No-Go Theorems for Quantum Resource Purification

TL;DR

This paper establishes fundamental quantum mechanical limits on the purification of noisy quantum resources, showing that perfect purification is impossible and providing bounds on resource distillation efficiency, especially for magic states in quantum computing.

Contribution

It proves universal no-go theorems that set lower bounds on resource purification, including the first explicit bounds for magic state distillation in quantum computation.

Findings

Perfect resource purification cannot be achieved, even probabilistically.

Universal lower bounds on the error of resource conversion protocols.

Explicit bounds on the resource cost of magic state distillation.

Abstract

The manipulation of quantum "resources" such as entanglement, coherence and magic states lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to "purify" the quantum resources, since they are inevitably contaminated by noise and thus often lose their power or become unreliable for direct usage. Here we prove fundamental limitations on how effectively generic noisy resources can be purified enforced by the laws of quantum mechanics, which universally apply to any reasonable kind of quantum resource. More explicitly, we derive nontrivial lower bounds on the error of converting any full-rank noisy state to any target pure resource state by any free protocol (including probabilistic ones)---it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widely used type of resource purification routine that underpins many key applications of quantum information science. In particular, this general result induces the first explicit lower bounds on the resource cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computation. Implications for the standard error-correction-based methods are specifically discussed.