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

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

2008-03-05

arxiv.org/abs/0803.0591v1

Internat.J.Math.19(2008),767-776

Mathematics

Operator Algebras

10 pages

Scientific paper

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.

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