Are quantum controlled non-unitary operations possible?
For instance can I define a controlled-reset where I reset a target qubit to $|0 \rangle$ if a controlled qubit is in the $|1\rangle$ state? How about if the control is in a superposition?
If the example above is problematic I'd be interested in other examples of non-unitary operations for which this makes sense.
If such a notion does make sense, how might I think about such an operation mathematically?