Testing the Wineland Criterion with Finite Statistics

TL;DR

This work formulates a rigorous hypothesis-testing framework to certify spin squeezing via the Wineland parameter $\xi_W^2$, explicitly accounting for finite statistics. By introducing the affine bound $\Gamma_c$ and a $p$-value bound derived from Bernstein-type inequalities, the authors provide upper and lower bounds on the probability that observed data could arise from non-spin-squeezed states. Applying these methods to a range of experiments, they show that most data do not yield statistically robust evidence for squeezing at the 5% level, and they quantify how many measurements would be required to achieve such evidence. The results offer a practical blueprint for designing future metrological experiments to reliably demonstrate spin squeezing and entanglement, with a framework extensible to other squeezing criteria.

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.

Figures (2)

• Figure 1: Experimental achievements together with their statistical analysis. The experiments that have been reported in the litterature are identified by a square whose position specifies the total number of experimental repetitions ($M$) that has been realized and the system size ($N$). The color of the square gives the observed value of the Wineland parameter. Every square is accompanied by a number used to identify the reference as specified below the graph. The significance of the experimental outcomes are quantified by the number of measurements that are necessary and sufficient to reach a p-value of 5%. Namely, the upper triangle (with the vertex pointing down) gives the total number of measurement repetitions that is sufficient to reject the hypothesis that non-spin squeezed states were measured. The lower triangle (with the vertex pointing up) specifies the number of measurements that is required in order to ensure that a specific non-spin squeezed state (according to the Wineland parameter) cannot reproduce the observed statistics.
• Figure 2: (Top) Estimation of the Wineland parameter from Monte Carlo samples of size $M$ sampled from a non-spin-squeezed state. Two data sets were sampled: one in the eigenbasis of the mean spin direction ($J_y$) and another in the eigenbasis of the squeezing axis. Each error bar corresponds to one standard deviation. (Bottom) P-value upper bound using the method proposed in the main text on the Monte-Carlo samples.