A new operator extension of strong subadditivity of quantum entropy

TL;DR

This paper introduces a new operator inequality related to the strong subadditivity of quantum entropy, extending weak monotonicity and its Rényi generalizations, with implications for quantum information theory.

Contribution

It presents a novel operator inequality that generalizes weak monotonicity and strong subadditivity of quantum entropy, including Rényi entropy extensions.

Findings

Derived a new operator inequality reducing to weak monotonicity

Extended the inequality to two independent density matrices

Provided Rényi-generalizations of the inequality

Abstract

Let $S(\rho)$ be the von Neumann entropy of a density matrix $\rho$. Weak monotonicity asserts that $S(\rho_{AB}) - S(\rho_A) + S(\rho_{BC}) - S(\rho_C)\geq 0$ for any tripartite density matrix $\rho_{ABC}$, a fact that is equivalent to the strong subadditivity of entropy. We prove an operator inequality, which, upon taking an expectation value with respect to the state $\rho_{ABC}$, reduces to the weak monotonicity inequality. Generalizations of this inequality to the one involving two independent density matrices, as well as their R\'enyi-generalizations, are also presented.