On Lower Semicontinuity of the Quantum Conditional Mutual Information and Its Corollaries

Abstract

We discuss the interconnections between basic correlation measures of a bipartite quantum state and basic information characteristics of a quantum channel, focusing on the benefits of these interconnections for solving specific problems concerning the characteristics of both types.We describe the basic properties of the (unoptimized and optimized) quantum discord in infinite-dimensional bipartite systems. In particular, using the generalized Koashi-Winter relation, a simple condition is obtained that guarantees that a state with zero quantum discord is quantum-classical.The generalized versions of Koashi-Winter and Xi-Lu-Wang-Li relations are used to obtain new continuity bounds for the output Holevo information of an ensemble of quantum states and for the Holevo capacity of a quantum channel in both finite-dimensional and infinite-dimensional cases.We also discuss the properties of quantum channels which are "doppelgangers" of the monotonicity of the quantum discord and the entropy reduction of a local measurement w.r.t. quantum channels acting on an unmeasured subsystem.Among others, it is shown that the entropy exchange of a channel does not decrease under concatenation with a channel that does not reduce the von Neumann entropy (in particular, with a bistochastic channel). Contents 1 Introduction 2 Preliminaries and basic notation 3 Entropy reduction of local measurements 4 The generalized versions of the Koashi-Winter and Xi-Lu-Wang-Li relations