R\'{e}nyi and Tsallis formulations of separability conditions in finite dimensions

TL;DR

This paper develops new separability criteria for bipartite quantum systems using Rényi and Tsallis entropies, convolution of probability distributions, and various measurement schemes, enhancing entanglement detection methods.

Contribution

It introduces a novel approach based on convolution of probability distributions and derives new separability conditions using entropic uncertainty and majorization relations.

Findings

Derived separability conditions for finite-dimensional systems.

Applied criteria to measurements like mutually unbiased bases.

Demonstrated relevance through several quantum system examples.

Abstract

Separability conditions for a bipartite quantum system of finite-dimensional subsystems are formulated in terms of R\'{e}nyi and Tsallis entropies. Entropic uncertainty relations often lead to entanglement criteria. We propose new approach based on the convolution of discrete probability distributions. Measurements on a total system are constructed of local ones according to the convolution scheme. Separability conditions are derived on the base of uncertainty relations of the Maassen-Uffink type as well as majorization relations. On each of subsystems, we use a pair of sets of subnormalized vectors that form rank-one POVMs. We also obtain entropic separability conditions for local measurements with a special structure, such as mutually unbiased bases and symmetric informationally complete measurements. The relevance of the derived separability conditions is demonstrated with several examples.