A Necessary and Sufficient Criterion for the Separability of Quantum State

TL;DR

This paper introduces a complete, operational criterion for determining whether a bipartite quantum state is separable, based on inequalities derived from the multiplicative Horn's problem, advancing the understanding of quantum entanglement detection.

Contribution

It provides the first necessary and sufficient set of inequalities for bipartite state separability using the multiplicative Horn's problem, building on Bloch vector representations.

Findings

Complete set of inequalities for separability

Application to Werner and isotropic states

Operational criterion for mixed states

Abstract

Quantum entanglement has been regarded as one of the key physical resources in quantum information sciences. However, the determination of whether a mixed state is entangled or not is generally a hard issue, even for the bipartite system. In this work we propose an operational necessary and sufficient criterion for the separability of an arbitrary bipartite mixed state, by virtue of the multiplicative Horn's problem. The work follows the work initiated by Horodecki {\it et. al.} and uses the Bloch vector representation introduced to the separability problem by J. De Vicente. In our criterion, a complete and finite set of inequalities to determine the separability of compound system is obtained, which may be viewed as trade-off relations between the quantumness of subsystems. We apply the obtained result to explicit examples, e.g. the separable decomposition of arbitrary dimension Werner state and isotropic state.