We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.
We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.
We consider Bayesian discrimination among multiple quantum states and establish a dimension-free one-shot upper bound on the minimum probability of error in terms of the sum of pairwise errors. This resolves a conjecture of Audenaert and Mosonyi [J. Math. Phys. 55 (2014)] and improves the multiple quantum Chernoff bound of Li [Ann. Statist. 44 (2016)] by removing its dimension-dependent prefactor. In the asymptotic many-copy regime, our bound proves the achievability of the multiple quantum Chernoff distance for arbitrary separable Hilbert spaces, thereby settling the previously open infinite-dimensional case, and further yields constant-factor sharp asymptotics for the optimal error probability.
In binary quantum hypothesis testing, we prove that the minimum error probability is characterized, up to universal constants, by a trace harmonic-mean quantity. Consequently, the optimal binary quantum error probability is within a factor of two of the optimal classical error probability for the associated Nussbaum-Szkoła distributions, complementing the lower bound of Nussbaum and Szkoła [Ann. Statist. 37 (2009)].
We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite correlated hypotheses. In particular, we introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relations. In the error exponent regime, we generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal Type-I error exponent, under an exponential constraint on the Type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse exponent regime, we establish a general lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence, and we prove a matching upper bound when the null hypothesis is a singleton, under additional assumptions. The generality of these results enables applications in various contexts, including (i) refining the generalized quantum Stein’s lemma by [Fang, Fawzi & Fawzi, 2024]; (ii) exhibiting counterexamples to the continuity of the regularized Petz R´enyi divergence and Hoeffding divergence; (iii) obtaining error exponents for adversarial channel discrimination and resource detection problems.
Quantum state exclusion is an operational task with application to ontological interpretations of quantum states. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, admitting the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, when there are two hypotheses, our result provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, when all channels are classical, we show that our upper bound is achievable by a parallel strategy, thus solving the exact error exponent of classical channel exclusion.
The trade-off between the two types of error probabilities in binary state discrimination may be quantified in the asymptotics by various error exponents. In the composite case, where the hypotheses consist of sets of states, any such exponent is upper bounded by the infimum of the corresponding pairwise exponents of discriminating individual members of the two sets. Attainability of this upper bound may depend on the type of exponents considered; whether the problem is classical or quantum; the cardinality and the geometric properties of the sets representing the hypotheses; and also on the dimensionality of the underlying Hilbert space. Our main contribution is clarifying this landscape considerably. We show that in the quantum case unattainability is the general behaviour for all the exponents, already in finite dimension, for a simple null-hypothesis and an alternative hypothesis consisting of only two states. Moreover, the upper bound may be strict even in the infinite-dimensional classical case if the alternative hypothesis contains at least countably infinitely many states. We also prove general attainability results, e.g., for classical adversarial and arbitrarily varying hypothesis testing, and for the quantum case where all states are pure, or the states commute with every state in the opposite hypothesis.
The ultimate limits of quantum state discrimination are often thought to be captured by asymptotic bounds that restrict the achievable error probabilities, notably the quantum Chernoff and Hoeffding bounds. Here we study hypothesis testing protocols that are permitted a probability of producing an inconclusive discrimination outcome, and investigate their performance when this probability is suitably constrained. We show that even by allowing an arbitrarily small probability of inconclusiveness, the limits imposed by the quantum Hoeffding and Chernoff bounds can be significantly exceeded. This completely circumvents the conventional trade-offs between error exponents in hypothesis testing while incurring only a vanishingly small overhead over conventional approaches. Such improvements over standard state discrimination are robust and can be obtained even when an exponentially vanishing probability of inconclusive outcomes is demanded. Relaxing the constraints on the inconclusive probability can enable even larger advantages, but this comes at a price. We show a 'strong converse' property of this setting: targeting error exponents beyond those achievable with vanishing inconclusiveness necessarily forces the probability of inconclusive outcomes to converge to one. By exactly quantifying the rate of this convergence, we give a complete characterisation of the trade-offs between error exponents and rates of conclusive outcome probabilities. Overall, our results provide a comprehensive asymptotic picture of how the allowance for inconclusive measurement outcomes reshapes optimal quantum hypothesis testing.
We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of the optimal binary error probabilities. By constructing a family of regular-simplex ensembles, we further show that the coefficient four is optimal, even for arbitrary global measurements. This result improves on the pairwise bounds established by Cheng and Liu [arXiv:2606.06246 (2026)] and entails an explicit, sharp guarantee for the standard PGM itself. Our proof is also simpler: rather than constructing sequential measurements and applying a union bound, our proof relies on purely matrix-analytic techniques, combining a direct block Gram matrix analysis of the PGM error and a refined inequality between quantum Hellinger distance and Bures χ2-divergence. Our analysis also yields a refined, one-shot pairwise Chernoff bound and explicit sufficient number of copies for a desired discrimination error.