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

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.