# How many Kraus operators are required to characterise a channel with different start and end dimensions?

If we have a quantum channel mapping from a $$d$$-dimensional state to a $$d$$-dimensional state, it can be described by at most $$d^2$$ Kraus operators. Suppose our channel maps instead from a $$d_1$$-dimensional state to a $$d_2$$-dimensional state, with $$d_1>d_2$$, e.g. with the quantum operation of taking the partial trace over a mode. What is the maximum required number of Kraus operators to characterise the channel? Is it $$d_1d_2$$, analogous to the case where $$d_1=d_2$$?

Yes. Choi's Theorem a priori uses different Hilbert spaces of potentially different dimensions $$d_1$$ and $$d_2$$. Then $$d_1=d_2$$ is a corollory. The proof is included there.