No-go theorems for quantum resource purification II: new approach and channel theory

TL;DR

This paper introduces a new method to analyze fundamental limits of quantum resource purification, providing tighter bounds and a robust no-purification theory for quantum channels, with implications for quantum error correction and circuit synthesis.

Contribution

The paper develops a novel approach for bounding quantum resource purification, extending the theory to quantum channels and improving existing limits on purification accuracy and efficiency.

Findings

Derived universal bounds on channel transformation error and cost.

Established a robust no-purification theory for quantum channels.

Connected results to quantum error correction and quantum Shannon theory.

Abstract

It has been recently shown that there exist universal fundamental limits to the accuracy and efficiency of the transformation from noisy resource states to pure ones (e.g.,~distillation) in any well-behaved quantum resource theory [Fang/Liu, Phys. Rev. Lett. 125, 060405 (2020)]. Here, we develop a novel and powerful method for analyzing the limitations on quantum resource purification, which not only leads to improved bounds that rule out exact purification for a broader range of noisy states and are tight in certain cases, but also enable us to establish a robust no-purification theory for quantum channel (dynamical) resources. More specifically, we employ the new method to derive universal bounds on the error and cost of transforming generic noisy channels (where multiple instances can be used adaptively, in contrast to the state theory) to some unitary resource channel under any free channel-to-channel map. We address several cases of practical interest in more concrete terms, and discuss the connections and applications of our general results to distillation, quantum error correction, quantum Shannon theory, and quantum circuit synthesis.