# Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing

arXiv: https://arxiv.org/abs/2406.18651 · HTML: https://mlsanity.com/paper/2406.18651/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 64 references that are in the map.
- First posted on arXiv 26 Jun 2024.

## Builds on (most cited first, 23 of 23)

- [RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response](https://mlsanity.com/paper/1407.6981/) (2014)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Quantum noise protects quantum classifiers against adversaries](https://mlsanity.com/paper/2003.09416/) (2020)
- [Differential Privacy as a Mutual Information Constraint](https://mlsanity.com/paper/1608.03677/) (2016)
- [Quantum Differential Privacy: An Information Theory Perspective](https://mlsanity.com/paper/2202.10717/) (2022)
- [Quantum machine learning with differential privacy](https://mlsanity.com/paper/2103.06232/) (2021)
- [The Structure of Optimal Private Tests for Simple Hypotheses](https://mlsanity.com/paper/1811.11148/) (2018)
- [Differentially Private Testing of Identity and Closeness of Discrete Distributions](https://mlsanity.com/paper/1707.05128/) (2017)
- [Quantum Differentially Private Sparse Regression Learning](https://mlsanity.com/paper/2007.11921/) (2020)
- [Certified Robustness of Quantum Classifiers against Adversarial Examples through Quantum Noise](https://mlsanity.com/paper/2211.00887/) (2022)
- [Local Differential Privacy Is Equivalent to Contraction of Eγ-Divergence](https://mlsanity.com/paper/2102.01258/) (2021)
- [Quantum Local Differential Privacy and Quantum Statistical Query Model](https://mlsanity.com/paper/2203.03591/) (2022)
- [Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities](https://mlsanity.com/paper/2206.02765/) (2022)
- [Private learning implies quantum stability](https://mlsanity.com/paper/2102.07171/) (2021)
- [Simple Binary Hypothesis Testing under Local Differential Privacy and Communication Constraints](https://mlsanity.com/paper/2301.03566/) (2023)
- [Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems](https://mlsanity.com/paper/2306.13054/) (2023)
- [Information-theoretic generalization bounds for learning from quantum data](https://mlsanity.com/paper/2311.05529/) (2023)
- [Contraction of Locally Differentially Private Mechanisms](https://mlsanity.com/paper/2210.13386/) (2022)
- [The Sample Complexity of Simple Binary Hypothesis Testing](https://mlsanity.com/paper/2403.16981/) (2024)
- [Verifying Fairness in Quantum Machine Learning](https://mlsanity.com/paper/2207.11173/) (2022)
- [A unifying framework for differentially private quantum algorithms](https://mlsanity.com/paper/2307.04733/) (2023)
- [An invitation to the sample complexity of quantum hypothesis testing](https://mlsanity.com/paper/2403.17868/) (2024)
- [Lemmas of Differential Privacy](https://mlsanity.com/paper/2211.11189/) (2022)

## Built on by (most cited first, 7 of 7)

- [Quantum Information Ordering and Differential Privacy](https://mlsanity.com/paper/2511.01467/) (2025)
- [Measured Hockey-Stick Divergence and its Applications to Quantum Pufferfish Privacy](https://mlsanity.com/paper/2501.12359/) (2025)
- [Query Complexity of Classical and Quantum Channel Discrimination](https://mlsanity.com/paper/2504.12989/) (2025)
- [Non-Linear Strong Data-Processing for Quantum Hockey-Stick Divergences](https://mlsanity.com/paper/2512.16778/) (2025)
- [Sample Complexity of Composite Quantum Hypothesis Testing](https://mlsanity.com/paper/2601.08588/) (2026)
- [Privacy Implies Stability: Information-Theoretic Generalization Bounds for Quantum Learning](https://mlsanity.com/paper/2602.01177/) (2026)
- [Privacy-Utility Tradeoffs in Quantum Information Processing](https://mlsanity.com/paper/2602.10510/) (2026)

## Abstract (arXiv)

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. With 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.