The hockey-stick divergence is a fundamental quantity characterizing several statistical privacy frameworks that ensure privacy for classical and quantum data. In such quantum privacy frameworks, the adversary is allowed to perform all possible measurements. However, in practice, there are typically limitations to the set of measurements that can be performed. To this end, here, we comprehensively analyze the measured hockeystick divergence under several classes of practically relevant measurement classes. We prove several of its properties, including data processing and convexity. We show that it is efficiently computable by semi-definite programming for some classes of measurements and can be analytically evaluated for Werner and isotropic states. Notably, we show that the measured hockeystick divergence characterizes optimal privacy parameters in the quantum pufferfish privacy framework. With this connection and the developed technical tools, we enable methods to quantify and audit privacy for several practically relevant settings. Lastly, we introduce the measured hockey-stick divergence of channels and explore its applications in ensuring privacy for channels.

A quantum generalized divergence by definition satisfies the data-processing inequality; as such, the relative decrease in such a divergence under the action of a quantum channel is at most one. This relative decrease is formally known as the contraction coefficient of the channel and the divergence. Interestingly, there exist combinations of channels and divergences for which the contraction coefficient is strictly less than one. Furthermore, understanding the contraction coefficient is fundamental for the study of statistical tasks under privacy constraints. To this end, here we establish upper bounds on contraction coefficients for the hockey-stick divergence under privacy constraints, where privacy is quantified with respect to the quantum local differential privacy (QLDP) framework, and we fully characterize the contraction coefficient for the trace distance under privacy constraints. Using the machinery developed, we also determine an upper bound on the contraction of both the Bures distance and quantum relative entropy relative to the normalized trace distance, under QLDP constraints. Next, we apply our findings to establish bounds on the sample complexity of quantum hypothesis testing under privacy constraints. Furthermore, we study various scenarios in which the sample complexity bounds are tight while providing order-optimal quantum channels that achieve those bounds. Lastly, we show how private quantum channels provide fairness and Holevo information stability in quantum learning settings.

We propose a versatile privacy framework for quantum systems, termedquantum 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 dataprocessing 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 maxrelative 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.

Pufferfish privacy (PP) is a generalization of differential privacy (DP), that offers flexibility in specifying sensitive information and integrates domain knowledge into the privacy definition. Inspired by the illuminating formulation of DP in terms of mutual information due to Cuff and Yu, this work explores PP through the lens of information theory. We provide an information-theoretic formulation of PP, termed mutual information PP (MI PP), in terms of the conditional mutual information between the mechanism and the secret, given the public information. We show that MI PP is implied by the regular PP and characterize conditions under which the reverse implication is also true, recovering the relationship between DP and its information-theoretic variant as a special case. We establish convexity, composability, and post-processing properties for MI PP mechanisms and derive noise levels for the Gaussian and Laplace mechanisms. The obtained mechanisms are applicable under relaxed assumptions and provide improved noise levels in some regimes. Lastly, applications to auditing privacy frameworks, statistical inference tasks, and algorithm stability are explored.

Pufferfish privacy (PP) is an appealing generalization of differential privacy (DP), that offers flexibility in specifying sensitive information and integrating domain knowledge into the privacy definition. Inspired by the illuminating equivalent formulation of DP in terms of mutual information proposed by Cuff and Yu [1], this work explores PP through the lens of information theory. We provide an equivalent information-theoretic formulation of PP as the conditional mutual information between the mechanism and the secret, given the public information. This formulation lends well for an information-theoretic analysis, and we use it to prove convexity, composability, and post-processing properties for PP mechanisms. We also leverage our formulation to derive noise levels for the Gaussian PP mechanisms. The obtained mechanisms are applicable under relaxed assumptions and provide improved noise levels in some regimes, compared to existing approaches,

Sliced mutual information (SMI) is defined as an average of mutual information (MI) terms between one-dimensional random projections of the random variables. It serves as a surrogate measure of dependence to classic MI that preserves many of its properties but is more scalable to high dimensions. However, a quantitative characterization of how SMI itself and estimation rates thereof depend on the ambient dimension, which is crucial to the understanding of scalability, remain obscure. This work provides a multifaceted account of the dependence of SMI on dimension, under a broader framework termed $k$-SMI, which considers projections to $k$-dimensional subspaces. Using a new result on the continuity of differential entropy in the 2-Wasserstein metric, we derive sharp bounds on the error of Monte Carlo (MC)-based estimates of $k$-SMI, with explicit dependence on $k$ and the ambient dimension, revealing their interplay with the number of samples. We then combine the MC integrator with the neural estimation framework to provide an end-to-end $k$-SMI estimator, for which optimal convergence rates are established. We also explore asymptotics of the population $k$-SMI as dimension grows, providing Gaussian approximation results with a residual that decays under appropriate moment bounds. All our results trivially apply to SMI by setting $k=1$. Our theory is validated with numerical experiments and is applied to sliced InfoGAN, which altogether provide a comprehensive quantitative account of the scalability question of $k$-SMI, including SMI as a special case when $k=1$.