Characterizing the dynamical semigroups that do not decrease a quantum entropy

TL;DR

This paper characterizes quantum dynamical semigroups that do not decrease various quantum entropies, providing insights into their generators and extending results to broader classes of maps and entropies.

Contribution

It offers a comprehensive characterization of entropy-preserving quantum dynamical semigroups and extends the analysis to positive maps and generalized entropies.

Findings

Characterization of semigroups preserving von Neumann, Tsallis, and Renyi entropies

Description of the infinitesimal generators of such semigroups

Extensions to positive trace-preserving maps and broader entropy classes

Abstract

In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Renyi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences --- in particular, a description of the associated infinitesimal generators --- are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.