Multi-dimensional Hermite polynomials in quantum optics

Pieter Kok and Samuel L. Braunstein

Journal of Physics A 34, 6185-6195 (2001)

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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 detectors and passive interferometry (beam-splitters, polarisation 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 photon polarisation can be modelled using this description.

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