An invitation to the sample complexity of quantum hypothesis testing - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2024

An invitation to the sample complexity of quantum hypothesis testing

Résumé

Quantum hypothesis testing has been traditionally studied from the informationtheoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of quantum hypothesis testing, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on quantum hypothesis testing, we characterize the sample complexity of binary quantum hypothesis testing in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple quantum hypothesis testing. In more detail, we prove that the sample complexity of symmetric binary quantum hypothesis testing depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary quantum hypothesis testing depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy, provided that the type II error probability is sufficiently small. We then provide lower and upper bounds on the sample complexity of multiple quantum hypothesis testing, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of quantum hypothesis testing is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study
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hal-04840569 , version 1 (16-12-2024)

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  • HAL Id : hal-04840569 , version 1

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Hao-Chung Cheng, Nilanjana Datta, Nana Liu, Theshani Nuradha, Robert Salzmann, et al.. An invitation to the sample complexity of quantum hypothesis testing. 2024. ⟨hal-04840569⟩
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