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Generalized Quantum Stein's Lemma and Second Law of Quantum Resource Theories

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arxiv 2408.02722 v4 pith:JCJY2LVH submitted 2024-08-05 quant-ph

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keywords quantumsecondlemmaresourceresourcesentropygeneralizedstein
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The second law of thermodynamics is the cornerstone of physics, characterizing the convertibility between thermodynamic states through a single function, entropy. Given the universal applicability of thermodynamics, 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. In 2008, a promising formulation was proposed, linking resource convertibility to the optimal performance of a variant of the quantum version of hypothesis testing. Central to this formulation was the generalized quantum Stein's lemma, which aimed to characterize this optimal performance by a measure of quantum resources, the regularized relative entropy of resource. If proven valid, the generalized quantum Stein's lemma would lead to the second law for quantum resources, with the regularized relative entropy of resource taking the role of entropy in thermodynamics. However, 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. In this work, we address this problem by developing alternative techniques to successfully prove the generalized quantum Stein's lemma under a smaller set of assumptions than the original analysis. Based on our proof, we reestablish and extend the second law of quantum resource theories, applicable to both static resources of quantum states and a fundamental class of dynamical resources represented by classical-quantum (CQ) channels. These results resolve the fundamental problem of bridging the analogy between thermodynamics and quantum information theory.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Generalized Quantum Stein's Lemma for Classical-Quantum Dynamical Resources

    quant-ph 2025-09 reject novelty 7.0 of 10

    A generalized quantum Stein's lemma is proven directly for classical-quantum channels, yielding a reversible resource theory for channel conversion whose rates are fixed by the regularized relative entropy.

  2. Quantum sequential parameter testing

    quant-ph 2026-08 conditional novelty 6.0 of 10

    A tolerance-based sequential test, the twin-peaks test, certifies a continuous parameter with asymptotic sample cost 2 log(1/epsilon)/(I(theta) delta^2) and is demonstrated for qubit phase and purity.

  3. Quantum Sequential Universal Hypothesis Testing

    quant-ph 2025-08 conditional novelty 6.0 of 10

    QSUT is an adaptive, sequential quantum hypothesis test for composite hypotheses with anytime-valid type I error control and reduced average copy complexity.

  4. Hypothesis testing and Stein's lemma in general probability theories with Euclidean Jordan algebra and its quantum realization

    quant-ph 2025-05 reject novelty 6.0 of 10

    Stein's lemma is claimed for all Euclidean-Jordan-algebra models of general probabilistic theories, but a key tensor-product claim in the proof is false.

  5. Predicting symmetries of quantum dynamics with optimal samples

    quant-ph 2025-02 conditional novelty 6.0 of 10

    Optimal failure probabilities for detecting identity, diagonal, and real symmetries of unknown qubit unitaries are exactly computed and achieved by parallel strategies.

  6. One-shot manipulation of coherence in dynamic quantum resource theory

    quant-ph 2025-02 conditional novelty 5.0 of 10

    One-shot dynamic coherence cost and distillation for the quantum Fourier transform are bounded by log-robustness and hypothesis-testing relative entropy, with a catalytic extension under approximately free superchannels.

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