Gaussian Matrix Product States

TL;DR

This paper introduces Gaussian Matrix Product States (GMPS), extending MPS to harmonic oscillator lattices, and explores their properties, including approximation, entanglement bounds, correlation functions, and relation to local Hamiltonian ground states.

Contribution

The paper defines GMPS, generalizing MPS to continuous-variable systems, and analyzes their properties and applications in quantum many-body physics.

Findings

GMPS can approximate arbitrary Gaussian states.

Entanglement in GMPS bonds can be bounded.

Correlation functions decay exponentially in 1D GMPS.

Abstract

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.