Free entropy with respect to a completely positive map

Given an element X in a tracial von Neumann algebra (M, τ), a unital subalgebra B ⊂ M, and a completely positive map η: B → B, we define the free Fisher information Φ*(X : B, η) of X relative to B with respect to η. The definition is a generalization of Voiculescu's definition of Φ*(X : B), which corresponds to η = τ. We show that many facts about Φ*(X : B) generalize to Φ*(X : B, η). As an application, we show that if B is commutative and X is singular with respect to B, then Φ*(X : B) is infinite. Φ*(X : B, η) is minimized when X is a B-valued semicircular variable with covariance η. Therefore, Φ*(· : B, η) is an appropriate quantity to study B-valued semicircular systems with arbitrary covariance. Associating to a pair of completely positive maps μ, η: B → B the quantity Φ*(Xμ : B, η), where Xμ is a B-valued semicircular variable with covariance μ, allows us to measure "absolute continuity" of μ and η. To a completely positive map μ we can naturally associate the number Φ*(Xμ : B, id). If μ is the conditional expectation onto a subfactor A ⊂ B of a II₁ factor B, we show that Φ*(X_E : B, id) is equal to the Jones' index [B ⊂ A].