Uhlmann’s theorem states that, for any two quantum states ρABand σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their reduced states ρAand σA. 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.