Probabilistic transformations of quantum resources

TL;DR

This paper introduces a new resource monotone for quantum resource theories that can determine the feasibility of probabilistic and deterministic state transformations, leading to improved bounds and operational insights.

Contribution

It develops a universal resource monotone that fully characterizes probabilistic transformations and enhances understanding of quantum resource manipulation limitations.

Findings

The monotone rules out all transformations between states in quantum resource theories.

It provides tighter bounds for errors and overheads in probabilistic distillation protocols.

The monotone offers a necessary and sufficient condition for state convertibility in broad resource classes.

Abstract

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 -- its monotonicity provides 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.