Regularization of the Umegaki relative entropy over polar-closed convex families is unnecessary if and only if a single-copy optimizer satisfies a phase-averaged supremum condition equal to 1.
α-z-R´ enyi relative entropies
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Additivity of quantum relative entropies as a single-copy criterion
Regularization of the Umegaki relative entropy over polar-closed convex families is unnecessary if and only if a single-copy optimizer satisfies a phase-averaged supremum condition equal to 1.