  In this paper, lower bounds on error probability in coding for discrete
classical and classical-quantum channels are studied. The contribution of the
paper goes in two main directions: i) extending classical bounds of Shannon,
Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new
framework for lower bounding the probability of error of channels with a
zero-error capacity in the low rate region. The relation between these two
problems is revealed by showing that Lov\'asz' bound on zero-error capacity
emerges as a natural consequence of the sphere packing bound once we move to
the more general context of classical-quantum channels. A variation of
Lov\'asz' bound is then derived to lower bound the probability of error in the
low rate region by means of auxiliary channels. As a result of this study,
connections between the Lov\'asz theta function, the expurgated bound of
Gallager, the cutoff rate of a classical channel and the sphere packing bound
for classical-quantum channels are established.
