Why quantum relative entropy is unique - Gilad Gour

Relative entropy is one of the central quantities in quantum information theory, appearing in distinguishability, thermodynamics, resource theories, and many-body physics. But why should this particular divergence be privileged among the many quantum divergences that satisfy data processing? I will describe a new operational route to this question based on binary guessing games. Given two quantum states, one can compare how useful they are for distinguishing hypotheses under all possible prior probabilities; this comparison gives a Lorenz-geometric order on pairs of states. I will explain how this order leads to a natural class of quantum Lorenz divergences, and how additivity singles out the Umegaki relative entropy among them. The result is genuinely quantum: in the classical case, additive monotones form mixtures of Rényi divergences, while quantum noncommutativity collapses this freedom to the ordinary relative entropy. Time permitting, I will also discuss connections with sharp dimension-independent bounds for smoothed quantum divergences.