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Probabilistic Entanglement Distillation: Error Exponents via Postselected Quantum Hypothesis Testing Against Separable States

Entanglement distillation and entanglement cost are fundamental tasks in quantum entanglement theory. This work studies both in the probabilistic setting and focuses on the asymptotic error exponent of probabilistic entanglement distillation when the operational model is $δ$-approximately nonentangling or $δ$-approximately dually nonentangling quantum instruments. While recent progress has clarified limitations of probabilistic transformations in general resource theories, an analytic formula for the error exponent of probabilistic entanglement distillation under approximately (dually) nonentangling operations has remained unavailable. Building on the framework of postselected quantum hypothesis testing, we establish a direct connection between probabilistic distillation and postselected hypothesis testing against the set of separable states. In particular, we derive an analytical characterization of the distillation error exponent under $δ$-approximately nonentangling quantum instruments. Besides, we relate the exponent to postselected hypothesis testing with measurements restricted to be separable. We further investigate probabilistic entanglement dilution and establish a relation between probabilistic entanglement costs under approximately nonentangling and approximately dually nonentangling instruments, together with a bound on the probabilistic entanglement cost under nonentangling instruments.