  Communication over a random-parameter quantum channel when the decoder is
required to reconstruct the parameter sequence is considered. We study
scenarios that include either strictly-causal, causal, or non-causal channel
side information (CSI) available at the encoder, and also when CSI is not
available. This model can be viewed as a form of quantum metrology, and as the
quantum counterpart of the classical rate-and-state channel with state
estimation at the decoder. Regularized formulas for the capacity-distortion
regions are derived. In the special case of measurement channels, single-letter
characterizations are derived for the strictly causal and causal settings.
Furthermore, in the more general case of entanglement-breaking channels, a
single-letter characterization is derived when CSI is not available. As a
consequence, we obtain regularized formulas for the capacity of
random-parameter quantum channels with CSI, generalizing previous results by
Boche et al. (2016) on classical-quantum channels.
