Quantum operations of photon addition and photon subtraction, and the wavefunction differentiator on their basis

We consider non-unitary operations of photon addition ($\rho \rightarrow a^{\dagger} \rho a$) and photon subtraction ($\rho \rightarrow a \rho a^{\dagger}$), which can be realized by selective measurements. Modifying the measurement, it is possible to implement a coherent superposition of those operations ($\rho \rightarrow (\mu a + \nu a^{\dagger}) \rho (\mu^{\ast} a^{\dagger} + \nu^{\ast} a)$) that reduces to the differentiator if $\mu=-\nu$. Further, we discuss the possibility of wavefunction reconstruction based on non-unitary tomography.