Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces

TL;DR

This paper introduces a unified framework for defining and analyzing monotone Riemannian metrics, relative entropy, and geodesic distances on non-commutative probability spaces using the relative modular operator, with key properties like contractivity under quantum operations.

Contribution

It provides a novel classification of convex operator functions into subsets that uniquely determine various quantum information geometric quantities.

Findings

Explicit formulas for special cases including logarithmic relative entropy

Demonstration of contractivity of these quantities under quantum channels

Introduction of a maximal contraction measure for these quantities

Abstract

We use the relative modular operator to define a generalized relative entropy for any convex operator function g on the positive real line satisfying g(1) = 0. We show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy, a unique family (parameterized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including the familiar logarithmic relative entropy. The relative entropies, Riemannian metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.