Fidelity for displaced squeezed thermal states and the oscillator semigroup

The fidelity for two displaced squeezed thermal states is computed using the fact that the corresponding density operators belong to the oscillator semigroup. A novel calculation technique is developed for the computation of the traces of product of Gaussian density operators and of square roots of Gaussian density operators. The method is exemplified for the one-mode case but it is also applicable to the multimode case.