  A Gaussian degree of entanglement for a symmetric two-mode Gaussian state can
be defined as its distance to the set of all separable two-mode Gaussian
states. The principal property that enables us to evaluate both Bures distance
and relative entropy between symmetric two-mode Gaussian states is the
diagonalization of their covariance matrices under the same beam-splitter
transformation. The multiplicativity property of the Uhlmann fidelity and the
additivity of the relative entropy allow one to finally deal with a single-mode
optimization problem in both cases. We find that only the Bures-distance
Gaussian entanglement is consistent with the exact entanglement of formation.
