In this manuscript, we establish a unified framework for analyzing the conditional error exponents of probabilistic resource distillation under approximately resource-nongenerating instruments. For generic quantum resource theories satisfying suitable structural conditions, we derive general oneshot bounds on the conditional distillation error exponents. By relating probabilistic distillation to postselected composite quantum hypothesis testing, we obtain bounds of the conditional error exponents for coherence distillation under finite blocklength and asymptotic zero-rate scenarios, we furthermore obtain analytical characterizations of the conditional error exponents for entanglement and magic distillation under finite blocklength and asymptotic zero-rate scenarios. For several representative families of states in entanglement and magic resource theories, these characterizations reduce to explicit closed-form formulas. Comparing them with the corresponding deterministic distillation exponents, we identify regimes in which postselection yields a strict improvement in the exponential decay rate of the conditional error. Our results reveal an operational advantage of postselection in quantum resource distillation and establish postselected composite hypothesis testing as a general tool for characterizing probabilistic resource-processing tasks.
In this manuscript, we establish a unified framework for analyzing the conditional error exponents of probabilistic resource distillation under approximately resource-nongenerating instruments. For generic quantum resource theories satisfying suitable structural conditions, we derive general oneshot bounds on the conditional distillation error exponents. By relating probabilistic distillation to postselected composite quantum hypothesis testing, we obtain bounds of the conditional error exponents for coherence distillation under finite blocklength and asymptotic zero-rate scenarios, we furthermore obtain analytical characterizations of the conditional error exponents for entanglement and magic distillation under finite blocklength and asymptotic zero-rate scenarios. For several representative families of states in entanglement and magic resource theories, these characterizations reduce to explicit closed-form formulas. Comparing them with the corresponding deterministic distillation exponents, we identify regimes in which postselection yields a strict improvement in the exponential decay rate of the conditional error. Our results reveal an operational advantage of postselection in quantum resource distillation and establish postselected composite hypothesis testing as a general tool for characterizing probabilistic resource-processing tasks.
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.
Abstract Probabilistic protocols can enlarge the set of feasible resource transformations by conditioning on successful outcomes, but in many practical setting there are fundamental limitations on the success probability and the error incurred in such transformation. We derive complementary one-shot converse bounds and their regularized asymptotic rate bounds. Successful outcomes are represented by branches of free superinstruments. While projective robustness yields an accuracy constraint independent of the success probability, generalized robustness directly bounds the probability itself. In particular, we establish the explicit relationship between the success probability and the error of the protocol using robustness measure. For a flagged erasure channel, postselection recovers the identity channel exactly. In this case, generalized robustness remains finite although projective robustness is infinite. Restricting the framework to preparation channels recovers state distillation and gives explicit bounds for coherence distillation from noisy coherent states. Together, these examples distinguish channel-level behavior from the state-theoretic case obtained by restricting the framework to preparation channels.
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 ......
The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over deterministic ones, significantly strengthening previous findings and extending recent no-go theorems. We apply our results to obtain a substantial improvement in bounds for the errors and overheads of probabilistic distillation protocols, directly applicable to tasks such as entanglement or magic state distillation, and computable through convex optimization. In broad classes of resources, we strengthen our results to show that the monotone completely governs probabilistic transformations-it serves as a necessary and sufficient condition for state convertibility. This endows the monotone with a direct operational interpretation, as it can exactly quantify the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol.
We develop two general approaches to characterising the manipulation of quantum states by means of probabilistic protocols constrained by the limitations of some quantum resource theory. First, we give a general necessary condition for the existence of a physical transformation between quantum states, obtained using a recently introduced resource monotone based on the Hilbert projective metric. In all affine quantum resource theories (e.g. coherence, asymmetry, imaginarity) as well as in entanglement distillation, we show that the monotone provides a necessary and sufficient condition for one-shot resource convertibility under resource-non-generating operations, and hence no better restrictions on all probabilistic protocols are possible. We use the monotone to establish improved bounds on the performance of both one-shot and many-copy probabilistic resource distillation protocols. Complementing this approach, we introduce a general method for bounding achievable probabilities in resource transformations under resource-non-generating maps through a family of convex optimisation problems. We show it to tightly characterise single-shot probabilistic distillation in broad types of resource theories, allowing an exact analysis of the trade-offs between the probabilities and errors in distilling maximally resourceful states. We demonstrate the usefulness of both of our approaches in the study of quantum entanglement distillation.
The overhead exponent -- characterizing the scaling of the number of noisy magic states with respect to the target distillation error -- has been a central quantity to benchmark magic state distillation protocols. On the other hand, a related but less investigated quantity motivated by an information-theoretic viewpoint is the asymptotic distillation rate, the largest ratio of output to input magic states such that error vanishes asymptotically. These two quantities are tightly related in the specific case -- the overhead exponent is zero if and only if the asymptotic distillation rate is linear. However, their relationship in other regimes has been unclear. Here, we show that their quantitative relation is generally not robust, by presenting a family of magic state distillation protocols with an overhead exponent not close to zero -- in fact, larger than one -- that still achieves the asymptotic rate arbitrarily close to the linear rate. This implies that the distillation rate is not constrained by the overhead exponent within the sublinear rate regime. Notably, our protocol is based on error checking by measurements of logical Clifford operators, which underlies the recent magic state cultivation protocol, suggesting the potential of this mechanism for asymptotic magic state distillation.