A Matrix Convexity Approach to Some Celebrated Quantum Inequalities

TL;DR

This paper introduces a matrix convexity framework to derive and extend fundamental quantum entropy inequalities, offering a unified algebraic approach that encompasses classical results and new inequalities.

Contribution

It develops a matrix perspective method for operator convex functions, providing a unified approach to quantum entropy inequalities and extending classical results like Lieb's inequality.

Findings

Derived quantum entropy inequalities from matrix perspectives

Extended Maréchal's perspectives to include new inequalities

Unified algebraic approach to quantum entropy inequalities

Abstract

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequality. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Mar\'{e}chal's extended perspectives provides additional inequalities, including a $p+q\leq 1$ result of Lieb.