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.
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.
Quantum resource distillation converts imperfect resource states into highly resourceful targets and is a central task in quantum information processing. When distillation is probabilistic, unfavorable outcomes may be discarded and the quality of the output is evaluated conditioned on success, leading to a notion of reliability that differs from the deterministic setting. Here we study the conditional error exponents of probabilistic quantum resource distillation under approximately resource-nongenerating instruments and derive its general connection to the postselected composite quantum hypothesis testing against the corresponding set of free states. This reduction leads naturally to scale-invariant projective distinguishability measures and provides finite-blocklength bounds for a broad class of resource theories. We apply the framework to entanglement, coherence, and magic. For entanglement and magic, we obtain exact zero-rate asymptotic characterizations of the conditional error exponent in terms of regularized projective distinguishability, while for coherence we derive general finite-blocklength and asymptotic bounds. For representative entangled and magic states, the resulting quantities can be evaluated analytically and reveal regimes in which postselection yields a strictly larger conditional error exponent than deterministic distillation. Furthermore, we also show that the conditional error exponents for probabilistic entanglement distillation under a strictly positive success probability constraint are smaller than those with no constraints on the success probability.
Historically, entanglement and coherence distillation focused on the distillable rate under full knowledge of the initial state. In this paper, we develop a unified finite-blocklength and asymptotic theory for universal distillation from convex compact sequences of possible states stable under tensor products. We first resolve an open question by showing that two regularized variants of the generalized quantum Hoeffding divergence need not coincide, and establish fundamental properties of the Hoeffding divergence and anti-divergence. For universal entanglement distillation, we connect finite-blocklength performance under non-entangling, PPT-preserving, and dually free operations to composite correlated hypothesis testing, yielding reliability functions for non-entangling and PPT-preserving operations characterized by regularized quantum Hoeffding divergences and strong converse bounds via the anti-divergence. We recover the entanglement concentration error exponent, zero-rate exponents under a continuity condition, and distillable entanglements, and characterize the dually free cases by regularized relative entropies under restricted measurements. Additionally, five concrete noise models justify the black-box assumptions for both resources. For coherence distillation under maximally incoherent and dephasing-covariant incoherent operations, we extend the one-shot semidefinite program characterization to universal inputs; relating a relaxed optimization to hypothesis testing yields the high-rate reliability function. A coherent-seed approach extends the single-state formula to the entire direct regime when the zero-error rate is positive. Finally, we completely characterize the states with positive zero-error coherence distillation rate, show that the rate equals the order-zero Petz Rényi relative entropy of coherence, and derive dimension-independent one-shot bounds ......
Computing the exact rate at which entanglement can be distilled from noisy quantum states is one of the longest-standing questions in quantum information. We give an exact solution for entanglement distillation under the set of dually non-entangling (DNE) operations—a relaxation of the typically considered local operations and classical communication, comprising all channels which preserve the sets of separable states and measurements. We show that the DNE distillable entanglement coincides with a modified version of the regularised relative entropy of entanglement in which the arguments are measured with a separable measurement. Ours is only the second known regularised formula for the distillable entanglement under any class of free operations in entanglement theory, after that given by Devetak and Winter for (one-way) local operations and classical communication. An immediate consequence of our finding is that, under DNE, entanglement can be distilled from any entangled state. As our second main result, we construct a general upper bound on the DNE distillable entanglement, using which we prove that the separably measured relative entropy of entanglement can be strictly smaller than the regularisation of the standard relative entropy of entanglement, solving an open problem posed by Li and Winter. Finally, we study also the reverse task of entanglement dilution and show that the restriction to DNE operations does not change the entanglement cost when compared with the larger class of non-entangling operations. This implies a strong form of irreversiblility of entanglement theory under DNE operations: even when asymptotically vanishing amounts of entanglement may be generated, entangled states cannot be converted reversibly. Computing how much entanglement can be extracted from imperfect quantum states is a long-standing problem in quantum resource theory. Here, Lami and Regula move away from the usual case in which the operations considered free to implement are local operations and classical communication, and find instead a solution for the case of dually non-entangling operations.
We recently proposed an entanglement distillation protocol based on local operations and classical communication that probabilistically and universally distills a maximally entangled Bell pair from multiple copies of an unknown two-qubit state [Phys. Rev. Lett. 135, 260202 (2025)]. Here, we investigate the robustness of this protocol against imperfections in its implementation. We consider several incoherent and coherent gate-error models, as well as measurement readout errors, and determine the regimes in which distillation enhances the fidelity with the target Bell state. We find that the protocol remains effective over substantial error ranges, although its robustness depends strongly on the type of error. We further analyze asymmetric errors between the two parties and identify conditions under which complementary postselection outcomes yield identical distillation dynamics, allowing the success probability to be enhanced.
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.