Resource theory of asymmetric distinguishability

We systematically develop the resource-theoretic perspective on distinguishability. The theory is a resource theory of asymmetric distinguishability, given that approximation is allowed for the first quantum state in general transformation tasks. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. We formally define the distillation and dilution tasks, and we find that the exact one-shot distillable distinguishability is equal to the min-relative entropy, the exact one-shot distinguishability cost is equal to the max-relative entropy, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. We also develop the resource theory of asymmetric distinguishability for quantum channels. For this setting, we prove that the exact distinguishability cost is equal to channel max-relative entropy and the distillable distinguishability is equal to the amortized channel relative entropy.