von Neumann entropy and relative position between subalgebras

Details von Neumann entropy and relative position between subalgebras von Neumann entropy and relative position between subalgebras

2010-07-07

arxiv.org/abs/1007.1037v1

Mathematics

Operator Algebras

8 pages

Scientific paper

We give a numerical characterization of mutual orthogonality (that is, complementarity) for subalgebras. In order to give such a characterization for mutually orthogonal subalgebras $A$ and $B$ of the $k \times k$ matrix algebra $M_k(\mathbb{C})$, where $A$ and $B$ are isomorphic to some $M_n(\mathbb{C})$ $(n \leq k)$, we consider a density matrix which is induced from the pair $\{A, B\}$. We show that $A$ and $B$ are mutually orthogonal if and only if the von Neumann entropy of the density matrix is the maximum value $2\log n$, which is the logarithm of the dimension of the subfactors.

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