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Lobachevskii Journal of Mathematics

On Optimization Problem for Positive Operator-Valued Measures

Alexander Semenovich Holevo

Published Jul 1, 2022• 9 citations
Publisher

Abstract

It is shown that a number of optimization problems in quantum information theory: the $$\chi$$ -capacity (called the Holevo capacity in literature) of a quantum channel; the classical capacity of quantum observable; entanglement of formation—can be recast as a generalization of a Bayes problem over the set of all quantum states. This allows us to consider it as a convex programming problem for which necessary and sufficient optimality conditions along with the dual problem can be formulated.

DOI
10.1134/s1995080222100158

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It is shown that a number of optimization problems in quantum information theory: the χ\chiχ -capacity (called the Holevo capacity in literature) of a quantum channel; the classical capacity of quantum observable; entanglement of formation—can be recast as a generalization of a Bayes problem over the set of all quantum states. This allows us to consider it as a convex programming problem for which necessary and sufficient optimality conditions along with the dual problem can be formulated.
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.
We identify necessary and sufficient conditions for a quantum channel to be optimal for any convex optimization problem in which the optimization is taken over the set of all quantum channels of a fixed size. Optimality conditions for convex optimization problems over the set of all quantum measurements of a given system having a fixed number of measurement outcomes are obtained as a special case. In the case of linear objective functions for measurement optimization problems, our conditions reduce to the well-known Holevo-Yuen-Kennedy-Lax measurement optimality conditions. We illustrate how our conditions can be applied to various state transformation problems having non-linear objective functions based on the fidelity, trace distance, and quantum relative entropy.
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.
This paper is devoted to further study of the Holevo capacity of infinite-dimensional quantum channels. The existence of a unique optimal average state for a quantum channel constrained by an arbitrary convex set of states is shown. The minimax expression for the Holevo capacity of a constrained channel is obtained. The χ\chiχ-function and the convex closure of the output entropy of an infinite-dimensional quantum channel are considered. It is shown that the χ\chiχ-function of an arbitrary channel is lower semicontinuous on the set of all states and has continuous restrictions to subsets of states with continuous output entropy. The explicit expression for the convex closure of the output entropy of an infinite-dimensional quantum channel is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the output entropy on the set of states with finite output entropy and, similarly to the χ\chiχ-function, it has continuous restrictions to subsets of states with continuous output entropy. The applications of the obtained results to the theory of entanglement are considered. The properties of the convex closure of the output entropy make it possible to generalize some results related to the additivity problem to the infinite-dimensional case.
Special sets of states, called optimal, which are related to the Holevo capacity and to the minimal output entropy of a quantum channel, are considered. By methods of convex analysis and operator theory, structural properties of optimal sets and conditions of their coincidence are explored for an arbitrary channel. It is shown that strong additivity of the Holevo capacity for two given channels provides projective relations between optimal sets for the tensor product of these channels and optimal sets for the individual channels.
We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average correct probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.