On PPT States in C K C M C N Composite Quantum Systems

Transcription

1 Commun. Theor. Phys. (Beijing, China) 42 (2004) pp c International Academic Publishers Vol. 42, No. 2, August 5, 2004 On PPT States in C K C M C N Composite Quantum Systems WANG Xiao-Hong, FEI Shao-Ming,,2 WANG Zhi-Xi, and WU Ke Department of Mathematics, Capital Normal University, Beijing 00037, China 2 Institute of Applied Mathematics, University of Bonn, 535 Bonn, Germany (Received December, 2003) Abstract We study the general representations of positive partial transpose (PPT) states in C K C M C N. For the PPT states with rank-n a canonical form is obtained, from which a sufficient separability condition is presented. PACS numbers: Hk, Ta, c Key words: separability, quantum entanglement, PPT state Due to the importance of quantum entangled states in quantum information and computation, [ 3] much effort has been done recently towards an operational characterization of separable states. [4 6] The manifestations of mixed-state entanglement can be very subtle. [7] Till now there is no general efficient criterion in judging the separability. The Bell inequalities, [8] Peres PPT criterion, [9] reduction criterion, [0,] majorization, [2] entanglement witnesses, [3,4] extension of Peres criterion, [5] matrix realignment, [6] generalized partial transposition criterion (GPT), [7] generalized reduced criterion, [8] give some necessary (and also sufficient for some special cases) [3] conditions for separability. The separability criterion in Ref. [3] is both necessary and sufficient but not operational. For low-rank density matrices there are also some necessary and sufficient operational criteria of separability. [9,20] In Refs. [2] and [22] the separability and entanglement of quantum mixed states in C 2 C 2 C N, C 2 C 3 C N, and C 2 C 2 C 2 C N composite quantum systems have been studied in terms of matrix analysis on tensor spaces. It is shown that all such quantum states ρ with positive partial transposes and rank r(ρ) N are separable. In this article we extend the results in Ref. [2] to the case of composite quantum systems in C K C M C N with general dimensions K, M, N N. We give a canonical form of PPT states in C K C M C N with rank N and present a sufficient separability criterion. A separable state in CA K CM B CN C is of the form where ρ ABC = i p i ρ i A ρ i B ρ i C, () p i =, 0 < p i, i ρ i α are density matrices associated with the subsystems α, α = A, B, C. In the following we denote by R(ρ), K(ρ), r(ρ), and k(ρ) the range, kernel, rank, and the dimension of the kernel of ρ, respectively. We first derive a canonical form of PPT states in CA 3 C3 B CN C with rank N, which allows for an explicit decomposition of a given state in terms of convex sum of projectors on product vectors. Let 0 A, A, 2 A ; 0 B, B, 2 B ; and 0 C (N ) C be some local bases of the sub-systems A, B, and C, respectively. Lemma Every PPT state ρ in C 3 A C3 B CN C r( 2 A, 2 B ρ 2 A, 2 B ) = r(ρ) = N such that can be transformed into the following canonical form by using a reversible local operation F [DB DA D CB CA C B A I ] [DB DA D CB CA C B A I ] F, (2) where A, B, C, D, F, and the identity I are N N matrices acting on C N C and satisfy the following relations [A, A ] = [B, B ] = [C, C ] = [D, D ] = [B, A] = [B, A ] = [C, A] = [C, A ] = [D, A] = [D, A ] = [C, B] = [C, B ] = [D, B] = [D, B ] = [D, C] = [D, C ] = 0 and F = F ( stands for the transposition and conjugate). Proof In the basis we considered, a density matrix ρ in C 3 A C3 B CN C with rank N can always be written as The project supported by National Natural Science Foundation of China under Grant Nos and 02708, and Natural Science Foundation of Beijing under Grant No

2 26 WANG Xiao-Hong, FEI Shao-Ming, WANG Zhi-Xi, and WU Ke Vol. 42 E E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 2 E 2 E 23 E 24 E 25 E 26 E 27 E 28 E 29 E 3 E 23 E 3 E 34 E 35 E 36 E 37 E 38 E 39 E 4 E 24 E 34 E 4 E 45 E 46 E 47 E 48 E 49 E 5 E 25 E 35 E 45 E 5 E 56 E 57 E 58 E 59, (3) E 6 E 26 E 36 E 46 E 56 E 6 E 67 E 68 E 69 E 7 E 27 E 37 E 47 E 57 E 67 E 7 E 78 E 79 E 8 E 28 E 38 E 48 E 58 E 68 E 78 E 8 E 89 E 9 E 29 E 39 E 49 E 59 E 69 E 79 E 89 E 9 where E s are N N matrices, r(e 9 ) = N. The projection 2 A ρ 2 A gives rise to a state E 7 E 78 E 79 2 A ρ 2 A = E 78 E 8 E 89, (4) E 79 E 89 E 9 which is a state in CB 3 CN C with r( ρ ) = r(ρ) = N. Let t α denote the partial transposition with respect to the subsystem α. As every principal minor determinant of ρ t A ( ρ t B ) is some principal minor determinant of ρ, the fact that ρ is PPT implies that ρ is also PPT, i.e., ρ 0. After performing a reversible local non-unitary filtering / E 9 on the third system and using Lemma 4 in Ref. [9] the matrix ρ can be written as B B B A B A B A A A, (5) B A I where [A, A ] = [B, B ] = [B, A] = [B, A ] = 0. Similarly, if we consider the projection 2 B ρ 2 B, for the same reasons as above we conclude that the resulting matrix E 3 E 36 E 39 D D D C D 2 B ρ 2 B = E 36 E 6 E 69 = C D C C C, E 39 E 69 E 9 D C I where [C, C ] = [D, D ] = [D, C] = [D, C ] = 0. Summarizing, after performing a local filtering operation / E 9 we can bring the matrix ρ to the form E E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 2 E 2 E 23 E 24 E 25 E 26 E 27 E 28 E 29 E 3 E 23 D D E 34 E 35 D C E 37 E 38 D E 4 E 24 E 34 E 4 E 45 E 46 E 47 E 48 E 49 E 5 E 25 E 35 E 45 E 5 E 56 E 57 E 58 E 59. (6) E 6 E 26 C D E 46 E 56 C C E 67 E 68 C E 7 E 27 E 37 E 47 E 57 E 67 B B B A B E 8 E 28 E 38 E 48 E 58 E 68 A B A A A E 9 E 29 D E 49 E 59 C B A I Owing to the fact that ρ 0, a vector v CA 3 C3 B CN C verified that the following vectors are in the kernel satisfying v ρ v = 0 is in the kernel of ρ. It is directly 2 f 22 A f, 20 g 22 B g, 2 h 22 C h, 02 k 22 D k, (7) for all vectors f, g, h, k CC N. This implies E 38 = D A, E 68 = C A, E 37 = D B, E 67 = C B, E 3 = E 9 D, E 23 = E 29 D, E 34 = E 49D, E 35 = E 59D, E i6 = E i9 C, E i7 = E i9 B, E i8 = E i9 A, i =, 2, 4, 5, (8) Substituting Eq. (8) into Eq. (6) and considering partial transposition of ρ with respect to the first sub-system A, we

3 No. 2 On PPT States in C K C M C N Composite Quantum Systems 27 have E E 2 E 3 E 4 E 24 E 49 D B E 9 B E 29 B D E 2 E 2 E 23 E 5 E 25 E 59 D A E 9 A E 29 A D E 3 E 23 D D C E 9 C E 29 C D E 9 E 29 D E 4 E 5 E 9 C E 4 E 45 E 49 C B E 49 B E 59 B C ρ t A = E 24 E 25 E 29 C E 45 E 5 E 59 C A E 49 A E 59 A C. (9) E 34 E 35 D C C E 49 C E 59 C C E 49 E 59 C E 9 B E 9 A E 9 E 49 B E 49 A E 49 B B B A B E 29 B E 29 A E 29 E 59 B E 59 A E 59 A B A A A D B D A D C B C A C B A I Since the partial transposition with respect to the sub-system A is positive, ρ t A 0, and it does not change 2 A ρ 2 A, we still have 20 g 22 B g, 2 f 22 A f k(ρ t A ). This gives rise to the following equalities E 9 = B D, E 29 = A D, E 49 = B C, E 59 = A C. (0) ρ is then of the following form E E 2 B D D E 4 E 5 B D C B D B B D A B D E 2 E 2 A D D E 24 E 25 A D C A D B A D A A D D DB D DA D D D CB D CA D C D B D A D E 4 E 24 B C D E 4 E 45 B C C B C B B C A B C E 5 E 25 A C D E 45 E 5 A C C A C B A C A A C. () C DB C DA C D C CB C CA C C C B C A C B DB B DA B D B CB B CA B C B B B A B A DB A DA A D A CB A CA A C A B A A A DB DA D CB CA C B A I and Set X = E 5 B D C B D B B D A B D E 25 A D C A D B A D A A D D CA D C D B D A D, Y = E 45 B C C B C B B C A B C ρ 5 = Σ + diag(, 0, 0, 0, 0), E E 2 B D D E 4 E 2 E 2 A D D E 24 D DB D DA D D D CB E 4 E 24 B C D E 4 where A C CA A C C A C B A C A A C C CA C C C B C A C Σ = B CA B C B B B A B A CA A C A B A A A CA C B A I, = E 5 A C CA (2) and diag(a, A 2,..., A m ) denotes a diagonal block matrix with blocks A, A 2,..., A m. ρ can then be written in the following partitioned matrix form ( ) Y X X. ρ 5 As Σ possesses the following 4N kernel vectors ( f, 0, 0, 0, f A C ) T, (0, g, 0, 0, g C ) T, (0, 0, h, 0, h B ) T, (0, 0, 0, i, i A ) T for arbitrary f, g, h, i C N C, the kernel K(Σ) has at least dimension 4N. On the other hand r(σ) + k(σ) = 5N, therefore r(σ) N. As the range of Σ has at least dimension N due to the identity entry on the diagonal, we have r(σ) = N. Notice that r(ρ 5 ) r(ρ) = N, it is easy to see that r(ρ 5 ) = N. To show that = 0, we make the following

4 28 WANG Xiao-Hong, FEI Shao-Ming, WANG Zhi-Xi, and WU Ke Vol. 42 elementary row transformations on the matrix ρ 5, I A C I 0 0 C I 0 B ρ 5 = (3) I A I CA C B A I As the rank of ρ 5 is N, from Eq. (3) we have = 0, and hence E 5 = A C CA. Now, notice that Ψ f ρ Ψ f = 0 for Ψ f = f 22 CA f and arbitrary f CC N. Since ρ 0 we have 0 = ρ Ψ f = 00 (E 5 B D CA) f + 0 (E 25 A D CA) f + 0 (E 45 B C CA) f, which, as f is arbitrary, leads to E 5 = B D CA, E 25 = A D CA, E 45 = B C CA, thus the matrix ρ becomes E E 2 B D D E 4 B D CA B D C B D B B D A B D E 2 E 2 A D D E 24 A D CA A D C A D B A D A A D D DB D DA D D D CB D CA D C D B D A D E 4 E 24 B C D E 4 B C CA B C C B C B B C A B C A C DB A C DA A C D A C CB A C CA A C C A C B A C A A C. (4) C DB C DA C D C CB C CA C C C B C A C B DB B DA B D B CB B CA B C B B B A B A DB A DA A D A CB A CA A C A B A A A DB DA D CB CA C B A I Similarly, we can derive E 4 = B C CB, E 4 = B D CB, E 24 = A D CB, E 2 = A D DA, E 2 = B D DA, E = B D DB. ρ then is of the following form B D DB B D DA B D D B D CB B D CA B D C B D B B D A B D A D DB A D DA A D D A D CB A D CA A D C A D B A D A A D D DB D DA D D D CB D CA D C D B D A D B C DB B C DA B C D B C CB B C CA B C C B C B B C A B C A C DB A C DA A C D A C CB A C CA A C C A C B A C A A C C DB C DA C D C CB C CA C C C B C A C B DB B DA B D B CB B CA B C B B B A B A DB A DA A D A CB A CA A C A B A A A DB DA D CB CA C B A I = [ DB DA D CB CA C B A I ] [DB DA D CB CA C B A I ]. The commutative relations [A, D] = [B, D] = [A, C] = [B, C] = [A, D ] = [B, D ] = [A, C ] = [B, C ] = 0 follow from the positivity of all partial transpositions of ρ. We first consider B D DB A D DB D DB B D CB A D CB D CB B D B A D B D B B D DA A D DA D DA B D CA A D CA D CA B D A A D A D A B D D A D D D D B D C A D C D C B D A D D B C DB A C DB C DB B C CB A C CB C CB B C B A C B C B ρ t B = B C DA A C DA C DA B C CA A C CA C CA B C A A C A C A. B C D A C D C D B C C A C C C C B C A C C B DB A DB DB B CB A CB CB B B A B B B DA A DA DA B CA A CA CA B A A A A B D A D D B C A C C B!A I Due to the positivity, the matrix ρ t B must possess the kernel vector 2 f 22 C f, 02 g 22 D g, which implies

5 No. 2 On PPT States in C K C M C N Composite Quantum Systems 29 that [A, C] = [B, C] = [A, D] = [B, D] = 0. The matrix ρ t B can be then written as D B D A D C B ρ t B C A = C ( B D C B A A D D B C A C C B A I ), I which implies automatically the positivity. From the positivity of ρ t AB, B D DB A D DB D DB B C DB A C DB C DB B DB A DB DB B D DA A D DA D DA B C DA A C DA C DA B DA A DA DA B D D A D D D D B C D A C D C D B D A D D B D CB A D CB D CB B C CB A C CB C CB B CB A CB CB ρ t AB = B D CA A D CA D CA B C CA A C CA C CA B CA A CA CA, B D C A D C D C B C C A C C C C B C A C C B D B A D B D B B C B A C B C B B B A B B B D A A D A D A B C A A C A C A B A A A A B D A D D B C A C C B A I we have that 2 f 22 C f, 02 g 22 D g are kernel vectors, which results in [A, D ] = [B, D ] = [A, C ] = [B, C ] = 0. ρ t AB is then of the form DB DA D CB ρ t AB = CA ( B D C B A A D D B C A C C B A I ). I This form assures positive definiteness, and concludes the proof of the Lemma. Using Lemma we can prove the following theorem. Theorem A PPT-state ρ in C 3 C 3 C N with r(ρ) = N is separable if there exists a product basis e A, f B such that r( e A, f B ρ e A, f B ) = N. Proof According to the Lemma the PPT state ρ can be written as B D A D D B C A C ( DB C B A DA D CB CA C B A I ). I

6 220 WANG Xiao-Hong, FEI Shao-Ming, WANG Zhi-Xi, and WU Ke Vol. 42 Since all A, A, B, B, C, C, D, and D commute, they have common eigenvectors f n. Let a n, b n, c n and d n be the corresponding eigenvalues of A, B, C, and D respectively. We have b nd n a nd n d n b nc n f n ρ f n = a nc n ( d n b n d n a n d n c n b n c n a n c n b n a n ) c n b n a n = d n c n b n a n (d n c n ) (b n a n ) = e A, f B e A, f B. We can thus write ρ as N ψ n ψ n φ n φ n f n f n, where n= ψ n = d n c n, φ n = b n a n. Because the local transformations are reversible, we can now apply the inverse transformations and obtain a decomposition of the initial state ρ in a sum of projectors onto product vectors. This proves the separability of ρ. The above approach can be extended to the case of higher dimensions like C 3 A CM B CN C. Let 0 A, A, 2 A ; 0 B,, (M ) B ; and 0 C,, (N ) C be some local bases of the sub-systems A, B, C respectively. Similar to the proof of Lemma, it is straightforward to prove the following conclusion. Lemma 2 Every PPT state ρ in C 3 A CM B CN C such that r( 2 A, (M ) B ρ 2 A, (M ) B ) = r(ρ) = N can be transformed into the following canonical form by using a reversible local operation: F T T F, (5) where T = (C B I) (A M A M 2 A I), A i, B, C, F, and the identity I are N N matrices acting on C N C and satisfy the following relations: [A i, A j ] = [A i, A j ] = [B, B ] = [C, C ] = [B, A i ] = [B, A i ] = [C, A i ] = [C, A i ] = 0, i, j =, 2,..., M and F = F. Extending Theorem to higher dimensions, we have the following theorem. Theorem 2 A PPT-state ρ in C 3 C M C N with r(ρ) = N is separable if there exists a product basis e A, f B such that r( e A, f B ρ e A, f B ) = N. By extending Lemma 2, Theorem 2 and the results in Refs. [2] and [22], we can give the canonical form of PPT states in CA K CM B CN C with rank N. Let 0 A,..., (K ) A ; 0 B,..., (M ) B ; and 0 C,, (N ) C be some local bases of the sub-systems A, B, and C respectively. Lemma 3 Every PPT state ρ in CA K CM B CN C such that r( (K ) A, (M ) B ρ (K ) A, (M ) B ) = r(ρ) = N, can be transformed into the following canonical form by using a reversible local operation: where F T T F, (6) T = (B K B K 2 B I) (A M A M 2 A I), A i, B j, F, and I are N N matrices acting on CC N and satisfy the following relations: [A i, A s ] = [A i, A s ] = [B t, B j ] = [B t, B j ] = [A i, B j ] = [A i, B j ] = 0 and F = F, i, s =, 2,..., M ; j, t =, 2,..., K. Proof In the basis we considered, a density matrix ρ in CA K CM B CN C with rank N can always be written as a KM KM partitioned matrix. Let E ij be the i, j- element of ρ. Denote E ii = E i. Every E s are N N- matrices and r(e KM ) = N. Because ρ is self-adjoint, we have E ij = E ji, i > j. The projection (K ) A ρ (K ) A gives rise to a state (K ) A ρ (K ) A, which is a state in CB M CN C with r( ρ ) = r(ρ) = N. The fact that ρ is PPT implies that ρ is also PPT, i.e., ρ 0. Using the Lemma 5 in Ref. [9] we have [A M,..., A, I] [A M,..., A, I], (7)

7 No. 2 On PPT States in C K C M C N Composite Quantum Systems 22 where [A i, A j ] = [A i, A j ] = 0, i, j =,..., M. Similar as above, if we consider the following projection (M ) B ρ (M ) B, we have [B K,, B, I] [B K,..., B, I], (8) where [B i, B j ] = [B i, B j ] = 0, i, j =,..., K. Owing to the fact that ρ 0, a vector v CA K CM B CC N satisfying v ρ v = 0 is in the kernel of ρ. It is directly verified that the following vectors are in the kernel: K, i f i K, M A M i f i, i = 0,,..., M 2, (9) j, K g j K, M B j g j, j = 0,,..., K 2 (20) for all vectors f i and g j CC N. This implies E i,km t = E i,km A t, t =, 2,..., M, E i,km sm = E i,km B s, s =, 2,..., K, i =, 2,..., KM M, (2) where D s lies in the (KM sm)-th column. From these kernel vectors of ρ, we observe that the E ij are dependent on the last column elements of ρ. If we write ρ as an K K partitioned matrix (A ij ), where A ij is an M M partitioned matrix, then ρ t A = (A ji ). From ρ t A 0 and that the partial transposition of ρ with respect to the first sub-system A does not change the positivity of (K ) A ρ (K ) A, we still have kernel vectors (9) belonging to k(ρ t A ), from which we can get the last column elements and hence the last row elements of ρ. Up to now the last M + rows and the last M + columns of (E ij ) can be obtained from Eq. (2). Then we can write ρ in the following partitioned matrix form ( ) Y X X, ρ 0 with ( Ek Z ρ 0 = Z W ), where Z and W are known, k = KM (M + ). Similar to the proof of Lemma, denoting ρ 0 = Σ + diag(, 0, 0,..., 0), where A B B A A B B A B A M A B A A B B B A B B B A M B A B A M Σ = B A A M B A M A M A M A A M, = E k A B B A (22) A B A A B A A M A A A B A B A M A I by proving that = 0 we get E k = A B B A. From E k and ρ 0, we can give all E ik, i < k. (Maybe some of E i, i < k are known, such as i = M.) By repeating the above procedure, we can calculate all the diagonal elements E j, j k and E ij, i < j. Therefore we can get all the E ij. The canonical form of ρ is presented. The rest commutative relations [A i, B j ] = [A i, B j ] = 0 follow from the positivity of all partial transpositions of ρ. As mentioned above we write ρ as a K K partitioned matrix (A ij ), where A ij is an M M partitioned matrix, then ρ t B = (A t ij ), in which At ij represent the transposition of A ij. Due to the positivity, the matrix ρ t B must possess the kernel vector (20), which implies that [A i, B j ] = 0. From the positivity of ρ t AB, we have j, K g j K, M B j g j, j = 0,,..., K 2 (23) are kernel vectors, which results in [A i, B j ] = 0. From the canonical form (6), we can obtain the following result. Theorem 3 A PPT-state ρ in C K C M C N with r(ρ) = N is separable if there exists a product basis e A, f B such that r( e A, f B ρ e A, f B ) = N. In the following we give some detailed examples related to our canonical form of PPT states and the separability criterion. i) An obvious separable mixed state on K M N is ( ) I/N 0 0 0, where I is an N N unit matrix. Obviously ρ is a PPT state with rank N, and there exist e A = 0 A, f B = 0 B such that e A, f B ρ e A, f B = I/N. Therefore r( e A, f B ρ e A, f B ) = r(ρ) = N. Thus the conditions in Theorem 3 are satisfied and ρ is separable. In fact, if we set ψ = ψ 2 = 0 A, φ = φ 2 = 0 B, ϕ = ( 0 C + C )/ 2, ϕ 2 = ( 0 C C )/ 2, p = p 2 = /2, then ρ can be written in a separated form: p ρ ρ 2 ρ 3 + p 2 ρ 2 ρ 22 ρ 23,

8 222 WANG Xiao-Hong, FEI Shao-Ming, WANG Zhi-Xi, and WU Ke Vol. 42 where ρ i = ψ i ψ i, ρ i2 = φ i φ i, ρ i3 = ϕ i ϕ i, i =, 2. ii) Consider a three-qubit state: ( A 0 ) ( /2 a ) 0 0 with A = a /2, a R. ρ is a mixed state as tr ρ 2 <. It is easily verified that ρ is PPT: ρ t A = ρ t B = ρ t C = ρ t AB = ρ t AC = ρ t BC = ρ, and r(ρ) = 2. Let e A = f B = 0. We have e A, f B ρ e A, f B = I/2. Therefore r( e A, f B ρ e A, f B ) = r(ρ) = 2. From Theorem 3 ρ is separable. In fact ρ has separable form p ρ ρ 2 ρ 3 + p 2 ρ 2 ρ 22 ρ 23, where ψ = ψ 2 = φ = φ 2 = 0, ϕ = ( 0 + )/ 2, ϕ 2 = ( 0 )/ 2, p = p 2 = /2, and ρ i = ψ i ψ i, ρ i2 = φ i φ i, ρ i3 = ϕ i ϕ i, i =, 2. iii) The biseparable three-qubit bound entangled state: 8 ( I 4 i= ) ψ i ψ i, where ψ i s are given by 0,, +,, +, 0, +,, 0,,, with ± = ( 0 ± )/ 2. ρ is a PPT state as ρ t A = ρ t B = ρ t C = ρ t AB = ρ t AC = ρ t BC = ρ. It is separable under any bipartite cut A BC, B CA, C AB. But it is entangled (not fully separable). As r(ρ) 2 this state does not satisfy the conditions of Theorem 3 and the corresponding conclusions could not be deduced. We have derived a canonical form of PPT states in C K C M C N with rank N and a sufficient separability criterion from this canonical form. For K 2, M 3, the separability criterion we can deduce is weaker, as PPT criterion is no longer sufficient and necessary for the separability of bipartite states. Nevertheless the canonical representation of PPT states can shed a light on studying the structure of bound entangles states which are PPT but not separable. References [] D.P. DiVincenzo, Science 270 (995) 255. [2] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70 (993) 895; G.M. D Ariano, P.Lo Presti, M.F. Sacchi, Phys. Lett. A272 (2000) 32; S. Albeverio and S.M. Fei, Phys. Lett. A276 (2000) 8; S. Albeverio and S.M. Fei, and W.L. Yang, Commun. Theor. Phys. (Beijing, China) 38 (2002) 30; Phys. Rev. A66 (2002) [3] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69 (992) 288; A. Ekert, Phys. Rev. Lett. 67 (99) 66; D. Deutsch, A. Ekert, P. Rozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77 (996) 2828; C.A. Fuchs, N. Gisin, R.B. Griffiths, C-S. Niu, and A. Peres, Phys. Rev. A56 (997) 63. [4] R.F. Werner, Phys. Rev. A40 (989) [5] G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rotteler, H. Weinfurter, R.F. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, Springer, New York (200). [6] D. Bruss, J.I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M. Lewenstein, and A. Sanpera, J. Mod. Opt. 49 (2002) 399; D. Bruss, J. Math. Phys. 43 (2002) [7] S. Popescu, Phys. Rev. Lett. 72 (994) 797; S. Popescu, Phys. Rev. Lett. 74 (995) 269. [8] J.S. Bell, Physics (N.Y.) (964) 95. [9] A. Peres, Phys. Rev. Lett. 76 (996) 43; K. Życzkowski and P. Horodecki, Phys. Rev. A58 (998) 883. [0] M. Horodecki and P. Horodecki, Phys. Rev. A59 (999) [] N.J. Cerf, C. Adami, and R.M. Gingrich, Phys. Rev. A60 (999) 898. [2] M.A. Nielsen and J. Kempe, Phys. Rev. Lett. 86 (200) 584. [3] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A223 (996). [4] B. Terhal, Phys. Lett. A27 (2000) 39; M. Lewenstein, B. Kraus, J.I. Cirac, and P. Horodecki, Phys. Rev. A62 (2000) [5] A.C. Doherty, P.A. Parrilo, and F.M. Spedalieri, Phys. Rev. Lett. 88 (2002) [6] O. Rudolph, quant-ph/02022; K. Chen and L.A. Wu, Quant. Inf. Comput. 3 (2003) 93. [7] K. Chen and L.A. Wu, Phys. Lett. A306 (2002) 4. [8] S. Albeverio, K. Chen, and S.M. Fei, Phys. Rev. A68 (2003) [9] P. Horodecki, M. Lewenstein, G. Vidal, and I. Cirac, Phys. Rev. A62 (2000) [20] P. Horodecki, J.A. Smolin, B.M. Terhal, and A.V. Thapliyal, Theor. Comp. Sci. 292 (2003) 589; S. Albeverio, S.M. Fei, and D. Goswami, Phys. Lett. A286 (200) 9; S.M. Fei, X.H. Gao, X.H. Wang, Z.X. Wang, and K. Wu, Phys. Lett. A300 (2002) 559; S.M. Fei, X.H. Gao, X.H. Wang, Z.X. Wang, and K. Wu, Int. J. Quant. Infor. (2003) 37. [2] S. Karnas and M. Lewenstein, Phys. Rev. A64 (200) 04233; S.M. Fei, X.H. Gao, X.H. Wang, Z.X. Wang, and K. Wu, Phys. Rev. A68 (2003) [22] S.M. Fei, X.H. Gao, X.H. Wang, Z.X. Wang, and K. Wu, Commun. Theor. Phys. (Beijing, China) 40 (2003) 55.