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arXiv:2506.06700 · Lobachevskii Journal of Mathematics

Quantum Accessible Information and Classical Entropy Inequalities

Alexander Semenovich Holevo, A. V. Utkin

Published Jun 1, 2026• 1 citation
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Abstract

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., 43, 1646–1650 (2022)), 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 57 (3), 218–226 (2010)) 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.

DOI
10.1134/s1995080226617704

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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., 43, 1646–1650 (2022)), 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 57 (3), 218–226 (2010)) 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.
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A number of optimization problems in quantum information theory, such as one-shot classical capacity of a quantum channel, entanglement of formation of a state, classical capacity of a quantum observable, and accessible information of an ensemble of quantum states, can be formulated as the calculation of the convex closure of the output entropy of a certain quantum channel. This general problem can be considered as a convex programming problem for which necessary and sufficient optimality conditions can be established along with a dual problem. These optimality conditions imply some tight lower bounds for entropy, either quantum or classical, which are related to the famous logarithmic Sobolev inequality.
This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb.
This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.
Given a quantum state ensemble E\mathcal{E}E and quantum observable M,\mathcal{M},M, one can define the Shannon information I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) and introduce the two fundamental information quantities. The accessible information A(E)A(\mathcal{E})A(E) of the ensemble E\mathcal{E}E is defined as supremum of I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) over all observables M\mathcal{M}M . The information capacity C(M)C(\mathcal{M})C(M) of the observable M\mathcal{M}M is essentially supremum of I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) over ensembles E\mathcal{E}E (where E\mathcal{E}E may be subject to certain additional constraints). Computation of these quantities presents important and rather involved optimization problems which are closely connected via the so called ensemble-observable duality. This paper is devoted to consideration of these two quantities for quantum Gaussian systems. In this case ensemble-observable duality admits quite an explicit description. The present paper surveys results recently obtained in these directions. It turns out that in both cases the maximizer is quantum Gaussian: the quantity I(E,M),I(\mathcal{E},\mathcal{M}),I(E,M), where E\mathcal{E}E is Gaussian ensemble, is maximized by a Gaussian observable, while I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) for Gaussian observable M\mathcal{M}M is maximized by a Gaussian state ensemble. Thus, we have here still another confirmation of the famous ‘‘hypothesis of quantum Gaussian optimizers’’.
Abstract In this work we apply the general convex programming approach of [A. S. Holevo, Lobachevskii J. Math. 43(7), 1646–1650 (2022)] to the problem of minimization of convex closure of the output differential entropy of quantum measurement channel, involved in computation of the classical capacity of the channel, and representing noise in the channel. We show that in the bosonic Gaussian case the optimality conditions derived from the convex programming approach lead to formulation of some new generalizations of the famous logarithmic Sobolev inequality which we call ‘‘homodyne’’ and ‘‘heterodyne’’ inequalities. We also show that the ‘‘heterodyne’’ inequality implies the ‘‘homodyne’’ one.