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arXiv:1203.6262 · Open Systems & Information Dynamics

Geometry of the Faces for Separable States Arising from Generalized Choi Maps

Kil-Chan Ha, Seung-Hyeok Kye

Published Jun 1, 2012• 6 citations
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Abstract

We exhibit examples of separable states which are on the boundary of the convex cone generated by all separable states but in the interior of the convex cone generated by all PPT states. We also analyze the geometric structures of the smallest face generated by those examples. As a byproduct, we obtain a large class of entangled states with positive partial transposes.

DOI
10.1142/s1230161212500096

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We exhibit examples of separable states which are on the boundary of the convex cone generated by all separable states but in the interior of the convex cone generated by all PPT states. We also analyze the geometric structures of the smallest face generated by those examples. As a byproduct, we obtain a large class of entangled states with positive partial transposes.
We construct faces of the convex set of all $2\ensuremath{\bigotimes}4$ bipartite separable states, which are affinely isomorphic to the simplex ${\ensuremath{\Delta}}_{9}$ with 10 extreme points. Every interior point of these faces is a separable state which has a unique decomposition into 10 product states, even though the ranks of the state and its partial transpose are 5 and 7, respectively. We also note that the number 10 is greater than $2\ifmmode\times\else\texttimes\fi{}4$, to disprove a conjecture on the lengths of qubit-qudit separable states. This face is inscribed in the corresponding face of the convex set of all PPT states so that subsimplices ${\ensuremath{\Delta}}_{k}$ of ${\ensuremath{\Delta}}_{9}$ share the boundary if and only if $k\ensuremath{\le}5$. This enables us to find a large class of $2\ensuremath{\bigotimes}4$ PPT entangled edge states with rank 5.
Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of CCC is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, kkk-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.
In the convex set of all 3 ⊗ 3 states with positive partial transposes, we show that one can take two extreme points whose convex combinations belong to the interior of the convex set. Their convex combinations may be even in the interior of the convex set of all separable states. In general, we need at least mn extreme points to get an interior point by their convex combination, for the case of the convex set of all m ⊗ n separable states. This shows a sharp distinction between PPT states and separable states. We also consider the same questions for positive maps and decomposable maps.
Power symmetric stochastic matrices introduced by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Hordeckis criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.
The convex cone V1\mathbb{V}_1V1​ generated by separable states is contained in the cone T\mathbb{T}T of all positive semi-definite block matrices whose block transposes are also positive semi-definite. We characterize faces of the cone V1\mathbb{V}_1V1​ induced by faces of the cone T\mathbb{T}T which are determined by pairs of subspaces of matrices.