Entropic and operational characterizations of dynamic quantum resources

Abstract: We provide new methods for characterizing general closed and convex quantum resource theories, including dynamic ones, based on entropic concepts and operational tasks. We propose a resource-theoretic generalization of the quantum conditional min-entropy, termed the free conditional min-entropy (FCME), in the sense that it quantifies an observer's ``subjective'' degree of uncertainty about a quantum system given that the observer's information processing is limited to free operations of the resource theory. This generalized concept gives rise to a complete set of entropic conditions for free convertibility between quantum states or channels in any closed and convex quantum resource theory. It also provides an information-theoretic interpretation for the resource global robustness of a state or a channel in terms of a mutual-information-like quantity derived from the FCME. Apart from this entropic approach, we also characterize dynamic resources by analyzing their performance in operational tasks. Based on such tasks, we construct operationally meaningful and complete sets of resource monotones, which enable faithful tests of free convertibility between quantum channels. Finally, we show that every well-defined robustness-based measure of a channel can be interpreted as an operational advantage of the channel over free channels in a communication task.