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.

翻译：量子广义散度根据定义满足数据处理不等式；因此，在量子信道作用下，此类散度的相对减少量至多为1。这种相对减少量在形式上被称为信道与散度的收缩系数。有趣的是，存在某些信道与散度的组合，其收缩系数严格小于1。此外，理解收缩系数对于研究隐私约束下的统计任务至关重要。为此，本文建立了隐私约束下曲棍球棒散度收缩系数的上界，其中隐私性通过量子局部差分隐私（QLDP）框架进行量化，并完整刻画了隐私约束下迹距离的收缩系数。利用所发展的工具，我们还确定了在QLDP约束下，Bures距离和量子相对熵相对于归一化迹距离的收缩上界。随后，我们将研究结果应用于建立隐私约束下量子假设检验样本复杂度的界限。此外，我们研究了多种样本复杂度界限达到紧致的场景，同时提供了实现这些界限的阶最优量子信道。最后，我们展示了隐私量子信道如何在量子学习环境中提供公平性和Holevo信息稳定性。