Approximating quantum channels by completely positive maps with small Kraus rank

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.