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.
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.
How can we analyze quantum correlations in large and noisy systems without quantum state tomography? An established method is to measure total angular momenta and employ the so-called spin-squeezing inequalities based on their expectations and variances. This allows detection of metrologically useful entanglement, but efficient strategies for estimating such nonlinear quantities have yet to be determined. In this paper we show that spin-squeezing inequalities can not only be evaluated by measurements of the total angular momentum but also by two-qubit correlations, either involving all pair correlations or randomly chosen pair correlations. Then we analyze the estimation errors of our approaches in terms of a hypothesis test. For this purpose, we discuss how error bounds can be derived for nonlinear estimators with the help of their variances, characterizing the probability of falsely detecting a separable state as entangled. We focus on the spin-squeezing inequalities in multiqubit systems. Our methods, however, can also be applied to spin-squeezing inequalities for qudits or for the statistical treatment of other nonlinear parameters of quantum states.
This work presents a theoretical, technical, statistical, and experimental framework for aquantum metrology system based on collective spin squeezing. The objective is to determineexperimentally whether an atomic ensemble can be prepared in a spin-squeezed state andwhether the resulting state provides a measurable metrological advantage for estimation of aweak physical parameter, such as a magnetic field.The central quantity is the Wineland spin-squeezing parameter, denoted here by ξ2R,ξ2R =N(∆S⊥)2|⟨S⟩|2,where S⊥ is the collective-spin component along the optimally squeezed transverse direc-tion. Values ξ2R < 1 indicate metrologically useful spin squeezing under the assumptions ofthe corresponding Ramsey-type measurement protocol.The v2.1 framework explicitly separates three concepts that are sometimes conflated: re-duction of collective-spin variance, metrological spin squeezing, and experimentally demon-strated parameter-estimation gain. It also distinguishes the quantum Fisher information of astate from the classical Fisher information obtained by a particular measurement.The protocol contains baseline characterization, squeezing generation, tomography of thetransverse covariance matrix, calibration of detection efficiency, estimation of systematic un-certainties, independent metrological testing, and reproducibility measurements. Negativecontrols are included to distinguish genuine quantum-projection-noise reduction from ordi-nary technical-noise suppression.The proposal is falsifiable. A failure to obtain a confidence interval for ξ2R below unity,failure to reproduce the effect on independent runs, or failure to demonstrate a sensitivityimprovement under a matched experimental comparison constitutes a negative result for thecorresponding hypothesis.No experimental result is claimed.
We present an experimentally practical method to reveal Einstein-Podolsky-Rosen (EPR) steering in non-Gaussian spin states by exploiting a connection to quantum metrology. Our criterion is based on the quantum Fisher information, and uses bounds derived from generalized spin-squeezing parameters that involve measurements of higher-order moments. This leads us to introduce the concept of conditional spin-squeezing parameters, which quantify the metrological advantage provided by conditional states, as well as detect the presence of an EPR paradox.
The work is organized in two main topics. At first we will outline the relation between spin squeezing, quantum metrology and entanglement detection, with a particular focus on the last. We will derive spin squeezing criteria for the detection of entanglement and its depth that outperform past approaches, especially for unpolarized states, recently produced in experiments and object of increasing interest in the community. Furthermore, we will extend the original definition of spin squeezed states by providing a new parameter that is thought to embrace different classes of states in a unified framework. Afterwards we consider a test of quantum principles in macroscopic objects originally designed by Leggett and Garg. In this case the scenario consists of a single party that is probed at different time instants and the quantum effect detected is the violation of Macrorealism (MR), due to strong correlations in time, rather than in space, between non-compatible observables. We will look at the problems of inconclusiveness of the LG tests arising from possible explanations of the results in terms of ``clumsy'' measurements, what has been termed ``clumsiness loophole''. We propose first a scheme to test and possibly falsify (MR) in macroscopic ensembles of cold atoms based on alternating Quantum Non-Demolition measurements and coherent, magnetically-driven collective spin rotations. Then we also propose a way to address the clumsiness loophole by introducing and computing an invasivity quantifier to add to the original LG expression. We provide numerical evidence that such a clumsiness-free test is feasible under state-of-the-art realistic experimental parameters and imperfections.
Abstract A key objective in conducting a Bell test is to quantify the statistical evidence against a local-hidden variable model (LHVM) given that we can collect only a finite number of trials in any experiment. The notion of statistical evidence is thereby formulated in the framework of hypothesis testing, where the null hypothesis is that the experiment can be described by an LHVM. The statistical confidence with which the null hypothesis of an LHVM is rejected is quantified by the so-called P value, where a smaller P value implies higher confidence. Establishing good statistical evidence is especially challenging if the number of trials is small, or the Bell violation very low. Here, we derive the optimal P value for a large class of Bell inequalities. What is more, we obtain very sharp upper bounds on the P value for all Bell inequalities. These values are easily computed from the experimental data, and are valid even if we allow arbitrary memory in the devices. Our analysis is able to deal with imperfect random number generators, and event-ready schemes, even if such a scheme can create different kinds of entangled states. Finally, we review requirements for sound data collection, and a method for combining P values of independent experiments. The methods discussed here are not specific to Bell inequalities. For instance, they can also be applied to the study of certified randomness or to tests of noncontextuality.
We apply spin-squeezing techniques to identify and quantify highly multi-partite photonic entanglement in polarization-squeezed light. We consider a practical single-mode scenario, and find that Wineland-criterion polarization squeezing implies entanglement of a macroscopic fraction of the total photons. A Glauber-theory computation of the observable N -photon density matrix, with N up to 100, finds that N -partite entanglement is observable despite losses and without post-selection. Genuine multi-partite entanglement up to at least is similarly confirmed. The preparation method can be made intrinsically permutation-invariant, allowing highly efficient state reconstruction. In this scenario, generation plus detection requires experimental resources, in stark contrast to the typical exponential scaling. We estimate existing detectors could observe 1000-partite entanglement from a few dB of polarization squeezing.