  We consider a cell-free hybrid massive multiple-input multiple-output (MIMO)
system with $K$ users and $M$ access points (APs), each with $N_a$ antennas and
$N_r< N_a$ radio frequency (RF) chains. When $K\ll M{N_a}$, efficient uplink
channel estimation and data detection with reduced number of pilots can be
performed based on low-rank matrix completion. However, such a scheme requires
the central processing unit (CPU) to collect received signals from all APs,
which may enable the CPU to infer the private information of user locations. We
therefore develop and analyze privacy-preserving channel estimation schemes
under the framework of differential privacy (DP). As the key ingredient of the
channel estimator, two joint differentially private noisy matrix completion
algorithms based respectively on Frank-Wolfe iteration and singular value
decomposition are presented. We provide an analysis on the tradeoff between the
privacy and the channel estimation error. In particular, we show that the
estimation error can be mitigated while maintaining the same privacy level by
increasing the payload size with fixed pilot size; and the scaling laws of both
the privacy-induced and privacy-independent error components in terms of
payload size are characterized. Simulation results are provided to further
demonstrate the tradeoff between privacy and channel estimation performance.
