Uhlmann’s theorem for relative entropies

Uhlmann’s theorem states that, for any two quantum states ρ AB and σ A , there exists an extension σ AB of σ A such that the fidelity between ρ AB and σ AB equals the fidelity between their reduced states ρ A and σ A . In this work, we generalize Uhlmann’s theorem to α-R´enyi relative entropies for α ∈ [1/2, ∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α = 1/2, α = 1, and α = ∞, respectively.