Integration with respect to operator-valued measures with applications to quantum estimation theory

Summary This paper is concerned with the development of an integration theory with respect to operator-valued measures which is required in the study of certain convex optimization problems. These convex optimization problems in their turn are rigorous formulations of detection theory in a quantum communication context, which generalise classical (Bayesian) detection theory. The integration theory which is developed in this paper is used in conjunction with convex analysis in Banach spaces to give necessary and sufficient conditions of optimality for this class of convex optimization problems.This research has been supported by the Air Force Office of Scientific Research under Grants AFOSR 77-3281D and AFOSR 82-0135 and the National Science Foundation under Grant NSF ENG 76-02860. A portion of this research was done while the first author was a CNR Visiting Professor at Istituto Matematico U. Dini, Università di Firenze, Italy.