Exchangeable, stationary, and entangled chains of Gaussian states

We explore conditions on the covariance matrices of a consistent chain of mean zero finite mode Gaussian states in order that the chain may be exchangeable or stationary. For an exchangeable chain, our conditions are necessary and sufficient. Every stationary Gaussian chain admits an asymptotic entropy rate. Whereas an exchangeable chain admits a simple expression for its entropy rate, in our examples of stationary chains, the same admits an integral formula based on the asymptotic eigenvalue distribution for Toeplitz matrices. An example of a stationary entangled Gaussian chain is given.