A note on the realignent criterion Chi-Kwong Li 1, Yiu-Tung Poon and Nung-Sing Sze 3 1 Departent of Matheatics, College of Willia & Mary, Williasburg, VA 3185, USA Departent of Matheatics, Iowa State University, Aes, IA 50011 3 Departent of Applied Matheatics, The Hong Kong Polytechnic University, Hung Ho, Hong Kong E-ail: ckli@ath.w.edu, ytpoon@iastate.edu and rayond.sze@inet.polyu.edu.hk PACS nubers: 03.67.-a, 03.67.Mn Abstract. For a quantu state in a bipartite syste represented as a density atrix, researchers used the realignent atrix and functions on its singular values to study the separability of the quantu state. We obtain bounds for eleentary syetric functions of singular values of realignent atrices. This answers soe open probles proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show that the proposed schee by these authors for testing separability would not work if the two subsystes of the bipartite syste have the sae diension.
A note on the realignent criterion 1. Introduction Quantu entangleent was first proposed by Einstein, Podolsky, and Rosen [3] and Schrödinger [17] as a strange phenoenon of quantu echanics, criticizing the copleteness of the quantu theory. Nowadays, entangleent is not only regarded as a key for the interpretation of quantu echanics or as a ere scientific curiosity, but also as a resource for various applications, like quantu cryptography [4], quantu teleportation [1], and quantu coputation [14]. Suppose quantu states of two quantu systes are represented by density atrices (positive seidefinite atrices with trace 1) of sizes and n, respectively. States of their bipartite coposition syste are represented by n n density atrices. Such a state is separable if there are positive nubers p j suing up to 1, density atrices ρ 1 j, and n n density atrices ρ j such that ρ = k p j ρ 1 j ρ j. j=1 A state is entangled if it is not separable. In quantu inforation science, it is iportant to deterine the separability of a state. However, the proble of characterizing separable states is NP-hard [5]. Therefore, researchers focus on finding effective criterion to deterine whether a density atrix is separable or not. A siple and strong criterion for separability of density atrix is the coputable cross nor or realignent (CCNR) criterion. The nae CCNR coes fro the fact that this criterion has been discovered in two different fors, naely, by cross nors [15, 16] and by realignent of density atrices []. To describe the realignent criterion, let M N be the set of N N coplex atrices. D(, n) will denote the set of all n n density atrices and D s (, n) the set of separable density atrices in D(, n). For any X = [x ij ] M n, let vec (X) = (x 11, x 1,..., x 1n, x 1, x,..., x n,..., x n1, x n,..., x nn ). If ρ = [X rs ] 1 r,s D(, n) with X rs M n, then the realignent of ρ is the n atrix ρ R with rows vec (X 11 ), vec (X 1 ),..., vec (X 1 ), vec (X 1 ),..., vec (X ),..., vec (X 1 ),... vec (X ).
A note on the realignent criterion 3 [ ] X For exaple, if (, n) = (, 3) and ρ = 11 X 1 D(, 3) with X rs M 3, then X 1 X ρ R = vec (X 11 ) vec (X 1 ) vec (X 1 ) vec (X ) The realignent criterion asserts that if ρ D s (, n) then the su of the singular values of ρ R is at ost 1. Recall that the singular values of an M N atrix A are the nonnegative square roots of the k = in{m, N} largest eigenvalues of the atrix AA. For convenience of notation, we assue that n in the following discussion. For ρ D(, n), let s 1 s can be stated as. be the singular values of ρ R. The realignent criterion s 1 + + s 1 for ρ D s (, n). In [10], Lupo, Aniello, and Scardicchio suggest further study of the syetric functions on the singular values of ρ R, in order to find conditions beyond the realignent criterion to identify entangleent. Let S(, n) = {(s 1,..., s ) : s 1 s are the singular values of ρ R, for soe ρ D(, n)} S s (, n) = {(s 1,..., s ) : s 1 s are the singular values of ρ R, for soe ρ D s (, n)}. For each 1 < l, define the l-th eleentary syetric function f l (s 1,..., s ) = Following [10], we define for each 1 < l, 1 i 1 < <i l Π l j=1s ij. B l (, n) = ax{f l (s) : s S(, n), s = (s 1,..., s ) with i=1 s i 1}, B l (, n) = ax{f l (s) : s S s (, n)}. The bounds B l (, n) and B l (, n) were introduced in [10] using different notations, naely, x l (d, D) and x l (d, D) with (d, D) = (, n ). It follows fro the definitions that if Bl (, n) > B l (, n), then there exists an entangled density atrix ρ such that the su of singular values of ρ R is at ost 1
A note on the realignent criterion 4 but f l (s 1,..., s ) > B l (, n). Therefore, the bound B l (, n) can be used to detect entangleent for which the realignent criterion fails. Nuerical estiations for these bounds were given for (, n) = (, ) and (, 3) in [10]. The nuerical results also suggest that B l (, ) = B l (, ) and B l (, 3) > B l (, 3). The authors of [10] raised the following two open probles in the search for criterion for entangleent beyond the realignent criterion: (P1) To deterine the actual values of the upper bounds B l (, n) and B l (, n). (P) To deterine if B l (, n) > B l (, n). In this paper, we study the singular values of ρ R for a density atrix ρ. We refine soe inequalities given in [10]. This leads to an explicit forula for B l (, n), for all n, except for 3 / < n < 3, that gives a partial solution to (P1). Furtherore, we show that B l (n, n) = B l (n, n) for all n and this iplies that the answer to (P) is negative if = n. We conclude this section with a reforulation of another siple and strong criterion for separability in ters of the singular values. Let X = [X rs ] 1 r,s D(, n) with X rs M n. The partial transpose of X with respect to the second subsyste is given by X T = [X t rs] 1 r,s, where X t rs is the transpose of X rs. The PPT criterion [1] states that if X D s (, n), then X T is positive sei-definite. For +n 5, PPT criterion is a necessary and sufficient condition for separability [7], i.e. X D s (, n) if and only if X T D(, n). For, n > 1 and +n > 5, the PPT criterion and the CCNR criterion are independent. Note that for X D(, n), X T is Heritian. So the singular values of X T are the absolute values of the eigenvalues of X T. Since the su of all eigenvalues of X T is equal to trace ( X T ) = trace (X) = 1, X T is positive sei-definite if and only if the su of the singular values of X T criterion shares a siilar for with the CCNR criterion.. Main results and their iplications is at ost 1, cf. [8, Corollary 1]. Thus the PPT In this section, we continue to use the notations introduced in Section 1 and assue that n. We will describe the results and their iplications. The proofs will be given in the next section. For any density atrix ρ, we obtain the following lower bound for the largest singular value for ρ R, the realigned atrix of ρ. Lea.1 Let s = (s 1,..., s ) S(, n). Then s 1 1 n.
A note on the realignent criterion 5 Recall that for two vectors x, y R N, x is ajorized by y, denotes by x y, if for all 1 k N, the su of the k largest entries of x is not larger than that of y, and the su of all entries of x is equal to that of y. A function f : R N R is Schur concave if f(y) f(x) whenever x y. Using Lea.1, we will show that if n 3, then the vector s in S(, n) always arojize a vector of the for (α, β,..., β). One can then apply the theory of ajorization and Schur concave functions (see [11]) to obtain the inequality f l (s) f l (α, β,..., β), as shown in Lea.. For 1 r N, ( ) N r will denote the binoial coefficient N!. r!(n r)! Lea. Suppose n 3 and s = (s 1,..., s ) S(, n) with i=1 s i 1. Let α = 1 n and β = 1 α 1 = n 1 n( 1). Then and for 1 < l, 1 {}}{ (α, β,..., β) 1 i=1 s (s 1,..., s ), i f l (s) f l (α, β,..., β) ( l ) ( ) l 1. Furtherore, (a) f l (s) = f l (α, β,..., β) if and only if s = (α, β,..., β); (b) f l (α, β,..., β) = ( ) ( 1 ) l if and only if n = 3. l It follows fro Lea. that B l (, n) ( ) ( 1 ) l for all n 3 and the equality holds if and only if n = 3, which has been shown in [10, Proposition 4]. The following result gives an explicit forula for B l (, n) for all n, except for 3 / < n < 3. This provides a partial solution to proble (P1). Theore.3 Suppose n 3 /. Then for 1 < l, l B l (, n) = f l (α, β,..., β), with α = 1 n and β = 1 α 1. If n 3, then B l (, n) = f l (1/,..., 1/ ) = ( ) ( 1 ) l. l
A note on the realignent criterion 6 Theore.3 gives the values of B l (, n) for all n, except for 3 / < n < 3. In particular, it holds for all n which is divisible by. In application, both n and are powers of. Therefore, n is always divisible by and B l (, n) is given by the above theore. When = n, following our proof of Theore.3 in the next section, one actually gives explicit forulas for B l (n, n) and B l (n, n). Theore.4 For any n and 1 l n, B l (n, n) = B l (n, n) = f l (α, β,..., β) with α = 1 n and β = n 1 n(n 1). Theore.4 provides partial solutions to both probles (P1) and (P). In particular, it gives a negative answer to proble (P) for the case when = n. As a result, if = n, the upper bounds of the eleentary syetric functions of realignent atrices cannot be used to derive new conditions for detecting separability beyond the realignent criterion. 3. Proofs Proof of Lea.1. Define x = (x 1,..., x ) t, y = (y 1,..., y n ) t by { 1 if i = k( + 1) + 1 for soe 0 k 1, x i = 0 otherwise, and { 1 if j = k(n + 1) + 1 for soe 0 k n 1, y j = 0 otherwise. Then 1 x and 1 n y are unit vectors and 1 x t ρ R 1 y = trace ρ = 1. n n n Because { } s 1 = ax u t ρ R v : u C and v C n are unit vectors, we conclude that s 1 1 n. Proof of Lea.. Note that n 3 n 4 n 1 1 β α.
A note on the realignent criterion 7 Suppose s = (s 1,..., s ) S(, n) with s = i=1 s i 1. Let s = (1/s) s. Then s 1 s 1 α. Therefore, (1/,..., 1/ ) (α, β,..., β) s. Since f l is strictly concave [11], we have f l (s) f l ( s) f l (α, β,..., β) f l (1/,..., 1/ ) = ( l ) ( ) l 1, and the equality f l (s) = f l (α, β,..., β) holds if and only if s = (α, β,..., β). proves (a). Assertion (b) follows readily fro (a). This Proof of Theore.3. We first consider the sipler case when n 3. It suffices to construct ρ D(, n) for which ρ R has singular values 1/,..., 1/. Suppose {E 1,1,..., E, } is the standard basis of atrices. For 1 k, l, let F k,l = (E k,l I ) O n 3. Then ρ = 1 E 3 k,l F k,l is an n n density atrix k, l=1 while ρ R has singular values 1/,..., 1/. Next, suppose n 3 /. By Lea., we have B l (, n) f l (α, β,..., β) for all 1 l. We will construct ρ D(, n) for which ρ R has singular values α, β,..., β. Suppose n = q + r with 0 r <. For 1 k, l, let F k,l = (E k,l I q ) O r. Define and ρ 1 = k, l=1 E k,l F k,l, ρ = I (I q O r ), and ρ 3 = I (O q I r ) ρ = s 1 ρ 1 + s ρ + s 3 ρ 3 with s 1 = β q, s = α β q, and s 3 = α. Denote J,n by the n atrix with all entries equal to one. Then the realigned atrix ρ R is (under perutation of rows and coluns) given by A = q ters {}}{ s 1 I + s J, s 1 I + s J, s 3 J,r O,( )q O O O s 1 I s 1 I }{{} q ters O. Note that AA = ( qs 1I + (qs 1 s + qs + rs 3)J, ) qs 1 I.
A note on the realignent criterion 8 Since J, has only one non-zero eigenvalue, a atrix of the for µi + νj, has eigenvalues µ + ν and µ with ultiplicity 1 and 1, respectively. As a result AA has one eigenvalue equal to qs 1 + (qs 1 s + qs + rs 3) = α 4 ( q + r) = α and 1 eigenvalues equal to qs 1 = β. Hence, taking square roots, we see that the atrix ρ R has the desired singular values α, β,..., β. It reains to show that ρ is a density atrix. Notice that trace (ρ) = s 1 (q) + s ( q) + s 3 (r) = α (q + r) = 1. Since ρ 1, ρ, and ρ 3 are all positive sei-definite and both s 1 and s 3 are nonnegative, ρ is a density atrix if s is nonnegative. Notice that s 0 1 n n 1 n q( 1) 1 n n q q. For a fixed, let f(q, r) = (q + r) q (q + r) q for q 1 and 0 r 1. Then f q = q r + q 3/ r q q(q + r) f > 0 and r = 1 1 q q q q + r > 0 for all q 1 and 0 r 1. Therefore, (a) f(q, r) f(, 1) for all 1 q and r 1; and (b) f( 1, r) f( 1, /) for all r /. So, it suffices to prove that (1) f(, 1) 1 and () f( 1, /) 1. To prove (1), since, we have 4 ( ) (( ) + 1) = 4 + 3 1 > 0.
A note on the realignent criterion 9 (( ) + 1) It follows that < and hence f(, 1) = To prove (), since 1 (( 1) + /) 1 and (( 1) + /) ( 1) (( ) + 1) (( ) + 1) ( ) 1. ( 1 ), i.e., ( 1 1 ), we have = 1 + = 1 1 + 1 + 1 1 1 + 1 + ( 1 1 ) + 1, ( ) 1 1 = 1 ( 1 + ). 1 Consequently, f( 1, /) = (( 1) + /) 1 ( 1 ) + (( 1) + /) ( 1) 1 1 ( 1 + ) = 1. 1 Reark The sallest values of, n which do not satisfy the conditions in Theore.3 are = 3 and n = 6. For these values, the proof in Theore.3 does not work because s < 0. In this case, the question about the exact value of B l (, n) is still open. Proof of Theore.4. Suppose = n. Then the atrix ρ constructed in the proof of Theore.3 has the for ρ = 1 ( In + xx t) n(n + 1) where x i = { 1 if i = k(n + 1) + 1 for soe 0 k n 1, 0 otherwise. It follows fro [13] that ρ is separable.
A note on the realignent criterion 10 4. Conclusion The ain goal of this paper is to investigate the open probles (P1) and (P) proposed in [10] in the search for a new criterion for separability. We study the singular values of the realignent of density atrices and obtain new bounds on the eleentary syetric functions. The results are applied to find explicit forulas for B l (, n), for all n, except 3 / < n < 3 and B l (n, n). This provides a partial answer to the open proble (P1). Furtherore, we show that B l (n, n) = B l (n, n) for all n so that one cannot use B l (, n) to differentiate separable atrices fro density atrices whose realignent atrix has trace nor at ost 1 when = n. This gives a negative answer to proble (P) when = n. For n, nuerical results in [10] suggested that B l (, n) > B l (, n). If this strict inequality holds, then we would have a new criterion for separability. Our explicit forula for B l (, n) will be useful in this study. Acknowledgents The research was done when Li was a 011 Fulbright Fellow at the Hong Kong University of Science and Technology. He is a Shanxi Hundred Talent Scholar of the Taiyuan University of Science and Technology, and is an honorary professor of the University of Hong Kong, and the Shanghai University. Research of the first two authors were supported in part by USA NSF. Research of the first and the third authors were supported in part by a HK RGC grant. References [1] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W.K. Wootters 1993 Teleporting an unknown quantu state via dual classical and Einstein-Podolsky-Rosen channels Phys. Rev. Lett. 70 1895 [] K. Chen and L.A. Wu 003 A atrix realignent ethod for recognizing entangleent, Quantu Inf. Coput. 3 193 [3] A. Einstein, B. Podolsky and N. Rosen 1935 Can quantu-echanical description of physical reality be considered coplete? Phys. Rev. 47 777 [4] A.K. Ekert 1991 Quantu cryptography based on Bell s theore, Phys. Rev. Lett. 67 661 [5] L. Gurvits 003 Classical deterinistic coplexity of Edonds proble and quantu entangleent, Proceedings of the 35th ACM Syposiu on Theory of Coputing (New York: ACM Press) p 10 [6] R. A. Horn and C. R. Johnson 1990 Matrix Analysis (Cabridge, UK: Cabridge University Press). [7] M. Horodecki, P. Horodecki and R. Horodecki 1996 Separability of ixed states: necessary and sufficient conditions, Phys. Lett. A 3 1
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