Probing quantum entanglement from magnetic-sublevels populations: beyond spin squeezing inequalities

TL;DR

This paper introduces a systematic method to detect quantum entanglement using Zeeman sublevel population measurements, extending beyond traditional spin squeezing inequalities, with applications to spinor Bose-Einstein condensates.

Contribution

It develops a new approach to construct entanglement criteria from Zeeman population data, applicable to any $d$-level quantum system, surpassing existing spin squeezing methods.

Findings

Demonstrated on spin-1 and spin-2 Bose-Einstein condensates

Allows inference of optimal permutationally-invariant entanglement witnesses

Provides a systematic framework for entanglement detection from collective measurements

Abstract

Spin squeezing inequalities (SSI) represent a major tool to probe quantum entanglement among a collection of few-level atoms, and are based on collective spin measurements and their fluctuations. Yet, for atomic ensembles of spin-$j$ atoms and ultracold spinor gases, many experiments can image the populations in all Zeeman sublevels $s=-j, -j+1, \dots, j$, potentially revealing finer features of quantum entanglement not captured by SSI. Here we present a systematic approach which exploits Zeeman-sublevel population measurements in order to construct novel entanglement criteria, and illustrate our approach on ground states of spin-1 and spin-2 Bose-Einstein condensates. Beyond these specific examples, our approach allows one to infer, in a systematic manner, the optimal permutationally-invariant entanglement witness for any given set of collective measurements in an ensemble of $d$-level quantum systems.