Entanglement and separability of quantum harmonic oscillator systems at finite temperature

TL;DR

This paper investigates how entanglement in quantum harmonic oscillator systems diminishes with increasing temperature, establishing bounds on the temperature where the system becomes fully separable, and analyzing specific topologies like rings.

Contribution

It provides new bounds on the critical temperature for entanglement disappearance and analyzes the entanglement behavior in systems with translation symmetry and ring topology.

Findings

Thermal states become fully separable at high temperatures.

Critical temperature bounds are tight for systems with translation symmetry.

Entanglement decreases as temperature approaches the energy scale of the strongest normal mode.

Abstract

In the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram.