  In this paper, we analyze classical data compression with quantum side
information (also known as the classical-quantum Slepian-Wolf protocol) in the
so-called large and moderate deviation regimes. In the non-asymptotic setting,
the protocol involves compressing classical sequences of finite length $n$ and
decoding them with the assistance of quantum side information. In the large
deviation regime, the compression rate is fixed, and we obtain bounds on the
error exponent function, which characterizes the minimal probability of error
as a function of the rate. Devetak and Winter showed that the asymptotic data
compression limit for this protocol is given by a conditional entropy. For any
protocol with a rate below this quantity, the probability of error converges to
one asymptotically and its speed of convergence is given by the strong converse
exponent function. We obtain finite blocklength bounds on this function, and
determine exactly its asymptotic value. In the moderate deviation regime for
the compression rate, the latter is no longer considered to be fixed. It is
allowed to depend on the blocklength $n$, but assumed to decay slowly to the
asymptotic data compression limit. Starting from a rate above this limit, we
determine the speed of convergence of the error probability to zero and show
that it is given in terms of the conditional information variance. Our results
complement earlier results obtained by Tomamichel and Hayashi, in which they
analyzed the so-called small deviation regime of this protocol.
