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arXiv:2501.10152 · IEEE Journal on Selected Areas in Communications

Quantum Advantage in Locally Differentially Private Hypothesis Testing

Seung-Hyun Nam, Hyun-Young Park, Si-Hyeon Lee, Joonwoo Bae

Published Jan 1, 2026• 0 citations
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Abstract

We consider a private hypothesis testing scenario, including both symmetric and asymmetric testing, based on classical data samples. The utility is measured by the error exponents, namely the Chernoff information and the relative entropy, while privacy is measured in terms of classical or quantum local differential privacy. In this scenario, we show a quantum advantage with respect to the optimal privacy-utility trade-off (PUT) in certain cases. Specifically, we focus on distributions referred to as smoothed point mass distributions, along with the uniform distribution, as hypotheses. We then derive upper bounds on the optimal PUTs achievable by classical privacy mechanisms, which are tight in specific instances. To show the quantum advantage, we propose a particular quantum privacy mechanism that achieves better PUTs than these upper bounds in both symmetric and asymmetric testing, specifically under stringent privacy constraints and small discrete data alphabet sizes ranging from 3 to 9. The proposed mechanism consists of a classical-quantum channel that prepares symmetric informationally complete (SIC) states, followed by a depolarizing channel.

DOI
10.1109/jsac.2026.3709807

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We consider a private hypothesis testing scenario, including both symmetric and asymmetric testing, based on classical data samples. The utility is measured by the error exponents, namely the Chernoff information and the relative entropy, while privacy is measured in terms of classical or quantum local differential privacy. In this scenario, we show a quantum advantage with respect to the optimal privacy-utility trade-off (PUT) in certain cases. Specifically, we focus on distributions referred to as smoothed point mass distributions, along with the uniform distribution, as hypotheses. We then derive upper bounds on the optimal PUTs achievable by classical privacy mechanisms, which are tight in specific instances. To show the quantum advantage, we propose a particular quantum privacy mechanism that achieves better PUTs than these upper bounds in both symmetric and asymmetric testing, specifically under stringent privacy constraints and small discrete data alphabet sizes ranging from 3 to 9. The proposed mechanism consists of a classical-quantum channel that prepares symmetric informationally complete (SIC) states, followed by a depolarizing channel.
For multiple hypothesis testing based on classical data samples, we demonstrate a quantum advantage in the optimal privacy-utility trade-off (PUT), where the privacy and utility measures are set to (quantum) local differential privacy and the pairwise-minimum Chernoff information, respectively. To show the quantum advantage, we consider some class of hypotheses that we coin smoothed point masses. For such hypotheses, we derive an upper bound of the optimal PUT achieved by classical mechanisms, which is tight for some cases, and propose a certain quantum mechanism which achieves a better PUT than the upper bound. The proposed quantum mechanism consists of a classical-quantum channel whose outputs are pure states corresponding to a symmetric informationally complete positive operator-valued measure (SIC-POVM), and a depolarizing channel.
We optimize the trade-off between privacy and utility in the high-privacy regime. We adopt local differential privacy (LDP) and its quantum extension, quantum local differential privacy (QLDP), for privacy protection, and investigate utility functions including the Holevo information (which reduces to the mutual information in the classical case) and the error exponents in symmetric and asymmetric hypothesis testing. Under the LDP and QLDP constraints, these utility functions have classical and quantum optimal values, which we denote simply by CCC and QQQ, respectively, in this abstract. In this paper, we provide optimal LDP and QLDP mechanisms achieving the classical and quantum optimal values in the high-privacy regime, and prove that the asymptotic ratio Q/CQ/CQ/C in this regime takes the same value regardless of the utility function. Our results reveal quantum advantages (specifically, Q/C≥3/2Q/C\ge3/2Q/C≥3/2) for the above utility functions when the private data take at least three possible values. In addition, conditional on a mathematical conjecture concerning equi-isoclinic tight fusion frames (EITFFs), our QLDP mechanisms also exhibit quantum advantages in a moderate-privacy regime.
When sensitive information is encoded in data, it is important to ensure the privacy of information when attempting to learn useful information from the data. There is a natural tradeoff whereby increasing privacy requirements may decrease the utility of a learning protocol. In the quantum setting of differential privacy, such tradeoffs between privacy and utility have so far remained largely unexplored. In this work, we study optimal privacy-utility tradeoffs for both generic and application-specific utility metrics when privacy is quantified by (ε,δ)(\varepsilon,δ)(ε,δ)-quantum local differential privacy. In the generic setting, we focus on optimizing fidelity and trace distance between the original state and the privatized state. We show that the depolarizing mechanism achieves the optimal utility for given privacy requirements. We then study the specific application of learning the expectation of an observable with respect to an input state when only given access to privatized states. We derive a lower bound on the number of samples of privatized data required to achieve a fixed accuracy guarantee with high probability. To prove this result, we employ existing lower bounds on private quantum hypothesis testing, thus showcasing the first operational use of them. We also devise private mechanisms that achieve optimal sample complexity with respect to the privacy parameters and accuracy parameters, demonstrating that utility can be significantly improved for specific tasks in contrast to the generic setting. In addition, we show that the number of samples required to privately learn observable expectation values scales as Θ((εβ)−2)Θ((\varepsilon β)^{-2})Θ((εβ)−2), where ε∈(0,1)\varepsilon \in (0,1)ε∈(0,1) is the privacy parameter and βββ is the accuracy tolerance. We conclude by initiating the study of private classical shadows, which promise useful applications for private learning tasks.
Differential privacy provides a rigorous framework for quantifying privacy leakage in data analysis, while its quantum extensions have become increasingly relevant with the development of quantum computing and quantum machine learning. In this work, we introduce and study quantum probabilistic local differential privacy, a relaxation of quantum local differential privacy in which the privacy constraint is allowed to fail on a spectral violation event with low probability. This quantity can be interpreted as the probability under the quantum superoperation of a quantum privacy-loss violation, and is closely related to the acceptance probability of the quantum Neyman-Pearson test at a small threshold. We investigate the basic structural properties of this privacy notion and clarify its relationship with existing forms of quantum differential privacy. We show the properties of quantum probabilistic local differential privacy under tensor-product composition and unitary post-processing, while it is in general neither convex nor closed under post-processing by arbitrary quantum channels. We further characterize when depolarizing noise satisfies quantum probabilistic local differential privacy under several representative scenarios. Finally, we connect quantum probabilistic privacy constraints with statistical inference by deriving a lower bound on probabilistically privatized contraction coefficients in terms of the hockey-stick divergence. As an application, we obtain sample complexity bounds of probabilistically privated asymmetric and symmetric quantum hypothesis testing. These results provide a systematic foundation for studying probabilistic privacy guarantees in quantum information processing and their operational consequences for private quantum statistical inference.
We study quantum differential privacy (QDP) by defining a notion of the order of informativeness between two pairs of quantum states. In particular, we show that if the hypothesis testing divergence of the one pair dominates over that of the other pair, then this dominance holds for every fff-divergence. This approach completely characterizes (ε,δ)(\varepsilon,δ)(ε,δ)-QDP mechanisms by identifying the most informative (ε,δ)(\varepsilon,δ)(ε,δ)-DP quantum state pairs. We apply this to study precise limits for privatized hypothesis testing and privatized quantum parameter estimation, including tight upper-bounds on the quantum Fisher information under QDP. Finally, we establish near-optimal contraction bounds for differentially private quantum channels with respect to the Hockey-Stick divergence.
Differential privacy provides a mathematical framework for guaranteeing privacy for sensitive data. In quantum information processing, the interaction of privacy constraints with quantum resources such as entanglement remains a question of interest. Given that the utility of many protocols, and often the presence of a quantum advantage, relies on quantum resources such as entanglement, it is crucial to understand when a privacy requirement for a quantum channel is compatible with the channel's ability to preserve entanglement. We study this question for quantum local differential privacy (QLDP). Our main result shows that every ε\varepsilonε-QLDP channel with a ddd-dimensional input is entanglement-breaking whenever ε≤log⁡dd−1\varepsilon\leq\log\frac{d}{d-1}ε≤logd−1d​. We also prove an approximate version for (ε,δ)(\varepsilon,δ)(ε,δ)-QLDP, where channels in the same high-privacy regime are close in diamond norm to an entanglement-breaking channel. We further prove a composition result for a collection of private quantum channels having entangled inputs and global measurements in the high-privacy regime. Finally, we apply our results to private quantum learning theory. We prove that any learning protocol using arbitrary quantum memory on copies of the output of an entanglement-breaking channel can be simulated by a protocol that measures the corresponding unprocessed input copies one at a time while storing only classical information. Combining this result with our high-privacy entanglement-breaking theorem, we show that under sufficiently private local noise, a learning protocol with quantum memory for purity testing and bipartite product testing is subject to the sample complexity lower bounds for protocols with single-copy measurements on the noiseless tasks. We also obtain stronger sample complexity lower bounds when a single highly private channel acts on the entire multipartite input.