Some general properties of unified entropies

TL;DR

This paper explores fundamental properties of unified entropies in finite-dimensional quantum systems, establishing bounds, continuity, stability, and key inequalities, thereby enhancing understanding of their mathematical structure and physical relevance.

Contribution

It provides new bounds, continuity, and stability results for unified entropies, including quantum Rényi entropies, and proves their subadditivity, triangle inequality, and monotonicity under projective measurements.

Findings

Unified entropies are continuous and stable in finite-dimensional quantum systems.

They satisfy subadditivity and triangle inequality for certain parameters.

All unified entropies increase under projective measurements.

Abstract

Basic properties of the unified entropies are examined. The consideration is mainly restricted to the finite-dimensional quantum case. Bounds in terms of ensembles of quantum states are given. Both the continuity in Fannes' sense and stability in Lesche's sense are shown for wide ranges of parameters. In particular, uniform estimates are obtained for the quantum R\'{e}nyi entropies. Stability properties in the thermodynamic limit are discussed as well. It is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters. Non-decreasing of all the unified entropies under projective measurement is proved.