Entanglement frustration for Gaussian states on symmetric graphs

We investigate the entanglement properties of multi-mode Gaussian states, which have some symmetry with respect to the ordering of the modes. We show how the symmetry constraints the entanglement between two modes of the system. In particular, we determine the maximal entanglement of formation that can be achieved in symmetric graphs like chains, 2d and 3d lattices, mean field models and the platonic solids. The maximal entanglement is always attained for the ground state of a particular quadratic Hamiltonian. The latter thus yields the maximal entanglement among all quadratic Hamiltonians having the considered symmetry.