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 $U={\phi }_{m,M}\left(S\right)$, derived from a known homomorphism, ${\phi }_{m,M}$, 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 ${\phi }_{m,M}^{-1}$ 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 between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

This article has results of four types. We show that the first eigenvalue \lambda_{1}(\Omega) of the weighted Laplacian of a bounded domain with smooth boundary can be obtained by S. Sato's iteration scheme of the Green operator, taking the limit \lambda_{1}(\Omega)=\lim_{k\to \infty} \Vert G^{k}(f)\Vert_{L^2}/\Vert G^{k+1}(f)\Vert_{L^2} for any f\in L^{2}(\Omega, \mu) , f > 0 . Then, we study the L^{1}(\Omega, \mu) -moment spectrum of \Omega in terms of iterates of the Green operator G , extending the work of McDonald–Meyers to the weighted setting. As corollary, we obtain the first eigenvalue of a weighted bounded domain in terms of the L^{1}(\Omega, \mu) -moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum \sigma^{\rm rad}(B_{h}(o,r)) of rotationally invariant geodesic balls B_{h}(o,r) of model manifolds. We prove an identity relating the radial eigenvalues of \sigma^{\rm rad}(B_{h}(o,r)) to an isoperimetric quotient, i.e., \sum 1/\lambda_{i}^{\rm rad} = \int V(s)/S(s) ds , V(s)={\rm vol}(B_{h}(o,s)) and S(s)={\rm vol}(\partial B_{h}(o,s)) . We then consider a proper minimal surface M\subset \mathbb{R}^{3} and the extrinsic ball \Omega=M\cap B_{\mathbb{R}^{3}}(o,r) . We obtain upper and lower estimates for the series \sum \lambda_i^{-2}(\Omega) in terms of the volume {\rm vol}(\Omega) and the radius r of the extrinsic ball \Omega .