Non-commutative f-divergence functional

Abstract

We introduce the non-commutative $f$-divergence functional $\Theta (\tilde{A},\tilde{B}):={\int }_T{B}_t^{\frac{1}{2}}f\left(B_t^{-\frac{1}{2}}{A}_t{B}_t^{-\frac{1}{2}}\right)B_t^{\frac{1}{2}}d\mu (t)$ for an operator convex function $f$, where $\tilde{A}=(A_t{)}_{t\in T}$ and $\tilde{B}=(B_t{)}_{t\in T}$ are continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function $f$ and the non-commutative $f$-divergence functional. In particular, an operator extension of Csiszár's result regarding $f$-divergence functional is presented. As some applications, we establish a refinement of the Choi--Davis--Jensen operator inequality, obtain some unitarily invariant norm inequalities and give some results related to the Kullback--Leibler distance.

22 pages, to appear in Math. Nachr

References in corpus (1)

• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities