Testing the Wineland Criterion with Finite Statistics

TL;DR

This paper develops a statistical hypothesis-testing framework to reliably detect spin squeezing using the Wineland parameter, accounting for finite measurement data, and applies it to experimental results.

Contribution

It introduces a rigorous statistical method to assess spin squeezing from the Wineland parameter considering finite statistics, improving experimental validation.

Findings

Most experiments could not reject the null hypothesis of no spin squeezing at 5% significance.

A non-spin squeezed state model reproduces observed data with p-value above 5%.

Provides a practical approach for future experiments to statistically confirm spin squeezing.

Abstract

The Wineland parameter aims at detecting metrologically useful entangled states, called spin-squeezed states, from expectations and variances of total angular momenta. {However, efficient strategies for estimating this parameter in practice have yet to be determined and in particular, the effects of a finite number of measurements remain insufficiently addressed. We formulate the detection of spin squeezing as a hypothesis-testing problem, where the null hypothesis assumes that the experimental data can be explained by non-spin-squeezed states. Within this framework, we derive upper and lower bounds on the p-value to quantify the statistical evidence against the null hypothesis.} By applying our statistical test to data obtained in multiple experiments, we are unable to reject the hypothesis that non-spin squeezed states were measured with a p-value of 5\% or less in most cases. We also find an explicit non-spin squeezed state according to the Wineland parameter reproducing most of the observed results with a p-value exceeding 5\%. More generally, our results provide a rigorous method to establish robust statistical evidence of spin squeezing from the Wineland parameter in future experiments,accounting for finite statistics.