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

TL;DR

This paper analyzes the measured hockey-stick divergence in quantum privacy, establishing its properties, computational methods, and applications to quantify and audit privacy in quantum and channel settings.

Contribution

It introduces the measured hockey-stick divergence, explores its properties, and demonstrates its role in quantum pufferfish privacy and channel privacy auditing.

Findings

Efficient semi-definite programming computation for certain measurement classes

Analytical evaluation for Werner and isotropic states

Characterizes optimal privacy parameters in quantum pufferfish privacy

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.