Transcribing quantum channels into quantum states

We introduce a quantum circuit that ``transcribes'' the elements of the process matrix $\ensuremath{\chi}$ of an arbitrary quantum channel onto the state of a pair of ancillary systems. The circuit works for an arbitrary quantum channel acting on a $d$-dimensional target system and returns as output not only the $\ensuremath{\chi}$ matrix encoded in the ${d}^{2}$-dimensional state of the ancilla systems but also the initial state of the target system. We compare the circuit to the well-known Choi circuit that outputs the Choi-Jamio\l{}kowski representation of the channel and the lesser-known Choi-Bell circuit that outputs the $\ensuremath{\chi}$ matrix. We show that the Choi-Bell circuit can be modified to transcribe quantum channels and also return the initial target state but requires more nonlocal gates. We briefly discuss some interesting features of the circuit and its eventual application in quantum process tomography and other computational tasks.