Criterion for SLOCC Equivalence of Multipartite Quantum States
aa r X i v : . [ qu a n t - ph ] S e p Criterion for SLOCC Equivalence of Multipartite Quantum States
Tinggui Zhang , Ming-Jing Zhao , and Xiaofen Huang ∗ School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, China Department of Mathematics, School of Science, Beijing InformationScience and Technology University, 100192, Beijing, China ∗ Corresponding author, e-mail address:[email protected]
We study the stochastic local operation and classical communication (SLOCC) equivalence forarbitrary dimensional multipartite quantum states. For multipartite pure states, we present anecessary and sufficient criterion in terms of their coefficient matrices. This condition can be usedto classify some SLOCC equivalent quantum states with coefficient matrices having the same rank.For multipartite mixed state, we provide a necessary and sufficient condition by means of therealignment of matrix. Some detailed examples are given to identify the SLOCC equivalence ofmultipartite quantum states.
PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w
I. INTRODUCTION
Quantum entanglement is not only a prime feature in quantum mechanics but also an important resource inquantum information processes [1, 2]. It can be used in quantum teleportation [3, 4], superdense coding [5, 6],quantum computation [7–10], quantum key distribution [11, 12] and etc. Therefore, it is important to understandwhat kind of entanglement a given quantum state has. One approach to classify entanglement is by means of Statisticlocal operations and classical communications (SLOCC) [13]. Entanglement in bipartite pure states has been wellunderstood, while many questions are still open for the mixed states and multipartite states.It has been shown that two pure states | ϕ i and | ψ i in H ⊗ H ⊗ · · · ⊗ H K , dim H i = n i , i = 1 , , · · · , K , areSLOCC equivalent if and only if they can be converted into each other with the tensor products of invertible localoperators(ILOs) | ϕ i = A ⊗ A ⊗ · · · ⊗ A K | ψ i . (1)Correspondingly, two mixed states ρ and ρ ′ belong to the same class under SLOCC if and only if they are convertedby ILOs with nonzero determinant, that is, ρ ′ = ( A ⊗ A ⊗ · · · ⊗ A K ) ρ ( A ⊗ A ⊗ · · · ⊗ A K ) † , (2)where A i is ILO in GL ( n i , C ) for each i [14]. Many researches have been conducted on entanglement classification underSLOCC since the beginning of this century [14–24]. In three-qubit system, all pure states are classified into six types[14]. This classification can be extended to three-qubit mixed states [15]. Even though, it is still a very difficult problemto find a SLOCC class of a given three-qubit mixed state except for a few rare case. For instance, a complete SLOCCclassification for the set of the GHZ-symmetric states was reported in Ref. [16]. In four-qubit case, all pure states areclassified into nine SLOCC inequivalent families using group theory [17]. For n -qubit system, Ref. [22] uses the ranksof the coefficient matrices to study SLOCC classification for pure state. Then Ref. [23] generalizes Li’s approachto n-qudit pure state. Recently, Ref. [24] shows that almost all SLOCC equivalent classes can be distinguished byratios of homogeneous SL-invariant polynomials of the same degree. Theoretically, their technique can be applied toany number of qudits in all dimensions. But, it is still a significant challenge to find a general scheme that is ableto completely identify the different entanglement classes and determine the transformation matrices connecting twoequivalent states under SLOCC for multipartite mixed states. In Ref. [25], we have constructed a nontrivial set ofinvariants for any multipartite mixed states under SLOCC.In this paper we present a general scheme for the SLOCC equivalence of arbitrary dimensional multipartite quantumpure or mixed states in terms of matrix realignment [26, 27]. In Sec. II, we recall some basic results, then we give thecriterion for how to judge a block invertible matrix can be decomposed as the tensor products of invertible matrices.In Sec. III, we give a necessary and sufficient criterion for the SLOCC equivalence of multipartite pure states. For themultipartite mixed states, we propose a similar criterion based on the density matrix itself in Sec. IV. These criteriaare shown to be still operational for general states, and we also give the explicit forms of the connecting matrix fortwo SLOCC equivalent states in specific examples. At last, we give the conclusions and remarks. II. TENSOR PRODUCTS DECOMPOSITION FOR BLOCK INVERTIBLE MATRIX
First we introduce the definitions for realignment of matrix [26, 27].
Definition 1:
For any M × N matrix A with entries a ij , vec ( A ) is defined by vec ( A ) ≡ [ a , · · · , a M , a · · · , a M , · · · , a N , · · · , a MN ] T , where T denotes transposition. Definition 2:
Let Z be an M × M block matrix with each block of size N × N , the realigned matrix R ( Z ) is definedby R ( Z ) ≡ [ vec ( Z ) , · · · , vec ( Z M ) , · · · , vec ( Z M ) , · · · , vec ( Z MM )] T . Based on the definitions of realignment, Ref. [28] shows a necessary and sufficient condition for the tensor productsdecomposition of invertible matrices for a matrix.
Lemma 1. An M N × M N invertible matrix A is expressed as the tensor product of an M × M invertible matrix A and an N × N invertible matrix A , i.e, A = A ⊗ A if and only if rank R ( A ) = 1 . For any N N · · · N K × N N · · · N K matrix A , we denote A i | b i the N i × N i block matrix with each block of size N N · · · N i − N i +1 · · · N K × N N · · · N i − N i +1 · · · N K . Namely, we view A as a bipartite partitioned matrix A i | b i withpartitions H i and H ⊗ H ...H i − ⊗ H i +1 ...H K . Accordingly, we have the realigned matrix R ( A i | b i ). Theorem 1.
Let A be an N N · · · N K × N N · · · N K invertible matrix, there exist N i × N i invertible matrices a i , i = 1 , , · · · , K , such that A = a ⊗ a ⊗ · · · ⊗ a K if and only if the rank ( R ( A i | b i )) = 1 for all i .Proof. First, if there exist N i × N i invertible matrices a i , i = 1 , , · · · , K , such that A = a ⊗ a ⊗ · · · ⊗ a K , by viewing A in bipartite partition and using Lemma 1, one has directly that rank( R ( A i | b i )) = 1 for all i .On the other hand, if rank( R ( A i | b i )) = 1, for any given i , we prove the conclusion by induction. First, for n = 3,from Lemma 1, we have A = a ⊗ a = a ⊗ a . Multiplying a − for the first subsystem from the left, it has ( a − ⊗ I ⊗ I ) A = I ⊗ a = a ⊗ (( a − ⊗ I ) a ). By tracing out the first subsystem, we get N a = a ⊗ T r (( a − ⊗ I ) a ),i.e, a = a ⊗ a ′ with invertible matrix a ′ = T r (( a − ⊗ I ) a ) /N . Assume that the conclusion is also true for K −
1, then for K , from Lemma 1, we have A = a ⊗ a b = a ⊗ a b = · · · = a K ⊗ a b K , where a i is an N i × N i invertiblematrix and a b i is an N N · · · N i − N i +1 · · · N K × N N · · · N i − N i +1 · · · N K invertible matrix, i = 1 , , · · · , K . Hence( I ⊗ · · · ⊗ I K − ⊗ a − K ) A = ( I ⊗ · · · ⊗ I K − ⊗ a − K )( a ⊗ a b ) = · · · = ( I ⊗ · · · ⊗ I K − ⊗ a − K )( a b K ⊗ a K ). Tracing out thelast subsystem we get a ⊗ T r K ( I ⊗ · · · ⊗ I N K − ⊗ a − K ) a b )) = · · · = T r K (( I ⊗ · · · ⊗ I K − ⊗ a − K ) ⊗ ( a K − ) = N K a b K .Based on the assumption, we know a b K can be written as the tensor products of local invertible operators. Therefore, A also can be written as the tensor products of local invertible operators, which completes the proof. III. CRITERION FOR MULTIPARTITE PURE STATES
First, we recall the notations of coefficient matrices of pure state [22, 23]. Let {| i i} n − i =0 , {| i i} n − i =0 , · · · , {| i K i} n K − i K =0 be orthnormal basis of K Hilbert spaces H , H , · · · , H K . For any K partite pure state | ψ i =Σ n − ,n − , ··· ,n K − i ,i , ··· ,i K =0 a i i , ··· ,i K | i i , · · · , i K i , Σ n − ,n − , ··· ,n K − i ,i , ··· ,i K =0 | a i i , ··· ,i K | = 1, we associate an m × n coefficientmatrix M ( | ψ i ) to it, m = n n · · · n t , n = n t +1 · · · n K , t = [ K ].For example, for three qubit pure state | ψ i = P i ,i ,i =0 a i i i | i i i i , we have the 2 × M ( | ψ i ) = (cid:18) a a a a a a a a (cid:19) . For four qubit pure state | ψ i = P s ,s ,s ,s =0 a s s s s | s s s s i , there is 4 × M ( | ψ i ) = a a a a a a a a a a a a a a a a . Using the rank of coefficient matrix M ( | ψ i ), Refs. [22, 23] classified multipartite pure states into different families.If the coefficient matrices of two pure states have different ranks, then these two pure states are not SLOCC equivalent.While the converse does not hold true, i.e. if the coefficient matrices have the same rank, then corresponding purestates are not necessarily SLOCC equivalent. Here we answer this question further when two states with the samerank of the coefficient matrices are equivalent under SLOCC. Theorem 2.
For two K -partite pure states | φ i and | ψ i , they are SLOCC equivalent if and only if for one pair ofcoefficient matrices M ( | φ i ) and M ( | ψ i ) , there are m × m unitary matrices X , X , invertible diagonal matrix B ,and n × n unitary matrices Y , Y , invertible diagonal matrix B , such that M ( | φ i ) = X B X † M ( | ψ i ) Y † B Y , (3) and rank [ R (( X B X † ) i | b i )] = 1 (4) and rank [ R (( Y † B Y ) j | b j )] = 1 , (5) i = 1 , , · · · , t , j = t + 1 , · · · , K .Proof. First, suppose | φ i and | ψ i are SLOCC equivalent, i.e. there exist invertible matrices C , C , · · · , C K such that | φ i = ( C ⊗ C ⊗ · · · ⊗ C K ) | ψ i . In matrix form, M ( | φ i ) = ( C ⊗ C ⊗ · · · ⊗ C t ) M ( | ψ i )( C t +1 ⊗ · · · ⊗ C K ) T . (6)For invertible matrices C ⊗ C ⊗ · · · ⊗ C t and ( C t +1 ⊗ · · · ⊗ C K ) T , by the singular value decomposition of a matrix,there exist m × m unitary matrices X , X , invertible diagonal matrix B , and n × n unitary matrices Y , Y , invertiblediagonal matrix B such that: C ⊗ C ⊗ · · · ⊗ C t = X B X † , ( C t +1 ⊗ · · · ⊗ C K ) T = Y B Y † . Inserting these decompositions into Eq. (6), one gets easily Eq. (3). By Lemma 1, we can get Eqs. (4) and (5), i = 1 , , · · · , t , j = t + 1 , · · · , K .On the other hand, suppose there exist one pair of coefficient matrices M ( | φ i ) and M ( | ψ i ) of | φ i and | ψ i satisfyingthe conditions mentioned in the Theorem. By Lemma 1, we know there are invertible matrices C , C , · · · , C K suchthat Eq. (6) holds true. Therefore | φ i = ( C ⊗ C ⊗ · · · ⊗ C k ) | ψ i , i.e. | φ i and | ψ i are SLOCC equivalent.Let us now take a closer look at equations in Theorem 2. Eq. (6) means if two pure states are SLOCC equivalent,then their coefficient matrices have the same rank. Eqs. (4) and (5) means if two pure states are SLOCC equivalent,then their coefficient matrices are connected by tensor products of invertible matrices. So if the coefficient matriceshave the same rank, then one needs to verify Eqs. (4) and (5) to check whether two pure states are SLOCC equivalentor not.Operationally, for two pure states | φ i and | ψ i , we first choose one kind of coefficient matrices M ( | φ i ) and M ( | ψ i ).If M ( | φ i ) and M ( | ψ i ) have different ranks, then | φ i and | ψ i are not SLOCC equivalent. If M ( | φ i ) and M ( | ψ i ) havethe same rank, then by the singular value decomposition, there are m × m unitary matrices X , X , diagonal matrixΛ , and n × n unitary matrices Y , Y , diagonal matrix Λ such that: M ( | φ i ) = X Λ Y (7)and M ( | ψ i ) = X Λ Y , (8)where Λ = diag ( λ , λ , · · · , λ r , , · · · , = diag ( µ , µ , · · · , µ r , , · · · , λ i and µ i are nonzero real num-bers. Let m × m invertible matrix B = diag ( q λ µ , q λ µ , · · · , q λ r µ r , , · · · ,
1) and n × n invertible matrix B = diag ( q λ µ , q λ µ , · · · , q λ r µ r , , · · · , = B Λ B and M ( | φ i ) = X B X † M ( | ψ i ) Y † B Y . Next oneneeds to calculate the ranks for the realignment of X B X † and Y † B Y under all partitions to see whether it is oneor not.For bipartite pure state | φ i = Σ n − ,n − i ,i =0 a i i | i i i , there is only one way to express its coefficients in matrix form, M ( | φ i ) = ( a i i ). Therefore, two bipartite pure states | φ i and | ψ i are SLOCC equivalence if and only if there existinvertible matrices C , C such that M ( | φ i ) = C M ( | ψ i ) C T . Or equivalently, two bipartite pure states | φ i and | ψ i are SLOCC equivalence if and only if their coefficient matriceshave the same rank. IV. CRITERION FOR MULTIPARTITE MIXED STATES
Theorem 3.
For two multipartite mixed quantum states ρ and ρ , they are SLOCC equivalent if and only if thereexist N N · · · N K × N N · · · N K unitary matrices X and Y , real diagonal invertible matrix B , such that ρ = XBY † ρ Y BX † , (9) and rank ( R ( XBY † ) i | b i ) = 1 , (10) for i = 1 , , · · · , K .Proof. If ρ and ρ are SLOCC equivalent, then there exist invertible matrices a , a , · · · , a K such that ( a ⊗ a ⊗· · · ⊗ a K ) ρ ( a ⊗ a ⊗ · · · ⊗ a K ) † = ρ . For matrix a ⊗ a ⊗ · · · ⊗ a K , by singular value decomposition, there exist N N · · · N K × N N · · · N K unitary matrices X , Y , real diagonal invertible matrix B , such that a ⊗ a ⊗ · · · ⊗ a K = XBY † . Then R ( XBY † ) = R ( a ⊗ a · · · ⊗ a n ). From Lemma 1, rank ( R ( XBY † ) i | b i ) = 1, for i = 1 , , · · · , K .On the other hand, if there exist N N · · · N K × N N · · · N K unitary matrices X and Y , real diagonal invertiblematrix B , such that Eq. (9) holds true and rank ( R ( XBY † ) i | b i ) = 1 for i = 1 , , · · · , K , then by Lemma 1, there existinvertible matrices a , a , · · · , a K such that XBY † = a ⊗ a ⊗ · · · ⊗ a n . Inserting this equation into Eq. (9), one gets( a ⊗ a ⊗ · · · ⊗ a n ) † ρ ( a ⊗ a ⊗ · · · ⊗ a n ) = ρ , which ends the proof.Eq. (9) means if two mixed states are SLOCC equivalent, then they have the same rank. Eq. (10) means if twomixed states are SLOCC equivalent, then they are connected by the tensor products of invertible matrices. Now weshow how to verify Theorem 3 explicitly. For two mixed states ρ and ρ , if they have different ranks, then they arenot SLOCC equivalent. Or else, if ρ and ρ have the same rank, then we first study their spectra decompositions, ρ = X Λ X † , ρ = Y Λ Y † , (11)where X = [ x , x , · · · , x N N ··· N K ] , Y = [ y , y , · · · , y N N ··· N K ], { x i } and { y i } are the normalized eigenvectors ofstates ρ and ρ . Λ = diag ( λ , λ , · · · , λ r , , · · · , = diag ( µ , µ , · · · , µ r , , · · · , λ i and µ i are nonzero realnumbers. For diagonal matrices Λ and Λ , there exists N N · · · N K × N N · · · N K invertible matrix B = diag ( s λ µ , s λ µ , · · · , s λ r µ r , s, · · · , t ) (12)such that Λ = B Λ B, where s, · · · , t are arbitrary nonzero numbers. Therefore, there exist N N · · · N K × N N · · · N K unitary matrices X and Y , real diagonal invertible matrix B , such that Eq. (9) holds true. Next we need to verify the rank of realignmentof XBY † to see whether ρ and ρ are SLOCC equivalent or not.Example 1. First, we consider two-qubit Bell-diagonal states in two-qubit system [29, 30]: ρ = X i =1 λ i | ψ i ih ψ i | , λ i ≥ , X i =1 λ i = 1 , i = 1 , , , ρ = X i =1 µ i | ψ i ih ψ i | , µ i ≥ , X i =1 µ i = 1 , i = 1 , , , | ψ i = √ ( | i + | i ), | ψ i = √ ( | i − | i ), | ψ i = √ ( | i + | i ), | ψ i = √ ( | i + | i ). Byspectra decomposition, we have X = Y = − − ; Λ = diag (2 λ , λ , λ , − λ − λ − λ ));Λ = diag (2 µ , µ , µ , − µ − µ − µ )). For simplicity, we consider only the non-degenerate case, whichmeans Λ and Λ are nonsingular. Let B = diag ( q λ µ , q λ µ , q λ µ , q λ + λ + λ − µ + µ + µ − ), then ρ and ρ satisfy Eq. (9).Next we need to study the rank of realignment matrix XBY † = B . We find if s λ µ : s λ µ = s λ µ : s λ + λ + λ − µ + µ + µ − , then rank ( R ( XBY † )) = 1. In this case, ρ and ρ are SLOCC equivalent.Example 2. Now we consider two mixed states in 2 ⊗ ⊗ ρ = 1 K a b c c b a
01 0 0 0 0 0 0 1 ,ρ = 1 M α β γ γ β α
01 0 0 0 0 0 0 1 , where the normalization factors K = 2 + a + b + c + a + b + c . M = 2 + α + β + γ + α + β + γ . First westudy the spectra decompositions of ρ and ρ . Here as in Eq. (11), Λ = K diag ( c , b , a , , a, b, c,
0) and Λ = M diag ( γ , β , α , , α, β, γ, a , b , c , α , β , γ take different values unequal to 0, 1, ,2. Then we can easily get X = Y = √ − √ √ √ . Let B = diag ( q MγKc , q MβKb , q MαKa , q MK , q MaKα , q MbKβ , q McKγ , C ) with C an arbitrary nonzero number. Then XBY † = B . Now we calculate the rank of the realignment of XBY † . If the coefficients of ρ and ρ satisfies thefollowing two condition,(1) p γc : p aα = q βb : q bβ = p αa : q cγ = 1 : C, (2) p γc : q βb = p αa : 1then rank ( R ( XBY † ) i | b i ) = 1 for i = 1 , ,
3. In this case, ρ and ρ are SLOCC equivalent. For instance, when p αa = √ q βb = 2; p γc = 2 √
2, one chooses C = . Then such two mixed states are SLOCC equivalent.Example 3. Let us consider another pair of mixed states in 2 ⊗ ⊗ ρ = 1 K a b c c b a ,ρ = 12 K b − b − / / a + c a − c − b b − / / a − c a + c c + a − a + c − − b + 1 0 − b − a + c c + a − b b , where the normalization factor K = + a + b + c + a + b + c . ρ and ρ have the same eigenvalues, Λ = Λ = K diag ( c , b , a , , a, b, c, ). Now we consider the case with different a , b , and c unequal to 0, 1, , , 2, , whichimplies that ρ and ρ are not degenerated. In such case, X = √ − √ √ √ ,Y † = √ √
00 0 0 0 0 √ √ − √ √ − √ √ − √ √ − √ √ − − − . Let B be the identity matrix. Then XBY † = 1 √ − − − − . It is easy to verify that rank ( R ( XBY † ) | ) = 1. Furthermore XBY = (cid:18) (cid:19) ⊗ √ (cid:18) − (cid:19) ⊗ (cid:18) (cid:19) . Henceall nondegenerated mixed states ρ and ρ are SLOCC equivalent.Now, we give one example for two quantum states non SLOCC equivalence. In fact, there are too many examplesfor two quantum states non SLOCC equivalence.Example 4. Suppose | ψ i = √ ( | i ) + | i , | ψ i = √ ( | i ) + | i .On one hand, the coefficient matrices of these two pure states have different ranks, by Theorem 2, we can easily todetermine that they are non SLOCC equivalence. On the other hand, we can also check their non SLOCC equivalenceby Theorem 3. Since these are pure states and their density matrices is rank one, therefore their density matrices haveonly one nonzero eigenvalue 1. In this case, we can choose B as identity matrix, the X and Y can easily respectivelyobtained. One has XBY † = 1 √ . It is easy to verify that rank ( R ( XBY ) | ) = 1. By Theorem 3, they are non SLOCC equivalence. V. CONCLUSIONS AND REMARKS
We have studied the SLOCC equivalence for arbitrary dimensional multipartite quantum states. Utilizing coefficientmatrix and realignment, we present necessary and sufficient criteria for multipartite pure states and mixed statesrespectively. These conditions can be used to classify some SLOCC equivalent quantum states having the same rank.Some detailed examples are given to identify the SLOCC equivalence or non SLOCC equivalence. However, ourmethods have to recognize its disadvantage in determining the SLOCC equivalence for degenerate state. The reasonis that the normalized eigenvectors of degenerate states can not be determined up to some unitary matrix. Thus thechoose of unitary matrices X and Y in Eq.(10) can not be determined up to some unknown unitary matrices, whichtakes infinite possibility. Therefore, to check Eq.(10) becomes terribly difficult since one should check all possiblechoices.Acknowledgments: We are very thankful to an anonymous referee for the helpful comments. This work is sup-ported by the NSF of China under Grant No. 11401032 and No. 11501153; the NSF of Hainan Province underGrant Nos.(20161006, 20151010, 114006); the Scientific Research Foundation for Colleges of Hainan Province underGrant No.Hnky2015-18; Scientific Research Foundation for the Returned Overseas Chinese Scholars, State EducationMinistry. [1] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.[2] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys , 865 (2009).[3] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895 (1993).[4] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Phys. Rev. Lett. , 1121 (1998).[5] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. , 2881 (1992).[6] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Phys. Rev. Lett. , 4656 (1996).[7] D. Deutsch and R. Jozsa, Proc. R. Soc. London, Ser. A , 553 (1992).[8] P. W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE Computer SocietyPress, Los Alamitos, CA, 1994), p. 124[9] A. Ekert and R. Jozsa,Rev. Mod. Phys. , 733 (1996).[10] L.K. Grover, Phys. Rev. Lett. , 325 (1997).[11] C.H. Bennett and G. Brassard, Proceedings of IEEE International Conference on Computers, Systems, and Signal Pro-cessing, Bangalore, India (IEEE, New York, 1984), pp. 175C179.[12] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. , 145 (2002)[13] C.H. Bennett, S. Popescu, D. Rohrlich, J.A. Smolin and A.V. Thapliyal, Phys. Rev. A , 012307 (2000). [14] W. Dur, G. Vidal, and J. I. Cirac, Phys. Rev. A , 062314 (2000).[15] A. Ac´ın D. Bruss, M. Lewenstein and A. Sanpera, Phys. Rev. Lett. , 040401 (2001).[16] C. Eltschka and J. Siewert, Phys. Rev. Lett. , 020502 (2012)[17] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A , 052112 (2002).[18] L. Chen and Y.X. Chen, Phys. Rev. A , 052310 (2006).[19] J.L. Li, S.Y. Li, and C.F. Qiao, Phys. Rev. A , 012301 (2012).[20] B. Li, L.C. Kwek, and H. Fan, J. Phys. A: Math. Theor. , 505301 (2012).[21] O. Viehmann, C. Eltschka, and J. Siewert, Phys. Rev. A , 052330 (2011).[22] X.R. Li and D. F. Li, Phys. Rev. Lett. , 180502 (2012); X.R. Li and D. F. Li, Phys. Rev. A , 042332 (2012).[23] S.H. Wang, Y. Lu, M. Gao, J.L. Cui, and J.L. Li, J. Phys. A: Math. Theor. , 105303 (2013); S.H. Wang, Y. Lu andG.L. Long, Phys. Rev. A , 062305 (2013)[24] G. Gour and N. R. Wallach, Phys. Rev. Lett. , 060502 (2013)[25] N. Jing, M. Li, X. Li-Jost, T. Zhang and S.M. Fei, J. Phys. A: Math. Theor. , 193 (2003)[28] L.L. Sun, J.L. Li and C.F. Qiao, arxiv.1401.6609v1. (2014)[29] R. Horodecki and M. Horodecki, Phys. Rev. A , 1838 (1996)[30] O. Rudolph, J. Math. Phys.45