Privacy Implies Stability: Information-Theoretic Generalization Bounds for Quantum Learning

We develop an information-theoretic framework connecting stability, privacy, and generalization for quantum learning algorithms. Learning procedures are modeled as quantum instruments with classical-quantum outputs, and losses are represented by observables. We prove that under a classical-quantum sub-Gaussian condition, an information-theoretic stability measure controls the expected generalization error. Furthermore, we establish a high-probability generalization bound using quantum R'enyi divergences to manage higher-order dependencies under non-commutativity. In the trusted Data Processor setting, quantum differential privacy (QDP) provides a mechanism for stability. We show that one-neighbor QDP strictly bounds the information leaked by the classical-quantum output. Combining this with our stability theorem yields a direct privacy-to-generalization guarantee. We also explore an untrusted Data Processor setting. Here, output privacy alone is insufficient since an adversarial processor could perform a highly informative procedure before applying noisy post-processing. To combat this, we introduce Information-Theoretic Admissibility (ITA), a certification condition ensuring the prescribed procedure is not just a degraded version of a strictly more informative, physically allowed operation on the encoded ensemble. We prove a fundamental separation: while admissibility and privacy are in strong tension in classical models, quantum non-orthogonality makes them compatible. A quantum measurement can be ITA - exhausting all relevant accessible information - without perfectly recovering the classical dataset. We illustrate this separation through a concrete quantum ITA example.