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arXiv · 1305.2023

Quantum marginal inequalities and the conjectured entropic inequalities

Lin Zhang·Hongjin He·Yuan-hong Tao
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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.

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Lin Zhang, Hongjin He, Yuan-hong Tao. 2013-05-09. Quantum marginal inequalities and the conjectured entropic inequalities. https://doi.org/10.1007/s10773-014-2093-x

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