Continuous ensembles and the capacity of infinite-dimensional quantum channels

This paper is devoted to the study of $\chi$-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the “maximal distance property” is obtained.