Covariance Dynamics and Entanglement in Translation Invariant Linear Quantum Stochastic Networks

TL;DR

This paper analyzes the stability and entanglement properties of translation invariant linear quantum stochastic networks, providing conditions for stability and quantum entanglement in large-scale and infinite networks using Fourier techniques and separability criteria.

Contribution

It introduces a framework for assessing stability and entanglement in large quantum networks with finite interaction range, extending results to infinite chains.

Findings

Derived a stability condition for finite-range networks.

Established necessary and sufficient conditions for entanglement.

Extended stability and entanglement results to infinite networks.

Abstract

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.