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

Entanglement distillation is a fundamental task in quantum entanglement theory. 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. This work considers the error exponents of probabilistic entanglement distillation under the finite block length scenario and zero rate scenario when the operational model is $δ$-approximately nonentangling or $δ$-approximately dually nonentangling quantum instruments. 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 $δ$-(dually) approximately nonentangling quantum instruments.