# Assessing degrees of entanglement of phonon states in atomic Bose gases through the measurement of commuting observables

## TL;DR

This paper demonstrates that measuring commuting observables in atomic Bose gases can effectively assess entanglement and steerability, using density fluctuation correlations to reveal quantum correlations in various experimental setups.

## Contribution

It introduces a method to evaluate entanglement and steerability through single observable measurements in cold atomic gases, simplifying experimental requirements.

## Key findings

- Single observable measurements can reveal entanglement.
- Density fluctuation correlations indicate quantum interferences.
- Application to experiments confirms practical relevance.

## Abstract

We show that measuring commuting observables can be sufficient to assess that a bipartite state is entangled according to either nonseparability or the stronger criterion of 'steerability'. Indeed, the measurement of a single observable might reveal the strength of the interferences between the two subsystems, as if an interferometer were used. For definiteness we focus on the two-point correlation function of density fluctuations obtained by in situ measurements in homogeneous one-dimensional cold atomic Bose gases. We then compare this situation to that found in transonic stationary flows mimicking a black hole geometry where correlated phonon pairs are emitted on either side of the sonic horizon by the analogue Hawking effect. We briefly apply our considerations to two recent experiments.