Bipartite quantum systems: on the realignment criterion and beyond

TL;DR

This paper introduces a new perspective on characterizing bipartite quantum states using Schmidt decomposition and symmetric polynomials, extending the realignment criterion to develop potentially stronger separability conditions.

Contribution

It proposes a novel family of necessary separability criteria based on Schmidt coefficients, extending the realignment criterion and exploring their implications for quantum channels.

Findings

Derived a family of necessary conditions for separability.

Numerical evidence supports the conjecture of stronger criteria.

Linked contraction rates of quantum maps to entanglement preservation.

Abstract

Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.