A remainder term for H\"older's inequality for matrices and quantum entropy inequalities

TL;DR

This paper establishes a sharp remainder term for H"older's inequality in matrix traces, leading to new bounds for quantum Renyi entropy and a refined quantum Pinsker inequality based on uniform convexity.

Contribution

It introduces a novel sharp remainder term for H"older's inequality for matrices and derives improved quantum entropy inequalities from it.

Findings

Sharp remainder term for H"older's inequality for matrices

New Pinsker type bounds for quantum Renyi entropy

Refined quantum Pinsker inequality from uniform convexity

Abstract

We prove a sharp remainder term for H\"older's inequality for traces as a consequence of the uniform convexity properties of the Schatten trace norms. We then show how this implies a novel family of Pinsker type bounds for the quantum Renyi entropy. Finally, we show how the sharp form of the usual quantum Pinsker inequality for relative entropy may be obtained as a fairly direct consequence of uniform convexity.