Testing the Wineland Criterion with Finite Statistics

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.