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

Abstract

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 = ϕ m,M (S), derived from a known homomorphism, ϕ 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 ϕ −1 m,M 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.

Optimal approximation to unitary quantum operators with linear optics

“…3 e −i 4π This evolution is impossible to achieve with lossless linear optics alone, which can be checked with the method in [19].…”

“…In a previous work, we have given an explicit inverse method to find the S corresponding to any U ∈ im(ϕ) which can be implemented using linear optics [19].…”

“…This basis is not necessarily orthonormal, but it can be orthogonalized and normalized. The details of the whole procedure can be found in [19]. This basis, together with the inner product of Eq.…”

“…The approximation is optimal in the local neighbourhood of the initial guess. Once we have the closest unitary that can be implemented, U , we can use a previous method [19] to obtain a scattering matrix S that gives the approximated evolution and there are multiple algorithms that give the physical setup corresponding to S using only beam splitters and phase shifters [4,7]. This gives a design method which starts from the desired evolution instead of the usual analysis methods which, given a classical description of the linear optical system, find out the evolution it induces on quantum states.…”

“…However, there are relatively few methods for the design of tailored quantum evolutions for multiple photons. We have previously presented an inverse method which can tell if any desired quantum evolution on n photons can be achieved with a linear optical system or not, giving the corresponding system when it is possible [19].…”