  We locate the set of pairs $(\rho_{1},\rho_{2})$ of Gaussian states of a
single mode electromagnetic field that exhibit maximal trace distance subject
to the energy constraint $\langle a^{\dagger}a \rangle_{\rho_{1}}=\langle
a^{\dagger}a \rangle_{\rho_{2}} = E$. Any such pair allows to achieve the
minimum possible error in the task of binary distinguishability of two single
mode, isoenergetic Gaussian quantum signals. In particular, we show that the
logarithm of the minimal error probability for distinguishing two maximally
trace distant, isoenergetic Gaussian states scales as $-E^{2}$, less than the
achievable scaling of the minimal error probability for distinguishing, e.g., a
pair of isoenergetic Heisenberg-Weyl coherent states with energy $E$ or a pair
of isoenergetic quadrature squeezed states with energy $E$. For the case of a
field consisting of $M>1$ modes, we locate the set of pairs of maximally trace
distant isoenergetic, isocovariant Gaussian states. These results have basic
applications in the theory of continuous variable quantum communications with
Gaussian states of light.
