  Shannon's channel coding theorem describes the maximum possible rate of
reliable information transfer through a classical noisy communication channel.
It, together with the source coding theorem, characterizes lossless channel
communication in the classical regime. Lossy compression scenarios require the
additional description provided by rate-distortion theory, which characterizes
the trade-off between compression rate and the distortion of the compressed
signal. Even in this context, the capacity characterizes the usefulness of a
channel -- a channel with more capacity will always outperform a channel with
less capacity. We show that this is no longer true when sending classical
information over a quantum channel. In particular, we find a pair of quantum
channels where the channel with the lower capacity causes less distortion than
the higher capacity channel when both are used at a fixed rate.

Quantum Physics