  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.
