  Gaussian states are at the heart of quantum mechanics and play an essential
role in quantum information processing. In this paper we provide approximation
formulas for the expansion of a general Gaussian symbol in terms of elementary
Gaussian functions. For this purpose we introduce the notion of a "phase space
frame" associated with a Weyl-Heisenberg frame. Our results give explicit
formulas for approximating general Gaussian symbols in phase space by phase
space shifted standard Gaussians as well as explicit error estimates and the
asymptotic behavior of the approximation.
