  We study the problem of approximating a quantum channel by one with as few
Kraus operators as possible (in the sense that, for any input state, the output
states of the two channels should be close to one another). Our main result is
that any quantum channel mapping states on some input Hilbert space
$\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be
compressed into one with order $d\log d$ Kraus operators, where
$d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than
$|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all
very mixed, this can be improved to order $d$. We discuss the optimality of
this result as well as some consequences.

Quantum Physics;Functional Analysis;Probability