Degree of separability of bipartite quantum states

TL;DR

This paper presents a systematic method using semidefinite relaxations to find the optimal convex decomposition of bipartite quantum states into separable and entangled parts, applicable to any finite dimension.

Contribution

It introduces a convergent procedure to compute the optimal Lewenstein-Sanpera decomposition for bipartite states of arbitrary finite dimension.

Findings

Method converges to the optimal decomposition

Applicable to bipartite states of any finite dimension

Numerical results demonstrate effectiveness

Abstract

We investigate the problem of finding the optimal convex decomposition of a bipartite quantum state into a separable part and a positive remainder, in which the weight of the separable part is maximal. This weight is naturally identified with the degree of separability of the state. In a recent work, the problem was solved for two-qubit states using semidefinite programming. In this paper, we describe a procedure to obtain the optimal decomposition of a bipartite state of any finite dimension via a sequence of semidefinite relaxations. The sequence of decompositions thus obtained is shown to converge to the optimal one. This provides, for the first time, a systematic method to determine the so-called optimal Lewenstein-Sanpera decomposition of any bipartite state. Numerical results are provided to illustrate this procedure, and the special case of rank-2 states is also discussed.