Entanglement in Gaussian matrix-product states

Gaussian matrix product states are obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. Replacing the projections by associated Gaussian states, the 'building blocks', we show that the entanglement range in translationally-invariant Gaussian matrix product states depends on how entangled the building blocks are. In particular, infinite entanglement in the building blocks produces fully symmetric Gaussian states with maximum entanglement range. From their peculiar properties of entanglement sharing, a basic difference with spin chains is revealed: Gaussian matrix product states can possess unlimited, long-range entanglement even with minimum number of ancillary bonds (M=1). Finally we discuss how these states can be experimentally engineered from N copies of a three-mode building block and N two-mode finitely squeezed states.