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arXiv:2607.24712

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

Dmitry Grinko, Ludovico Lami

quant-phcs.ITmath-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations
v1 · Jul 27, 2026v2 · Aug 19, 2026v3 · Oct 30, 2026
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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 $δ\to 0$.

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Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate nnn copies of a given resourceful state ρρρ 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-nnn 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 δδδ can be achieved with n=O(log⁡(1/δ)D∞(ρ∥F))n=O\left(\frac{\log(1/δ)}{D^\infty(ρ\|F)}\right)n=O(D∞(ρ∥F)log(1/δ)​) copies of ρρρ, in the limit where δ→0δ\to 0δ→0.
The performance of quantum resource manipulation protocols, including key examples such as distillation of quantum entanglement, is measured in terms of the rate at which desired target states can be produced from a given noisy state. However, to achieve optimal rates, known protocols require precise tailoring to the quantum state in question, demanding a perfect knowledge of the input and allowing no errors in its preparation. Here we show that distillation of quantum resources in the framework of resource non-generating operations can be performed universally: optimal rates of distillation can be achieved with no knowledge of the input state whatsoever, certifying the robustness of quantum resource distillation. The findings apply in particular to the purification of quantum entanglement under non-entangling maps, where the optimal rates are governed by the regularised relative entropy of entanglement. Our result relies on an extension of the generalised quantum Stein's lemma in quantum hypothesis testing to a composite setting where the null hypothesis is no longer a fixed quantum state, but is rather composed of i.i.d. copies of an unknown state. The solution of this asymptotic problem is made possible through new developments in one-shot quantum information and a refinement of the blurring technique from [Lami, arXiv:2408.06410].
The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.
We solve the generalised quantum Stein's lemma, proving that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing betweennnncopies of an entangled stateρAB\rho_{A B}ρAB​and a generic separable stateσAn:Bn\sigma_{A^{n}: B^{n}}σAn:Bn​, equals the regularised relative entropy of entanglement. Not only does this determine the ultimate performance of entanglement testing, but it also establishes the reversibility of all quantum resource theories under asymptotically resource non-generating operations, with the regularised relative entropy of resource governing the asymptotic transformation rate between any two quantum states. To solve the problem we introduce two techniques. The first is a procedure that we call ‘blurring’, which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. Blurring alone suffices to prove the generalised Stein's lemma in the fully classical case, but not in the quantum case. Our second technical innovation, therefore, is to perform a second quantisation step to lift the problem to an infinite-dimensional bosonic quantum system; we then solve it there by using techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel. A careful examination of this second quantisation step is the core of our quantum solution. Longer version [1] at [arXiv:2408.06410].
We solve the generalised quantum Stein’s lemma, proving that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing betweenn\boldsymbol {n}ncopies of an entangled stateρAB\boldsymbol {\rho _{AB}}ρAB​and a generic separable stateσAn:Bn\boldsymbol {\sigma _{A^{n}:B^{n}}}σAn:Bn​, equals the regularised relative entropy of entanglement. Not only does this determine the ultimate performance of entanglement testing, but it also establishes the reversibility of all quantum resource theories under asymptotically resource non-generating operations, with the regularised relative entropy of resource governing the asymptotic transformation rate between any two quantum states. As a by-product, we prove that the same Stein exponent can also be achieved when the null hypothesis is only approximately i.i.d., in the sense that it can be modelled by an ‘almost power state’. To solve the problem we introduce two techniques. The first is a procedure that we call ‘blurring’, which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. Blurring alone suffices to prove the generalised Stein’s lemma in the fully classical case, but not in the quantum case. Our second technical innovation, therefore, is to perform a second quantisation step to lift the problem to an infinite-dimensional bosonic quantum system; we then solve it there by using techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel. A careful examination of this second quantisation step is the core of our quantum solution.
A common quantum hypothesis testing task is to decide whether a given state is entangled or not. Given more copies, it is possible to exponentially reduce the chance of making an error. The generalized Stein’s lemma connects this rate of decay to the relative entropy of entanglement of this state. It extends the result in the i.i.d. case, which itself relies on the original classical result from Chernoff and Stein. After a mistake was found in its original proof, the generalized theorem was recently proven independently by Lami and Hayashi. The former developed two new techniques, namely the blurring map and the bosonic lifting procedure, while the latter made use of the pinching inequality. The goal of this report is to present the problem, and build intuition around these tools. After reviewing the proofs, we also consider some possible extensions of the result with the aim to find “one-shot” bounds, effectively bounding the rate of convergence. We finally compare the two rather different approaches from Lami and Hayashi and try to build connections between the tools used in both proofs.
We extend the recent proof of the Generalized Quantum Stein's Lemma by Hayashi and Yamasaki [arXiv:2408.02722] to classical-quantum (c-q) channels. We analyze the composite hypothesis testing problem of testing a c-q channel E⊗n\mathcal{E}^{\otimes n}E⊗n against a sequence of sets of c-q channels (Sn)n(\mathcal{S}_n)_n(Sn​)n​ (satisfying certain natural assumptions), under parallel strategies. We prove that the optimal asymptotic asymmetric error exponent is given by the regularization of Umegaki channel divergence, minimized over Sn\mathcal{S}_nSn​. This allows us to prove the reversibility of resource theories of classical-quantum channels in a natural framework, where the distance between channels (and hence also the notion of approximate interconvertibility of channels) is measured in diamond norm, and the set of free operations is the set of all asymptotically resource non-generating superchannels. The results we obtain are similar to the ones in the concurrent and independent work by Hayashi and Yamasaki [arXiv:2509.07271]. However the proof of the direct part of the GQSL uses different arguments and techniques to deal with the challenges that arise from dealing with c-q channels.