Unique decompositions, faces, and automorphisms of separable states

被引:36
作者
Alfsen, Erik [1 ]
Shultz, Fred [2 ]
机构
[1] Univ Oslo, Dept Math, N-1053 Oslo, Norway
[2] Wellesley Coll, Dept Math, Wellesley, MA 02481 USA
关键词
ENTANGLEMENT; VOLUME; CRITERION; ENTROPY;
D O I
10.1063/1.3399808
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
Let S-k be the set of separable states on B(C-m circle times C-n) admitting a representation as a convex combination of k pure product states, or fewer. If m > 1, n > 1, and k <= max (m, n), we show that Sk admits a subset V-k such that V-k is dense and open in S-k, and such that each state in V-k has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains V-k. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of B(C-m circle times C-n) that preserve entanglement and separability. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3399808]
引用
收藏
页数:13
相关论文
共 33 条
[1]  
ALFSEN EM, 1971, ERGEBNISSE MATH GREN, V57
[2]  
[Anonymous], 1965, Topology
[3]   Tensor products of convex sets and the volume of separable states on N qudits -: art. no. 022109 [J].
Aubrun, G ;
Szarek, SJ .
PHYSICAL REVIEW A, 2006, 73 (02)
[4]   Unextendible product bases and bound entanglement [J].
Bennett, CH ;
DiVincenzo, DP ;
Mor, T ;
Shor, PW ;
Smolin, JA ;
Terhal, BM .
PHYSICAL REVIEW LETTERS, 1999, 82 (26) :5385-5388
[5]   Optimal decompositions of barely separable states [J].
Divincenzo, DP ;
Terhal, BM ;
Thapliyal, AV .
JOURNAL OF MODERN OPTICS, 2000, 47 (2-3) :377-385
[6]   Separability of mixed states: Necessary and sufficient conditions [J].
Horodecki, M ;
Horodecki, P ;
Horodecki, R .
PHYSICS LETTERS A, 1996, 223 (1-2) :1-8
[7]   Separability criterion and inseparable mixed states with positive partial transposition [J].
Horodecki, P .
PHYSICS LETTERS A, 1997, 232 (05) :333-339
[8]   Operational criterion and constructive checks for the separability of low-rank density matrices [J].
Horodecki, P ;
Lewenstein, M ;
Vidal, G ;
Cirac, I .
PHYSICAL REVIEW A, 2000, 62 (03) :10
[9]   Information-theoretic aspects of inseparability of mixed states [J].
Horodecki, R ;
Horodecki, M .
PHYSICAL REVIEW A, 1996, 54 (03) :1838-1843
[10]   A COMPLETE CLASSIFICATION OF QUANTUM ENSEMBLES HAVING A GIVEN DENSITY-MATRIX [J].
HUGHSTON, LP ;
JOZSA, R ;
WOOTTERS, WK .
PHYSICS LETTERS A, 1993, 183 (01) :14-18