A quantum channel is a completely positive trace-preserving map. Given a quantum state $\rho$ and channel $N$, let the output be $N(\rho)$.
Given the Kraus operators of the channel, how can one find its kernel?
To take an explicit example, let $N: A\rightarrow B$ be a projection from a 2-qubit space to a 1-qubit space such as a channel with Kraus operators $\{\vert 0\rangle\langle 00\vert, \vert 0\rangle\langle 11\vert, \vert 1\rangle\langle 01\vert, \vert 1\rangle\langle 10\vert \}$.
We obviously lost the ability to tell $\vert 00\rangle$ apart from $\vert 11\rangle$ so it is easy to say by inspection that $\vert 00\rangle\langle 00\vert - \vert 11\rangle\langle 11\vert$ is in its kernel. How can one do this systematically for an arbitrary channel only from its Kraus operators $\{K_i\}$?
Note: Question edited after comment by gIS.
