  We introduce Gaussian Matrix Product States (GMPS), a generalization of
Matrix Product States (MPS) to lattices of harmonic oscillators. Our definition
resembles the interpretation of MPS in terms of projected maximally entangled
pairs, starting from which we derive several properties of GMPS, often in close
analogy to the finite dimensional case: We show how to approximate arbitrary
Gaussian states by MPS, we discuss how the entanglement in the bonds can be
bounded, we demonstrate how the correlation functions can be computed from the
GMPS representation, and that they decay exponentially in one dimension, and
finally relate GMPS and ground states of local Hamiltonians.
