Relative entropy for maximal abelian subalgebras of matrices and the entropy of unistochastic matrices

TL;DR

This paper explores the entropy relationships between maximal abelian subalgebras of matrices and unistochastic matrices, introducing new entropy constants and linking them to unistochastic matrix entropy.

Contribution

It introduces modified relative entropy measures for subalgebras and connects these to the entropy of unistochastic matrices, providing new insights into their structure.

Findings

h(A | B) equals the entropy of the unistochastic matrix b(u)

h(A | B) reaches maximum log n when A and B are orthogonal

Computed h_φ(A | B) explicitly for the modified entropy measures

Abstract

Let $A$ and $B$ be two maximal abelian *-subalgebras of the $n\times n$ complex matrices $M_n(\mathbb{C}).$ To study the movement of the inner automorphisms of $M_n(\mathbb{C}),$ we modify the Connes-St$\o$rmer relative entropy $H(A | B)$ and the Connes relative entropy $H_\phi(A | B)$ with respect to a state $\phi,$ and introduce the two kinds of the constant $h(A | B)$ and $h_\phi(A | B).$ For the unistochastic matrix $b(u)$ defined by a unitary $u$ with $B = uAu^*,$ we show that $h(A | B)$ is the entropy $H(b(u))$ of $b(u).$ This is obtained by our computation of $h_\phi(A | B).$ The $h(A | B)$ attains to the maximal value $\log n$ if and only if the pair $\{A, B\}$ is orthogonal in the sense of Popa.