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 an d squeezers, together with photo-detectors and passive interferometry (beam splitters, polarization rotations, phase-shifters, etc). We show that the o utgoing state leaving the optical circuit can be expressed in terms of so-c alled multi-dimensional Hermite polynomials and give their recursion and or thogonality relations. We show how quantum teleportation of single-photon p olarization states can be modelled using this description.