 Controlled-transfer gate

A controlled rotation gate followed by computational-basis dephasing mixes the two target populations in the control-one subspace. If they are $c,d$, their sum is preserved and their difference is multiplied by $\cos \theta$. The operation is not unitary after dephasing. On a two-qubit diagonal state, the population of $|00⟩$ is unchanged when either qubit is the target, because the other control bit is zero.