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Comment on: “Construction of bound entangled edge states with special ranks” [Phys. Lett. A 359 (2006) 603]
Physics Letters A 372 (2008) 2336–2338 www.elsevier.com/locate/pla
Comment
Comment on: “Construction of bound entangled edge states with special ranks” [Phys. Lett. A 359 (2006) 603] Woo Chan Kim ∗,1 Department of Mathematics, Seoul National University, Seoul 151-742, South Korea Received 5 October 2007; received in revised form 14 November 2007; accepted 14 November 2007 Available online 19 November 2007 Communicated by P.R. Holland
Abstract Recently, Clarisse [L. Clarisse, Phys. Lett. A 359 (2006) 603] and Ha [K.-C. Ha, Phys. Lett. A 361 (2007) 515] found examples of types (5, 5) and (6, 6) entangled states with positive partial transposes. In this Letter, we show that their examples have the Schmidt number as 2. © 2007 Elsevier B.V. All rights reserved. PACS: 03.65.Bz; 03.67.-a; 03.67.Hk MSC: 81P15; 46L05; 15A30 Keywords: Entangled state; PPTES; Schmidt number
1. Introduction The theory of quantum entanglement has been extensively studied as a resource in quantum information and computation theory. One of the important tasks in the theory is the characterization of entangled states. The important tools for this characterization are the PPT criterion [3,4] and the notion of Schmidt decomposition of pure states [5]. mThe PPT criterion states +that a density matrix A = i,j =1 aij ⊗ eij in (Mn ⊗ Mm ) preserves positivity under partial transposition if A is separable; the partial transpose of A, denoted by Aτ , is given by m a j i,j =1 i ⊗ eij . Since the converse of the PPT criterion has been known to be a false argument in general [3,6,7], the class of entangled states with positive partial transposes (PPTES) has been of great interest. The notion
DOI of original article: 10.1016/j.physleta.2006.07.045. * Tel.: +82 11 591 9317; fax: +82 2 887 4694.
E-mail address: [email protected]. 1 This work was partially supported by a KRF grant (R14-2003-006-
01002-0). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.023
of PPTES is related not only to bound entangled states [8] but also to the structure of indecomposable maps between matrix algebras [9–11]. In particular edge PPTES [12] are important as they mark the boundary between PPTES and states with a negative partial transpose. Edge PPTES may be classified by their range dimensions, as shown in [13]. An edge PPTES A is said to be of the type (s, t) if the range dimensions of A and Aτ are s and t, respectively. In the case of a 3 ⊗ 3 system, there are many examples of (4, 4) and (7, 6) edge PPTES in the literature; PPTES of types (7, 5), (6, 5), and (8, 5) were found in [10]. Recently, (5, 5) and (6, 6) edge PPTES were also found by Clarisse [1] and Ha [2] independently, and the extremal properties of (5, 5) states were discussed in [14]. To generalize the Schmidt decomposition, the authors in [15] introduced the Schmidt number for density matrices. The Schmidt number of a density matrix A is defined by the minimum, taken from among all the decompositions of A, of the maximum Schmidt rank of pure state in the decomposition. As the Schmidt decomposition of pure states, A is entangled if A has a Schmidt number greater than or equal to 2.
W.C. Kim / Physics Letters A 372 (2008) 2336–2338
In [13], the authors conjectured that every 3 ⊗ 3 PPTES has the Schmidt number 2 and showed that this is the case for the type (4, 4) edge PPTES. The authors in [9] also showed that a family of (4, 4) PPTES has the Schmidt number 2 by providing direct arguments. From the duality theory [16] that involves positive linear maps and entanglements, this conjecture is equivalent to claiming that every 2-positive linear map in M3 is decomposable; this conjecture has also been studied by operator algebraists (see [17,18]). In the cases of PPTES of the types (6, 5), (7, 5), (8, 5) and (7, 6) in [7,10,19], the states can be easily decomposed to the form A1 + A2 , where A1 is of type (4, 4) and A2 has a Schmidt number less than or equal to 2. Hence, these PPTES also have the Schmidt number as 2. The purpose of the present Letter is to show that the examples of Clarisse [1] and Ha [2] concerning PPTES of types (5, 5) and (6, 6) also have the Schmidt number 2 by providing explicit decompositions. This is supporting evidence for the conjecture in Ref. [13]. 2. Schmidt number We introduce the following notations, which were defined in [2,16]. For an m × n matrix z = [zik ] ∈ Mm×n , we define zi =
n
zik ek ∈ C , n
i = 1, 2, . . . , m,
k=1
z˜ =
m
zi ⊗ ei ∈ Cn ⊗ Cm .
i=1
Then, z → z˜ defines an inner product isomorphism from Mm×n onto Cn ⊗ Cm . Therefore every density matrix A =∗ m + can be written as A = a ⊗ e ∈ (M ⊗ M ) ij n m i,j =1 ij i z˜ i z˜ i for some zi s in Mm×n , where z˜ i is an mn × 1 matrix and z˜ i∗ is a 1 × mn matrix whose entries are complex conjugates of those of z˜ i . We also note that z˜ z˜ ∗ is a positive semi-definite matrix in Mn ⊗ Mm of rank one. We consider the convex cones Vs = conv z˜ z˜ ∗ : rank z s , τ Vs = conv z˜ z˜ ∗ : rank z s for s = 1, 2, . . . , m ∧ n, where m ∧ n is the minimum of m and n. Then, a density matrix A in (Mn ⊗ Mm )+ has the Schmidt number k if and only if A is present in Vk \ Vk−1 . The above-mentioned conjecture in [13] is equivalent to stating V3 ∩ V3 ⊂ V2 in the 3 ⊗ 3 system. Let {eij } be the matrix unit in M3×3 . We consider √ D1 = {e12 , e12 − e13 + e33 , e32 − e33 , 2e21 , e21 − e23 − e31 }, E1 = e21 − e23 , e21 − e31 , e21 + e33 , √ √ √ 1 1 2e12 − √ e13 , 2e33 − √ e13 − 2e32 , 2 2
D2 =
2337
1 (e11 − e12 + e21 − e22 + e32 + 2e33 ), 2
1 (e11 + e12 − e21 − e22 − e32 + 2e33 ), 2 1 (e11 + e12 + e21 − e22 + e32 ), 2 1 (−e11 + e12 + e21 + e22 + e32 ), 2 e12 , e13 − e31 , e23 + e31 , e33 , √ 1 E2 = e11 − e33 , 2e12 − √ e21 , 2 √ 1 e13 + e31 , √ e21 + 2e33 , e22 + e31 , e23 − e32 , 2 1 D3 = (e11 + e12 + e21 + e22 + e32 + 2e33 ), 2 1 (e11 − e12 − e21 + e22 − e32 + 2e33 ), 2 1 (e11 − e12 + e21 + e22 + e32 ), 2 1 (−e11 − e12 + e21 − e22 + e32 ), 2 e12 , e13 − e31 , e23 + e31 , e33 , √ 1 E3 = e11 − e33 , 2e12 + √ e21 , 2 √ 1 e13 + e31 , √ e21 + 2e33 , e22 + e31 , e23 + e32 . 2 Now, wecan construct the entangled states Ak = i z˜ i z˜i ∗ and Aτk = i w˜ j w˜ j∗ for zi ∈ Dk and wj ∈ Ek , respectively. Then, A1 and A2 are the states of Clarisse of type (5, 5) and (6, 6), respectively, in [1]. Moreover, A3 is the state of Ha of type (6, 6) in [2]. Since the rank of every 3 × 3 matrix in the above-mentioned list is less than or equal to 2, A1 , A2 , and A3 and their partial transposes have the Schmidt number 2. The type (5, 5) state of Ha [2] also has a Schmidt number of 2 by the construction itself. Acknowledgement The authors are grateful to the referee for valuable comments. References [1] L. Clarisse, Phys. Lett. A 359 (2006) 603. [2] K.-C. Ha, Phys. Lett. A 361 (2007) 515. [3] M.-D. Choi, Operator Algebras and Applications, Part 2, Kingston, 1980, Proceedings of Symposia in Pure Mathematics, vol. 38, Amer. Math. Soc., 1982, p. 583. [4] A. Peres, Phys. Rev. Lett. 77 (1996) 1413. [5] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, England, 2000. [6] S.L. Woronowicz, Rep. Math. Phys. 10 (1976) 165. [7] P. Horodecki, Phys. Lett. A 232 (1997) 333.
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W.C. Kim / Physics Letters A 372 (2008) 2336–2338
[8] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 80 (1998) 5239. [9] K.-C. Ha, S.-H. Kye, Y.S. Park, Phys. Lett. A 313 (2003) 163. [10] K.-C. Ha, S.-H. Kye, J. Phys. A 38 (41) (2005) 9039. [11] B.M. Terhal, Linear Algebra Appl. 323 (2001) 61. [12] M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 62 (2000) 052310. [13] A. Sanpera, D. Bruß, M. Lewenstein, Phys. Rev. A 63 (2001) 050301.
[14] [15] [16] [17] [18] [19]
W.C. Kim, S.-H. Kye, Phys. Lett. A 369 (2007) 16. B.M. Terhal, P. Horodecki, Phys. Rev. A 61 (2000) 040301. M.-H. Eom, S.-H. Kye, Math. Scand. 86 (2000) 130. S.-J. Cho, S.-H. Kye, S.G. Lee, Linear Algebra Appl. 171 (1992) 213. K.-C. Ha, Publ. RIMS Kyoto Univ. 34 (1998) 591. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 82 (1999) 1056.
Comment
Comment on: “Construction of bound entangled edge states with special ranks” [Phys. Lett. A 359 (2006) 603] Woo Chan Kim ∗,1 Department of Mathematics, Seoul National University, Seoul 151-742, South Korea Received 5 October 2007; received in revised form 14 November 2007; accepted 14 November 2007 Available online 19 November 2007 Communicated by P.R. Holland
Abstract Recently, Clarisse [L. Clarisse, Phys. Lett. A 359 (2006) 603] and Ha [K.-C. Ha, Phys. Lett. A 361 (2007) 515] found examples of types (5, 5) and (6, 6) entangled states with positive partial transposes. In this Letter, we show that their examples have the Schmidt number as 2. © 2007 Elsevier B.V. All rights reserved. PACS: 03.65.Bz; 03.67.-a; 03.67.Hk MSC: 81P15; 46L05; 15A30 Keywords: Entangled state; PPTES; Schmidt number
1. Introduction The theory of quantum entanglement has been extensively studied as a resource in quantum information and computation theory. One of the important tasks in the theory is the characterization of entangled states. The important tools for this characterization are the PPT criterion [3,4] and the notion of Schmidt decomposition of pure states [5]. mThe PPT criterion states +that a density matrix A = i,j =1 aij ⊗ eij in (Mn ⊗ Mm ) preserves positivity under partial transposition if A is separable; the partial transpose of A, denoted by Aτ , is given by m a j i,j =1 i ⊗ eij . Since the converse of the PPT criterion has been known to be a false argument in general [3,6,7], the class of entangled states with positive partial transposes (PPTES) has been of great interest. The notion
DOI of original article: 10.1016/j.physleta.2006.07.045. * Tel.: +82 11 591 9317; fax: +82 2 887 4694.
E-mail address: [email protected]. 1 This work was partially supported by a KRF grant (R14-2003-006-
01002-0). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.023
of PPTES is related not only to bound entangled states [8] but also to the structure of indecomposable maps between matrix algebras [9–11]. In particular edge PPTES [12] are important as they mark the boundary between PPTES and states with a negative partial transpose. Edge PPTES may be classified by their range dimensions, as shown in [13]. An edge PPTES A is said to be of the type (s, t) if the range dimensions of A and Aτ are s and t, respectively. In the case of a 3 ⊗ 3 system, there are many examples of (4, 4) and (7, 6) edge PPTES in the literature; PPTES of types (7, 5), (6, 5), and (8, 5) were found in [10]. Recently, (5, 5) and (6, 6) edge PPTES were also found by Clarisse [1] and Ha [2] independently, and the extremal properties of (5, 5) states were discussed in [14]. To generalize the Schmidt decomposition, the authors in [15] introduced the Schmidt number for density matrices. The Schmidt number of a density matrix A is defined by the minimum, taken from among all the decompositions of A, of the maximum Schmidt rank of pure state in the decomposition. As the Schmidt decomposition of pure states, A is entangled if A has a Schmidt number greater than or equal to 2.
W.C. Kim / Physics Letters A 372 (2008) 2336–2338
In [13], the authors conjectured that every 3 ⊗ 3 PPTES has the Schmidt number 2 and showed that this is the case for the type (4, 4) edge PPTES. The authors in [9] also showed that a family of (4, 4) PPTES has the Schmidt number 2 by providing direct arguments. From the duality theory [16] that involves positive linear maps and entanglements, this conjecture is equivalent to claiming that every 2-positive linear map in M3 is decomposable; this conjecture has also been studied by operator algebraists (see [17,18]). In the cases of PPTES of the types (6, 5), (7, 5), (8, 5) and (7, 6) in [7,10,19], the states can be easily decomposed to the form A1 + A2 , where A1 is of type (4, 4) and A2 has a Schmidt number less than or equal to 2. Hence, these PPTES also have the Schmidt number as 2. The purpose of the present Letter is to show that the examples of Clarisse [1] and Ha [2] concerning PPTES of types (5, 5) and (6, 6) also have the Schmidt number 2 by providing explicit decompositions. This is supporting evidence for the conjecture in Ref. [13]. 2. Schmidt number We introduce the following notations, which were defined in [2,16]. For an m × n matrix z = [zik ] ∈ Mm×n , we define zi =
n
zik ek ∈ C , n
i = 1, 2, . . . , m,
k=1
z˜ =
m
zi ⊗ ei ∈ Cn ⊗ Cm .
i=1
Then, z → z˜ defines an inner product isomorphism from Mm×n onto Cn ⊗ Cm . Therefore every density matrix A =∗ m + can be written as A = a ⊗ e ∈ (M ⊗ M ) ij n m i,j =1 ij i z˜ i z˜ i for some zi s in Mm×n , where z˜ i is an mn × 1 matrix and z˜ i∗ is a 1 × mn matrix whose entries are complex conjugates of those of z˜ i . We also note that z˜ z˜ ∗ is a positive semi-definite matrix in Mn ⊗ Mm of rank one. We consider the convex cones Vs = conv z˜ z˜ ∗ : rank z s , τ Vs = conv z˜ z˜ ∗ : rank z s for s = 1, 2, . . . , m ∧ n, where m ∧ n is the minimum of m and n. Then, a density matrix A in (Mn ⊗ Mm )+ has the Schmidt number k if and only if A is present in Vk \ Vk−1 . The above-mentioned conjecture in [13] is equivalent to stating V3 ∩ V3 ⊂ V2 in the 3 ⊗ 3 system. Let {eij } be the matrix unit in M3×3 . We consider √ D1 = {e12 , e12 − e13 + e33 , e32 − e33 , 2e21 , e21 − e23 − e31 }, E1 = e21 − e23 , e21 − e31 , e21 + e33 , √ √ √ 1 1 2e12 − √ e13 , 2e33 − √ e13 − 2e32 , 2 2
D2 =
2337
1 (e11 − e12 + e21 − e22 + e32 + 2e33 ), 2
1 (e11 + e12 − e21 − e22 − e32 + 2e33 ), 2 1 (e11 + e12 + e21 − e22 + e32 ), 2 1 (−e11 + e12 + e21 + e22 + e32 ), 2 e12 , e13 − e31 , e23 + e31 , e33 , √ 1 E2 = e11 − e33 , 2e12 − √ e21 , 2 √ 1 e13 + e31 , √ e21 + 2e33 , e22 + e31 , e23 − e32 , 2 1 D3 = (e11 + e12 + e21 + e22 + e32 + 2e33 ), 2 1 (e11 − e12 − e21 + e22 − e32 + 2e33 ), 2 1 (e11 − e12 + e21 + e22 + e32 ), 2 1 (−e11 − e12 + e21 − e22 + e32 ), 2 e12 , e13 − e31 , e23 + e31 , e33 , √ 1 E3 = e11 − e33 , 2e12 + √ e21 , 2 √ 1 e13 + e31 , √ e21 + 2e33 , e22 + e31 , e23 + e32 . 2 Now, wecan construct the entangled states Ak = i z˜ i z˜i ∗ and Aτk = i w˜ j w˜ j∗ for zi ∈ Dk and wj ∈ Ek , respectively. Then, A1 and A2 are the states of Clarisse of type (5, 5) and (6, 6), respectively, in [1]. Moreover, A3 is the state of Ha of type (6, 6) in [2]. Since the rank of every 3 × 3 matrix in the above-mentioned list is less than or equal to 2, A1 , A2 , and A3 and their partial transposes have the Schmidt number 2. The type (5, 5) state of Ha [2] also has a Schmidt number of 2 by the construction itself. Acknowledgement The authors are grateful to the referee for valuable comments. References [1] L. Clarisse, Phys. Lett. A 359 (2006) 603. [2] K.-C. Ha, Phys. Lett. A 361 (2007) 515. [3] M.-D. Choi, Operator Algebras and Applications, Part 2, Kingston, 1980, Proceedings of Symposia in Pure Mathematics, vol. 38, Amer. Math. Soc., 1982, p. 583. [4] A. Peres, Phys. Rev. Lett. 77 (1996) 1413. [5] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, England, 2000. [6] S.L. Woronowicz, Rep. Math. Phys. 10 (1976) 165. [7] P. Horodecki, Phys. Lett. A 232 (1997) 333.
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W.C. Kim / Physics Letters A 372 (2008) 2336–2338
[8] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 80 (1998) 5239. [9] K.-C. Ha, S.-H. Kye, Y.S. Park, Phys. Lett. A 313 (2003) 163. [10] K.-C. Ha, S.-H. Kye, J. Phys. A 38 (41) (2005) 9039. [11] B.M. Terhal, Linear Algebra Appl. 323 (2001) 61. [12] M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 62 (2000) 052310. [13] A. Sanpera, D. Bruß, M. Lewenstein, Phys. Rev. A 63 (2001) 050301.
[14] [15] [16] [17] [18] [19]
W.C. Kim, S.-H. Kye, Phys. Lett. A 369 (2007) 16. B.M. Terhal, P. Horodecki, Phys. Rev. A 61 (2000) 040301. M.-H. Eom, S.-H. Kye, Math. Scand. 86 (2000) 130. S.-J. Cho, S.-H. Kye, S.G. Lee, Linear Algebra Appl. 171 (1992) 213. K.-C. Ha, Publ. RIMS Kyoto Univ. 34 (1998) 591. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 82 (1999) 1056.