Generalized quantum Stein's lemma for mixed sources

• URL: http://arxiv.org/abs/2605.20776v1
• Date: Wed, 20 May 2026 06:17:23 GMT
• Title: Generalized quantum Stein's lemma for mixed sources
• Abstract summary: We develop techniques for the non-commutative quantum setting inspired by classical information-spectrum analysis.<n>We show that the concisely characterization does not hold in general for a fixed nonzero type-I error threshold.<n>These results clarify the applicability of the generalized quantum Stein's lemma to highly non-IID null hypotheses.
• Score: 1.0742675209112622
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The generalized quantum Stein's lemma characterizes the optimal asymptotic exponent of the type-II error in quantum hypothesis testing for an independent and identically distributed (IID) null hypothesis against a composite alternative hypothesis. Classically, a probabilistic mixture of IID sources arises as a natural generalization of IID sources, and, in the non-composite setting, the optimal type-II error exponent in hypothesis testing for such classical mixed sources is known to be characterized concisely by the worst-case component of the mixture. In this work, we extend these foundational results to composite quantum hypothesis testing where the null hypothesis is a mixed source, i.e., a probabilistic mixture of IID quantum states, and the alternative hypothesis is composite as in the generalized quantum Stein's lemma. When the type-I error vanishes asymptotically, we characterize the optimal type-II error exponent of this composite quantum hypothesis testing problem in terms of the worst-case component of the mixture, by developing techniques for the non-commutative quantum setting inspired by the classical information-spectrum analysis. We also show that the analogous characterization does not hold in general for a fixed nonzero type-I error threshold, by providing a counterexample beyond the vanishing type-I error regime. These results clarify the applicability of the generalized quantum Stein's lemma to highly non-IID null hypotheses arising from arbitrary finite probabilistic mixtures of IID quantum states.

Related papers

• Generalised quantum Stein's lemma more robust than ever [40.77349008361407]
  generalised quantum Stein's lemma is a key result in quantum hypothesis testing.<n>We establish a version of generalised quantum Stein's lemma that is conceptually much simpler and general.
  arXiv  Detail & Related papers  (2026-09-15T15:19:27Z)
• A doubly composite Chernoff-Stein lemma and its applications [13.283296551055734]
  Given a sequence of random variables, discriminating between two hypotheses on the underlying probability distribution is a key task in statistics and information theory.<n>Generalisations of this result exist in the case of composite hypotheses, but mostly to settings where the probability distribution of $Xn$ is not genuinely correlated.<n>Here, we establish a general Chernoff-Stein lemma that applies to the setting where both hypotheses are composite and genuinely correlated.
  arXiv  Detail & Related papers  (2025-10-07T18:06:30Z)
• Generalised quantum Sanov theorem revisited [13.283296551055734]
  Given two families of quantum states $A$ and $B$, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to $A$ or $B$.<n>In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the rate of decay of the type II error probability for a given threshold on the type I error probability.<n>In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite.
  arXiv  Detail & Related papers  (2025-10-07T18:05:25Z)
• Quantum Sequential Universal Hypothesis Testing [62.751483592497806]
  Quantum hypothesis testing (QHT) concerns the statistical inference of unknown quantum states.<n>We introduce the quantum sequential universal test (QSUT), a novel framework for sequential QHT in the general case of composite hypotheses.<n> QSUT builds on universal inference, and it alternates between adaptive local measurements aimed at exploring the hypothesis space and joint measurements optimized for maximal discrimination.
  arXiv  Detail & Related papers  (2025-08-29T12:50:04Z)
• Error exponents of quantum state discrimination with composite correlated hypotheses [40.82628972269358]
  We study the error exponents in quantum hypothesis testing between two sets of quantum states.<n>We introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states.
  arXiv  Detail & Related papers  (2025-08-18T13:04:06Z)
• Towards Quantum Universal Hypothesis Testing [11.32644286794219]
  We introduce a quantum universal hypothesis testing framework that serves as a quantum analog to Hoeffding's UHT.<n>We employ quantum state tomography to reconstruct the unknown state prior to forming the test statistic.
  arXiv  Detail & Related papers  (2025-04-22T22:41:08Z)
• Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints [14.29947046463964]
  We study a distributed binary hypothesis testing problem to infer a bipartite quantum state shared between two remote parties.<n>We show that the Stein's exponent when one of the parties communicates classically at zero-rate is given by a multi-letter expression.<n>As a key tool for proving the converse direction of our results, we develop a quantum version of the blowing-up lemma.
  arXiv  Detail & Related papers  (2024-10-11T16:03:10Z)
• Generalized Quantum Stein's Lemma and Second Law of Quantum Resource Theories [47.02222405817297]
  A fundamental question in quantum information theory is whether an analogous second law can be formulated to characterize the convertibility of resources for quantum information processing by a single function.<n>In 2008, a promising formulation was proposed, linking resource convertibility to the optimal performance of a variant of the quantum version of hypothesis testing.<n>In 2023, a logical gap was found in the original proof of this lemma, casting doubt on the possibility of such a formulation of the second law.
  arXiv  Detail & Related papers  (2024-08-05T18:00:00Z)
• To Believe or Not to Believe Your LLM [51.2579827761899]
  We explore uncertainty quantification in large language models (LLMs) We derive an information-theoretic metric that allows to reliably detect when only epistemic uncertainty is large. We conduct a series of experiments which demonstrate the advantage of our formulation.
  arXiv  Detail & Related papers  (2024-06-04T17:58:18Z)
• Testing trajectory-based determinism via time probability distributions [41.99844472131922]
  We introduce a prescription for constructing an arrival-time probability distribution within generic trajectory-equipped theories.<n>We derive a conditional probability distribution that is unreachable by quantum mechanics.<n>Our results can be tested experimentally, thereby assessing the validity of trajectory-based determinism.
  arXiv  Detail & Related papers  (2024-04-15T11:36:38Z)
• Postselected quantum hypothesis testing [9.131273927745731]
  We study a variant of quantum hypothesis testing wherein an additional 'inconclusive measurement outcome' is added. The error probabilities are conditioned on a successful attempt, with inconclusive trials disregarded. We prove that the error exponent of discriminating any two quantum states $rho$ and $sigma$ is given by the Hilbert projective metric $D_max(|sigma) + D_max(sigma | rho)$ in asymmetric hypothesis testing.
  arXiv  Detail & Related papers  (2022-09-21T18:00:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.