Quantum marginal inequalities and the conjectured entropic inequalities

TL;DR

This paper investigates a conjectured quantum entropy inequality, providing numerical evidence that it holds for all qubit pairs, thus contributing to understanding quantum entropic inequalities.

Contribution

It offers the first numerical study supporting the modified super-additivity inequality of relative entropy in quantum systems.

Findings

The inequality appears to hold for all tested qubit pairs.

Numerical results support the conjecture's validity in low-dimensional systems.

Provides evidence for the conjecture's potential general applicability.

Abstract

A conjecture -- \emph{the modified super-additivity inequality} of relative entropy -- was proposed in \cite{Zhang2012}: There exist three unitary operators $U_A\in \unitary{\cH_A},U_B\in \unitary{\cH_B}$, and $U_{AB}\in \unitary{\cH_A\ot \cH_B}$ such that $$ \rS(U_{AB}\rho_{AB}U^\dagger_{AB}||\sigma_{AB}) \geqslant \rS(U_A\rho_AU^\dagger_A||\sigma_A) + \rS(U_B\rho_BU^\dagger_B||\sigma_B), $$ where the reference state $\sigma$ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.