Generalised purity as an entanglement measure for two- and three-well Bose-Hubbard models

TL;DR

This paper critically examines the use of purity measures based on SU(2) and SU(3) Lie algebras as indicators of entanglement in two- and three-well Bose-Hubbard models, revealing limitations and state-dependence.

Contribution

It demonstrates that these purity-based measures are not reliable indicators of entanglement, as they depend on the quantum states of the modes and can be misleading.

Findings

Purities depend on the quantum statistical states of modes.

Separable states can exhibit purity values suggesting entanglement.

Purity measures are not unambiguous indicators of entanglement.

Abstract

The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two- and three-well Bose-Einstein condensates, with the group operators being expressed in terms of bosonic annihilation and creation operators at each site. The success of these representations has led to the purities associated with these algebras to be promoted as a measure of entanglement between the atomic modes in each well. In this report, we show that these purities do not provide an unambiguous measure, but instead give results which depend on the quantum statistical states of each atomic mode. Using the example of totally uncoupled modes which have never interacted, we quantify these purities for different states and show that completely separable states can give values which have been claimed to indicate the presence of entanglement.