Random Number Conversion and LOCC Conversion via Restricted Storage

We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of first- and second- order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behaviour of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy.