Unifying several separability conditions using the covariance matrix criterion

TL;DR

This paper introduces a unifying framework using covariance matrices to determine quantum state separability, effectively detecting entanglement where traditional criteria like PPT fail, and showing many existing criteria are special cases.

Contribution

The authors develop the covariance matrix criterion as a comprehensive tool for quantum separability, unifying and extending previous entanglement detection methods.

Findings

The covariance matrix criterion detects entanglement beyond PPT.

Many existing criteria are special cases of the covariance matrix criterion.

The framework simplifies the understanding of various separability conditions.

Abstract

We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion - the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others - are in fact a corollary of the covariance matrix criterion.