Tight constraints on probabilistic convertibility of quantum states

TL;DR

This paper introduces new theoretical methods to precisely characterize the limits of probabilistic quantum state transformations within resource theories, providing tight bounds and necessary conditions for resource distillation.

Contribution

It develops a resource monotone based on the Hilbert projective metric and a convex optimization framework to tightly bound probabilistic quantum state transformations, advancing understanding in resource theories.

Findings

The resource monotone provides necessary and sufficient conditions for one-shot convertibility.

The methods establish tight bounds on probabilistic resource distillation performance.

Application to entanglement distillation demonstrates practical relevance.

Abstract

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.