Non-commutative f-divergence functional

TL;DR

This paper introduces a new non-commutative $f$-divergence functional for operator convex functions, explores its properties, and applies it to refine inequalities and analyze distances like the Kullback-Leibler divergence.

Contribution

It defines a novel non-commutative $f$-divergence functional and establishes its properties, relations to perspectives, and applications to operator inequalities and divergence measures.

Findings

Established properties of the non-commutative $f$-divergence functional.

Provided an operator extension of Csiszár's $f$-divergence result.

Refined the Choi--Davis--Jensen inequality and derived norm inequalities.

Abstract

We introduce the non-commutative $f$-divergence functional $\Theta(\widetilde{A},\widetilde{B}):=\int_TB_t^{\frac{1}{2}}f\left(B_t^{-\frac{1}{2}} A_tB_t^{-\frac{1}{2}}\right)B_t^{\frac{1}{2}}d\mu(t)$ for an operator convex function $f$, where $\widetilde{A}=(A_t)_{t\in T}$ and $\widetilde{B}=(B_t)_{t\in T}$ 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 Csisz\'{a}r'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.