Uhlmann's theorem states that, for any two quantum states ρAB and σA, there exists an extension σAB of σA such that the fidelity between ρAB and σAB equals the fidelity between their reduced states ρA and σA. In this work, we generalize Uhlmann's theorem to α-Rényi relative entropies for α∈[12,∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α=12, α=1, and α=∞, respectively. Based on joint work with Giulia Mazzola and Renato Renner.