No-Go Theorems for Quantum Resource Purification

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.

Figure 1

Interplay between the transformation error $\epsilon$ and success probability $p$. The lower right corner represents the most ideal scenario where $\epsilon$ is small and $p$ is large. The red region and solid lines represent the forbidden regime such that no purification protocol with the corresponding parameters can exist. $ϵ(\rho ,\psi )={\lambda }_{\min }(\rho )(1-f_{\psi })$.

Figure 2

A qubit coherence theory example illustrated using the Bloch sphere. Here $\rho$ is a mixed state close to the maximally coherent state $|+⟩$, and $\psi$ is a pure state close to the basis state $|1⟩$. Our no-go theorems indicate that an arbitrarily accurate probabilistic transformation from $\rho$ to $\psi$ is impossible.