Multi-dimensional Hermite polynomials in quantum optics

We study a class of optical circuits with vacuum input states consisting of Gaussian sources without coherent displacements such as down-converters and squeezers, together with photo-detectors and passive interferometry (beamsplitters, polarization rotations, phase-shifters, etc). We show that the outgoing state leaving the optical circuit can be expressed in terms of so-called multi-dimensional Hermite polynomials and give their recursion and orthogonality relations. We show how quantum teleportation of single-photon polarization states can be modelled using this description.