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The evolution of quantum light through linear optical devices can bedescribed by the scattering matrix $S$ of the system. For linear opticalsystems with $m$ possible modes, the evolution of $n$ input photons is given bya unitary matrix $U=\varphi_{m,M}(S)$ given by a known homomorphism,$\varphi_{m,M}$, which depends on the size of the resulting Hilbert space ofthe possible photon states, $M$. We present a method to decide whether a givenunitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linearoptics or not and the inverse transformation $\varphi_{m,M}^{-1}$ when thetransformation can be implemented. Together with previous results, the methodcan be used to find a simple optical system which implements any quantumoperation within the reach of linear optics. The results come from studying theadjoint map bewtween the Lie algebras corresponding to the Lie groups of therelevant unitary matrices.

A method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe