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.
τ σ − 1 2 1− 1−α z +it 0 χα,z(ρ, σ0)σ − 1 2 1− 1−α z −it 0 # =Q α,z(ρ∥σ0).(109) Proof.We derive the equivalence between (108) and sup t∈R,τ∈F 1 Tr
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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.