Infinitely many constrained inequalities for the von Neumann entropy
Josh Cadney, Noah Linden, Andreas Winter
TL;DR
The paper extends Makarychev et al.'s constrained Shannon-type entropy inequalities to the quantum von Neumann setting by proving an infinite family of inequalities that hold under zero conditional mutual information constraints I(A:C|B)=I(B:C|A)=0. It provides a detailed proof using the HJPW decomposition, a local A-measurement that introduces a classical register, and a Makarychev-style analysis on an augmented state, then transfers the result back to the original state. Crucially, the authors establish independence of these inequalities from each other and from the basic inequalities, and also demonstrate related variants and 4-party special cases. They discuss the implications for the quantum entropy cone, the potential universality of balanced inequalities across classical and quantum domains, and open questions about unconstrained inequalities, supported by numerical checks in small dimensions.
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.