Equivalence of Privacy and Stability with Generalization Guarantees in Quantum Learning

Abstract: We present a unified information-theoretic framework to analyze the generalization performance of differentially private (DP) quantum learning algorithms. By leveraging the connection between privacy and algorithmic stability, we establish that $(\varepsilon, δ)$-Quantum Differential Privacy (QDP) imposes a strong constraint on the mutual information between the training data and the algorithm's output. We derive a rigorous, mechanism-agnostic upper bound on this mutual information for learning algorithms satisfying a 1-neighbor privacy constraint. Furthermore, we connect this stability guarantee to generalization, proving that the expected generalization error of any $(\varepsilon, δ)$-QDP learning algorithm is bounded by the square root of the privacy-induced stability term. Finally, we extend our framework to the setting of an untrusted Data Processor, introducing the concept of Information-Theoretic Admissibility (ITA) to characterize the fundamental limits of privacy in scenarios where the learning map itself must remain oblivious to the specific dataset instance.