Asymptotic quantification of entanglement with a single copy

Despite the central importance of quantum entanglement in quantum technologies, the understanding of 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 entanglement detection in the many-copy regime, and entanglement distillation, 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 generalised quantum Sanov’s theorem, enabling the exact evaluation of 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.

💡 Research Summary

The paper tackles two cornerstone problems in quantum information theory—entanglement testing and entanglement distillation—by shifting the usual performance metrics and by working within the broad class of non‑entangling operations. Traditionally, entanglement testing has been studied through the type‑II error exponent (the Stein exponent), while distillation has been evaluated by the asymptotic yield under LOCC. Both approaches lead to regularised formulas that require limits over many copies of the state, making them analytically intractable for most states.

The authors instead focus on the type‑I error exponent (the Sanov exponent) for testing, i.e., the exponential rate at which the probability of falsely declaring an entangled source as separable decays when many copies are available. For distillation they replace the yield with the error exponent that quantifies how fast the failure probability of a protocol can be driven to zero under non‑entangling operations. This mirrors the hypothesis‑testing viewpoint where the exponent of the error probability is the natural figure of merit.

The central technical contribution is a generalized quantum Sanov theorem. By treating the hypothesis test as a composite one—null hypothesis being the whole set of separable states and alternative hypothesis the n‑fold tensor product of the target state—the theorem shows that the optimal asymptotic type‑I error exponent equals the reverse relative entropy of entanglement: \