Relative entropy for pairs of subalgebras and invariants arising from automorphisms

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For a pair {A, B} of subalgebra of an operator algebra M, by modifying the Connes-Stormer relative entropy H(A|B), we defined h(A|B), and measured of distance between two algebras. For examples, in the case where M is the matrix algebra with the size n, then { h(A,B) ; A and B are maximal abelian subalgebras M}=[0.log n] and h(A,B)=log n iff A and B are mutually orthogonal. Similar result holds for pairs of subfactor with index 2 of type II_1 factors.