  We provide a rate distortion interpretation of the problem of quantum data
compression of ensembles of mixed states with commuting density operators.
There are two versions of this problem. In the visible case the sequence of
states is available to the encoder and in the blind or hidden case the encoder
may access only a sequence of measurements. We find the exact optimal
compression rates for both the visible and hidden cases. Our analysis includes
the scenario in which asymptotic reconstruction is imperfect.
