Exchangeable, stationary and entangled chains of Gaussian states

TL;DR

This paper investigates the mathematical conditions under which sequences of Gaussian quantum states are exchangeable or stationary, deriving formulas for their entropy rates and providing an example of an entangled stationary chain.

Contribution

It establishes necessary and sufficient conditions for exchangeability and stationary properties of Gaussian state chains, and derives entropy rate formulas including an example of entanglement.

Findings

Exchangeability conditions are necessary and sufficient.

Stationary chains have an asymptotic entropy rate expressed via eigenvalue distributions.

An example of a stationary entangled Gaussian chain is provided.

Abstract

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.