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IEEE International Symposium on Information Theory (ISIT)

Characterizing Acceptance in Post-Selection One-Shot Quantum Hypothesis Testing

Saurabh Kumar Gupta, Abhishek Kr. Gupta

Published Jun 25, 2023• 0 citations
Publisher

Abstract

In post-selection hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In this paradigm, minimum error for various cases is characterized in literature conditioned on the fact that one of the selected outcomes occurs. However, not selecting any of the hypotheses decimates the quality of testing. There is no discussion on the quality of post-selection hypothesis testing. In this work, we characterize the quality of post-selection hypothesis testing by defining a new metric acceptance. We find out measurements which lead to optimal post-selection error. For these measurements, we have derived the expression for acceptance in post-selection symmetric and asymmetric hypothesis testing problems. We have also given closed-form expressions in some specific cases.

DOI
10.1109/isit54713.2023.10206804

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In post-selection hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In this paradigm, minimum error for various cases is characterized in literature conditioned on the fact that one of the selected outcomes occurs. However, not selecting any of the hypotheses decimates the quality of testing. There is no discussion on the quality of post-selection hypothesis testing. In this work, we characterize the quality of post-selection hypothesis testing by defining a new metric acceptance. We find out measurements which lead to optimal post-selection error. For these measurements, we have derived the expression for acceptance in post-selection symmetric and asymmetric hypothesis testing problems. We have also given closed-form expressions in some specific cases.
In hypothesis testing with quantum states, given a black box containing one of the two possible states, measurement is performed to detect in favor of one of the hypotheses. In postselected hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In postselected scenario, minimum error one-shot symmetric hypothesis testing is characterized in literature conditioned on the fact that one of the selected outcomes occur. We proceed further in this direction to give the set of all possible measurements that lead to the minimum error. We have given an arbitrary error-minimizing measurement in a parametric form. Note that not selecting any of the hypotheses decimates the quality of testing. We further give an example to show that these measurements vary in quality. There is a need to discuss the quality of postselected hypothesis testing. We then characterize the quality of postselected hypothesis testing by defining a new metric acceptance and give expression of acceptance for an arbitrary error-minimizing measurement in terms of some parameters of the measurement. On the set of measurements that achieve minimum error, we have maximized the acceptance, and given an example which achieves that, thus giving an example of the best possible measurement in terms of acceptance.
We study a variant of quantum hypothesis testing wherein an additional ‘inconclusive’ measurement outcome is added, allowing one to abstain from attempting to discriminate the hypotheses. The error probabilities are then conditioned on a successful attempt, with inconclusive trials disregarded. We completely characterise this task in both the single-shot and asymptotic regimes, providing exact formulas for the optimal error probabilities. In particular, we prove that the asymptotic error exponent of discriminating any two quantum states$\rho andandand\sigma isgivenbytheHilbertprojectivemetricis given by the Hilbert projective metricisgivenbytheHilbertprojectivemetricD_{\max }(\rho \|\sigma ) + D_{\max }(\sigma \| \rho )inasymmetrichypothesistesting,andbytheThompsonmetricin asymmetric hypothesis testing, and by the Thompson metricinasymmetrichypothesistesting,andbytheThompsonmetric\max \! \big \{ D_{\max }(\rho \|\sigma ),\, D_{\max }(\sigma \| \rho ) \big \}$in symmetric hypothesis testing. This endows these two quantities with fundamental operational interpretations in quantum state discrimination. Our findings extend to composite hypothesis testing, where we show that the asymmetric error exponent with respect to any convex set of density matrices is given by a regularisation of the Hilbert projective metric. We apply our results also to quantum channels, showing that no advantage is gained by employing adaptive or even more general discrimination schemes over parallel ones, in both the asymmetric and symmetric settings. Our state discrimination results make use of no properties specific to quantum mechanics and are also valid in general probabilistic theories.
This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.
This lecture notes gives a basic survey of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.
Can adaptive strategies outperform non-adaptive ones for quantum hypothesis selection? We exhibit problems where adaptive strategies provably reduce the number of required samples by a factor four in the worst case, and possibly more when the actual difficulty of the problem makes it possible. In addition, we exhibit specific hypotheses classes for which there is a provable polynomial separation between adaptive and non-adaptive strategies-a specificity of the quantum framework that does not appear in classical testing.
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. 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. 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 and contribute to the problem of sample complexity of quantum hypothesis testing, and we outline a number of open directions for future research.