d −1

1 2

(q0i 2 ···in (2k) − q1i 2 ···in (2k) + q0i 2 ···in (2k+1) + q1i 2 ···in (2k+1) ),

k =0 d −1

1 2

(q0i 2 ···in (2k+1) + q1i 2 ···in (2k+1) − q0i 2 ···in (2k) + q1i 2 ···in (2k) ),

k =0

where i 2 , . . . , in = 0 or 1. The eigenvectors of

TA

n+1 ρ2×···× are 2×2d

(1, 0, . . . , 0, 0, . . . , 0, 1, 0, . . . , . . . , 0)T , (0, 1, . . . , 0, 0, . . . , 1, 0, 0, . . . , . . . , 0)T ,

··· (0 , . . . , 0 , 0 , . . . , 1 , 0 , . . . , 0 , 1 , . . . , 0 ) T , (0, . . . , 0, 0, . . . , 0, 1, . . . , 1, 0, . . . , 0)T ,

(1, 0, . . . , 0, 0, . . . , 0, −1, 0, . . . , . . . , 0)T ,

(0, 1, . . . , 0, 0, . . . , −1, 0, 0, . . . , . . . , 0)T ,

(0, . . . , 0, 0, . . . , 1, 0, . . . , 0, −1, . . . , 0)T ,

(0, . . . , 0, 0, . . . , 0, 1, . . . , −1, 0, . . . , 0)T .

We divide these eigenvalues into 2n−1 sets. Each set has parameters q0i 2 ···in (2k) , q1i 2 ···in (2k) , q0i 2 ···in (2k+1) , and q1i 2 ···in (2k+1) . Therefore the density matrix ρ2×···×2×2d can be represented by a convex combination of 2n−1 sub-density matrices, i.e.,

ρ2×···×2×2d = λ00···00 ρ00···00 + λ00···01 ρ00···01 + · · · + λ11···11 ρ11···11 ,

H. Zhao / Physics Letters A 373 (2009) 3924–3930

3929

where d −1 (q0i 2 ···in (2k) + q0i 2 ···in (2k+1) + q1i 2 ···in (2k) + q1i 2 ···in (2k+1) ),

λi 2 ···in =

d −1 q0i

2 ···i n (2k)

λi 2 ···in

k =0

q1i 2 ···in (2k)

+

λi 2 ···in = 1,

i 2 ,...,in =0

k =0

ρi2 ···in =

1

λi 2 ···in

|ψ0···in (2k) ψ0···in (2k) | +

q0i 2 ···in (2k+1)

λi 2 ···in

q1i 2 ···in (2k+1)

|ψ1···in (2k) ψ1···in (2k) | +

λi 2 ···in

|ψ0···in (2k+1) ψ0···in (2k+1) |

|ψ1···in (2k+1) ψ1···in (2k+1) | .

If ρ is a genuinely entangled state, there exists at least one entangled state among state, it can be written as a convex combination of the following bases

1 |0 0 ± |1 1 , 2

√

ρ00···00 , . . . , ρ11···11 . Suppose that ρi2 ···in is an entangled

1 |0 1 ± |1 0 , 2

√

where

|0 = |0i 2 · · · in ,

|1 = |1i 2 · · · in ,

with ik |ik = 0, k ∈ (1, 2, . . . , n). By Theorem 1, eigenvalues of

TA

TA

TA

ρi2 ···n+in1 is negative. Since multiplying the eigenvalues of ρi2 ···n+in1 by λi2 ···in equal the

n +1 ρ2×···× , ρ2×···×2×2d is negative under partial transposition for the (n + 1)-th qudit. 2 2×2d

Remark 3. In Ref. [30] a necessary condition for genuine entanglement of Bell diagonal states for 2 × 2 × · · · × 2 systems was presented. Their result is a special case of our Theorem 3. We generalize their result to the case of 2 × 2 × · · · × N systems. Remark 4. If we choose appropriate shift operator and phase module for arbitrary dimensions Bell diagonal mixed states, we can construct a class of Bell diagonal mixed states which are similar to ρ2×···×2× N . For example, consider Bell diagonal mixed states for k1 × k2 × · · · × kn × kn+1 systems, where ki = 2di ,

ρk1 ×k2 ×···×kn ×kn+1 =

qi 1 ···in+1 |ψi 1 ···in+1 ψi 1 ···in+1 |,

(7)

its Bell basis |ψi 1 ···in+1 can be constructed as follows:

|ψi 1 i 2 = (Ω)i 1 ⊗ ( S )i 2 ⊗ · · · ⊗ ( S )in+1 |ψ00 , where |ψ0···0 = √1

1

k=0 |k · · · |k,

2

⎛

1

0

0

...

Ω and S denote phase module and shift operator, respectively, 0

⎞

⎛

⎜ 0 −1 0 · · · 0 ⎟ ⎟ ⎜ ⎟ ⎜ Ω =⎜ 0 0 1 ··· 0 ⎟, ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠ 0

0

0

···

(8)

0

1

0

···

0

···

1

⎞

⎜ 1 0 0 ··· 0 ⎟ ⎟ ⎜ ⎟ ⎜ S =⎜ 0 0 1 ··· 0 ⎟. ⎟ ⎜ ⎝··· ··· ··· ··· ···⎠

1

0

0

0

Eq. (8) can be rewritten equivalently as 1 1

|ψi 1 ···in+1 = √

2 j =0

exp 2π i

ji 1 2

n +1 ( j + il ) mod 2 (i 1 = 0, 1, . . . , 2d1 − 1; . . . ; in+1 = 0, 1, . . . , 2dn+1 − 1). | j 1 l

(9)

l =2

Using the similar method as above and applying Theorem 2, we can obtain the following theorem. Theorem 4. If the density matrix ρk1 ×k2 ×···×kn ×kn+1 of the form Eq. (7) is a genuinely entangled state, it must be negative under partial transposition for the (n + 1)-th qudit. 4. Conclusion We have investigated entanglement of Bell diagonal mixed states in bipartite and multipartite quantum systems. A suﬃcient and necessary condition for separability of Bell diagonal mixed states for bipartite systems in higher dimensions has been presented. Moreover, we have presented a necessary condition for genuine entanglement of Bell diagonal mixed states in higher dimensions for multipartite systems.

3930

H. Zhao / Physics Letters A 373 (2009) 3924–3930

Acknowledgements We greatly appreciate the referees’ valuable suggestions for improving the original version. We also greatly thank professor Jia-nai Chen and professor Zhi-Xi Wang for their advice. This work is supported by the National Natural Science Foundation of China under Nos. 10826086 and 10871227, the Natural Science Foundation of Beijing under No. 1092008, Scientiﬁc Research Common Program of Beijing Municipal Commission of Education under No. KM200810028003. References [1] [2] [3] [4] [5] [6] [7] [8]

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[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

[30] [31] [32]

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