  We analyze the Uhlmann fidelity of a pair of $n$-mode Gaussian states of the
quantum radiation field. This quantity is shown to be the product of an
exponential function depending on the relative average displacement and a
factor fully determined by the symplectic spectrum of the covariance matrix of
a specific Gaussian state. However, it is difficult to handle our general
formula unless the Gaussian states commute or at least one of them is pure. On
the contrary, in the simplest cases $n=1$ and $n=2$, it leads to explicit
analytic formulae. Our main result is a calculable expression of the fidelity
of two arbitrary two-mode Gaussian states. This can be applied to build
reliable measures of quantum correlations between modes in various branches of
quantum physics.
