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arXiv:2411.14292 · Physical Review A

Hypothesis testing of symmetry in quantum dynamics

Yu-Ao Chen, Chenghong Zhu, Keming He, Yingjian Liu, Xin Wang

Published May 5, 2025• 4 citations
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Abstract

Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, we develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry ($T$ symmetry) and diagonal symmetry ($Z$ symmetry) with limited queries. Contrasting with the advantages of indefinite causal-order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal-order strategies have equal power for our tasks. We establish optimal protocols for $T$ symmetry testing and $Z$ symmetry testing for six and five queries, respectively, from which we infer that the type-II error exhibits a decay rate of $O({m}^{\ensuremath{-}2})$ with respect to the number of queries $m$. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of $O({m}^{\ensuremath{-}1})$.

DOI
10.1103/physreva.111.052406

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Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, we develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry (TTT symmetry) and diagonal symmetry (ZZZ symmetry) with limited queries. Contrasting with the advantages of indefinite causal-order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal-order strategies have equal power for our tasks. We establish optimal protocols for TTT symmetry testing and ZZZ symmetry testing for six and five queries, respectively, from which we infer that the type-II error exhibits a decay rate of O(m\ensuremath−2)O({m}^{\ensuremath{-}2})O(m\ensuremath−2) with respect to the number of queries mmm. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of O(m\ensuremath−1)O({m}^{\ensuremath{-}1})O(m\ensuremath−1).
Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by δδδ, we establish the explicit optimal sample complexity which scales as O(δ−1/3)\mathcal{O}(δ^{-1/3})O(δ−1/3) for identity testing and O(δ−1/2)\mathcal{O}(δ^{-1/2})O(δ−1/2) for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.
Can a symmetry be tested optimally without using the quantum resource naturally associated with it? For parallel subgroup-versus-Haar testing, we derive a quantitative representation-theoretic converse for every protocol with zero type-I error: the excess type-II error is bounded below by the sum of two nonnegative penalties, one for weight on suboptimal subgroup types and one for deviations of within-type Schur profiles from their typewise extremal profiles. At optimality, both penalties vanish, yielding support and weight locking for every optimal system marginal. Disjointness of the resulting locked set and the free marginal set rules out resource-free optimality, whereas overlap alone does not establish attainability. For the diagonal torus in dimension d≥2d\geq 2d≥2, optimal testing without coherence is possible only with a single query. We determine both the exact optimal type-II error over all protocols with incoherent system marginals and the minimum system-marginal coherence required to attain the unrestricted optimum. For the orthogonal subgroup in dimension d≥2d\geq 2d≥2, a real protocol attains the optimum at every query number. For the qutrit Clifford group, Wigner-positive optimal protocols exist at three and four queries but not from five through nine; whether they exist beyond nine queries remains open. Together, these results relate the subgroup- and query-dependent resource requirements of optimal symmetry testing to structural constraints on optimal system marginals.
The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say ρρρ and σσσ. This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call sss-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter sss. In particular, sss-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of sss-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as ξs(ρ∥σ)ξ_s(ρ\|σ)ξs​(ρ∥σ), and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for s=1s = 1s=1, and asymmetric hypothesis testing (the Umegaki relative entropy) for s=0s = 0s=0.
We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds.
Quantum networks play a key role in many scenarios of quantum information theory. Here we consider the quantum causal networks in the manner of entropy. First we present a revised smooth max-relative entropy of quantum combs, then we present a lower and upper bound of a type II error of the hypothesis testing. Next we present a lower bound of the smooth max-relative entropy for the quantum combs with asymptotic equipartition. At last, we consider the score to quantify the performance of an operator. We present a quantity equaling to the smooth asymptotic version of the performance of a quantum positive operator.
One of the key tasks in physics is to perform measurements in order to determine the state of a system. Often, measurements are aimed at determining the values of physical parameters, but one can also ask simpler questions, such as ``is the system in state A or state B?''. In quantum mechanics, the latter type of measurements can be studied and optimized using the framework of quantum hypothesis testing. In many cases one can explicitly find the optimal measurement in the limit where one has simultaneous access to a large number n of identical copies of the system, and estimate the expected error as n becomes large. Interestingly, error estimates turn out to involve various quantum information theoretic quantities such as relative entropy, thereby giving these quantities operational meaning. In this paper we consider the application of quantum hypothesis testing to quantum many-body systems and quantum field theory. We review some of the necessary background material, and study in some detail the situation where the two states one wants to distinguish are parametrically close. The relevant error estimates involve quantities such as the variance of relative entropy, for which we prove a new inequality. We explore the optimal measurement strategy for spin chains and two-dimensional conformal field theory, focusing on the task of distinguishing reduced density matrices of subsystems. The optimal strategy turns out to be somewhat cumbersome to implement in practice, and we discuss a possible alternative strategy and the corresponding errors.