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

arXiv: https://arxiv.org/abs/2602.01177 · HTML: https://mlsanity.com/paper/2602.01177/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 42 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Builds on (most cited first, 11 of 11)

- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [Algorithmic Stability for Adaptive Data Analysis](https://mlsanity.com/paper/1511.02513/) (2015)
- [Generalization Error Bounds Via R'enyi-, f-Divergences and Maximal Leakage](https://mlsanity.com/paper/1912.01439/) (2019)
- [Quantum Differential Privacy: An Information Theory Perspective](https://mlsanity.com/paper/2202.10717/) (2022)
- [Quantum Differentially Private Sparse Regression Learning](https://mlsanity.com/paper/2007.11921/) (2020)
- [Quantum Local Differential Privacy and Quantum Statistical Query Model](https://mlsanity.com/paper/2203.03591/) (2022)
- [Information-theoretic generalization bounds for learning from quantum data](https://mlsanity.com/paper/2311.05529/) (2023)
- [Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing](https://mlsanity.com/paper/2406.18651/) (2024)
- [Upper Bounds on the Generalization Error of Private Algorithms for Discrete Data](https://mlsanity.com/paper/2005.05889/) (2020)
- [Quantum Information Ordering and Differential Privacy](https://mlsanity.com/paper/2511.01467/) (2025)
- [Generalization Bounds for Quantum Learning via R'enyi Divergences](https://mlsanity.com/paper/2505.11025/) (2025)

## Abstract (arXiv)

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.