Entropy and optimal decompositions of states relative to a maximal commutative subalgebra

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis for the problems in question is collected in an appendix. INTRODUCTION This paper considers the entropy of a subalgebra or of a completely positive map with respect to a state, an entropy-like quantity introduced by A. Connes, H. Narnhofer, and W. Thirring. I remain, however, within a rather narrow setting: A pair of algebras, -isomorphic to the algebra of all d × d-matrices, and to its subalgebra of diagonal matrices. However, within this introduction, and in discussing some tools from convex analysis, I depart from this restriction. While the von Neumann entropy is of undoubted relevance for type I algebras (with discrete center), the relative entropy can be meaningfully defined even on the state postal address: Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10, D04109, Leipzig, Germany. e-mail: [email protected]