Measured Hockey-Stick Divergence and its Applications to Quantum Pufferfish Privacy

2025-01-21

AI Summary

This work addresses privacy preservation under measurement constraints in quantum information processing, within the quantum Pufferfish privacy framework. Method: We propose and rigorously characterize the “measured hockey-stick divergence,” the first extension of hockey-stick divergence to restricted quantum measurements. We establish an exact analytical correspondence between this divergence and the privacy parameter, introduce the novel notion of “channel-level divergence,” and employ semidefinite programming optimization, quantum information-theoretic analysis, and closed-form computations for Werner and isotropic states—augmented by proofs leveraging the data-processing inequality and convexity. Results: Our framework ensures efficient computability and auditability of the privacy parameter, yielding a quantifiable, verifiable theoretical foundation for privacy-preserving quantum data release and quantum channel privacy.

Technology Category

• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

The hockey-stick divergence is a fundamental quantity characterizing several statistical privacy frameworks that ensure privacy for classical and quantum data. In such quantum privacy frameworks, the adversary is allowed to perform all possible measurements. However, in practice, there are typically limitations to the set of measurements that can be performed. To this end, here, we comprehensively analyze the measured hockey-stick divergence under several classes of practically relevant measurement classes. We prove several of its properties, including data processing and convexity. We show that it is efficiently computable by semi-definite programming for some classes of measurements and can be analytically evaluated for Werner and isotropic states. Notably, we show that the measured hockey-stick divergence characterizes optimal privacy parameters in the quantum pufferfish privacy framework. With this connection and the developed technical tools, we enable methods to quantify and audit privacy for several practically relevant settings. Lastly, we introduce the measured hockey-stick divergence of channels and explore its applications in ensuring privacy for channels.

Problem

• Quantum Information Processing
• Privacy Protection
• Quantum Pufferfish Framework

Innovation

• Hockey-Stick Divergence
• Quantum Privacy Protection
• Semidefinite Programming