Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing

TL;DR

This paper investigates how privacy constraints affect the contraction of quantum divergences and distances, providing bounds on contraction coefficients, sample complexity in quantum hypothesis testing, and implications for quantum learning.

Contribution

It establishes upper bounds on contraction coefficients for quantum divergences under privacy constraints and characterizes these bounds for key distances, advancing understanding of privacy in quantum information.

Findings

Upper bounds on contraction coefficients for hockey-stick divergence under privacy constraints

Full characterization of contraction coefficient for trace distance under privacy constraints

Bounds on sample complexity for quantum hypothesis testing under privacy constraints

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.