Abstract
Let \(S(\rho )\) be the von Neumann entropy of a density matrix \(\rho \). Weak monotonicity asserts that \(S(\rho _{AB}) - S(\rho _A) + S(\rho _{BC}) - S(\rho _C)\ge 0\) 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 Rényi-generalizations, are also presented.
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Acknowledgements
IK thanks Mark Wilde for helpful discussions. MHH thanks Marco Tomamichel for helpful discussions. TCL thanks John McGreevy, Bowen Shi, and Xiang Li for helpful discussions. We thank Geoff Penington for pointing out Ref. [17] and Andreas Winter for helpful comments. We thank the anonymous reviewers for their helpful comments and corrections to the citations.
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Lin, TC., Kim, I.H. & Hsieh, MH. A new operator extension of strong subadditivity of quantum entropy. Lett Math Phys 113, 68 (2023). https://doi.org/10.1007/s11005-023-01688-6
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DOI: https://doi.org/10.1007/s11005-023-01688-6