  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.
