Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators (U_{A}in mathrm {U}(mathcal {H}_{A}), U_{B}in mathrm {U}(mathcal {H}_{B})) , and (U_{AB}in mathrm {U}(mathcal {H}_{A}otimes mathcal {H}_{B})) such that $$mathrm{S}left(U_{AB}rho_{AB}U^{dagger}_{AB}||sigma_{AB}right)geqslant mathrm{S}left(U_{A}rho_{A}U^{dagger}_{A}||sigma_{A}right) + mathrm{S}left(U_{B}rho_{B}U^{dagger}_{B}||sigma_{B}right), $$ where the reference state σ 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.