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. Řeháček, 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.

翻译：计算量子态系综的可访问信息是量子信息论中的一个基本问题。我们证明，最近获得的最优性判据（A.S. Holevo, Lobachevskii J. Math., \textbf{43}:7 (2022), 1646-1650）应用于特定态系综时，会导出一系列非平凡的紧致熵不等式，这些不等式是著名对数索博列夫不等式的离散形式关联式。由此视角出发，我们重新审视了关于等角等概率态系综（量子棱锥体）的全局信息最优测量假设（B.-G. Englert and J. Řeháček, J. Mod. Optics \textbf{57 }N3 (2010) 218-226），并提出了相应的熵不等式。通过该最优性判据，这为证明关于量子棱锥体全局信息最优观测量的猜想提供了一种可能途径。