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.
Conditional Entanglement
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Based on the ideas of {\it quantum extension} and {\it quantum conditioning}, we propose a generic approach to construct a new kind of entanglement measures called {\it conditional entanglement}. The new measures, built from the known entanglement measures, are convex, automatically {\it super-additive}, and even smaller than the regularized versions of the generating measures. More importantly, new measures can also be built directly from measures of correlations, enabling us to introduce an {\it additive} measure and generalize it to a multipartite entanglement measure.
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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.