A method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe. (arXiv:1901.06178v2 [quant-ph] UPDATED)

The evolution of quantum light through linear optical devices can be
described by the scattering matrix $S$ of the system. For linear optical
systems with $m$ possible modes, the evolution of $n$ input photons is given by
a 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 of
the possible photon states, $M$. We present a method to decide whether a given
unitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linear
optics or not and the inverse transformation $\varphi_{m,M}^{-1}$ when the
transformation can be implemented. Together with previous results, the method
can be used to find a simple optical system which implements any quantum
operation within the reach of linear optics. The results come from studying the
adjoint map bewtween the Lie algebras corresponding to the Lie groups of the
relevant unitary matrices.

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