Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols

TL;DR

This paper demonstrates that allowing probabilistic protocols in quantum state transformations surpasses traditional entropic limits, introducing a new entropic measure and establishing tight bounds for resource distillation.

Contribution

It introduces a new entropic quantity based on the Hilbert projective metric for probabilistic quantum state transformations, extending the understanding beyond deterministic protocols.

Findings

Probabilistic protocols can achieve higher transformation rates than relative entropy limits.

A new asymptotic equipartition property for the projective relative entropy is established.

Strong converse bounds for probabilistic distillation are derived and shown to be tight.

Abstract

The quantum relative entropy is known to play a key role in determining the asymptotic convertibility of quantum states in general resource-theoretic settings, often constituting the unique monotone that is relevant in the asymptotic regime. We show that this is no longer the case when one allows stochastic protocols that may only succeed with some probability, in which case the quantum relative entropy is insufficient to characterize the rates of asymptotic state transformations, and a new entropic quantity based on a regularization of the Hilbert projective metric comes into play. Such a scenario is motivated by a setting where the cost associated with transformations of quantum states, typically taken to be the number of copies of a given state, is instead identified with the size of the quantum memory needed to realize the protocol. Our approach allows for constructing transformation protocols that achieve strictly higher rates than those imposed by the relative entropy. Focusing on the task of resource distillation, we give broadly applicable strong converse bounds on the asymptotic rates of probabilistic distillation protocols, and show them to be tight in relevant settings such as entanglement distillation with non-entangling operations. This generalizes and extends previously known limitations that only applied to deterministic protocols. Our methods are based on recent results for probabilistic one-shot transformations as well as a new asymptotic equipartition property for the projective relative entropy.