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arXiv:2011.10773 · Physical Review Letters

Quantum Sequential Hypothesis Testing

Esteban Martínez-Vargas, Christoph Hirche, Gael Sentís, Michalis Skotiniotis, Marta Carrizo, R. Muñoz-Tapia, John Calsamiglia

Published May 6, 2021• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.126.180502

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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.
Quantum hypothesis testing (QHT) concerns the statistical inference of unknown quantum states. In the general setting of composite hypotheses, the goal of QHT is to determine whether an unknown quantum state belongs to one or another of two classes of states based on the measurement of a number of copies of the state. Prior art on QHT with composite hypotheses focused on a fixed-copy two-step protocol, with state estimation followed by an optimized joint measurement. However, this fixed-copy approach may be inefficient, using the same number of copies irrespective of the inherent difficulty of the testing task. To address these limitations, we introduce the quantum sequential universal test (QSUT), a novel framework for sequential QHT in the general case of composite hypotheses. 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. QSUT is proven to rigorously control the type I error under minimal assumptions about the hypothesis structure. We present two practical instantiations of QSUT, one based on the Helstrom-Holevo test and one leveraging shallow variational quantum circuits. Empirical results across a range of composite QHT tasks demonstrate that QSUT consistently reduces copy complexity relative to state-of-the-art fixed-copy strategies.
Sequential methods for quantum hypothesis testing offer significant advantages over fixed-length approaches, which rely on a predefined number of state copies. Despite their potential, these methods remain underexplored for unambiguous discrimination. In this work, we derive performance bounds for such methods when applied to the discrimination of a set of pure states. The performance is evaluated based on the expected number of copies required. We establish a lower bound applicable to any sequential method and an upper bound on the optimal sequential method. The upper bound is derived using a novel and simple non-adaptive method. Importantly, the gap between these bounds is minimal, scaling logarithmically with the number of distinct states.
Sequential strategies in hypothesis testing use a variable number of measurement rounds, allowing a decision to be made as soon as the observed data provide a prescribed level of error tolerance. Although sequential testing is well established for a discrete set of hypotheses, extending this framework to a continuous parameter space poses additional challenges and has remained largely unexplored. In this article, we introduce sequential \textit{parameter testing}, a framework for determining an unknown parameter up to a prescribed tolerance by ruling out sufficiently distant competing values with a target error tolerance. We clarify its operational distinction from conventional parameter estimation and develop sequential tests for continuous families of hypotheses, introducing the \textit{twin-peaks test} as a natural and computationally efficient analog of sequential likelihood-ratio testing. We apply the framework to two paradigmatic quantum tasks: testing the phase and the purity of a qubit. For phase testing, we show numerically that adaptive projective measurements achieve the same average sample cost as collective covariant measurements with a fixed number of copies. For purity testing, local measurements are optimal, and sequential parameter testing yields significant average sample savings over fixed sample-size protocols. Our results establish parameter testing as an operationally meaningful framework for resource-efficient certification tasks involving continuous parameters.
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.
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.
Quantum state discrimination is a fundamental information processing task that serves as a key component in many applications while also carrying foundational significance. In this work, we consider minimum error discrimination of multi-copy states, where instead of preparing a single system we assume that multiple instances of the same state are prepared. Now the discrimination allows for measurements from multiple parties with different measurement strategies varying from global measurement strategy to ones restricted to different forms of local operations and classical communication strategies. By comparing the average success probabilities in quantum and classical cases, we find a qubit strategy that outperforms all the bit strategies. On the other hand, we show that the classical measurement strategy does not give benefit in qubit over bit. However, we find that there are other (qu)bit-like operational theories which can outperform the best qubit strategies even with a classical measurement strategy and we are able to identify instances of different theories where different measurement strategies are optimal. In this way, we are able to find instances of nonlocality without entanglement as well as provide general bounds for bit-like operational theories.