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

TL;DR

This paper investigates the entropy of states relative to a maximal commutative subalgebra in finite-dimensional matrix algebras, using convexity and symmetry to analyze optimal decompositions and their properties.

Contribution

It introduces general properties of entropy relative to subalgebras in finite dimensions and applies convex analysis tools to understand optimal decompositions in this context.

Findings

Characterization of entropy and optimal decompositions for maximal commutative subalgebras

Development of convex analysis tools for entropy optimization problems

Insights into the structure of entanglement related to subfactors

Abstract

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.