Sample complexity of quantum resource testing via one-shot quantum blurring

2026-07-27

AI Summary

This work addresses the lack of finite-sample performance guarantees in quantum resource detection by proposing an analytical framework based on one-shot quantum hypothesis testing. It establishes, for the first time, a rigorous upper bound on the number of samples required to distinguish resourceful states from free states. By integrating the generalized quantum Stein’s lemma, Rényi relative entropies, and quantum resource theory, the study proves the convergence of the regularized Rényi relative entropy and resolves an open problem posed by Fang and Hayashi. The main contribution is the derivation of the sample complexity for asymmetric hypothesis testing, given by \( n = O\left(\frac{\log(1/\delta)}{D^{\infty}(\rho\| \mathcal{F})}\right) \), which precisely quantifies the number of copies of a quantum state needed to achieve a false-negative error probability no greater than \(\delta\).

Abstract

Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate $n$ copies of a given resourceful state $ρ$ from all free (i.e., resourceless) states; key instances for applications are entanglement testing and quantum magic testing. The asymptotic characterisation relies on the recently proven generalised quantum Stein's lemma, which establishes the rate of decay of the false negative error probability for a fixed false positive error probability. This result, however, is intrinsically asymptotic and thus can provide no finite-resource guarantees, which makes its practical implications unclear. Here, we establish the first rigorous finite-$n$ bounds on quantum resource testing and hence quantum resource manipulation, providing explicit estimates on the number of copies needed to achieve a prescribed performance. As notable consequences, we obtain (a) the convergence of the regularised Rényi relative entropies of a resource, which settles the important open problem from [Fang/Hayashi, IEEE ToIT 72:6, 2026]; and (b) the first sample-complexity bound for asymmetric resource testing: for any fixed false positive error probability, a false negative error probability of at most $δ$ can be achieved with $n=O\left(\frac{\log(1/δ)}{D^\infty(ρ\|F)}\right)$ copies of $ρ$, in the limit where $δ\to 0$.

Problem

• quantum resource testing
• sample complexity
• finite-sample bounds
• quantum information processing
• asymmetric hypothesis testing

Innovation

• quantum resource testing
• sample complexity
• finite-n bounds
• Rényi relative entropy
• quantum Stein's lemma