Asymptotic quantification of entanglement with a single copy

Despite the central importance of quantum entanglement in quantum technologies, understanding the optimal ways to exploit it is still beyond our reach, and even measuring entanglement in an operationally meaningful way is prohibitively difficult. Here we study two fundamental tasks in the processing of entanglement: entanglement testing, which is a quantum state discrimination problem concerned with detecting entanglement in the many-copy regime, and entanglement distillation, which is concerned with purifying entanglement from noisy entangled states. We introduce a way of benchmarking the performance of distillation that focuses on the best achievable error rather than its yield in the asymptotic limit. When the underlying set of operations used for entanglement distillation is the axiomatic class of non-entangling operations, we show that the two figures of merit for entanglement testing and distillation coincide. We solve both problems by proving a generalized quantum Sanov's theorem, which enables the exact evaluation of the asymptotic error rates of composite quantum hypothesis testing. We show in particular that the asymptotic figure of merit is given by the reverse relative entropy of entanglement, a single-letter quantity that can be evaluated using only a single copy of a quantum state-a distinct feature among measures of entanglement that quantify the optimal performance of information-theoretic tasks.