Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing

2024-06-26

IEEE Transactions on Information Theory

AI Summary

This work investigates the contraction behavior of quantum channels under quantum local differential privacy (QLDP) constraints, focusing on multiple quantum divergences—including trace distance, hockey-stick divergence, Bures distance, and quantum relative entropy. Method: We derive exact characterizations and tight upper bounds for contraction coefficients under QLDP, and systematically apply these to private quantum hypothesis testing and private quantum learning. Contribution/Results: We fully characterize the optimal trace-distance contraction coefficient under QLDP and establish tight upper bounds for hockey-stick and other divergences. Applying contraction theory, we obtain matching upper and lower bounds on the sample complexity of private quantum hypothesis testing and construct first-order optimal private quantum channels achieving these bounds. Furthermore, we reveal how contraction coefficients fundamentally govern fairness guarantees and Holevo information stability in private quantum learning, thereby providing a unified information-theoretic foundation for the privacy–utility trade-off in quantum data analysis.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

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.

Problem

• Quantum Divergence Contraction
• Quantum Hypothesis Testing
• Quantum Learning Fairness

Innovation

• Quantum Local Differential Privacy (QLDP)
• Sample Complexity Bounds
• Private Quantum Channels