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

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