Information Capacity of State Ensembles and Observables

• Published in: Lobachevskii journal of mathematics Vol. 45; no. 6; pp. 2509 - 2526
• Main Author: Holevo, A. S.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.06.2024; Springer Nature B.V
• Subjects: Algebra; Analysis; Geometry; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; observable; positive operator-valued measure; accessible information; Gaussian maximizers; quantum state ensemble; capacity
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S199508022460314X

Given a quantum state ensemble and quantum observable one can define the Shannon information and introduce the two fundamental information quantities. The accessible information of the ensemble is defined as supremum of over all observables . The information capacity of the observable is essentially supremum of over ensembles (where 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 where is Gaussian ensemble, is maximized by a Gaussian observable, while for Gaussian observable is maximized by a Gaussian state ensemble. Thus, we have here still another confirmation of the famous ‘‘hypothesis of quantum Gaussian optimizers’’.