A family of generalized quantum entropies: definition and properties

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.Comment: 26 pages, 1 figure. Close to published versio

Approximate transformations of bipartite pure-state entanglement from the majorization lattice

“…In particular, majorization arises naturally in several problems of quantum information theory. A nonexhaustive list includes entanglement criteria [2,3], characterizing mixing and quantum measurements [4][5][6], majorization uncertainty relations [7][8][9][10][11][12] and quantum entropies [13][14][15][16], among others [17][18][19][20][21][22][23] (see also [24,25] for reviews of some of these topics).…”

Group theory, entropy and the third law of thermodynamics

“…These generalized entropies aim to explain the non-equilibrium stationary metastable states through the deformation in the underlying entropic structure. Along this direction, many important applications were reported in the fields of generalized reaction rates [4][5][6][7], quantum information [8][9][10][11][12][13][14], plasma physics [15][16][17], high energy physics [18][19][20] and the rigid rotators in modelling the molecular structure [21,22]. The common feature of these entropies is to yield inverse power law distributions through the entropy maximization [1,3,23,24].…”

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

“…Previous works have focused on some entropic measures in the setting of generalized probabilities [49,66,67]. In this paper, we extend a new family of entropies based on the (h, φ)-entropies to the general probabilistic setting [68,69]. These measures include the previous ones studied in the literature as particular cases.…”