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Quantum channel discrimination has been studied from an information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of unknown channel accesses. In this paper, we study the query complexity of quantum channel discrimination, wherein the goal is to determine the minimum number of channel uses needed to reach a desired error probability. To this end, we show that the query complexity of binary channel discrimination depends logarithmically on the inverse error probability and inversely on the negative logarithm of the (geometric and Holevo) channel fidelity. As special cases of these findings, we precisely characterize the query complexity of discriminating two classical channels and two classical–quantum channels. Furthermore, by obtaining an optimal characterization of the sample complexity of quantum hypothesis testing when the error probability does not exceed a fixed threshold, we provide a more precise characterization of query complexity under a similar error probability threshold constraint. We also provide lower and upper bounds on the query complexity of binary asymmetric channel discrimination and multiple quantum channel discrimination. For the former, the query complexity depends on the geometric Rényi and Petz–Rényi channel divergences, while for the latter, it depends on the negative logarithm of the (geometric and Uhlmann) channel fidelity. For multiple channel discrimination, the upper bound scales as the logarithm of the number of channels. 

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. 

The bivariate classical fidelity is a widely used measure of the similarity of two probability distributions. There exist a few extensions of the notion of the bivariate classical fidelity to more than two probability distributions; herein we call these extensions multivariate classical fidelities, with some examples being the Matusita multivariate fidelity and the average pairwise fidelity. Hitherto, quantum generalizations of multivariate classical fidelities have not been systematically explored, even though there are several well known generalizations of the bivariate classical fidelity to quantum states, such as the Uhlmann and Holevo fidelities. The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose several variants that reduce to the average pairwise fidelity for commuting states, including the average pairwisez-fidelities, the multivariate semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All of these variants satisfy the following properties: (i) reduction to multivariate classical fidelities for commuting states, (ii) the data-processing inequality, (iii) invariance under permutations of the states, (iv) its values are in the interval $[0,1]$; they are faithful, that is, their values are equal to one if and only if all the states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the states are mutually orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and (vii) uniform continuity bounds under certain conditions. Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. We also introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and it has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis. Lastly, we propose multivariate generalizations of Matsumoto’s geometric fidelity and establish several properties of them. 

We propose a versatile privacy framework for quantum systems, termed quantum pufferfish privacy (QPP). Inspired by classical pufferfish privacy, our formulation generalizes and addresses limitations of quantum differential privacy by offering flexibility in specifying private information, feasible measurements, and domain knowledge. We show that QPP can be equivalently formulated in terms of the Datta–Leditzky information spectrum divergence, thus providing the first operational interpretation thereof. We reformulate this divergence as a semi-definite program and derive several properties of it, which are then used to prove convexity, composability, and post-processing of QPP mechanisms. Parameters that guarantee QPP of the depolarization mechanism are also derived. We analyze the privacy-utility tradeoff of general QPP mechanisms and, again, study the depolarization mechanism as an explicit instance. The QPP framework is then applied to privacy auditing for identifying privacy violations via a hypothesis testing pipeline that leverages quantum algorithms. Connections to quantum fairness and other quantum divergences are also explored and several variants of QPP are examined. 

The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the data-processing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched Rényi relative entropy, of which the sandwiched Rényi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched Rényi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical-quantum states. Finally, we establish semi-definite programs for smooth max-relative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first. 

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out psi-epistemic ontological models of quantum mechanics (Pusey et al. in Nat Phys 8(6):475–478, 2012). Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent—the rate at which the optimal error probability vanishes to zero asymptotically—for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.