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. Here, we develop two general approaches to this problem. First, we introduce a new resource monotone based on the Hilbert projective metric and we show that it 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 lower 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 show that no better restrictions on probabilistic protocols are possible -- our monotone can provide a necessary and sufficient condition for probabilistic resource transformations, thus allowing us to quantify exactly the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol. Complementing this approach, we introduce a general method for bounding achievable probabilities in resource transformations through a family of convex optimization problems. We show it to tightly characterize 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.
تحميل البحث