Monotonicity versions of Epstein's Concavity Theorem and related inequalities

TL;DR

This paper establishes new monotonicity theorems related to Epstein's concavity theorem, enhancing understanding of trace inequalities and their physical applications, with implications for quantum information theory.

Contribution

It introduces novel monotonicity results that extend classical concavity theorems, providing new proofs and duality-based approaches.

Findings

Proved a new monotonicity theorem as a corollary to Epstein's concavity.

Extended Lieb Concavity and Lieb Convexity theorems with monotonicity versions.

Provided multiple proofs using interpolation and duality arguments.

Abstract

Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.