# Optimal entanglement witnesses in a split spin-squeezed Bose-Einstein condensate

## TL;DR

This paper develops optimal entanglement witnesses for split spin-squeezed Bose-Einstein condensates, enabling detection of quantum correlations with minimal measurement complexity, even under noise, advancing many-body Bell tests.

## Contribution

It derives the first and second order moment-based optimal entanglement witnesses for split spin-squeezed states, considering noise resilience and measurement statistics.

## Key findings

- Optimal witnesses are robust against local white noise.
- Second order moments provide more noise resistance than first order.
- Measurement requirements are feasible for current experimental setups.

## Abstract

How to detect quantum correlations in bi-partite scenarios using a split many-body system and collective measurements on each party? We address this question by deriving entanglement witnesses using either only first or first and second order moments of local collective spin components. In both cases, we derive optimal witnesses for spatially split spin squeezed states in the presence of local white noise. We then compare the two optimal witnesses with respect to their resistance to various noise sources operating either at the preparation or at the detection level. We finally evaluate the statistics required to estimate the value of these witnesses when measuring a split spin-squeezed Bose-Einstein condensate. Our results can be seen as a step towards Bell tests with many-body systems.