  This paper is concerned with a translation invariant network of identical
quantum stochastic systems subjected to external quantum noise. Each node of
the network is directly coupled to a finite number of its neighbours. This
network is modelled as an open quantum harmonic oscillator and is governed by a
set of linear quantum stochastic differential equations. The dynamic variables
of the network satisfy the canonical commutation relations. Similar large-scale
networks can be found, for example, in quantum metamaterials and optical
lattices. Using spatial Fourier transform techniques, we obtain a sufficient
condition for stability of the network in the case of finite interaction range,
and consider a mean square performance index for the stable network in the
thermodynamic limit. The Peres-Horodecki-Simon separability criterion is
employed in order to obtain sufficient and necessary conditions for quantum
entanglement of bipartite systems of nodes of the network in the Gaussian
invariant state. The results on stability and entanglement are extended to the
infinite chain of the linear quantum systems by letting the number of nodes go
to infinity. A numerical example is provided to illustrate the results.
