Quantum accessible information and classical entropy inequalities

TL;DR

This paper explores the computation of accessible information in quantum ensembles, deriving new entropy inequalities related to optimal measurements, and revisits the hypothesis of globally information-optimal measurements for quantum pyramids.

Contribution

It introduces tight entropy inequalities derived from optimality criteria and proposes an approach to prove conjectures on globally information-optimal measurements for quantum pyramids.

Findings

Derived nontrivial entropy inequalities related to quantum state ensembles.

Reconsidered the hypothesis of globally information-optimal measurements for quantum pyramids.

Suggested an approach to prove conjectures on optimal observables for quantum pyramids.

Abstract

Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the recently obtained optimality criterion (A.S. Holevo, Lobachevskii J. Math., \textbf{43}:7 (2022), 1646-1650), when applied to specific ensembles of states leads to nontrivial tight entropy inequalities that are discrete relatives of the famous log-Sobolev inequality. In this light, the hypothesis of globally information-optimal measurement for an ensemble of equiangular equiprobable states (quantum pyramids) (B.-G. Englert and J. \v{R}eh\'{a}\v{c}ek, J. Mod. Optics \textbf{57 }N3 (2010) 218-226) is reconsidered and the corresponding entropy inequalities are proposed. Via the optimality criterion, this suggests also an approach to the proof of the conjectures concerning globally information-optimal observables for quantum pyramids.