Entropy behaviour under completely positive maps

Göran Lindblad

Ichiro Fujimoto

Chris Heunen, Jamie Vicary

Abstract Completely positive maps give a notion of quantum process that allows the description of probabilistic effects in quantum theory. We show how to describe this categorically in terms of the CP construction and contrast the behaviour of classical and quantum information in this setting. We also study the behaviour of duality and linear structure under the CP construction.

Andrzej Kossakowski, M. Ohya, N. WATANABE

Максим Евгеньевич Широков

Global and local continuity conditions for the output von Neumann entropy for positive maps between Banach spaces of trace-class operators in separable Hilbert spaces are obtained. Special attention is paid to completely positive maps: infinite dimensional quantum channels and operations. It is shown that as a result of some specific properties of the von Neumann entropy (as a function on the set of density operators) several results on the output entropy of positive maps can be obtained, which cannot be derived from the general properties of entropy type functions. In particular, it is proved that global continuity of the output entropy of a positive map follows from its finiteness. A characterization of positive linear maps preserving continuity of the entropy (in the following sense: continuity of the entropy on an arbitrary subset of input operators implies continuity of the output entropy on this subset) is obtained. A connection between the local continuity properties of two completely positive complementary maps is considered. Bibliography: 21 titles.

Garani Ananthakrishna, E. C. G. Sudarshan, Vittorio Gorini