  Rate-distortion theory provides bounds for compressing data produced by an
information source to a specified encoding rate that is strictly less than the
source's entropy. This necessarily entails some loss, or distortion, between
the original source data and the best approximation after decompression. The
so-called Information Bottleneck Method is designed to compress only `relevant'
information. Which information is relevant is determined by the correlation
between the data being compressed and a variable of interest, so-called side
information. In this paper, an Information Bottleneck Method is introduced for
the compression of quantum data. The channel communication picture is used for
compression and decompression. The rate of compression is derived using an
entanglement assisted protocol with classical communication, and under an
unproved conjecture that the rate function is convex in the distortion
parameter. The optimum channel achieving this rate for a given input state is
characterised. The conceptual difficulties arising due to differences in the
mathematical formalism between quantum and classical probability theory are
discussed and solutions are presented.
