One-shot and asymptotic classical capacity in general physical theories

With the recent development of quantum information theory, some attempts have been made to construct information theory beyond quantum theory. Here, we consider hypothesis-testing relative entropy and one-shot classical capacity, that is, the optimal rate of classical information transmitted by using a single channel under the constraint of a certain error probability, in general physical theories where states and measurements are operationally defined. Then we obtain the upper bound of the one-shot classical capacity by generalizing the method given by Wang and Renner Phys. Rev. Lett. 108, 200501 (2012). Also, we derive the lower bound of the capacity by showing the existence of a good code that can transmit classical information with a certain error probability. Applying the above two bounds, we prove the asymptotic equivalence between classical capacity and hypothesis-testing relative entropy in any general physical theory.