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arXiv:quant-ph/9704017

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

Armin Uhlmann

Published Apr 8, 1997• 39 citations
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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.

DOI
10.48550/arxiv.quant-ph/9704017

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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.
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This is an expository survey article based on a talk for the Spe- cial Session (Operator Algebra) in AMC2009. For two subalgebras A and B of a finite von Neumann algebra M, we discuss a modified version h(A|B) of the Connes-Stormer relative entropy H(A|B). As one of the most elementary examples, we pick two maximal abelian *-subalgebras A, B of the n × n com- plex matrices Mn(C), and show relations between h(A|B) and the entropy for unistochastic matrices. Also we show related results in the case of a pair of subfactors of a type II1 factor.
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