The output entropy of quantum channels and operations

TL;DR

This paper investigates the continuity properties of the output entropy of quantum channels and operations, providing conditions under which the entropy remains continuous and exploring implications for quantum information theory.

Contribution

It introduces new continuity conditions for the output entropy of quantum channels and operations, with a focus on Kraus representations and preservation of entropy continuity.

Findings

Finiteness of output entropy implies continuity of positive maps.

Conditions for entropy continuity are expressed via Kraus operators.

Relations between continuity properties of complementary quantum operations are established.

Abstract

Continuity properties of the output entropy of positive linear maps between Banach spaces of trace class operators are investigated with the special attention to the classes of quantum channels and operations. It is shown that finiteness of the output entropy of a positive map on the whole input state space implies its continuity. Sufficient conditions for this property expressed in terms of the Kraus representation of quantum channels and operations are presented. The characterization of a positive map "preserving continuity of the entropy" (in the sense that continuity of the entropy on a set of input states implies continuity of its output entropy on this set) is obtained and its applications to the class of quantum operations are considered. The special relation between continuity properties of the output entropies of complementary quantum operations is established and its corollaries are discussed. Continuity conditions for the output entropy of positive maps considered as a function of a pair (map, input state) are also obtained. Several applications to the quantum information theory are presented in the last part of the paper.