Quantum accessible information and classical entropy inequalities
arXiv:2506.06700
The paper applies a recent optimality criterion for quantum measurements to specific ensembles, deriving tight entropy inequalities related to the log‑Sobolev inequality and revisiting conjectures about globally optimal measurements for equiangular quantum states (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. ÅeháÄek, 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.
45 pages, no figures. Postscriptum added including new references (in particular, the ones confirming the conjectured inequalities) and a clarification comcerning the optimality conditions