Generalized entropic criterion for separability

TL;DR

This paper explores a generalized entropic criterion for quantum state separability, establishing necessary conditions based on non-additive entropies and analyzing their limitations through examples and eigenvalue relations.

Contribution

It introduces a generalized entropic criterion for separability applicable to non-additive entropies and discusses its limitations and relation to eigenvalues.

Findings

The generalized entropy of a separable state is not smaller than that of its subsystems.

The criterion is necessary but not sufficient for separability.

Positivity of conditional Tsallis entropy is necessary but not sufficient for separability.

Abstract

We discuss the entropic criterion for separability of compound quantum systems for general non-additive entropic forms based on arbitrary concave functions $f$. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of $f$, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all $q$, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.