We discuss the asymptotic behavior of conversions between two independent andidentical distributions up to the second-order conversion rate when theconversion is produced by a deterministic function from the input probabilityspace to the output probability space. To derive the second-order conversionrate, we introduce new probability distributions named Rayleigh-normaldistributions. The family of Rayleigh-normal distributions includes a Rayleighdistribution and coincides with the standard normal distribution in the limitcase. Using this family of probability distributions, we represent theasymptotic second-order rates for the distribution conversion. As anapplication, we also consider the asymptotic behavior of conversions betweenthe multiple copies of two pure entangled states in quantum systems when onlylocal operations and classical communications (LOCC) are allowed. This problemcontains entanglement concentration, entanglement dilution and a kind ofcloning problem with LOCC restriction as special cases.
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