Infinitely many constrained inequalities for the von Neumann entropy

TL;DR

This paper introduces infinitely many new constrained inequalities for the von Neumann entropy, extending classical Shannon entropy inequalities to the quantum domain and establishing their independence from known inequalities.

Contribution

It extends classical entropy inequalities to the quantum setting, proving their independence and providing an infinite family of new inequalities for von Neumann entropy.

Findings

Infinite new inequalities for von Neumann entropy

Independence of these inequalities from known ones

Extension of classical Shannon entropy inequalities to quantum domain

Abstract

We exhibit infinitely many new, constrained inequalities for the von Neumann entropy, and show that they are independent of each other and the known inequalities obeyed by the von Neumann entropy (basically strong subadditivity). The new inequalities were proved originally by Makarychev et al. [Commun. Inf. Syst., 2(2):147-166, 2002] for the Shannon entropy, using properties of probability distributions. Our approach extends the proof of the inequalities to the quantum domain, and includes their independence for the quantum and also the classical cases.