Entanglement Frustration for Gaussian States on Symmetric Graphs

We investigate the entanglement properties of multimode Gaussian states, which have some symmetry with respect to the ordering of the modes. We show how the symmetry constrains 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, $\mathrm{2}\mathrm{D}$ and $\mathrm{3}\mathrm{D}$ 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.

Figure 1

(a) Apart from chains, cubic lattices, and mean field clusters there are several familiar symmetric graphs. Examples are the five platonic solids (e.g., the dodecahedron), and hexagonal or trigonal lattices. (b): Maximal nearest neighbor entanglement $E_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (ebits) in a ring of $N$ harmonic oscillators. The dotted curves represent the envelopes corresponding to Eq. (10).