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arXiv:2608.21321

Hypothesis testing between quantum ensembles

Jian Yao, Quntao Zhuang

quant-phSubmitted Aug 21, 2026• Announced Aug 24, 2026• 0 citations
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Abstract

Quantum state ensembles are important in quantum information processing. For example, quantum $t$-designs model highly entangled states in complex systems, while projected ensembles appear in generative quantum machine learning and studies of thermalization. With their sample state accompanied by a classical label, these ensembles contain operational information beyond their average density operators. Yet an ensemble differs from a classical-quantum state because it is invariant under permutations of labels. We formulate binary hypothesis testing between finite quantum ensembles and derive fundamental limits on error probability. Given an observed label pattern, we show that the joint sampled state can be described by power-weighted ensemble moments. This yields the Bayes-optimal measurement and exact finite-sample error, revealing that discrimination is governed by the full moment hierarchy up to the number of samples. In the many-sample limit, we derive Chernoff bounds and obtain exact error exponents for finite uniform pure-state ensembles. We apply these results to optical communication and $t$-designs. For finite uniform pure-state $t$-designs with large $t$, the maximal discrimination exponent scales sharply as $\sim t^{-2}$, while equal-prior fixed-error testing requires $\sim t^2$ samples.

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5+25 pages, 2 figures

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We consider sequential hypothesis testing among multiple samples of one amongMMMpure quantum states in an equidistant ensemble, i.e., those with identical pair-wise inner products. Each measurement in the sequence is a binary projective measurement that collapses the sample of the state measured at that instant into a linear span of a collection of states from the ensemble or its orthogonal complement. The algorithm adaptively decides if additional samples are needed or sufficient observation has been gathered. We show that our sequential measurement algorithm outperforms the sequential testing (ST) receiver, whose error-probability exponent is known to achieve the quantum Chernoff bound asymptotically in the limit of large (and fixed) number of samples. Even though our algorithm does not attain the quantum limit of minimum error probability (the Helstrom limit), it paves the way for future research on more advanced sequential quantum hypothesis tests, e.g., those that go beyond binary projective measurements on each sample.
We introduce sequential analysis in quantum information processing, by focusing on the fundamental task of quantum hypothesis testing. In particular, our goal is to discriminate between two arbitrary quantum states with a prescribed error threshold ε when copies of the states can be required on demand. We obtain ultimate lower bounds on the average number of copies needed to accomplish the task. We give a block-sampling strategy that allows us to achieve the lower bound for some classes of states. The bound is optimal in both the symmetric as well as the asymmetric setting in the sense that it requires the least mean number of copies out of all other procedures, including the ones that fix the number of copies ahead of time. For qubit states we derive explicit expressions for the minimum average number of copies and show that a sequential strategy based on fixed local measurements outperforms the best collective measurement on a predetermined number of copies. Whereas for general states the number of copies increases as log1/ε, for pure states sequential strategies require a finite average number of samples even in the case of perfect discrimination, i.e., ε=0.
We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of the optimal binary error probabilities. By constructing a family of regular-simplex ensembles, we further show that the coefficient four is optimal, even for arbitrary global measurements. This result improves on the pairwise bounds established by Cheng and Liu [arXiv:2606.06246 (2026)] and entails an explicit, sharp guarantee for the standard PGM itself. Our proof is also simpler: rather than constructing sequential measurements and applying a union bound, our proof relies on purely matrix-analytic techniques, combining a direct block Gram matrix analysis of the PGM error and a refined inequality between quantum Hellinger distance and Bures χ2χ^2χ2-divergence. Our analysis also yields a refined, one-shot pairwise Chernoff bound and explicit sufficient number of copies for a desired discrimination error.
We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.
We consider the problem of testing multiple quantum hypotheses {ρ1⊗n,…,ρr⊗n}\{\rho_{1}^{\otimes n},\ldots,\rho_{r}^{\otimes n}\}{ρ1⊗n​,…,ρr⊗n​}, where an arbitrary prior distribution is given and each of the rrr hypotheses is nnn copies of a quantum state. It is known that the minimal average error probability PeP_{e}Pe​ decays exponentially to zero, that is, Pe=exp⁡{−ξn+o(n)}P_{e}=\exp\{-\xi n+o(n)\}Pe​=exp{−ξn+o(n)}. However, this error exponent ξ\xiξ is generally unknown, except for the case that r=2r=2r=2. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkoła’s conjecture that ξ=min⁡i≠jC(ρi,ρj)\xi=\min_{i\neq j}C(\rho_{i},\rho_{j})ξ=mini=j​C(ρi​,ρj​). The right-hand side of this equality is called the multiple quantum Chernoff distance, and C(ρi,ρj):=max⁡0≤s≤1{−log⁡Tr⁡ρisρj1−s}C(\rho_{i},\rho_{j}):=\max_{0\leq s\leq1}\{-\log\operatorname{Tr}\rho_{i}^{s}\rho_{j}^{1-s}\}C(ρi​,ρj​):=max0≤s≤1​{−logTrρis​ρj1−s​} has been previously identified as the optimal error exponent for testing two hypotheses, ρi⊗n\rho_{i}^{\otimes n}ρi⊗n​ versus ρj⊗n\rho_{j}^{\otimes n}ρj⊗n​. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkoła’s lower bound. Specialized to the case r=2r=2r=2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.
Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-kkk, which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order kkk increases. For pure-state ensembles of size NNN, estimating MMD-kkk with arbitrary measurement schemes requires Θ(N1−1/k)Θ(N^{1-1/k})Θ(N1−1/k) samples for constant kkk. At the same time, we prove that any stable distance metric with full discriminative power admits an O(Nlog⁡N)O(N\log N)O(NlogN) upper bound and an Ω(N)Ω(N)Ω(N) instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an O(Nlog⁡N)O(N\log N)O(NlogN) upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.