von Neumann entropy and relative position between subalgebras

Abstract

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 $M_n(\mathbb{C})$ $(n\le k)$, 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.

8 pages