von Neumann entropy and relative position between subalgebras

TL;DR

This paper provides a numerical method to characterize the orthogonality of subalgebras in matrix algebras using von Neumann entropy, establishing a clear criterion for their mutual orthogonality.

Contribution

It introduces a new entropy-based criterion for mutual orthogonality of subalgebras in matrix algebras, linking entropy maximization to subalgebra complementarity.

Findings

Mutual orthogonality corresponds to maximum von Neumann entropy of the induced density matrix.

The entropy reaches $2\log n$ if and only if the subalgebras are mutually orthogonal.

Provides a numerical characterization of subalgebra complementarity in finite-dimensional matrix algebras.

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