A family of generalized quantum entropies: definition and properties

TL;DR

This paper introduces a broad family of quantum entropies based on the classical $(h,\,\phi)$-entropies, explores their properties, and demonstrates their applications in quantum entanglement detection.

Contribution

It defines a new general class of quantum entropies that unify many existing entropies and analyzes their properties and applications in quantum information theory.

Findings

Quantum $(h,\,\phi)$-entropies unify known entropies like von Neumann, Rényi, and Tsallis.

Majorization explains common features of these entropies.

Generalized entropies can be used to detect quantum entanglement.

Abstract

We present a quantum version of the generalized $(h,\phi)$-entropies, introduced by Salicr\'u \textit{et al.} for the study of classical probability distributions. We establish their basic properties, and show that already known quantum entropies such as von Neumann, and quantum versions of R\'enyi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicr\'u form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum $(h,\phi)$-entropies under the action of quantum operations, and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement, and introduce a discussion on possible generalized conditional entropies as well.