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.
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.
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-QLDP channel with a d-dimensional input is entanglement-breaking whenever ε≤logd−1d. We also prove an approximate version for (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.
Differential privacy provides a robust framework for protecting sensitive data, while maintaining its utility for computation. In essence, a differentially private algorithm takes as input the data of multiple parties, and returns an output disclosing minimal information about any individual party. Previous research has introduced several quantum extensions of differential privacy, with applications ranging from quantum machine learning on private classical data to quantum shadow tomography. However, the local model of quantum differential privacy – where each party is responsible for privatizing their own data at a local level – has received limited attention. This work delves into locally differentially private quantum measurements. Although any measurement can be made locally differentially private by adding noise to the outcome, we demonstrate that certain quantum measurements inherently satisfy some degree of local differential privacy for specific classes of input states. This finding has two significant implications: first, limiting the analysis to classical noise injection mechanisms may lead to suboptimal privacy-utility trade-offs for quantum data; second, the theory of differential privacy can be harnessed to further investigate the capabilities of quantum measurements. Motivated by these insights, we establish strong data processing inequalities for the quantum relative entropy under local differential privacy and apply these results to asymmetric hypothesis testing of quantum states with restricted measurements. Additionally, we prove an equivalence between quantum statistical queries and quantum differential privacy in the local model, thereby addressing an open question posed by Arunachalam et al. (2021). Finally, we consider the task of quantum multi-party computation under local differential privacy, demonstrating that parity functions can be efficiently learned in this model, whereas the corresponding classical task requires exponentially many samples.
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 f-divergence. This approach completely characterizes (ε,δ)-QDP mechanisms by identifying the most informative (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.
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)-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), where ε∈(0,1) is the privacy parameter and β is the accuracy tolerance. We conclude by initiating the study of private classical shadows, which promise useful applications for private learning tasks.
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.
Composition is a cornerstone of classical differential privacy, enabling strong end-to-end guarantees for complex algorithms through composition theorems (e.g., basic and advanced). In the quantum setting, however, privacy is defined operationally against arbitrary measurements, and classical composition arguments based on scalar privacy-loss random variables no longer apply. As a result, it has remained unclear when meaningful composition guarantees can be obtained for quantum differential privacy (QDP).
In this work, we clarify both the limitations and possibilities of composition in the quantum setting. We first show that classical-style composition fails in full generality for POVM-based approximate QDP: even quantum channels that are individually perfectly private can completely lose privacy when combined through correlated joint implementations. We then identify a setting in which clean composition guarantees can be restored. For tensor-product channels acting on product neighboring inputs, we introduce a quantum moments accountant based on an operator-valued notion of privacy loss and a matrix moment-generating function. Although the resulting Rényi-type divergence does not satisfy a data-processing inequality, we prove that controlling its moments suffices to bound measured Rényi divergence, yielding operational privacy guarantees against arbitrary measurements. This leads to advanced-composition-style bounds with the same leading-order behavior as in the classical theory.
Our results demonstrate that meaningful composition theorems for quantum differential privacy require carefully articulated structural assumptions on channels, inputs, and adversarial measurements, and provide a principled framework for understanding which classical ideas do and do not extend to the quantum setting.