Recent experiments demonstrate the production of many thousands of neutral atoms entangled in their spin degrees of freedom. We present a criterion for estimating the amount of entanglement based on a measurement of the global spin. It outperforms previous criteria and applies to a wider class of entangled states, including Dicke states. Experimentally, we produce a Dicke-like state using spin dynamics in a Bose-Einstein condensate. Our criterion proves that it contains at least genuine 28-particle entanglement. We infer a generalized squeezing parameter of -11.4(5)  dB.

Entanglement is an important resource that allows quantum technologies to go beyond the classically possible. There are many ways quantum systems can be entangled, ranging from the archetypal two-qubit case to more exotic scenarios of entanglement in high dimensions or between many parties. Consequently, a plethora of entanglement quantifiers and classifiers exist, corresponding to different operational paradigms and mathematical techniques.However, for most quantum systems, exactly quantifying the amount of entanglement is extremely demanding, if at all possible. This is further exacerbated by the difficulty of experimentally controlling and measuring complex quantum states. Consequently, there are various approaches for experimentally detecting and certifying entanglement when exact quantification is not an option, with a particular focus on practically implementable methods and resource efficiency. The applicability and performance of these methods strongly depends on the assumptions one is willing to make regarding the involved quantum states and measurements, in short, on the available prior information about the quantum system. In this review we discuss the most commonly used paradigmatic quantifiers of entanglement. For these, we survey state-of-the-art detection and certification methods, including their respective underlying assumptions, from both a theoretical and experimental point of view.In the early twentieth century, the phenomenon of quantum entanglement rose to prominence as a central feature of the famous thought experiment by Einstein, Podolsky, and Rosen [1]. Initially disregarded as a mathematical artefact that showcases the incompleteness of quantum theory, the properties of entanglement were largely ignored until 1964, when John Bell famously proposed an experimentally testable inequality able to distinguish between the predictions of quantum mechanics and those of any local-realistic theory [2]. With the advent of the first experimental tests [3], spearheaded by , emerged the realisation that entanglement constitutes a resource for information processing *

Modern quantum technologies in the fields of quantum computing, quantum simulation, and quantum metrology require the creation and control of large ensembles of entangled particles. In ultracold ensembles of neutral atoms, nonclassical states have been generated with mutual entanglement among thousands of particles. The entanglement generation relies on the fundamental particle-exchange symmetry in ensembles of identical particles, which lacks the standard notion of entanglement between clearly definable subsystems. Here, we present the generation of entanglement between two spatially separated clouds by splitting an ensemble of ultracold identical particles prepared in a twin Fock state. Because the clouds can be addressed individually, our experiments open a path to exploit the available entangled states of indistinguishable particles for quantum information applications.

Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete problems.

We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.PACS numbers: 03.67. Mn, 05.50.+q, 42.50.Dv, With an interest towards fundamental questions in quantum physics, as well as applications, larger and larger entangled quantum systems have been realized with photons, trapped ions and cold atoms [1]. Quantum entanglement can be used as a resource for certain quantum information processing tasks [1], and it is also necessary for a wide range of interferometric schemes to achieve the maximum sensitivity in metrology [2]. Hence, the verification of the presence of entanglement is a crucial but exceedingly challenging task, especially in an ensemble of many, say 10 6 −10 12 , particles. In such systems, typically the particles are not accessible individually and only collective operators can be measured. A ubiquitous entanglement criterion in this context is the spin squeezing inequality [3]where N is the number of spin-for l = x, y, z are the collective angular momentum components and j (n) l are the single spin angular momentum components acting on the n th particle. If a state violates Eq. (1), then it is entangled (i.e., not fully separable [4]). Such spin squeezed states [5] have been created in numerous experiments with cold atoms and trapped ions [1,6], and can be used, for instance, in atomic clocks to achieve a precision higher than the shot noise limit [5].Recently, after several generalized spin squeezing inequalities (SSIs) for the detection of entanglement appeared in the literature [7][8][9] and were used experimentally [10], a complete set of such entanglement conditions has been presented in Ref. [11]. However, all of the above mentioned conditions are for spin-1/2 particles (qubits), and so far the literature on systems of particles with j > 1 2 is limited to a small number of conditions, specialized for certain quantum states or particles with a low dimension [7,12,13]. At this point the question arises: Could one obtain a complete set of inequalities for j > 1 2 ? Such conditions would be very relevant from the practical point of view since in most of the experiments the physical spin of the particles is larger than 1 2 and the spin-1 2 subsystems are created artificially. Thus, knowing the full set of entanglement criteria for j > 1 2 , many experiments for realizing large scale entanglement could be technologically less demanding, and fundamentally new experiments could also be carried out. The solution is not simple: Known methods for detecting entanglement for spin-1 2 particles by spin-squeezing cannot straightforward...

A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-1 2 particles to entanglement conditions for spin-j particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in [A. Sørensen et al., Nature 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.

We show how a test of macroscopic realism based on Leggett-Garg inequalities (LGIs) can be performed in a macroscopic system. Using a continuous-variable approach, we consider quantum nondemolition (QND) measurements applied to atomic ensembles undergoing magnetically driven coherent oscillation. We identify measurement schemes requiring only Gaussian states as inputs and giving a significant LGI violation with realistic experimental parameters and imperfections. The predicted violation is shown to be due to true quantum effects rather than to a classical invasivity of the measurement. Using QND measurements to tighten the "clumsiness loophole" forces the stubborn macrorealist to recreate quantum backaction in his or her account of measurement.

We present criteria to detect the depth of entanglement in macroscopic ensembles of spin-j particles using the variance and second moments of the collective spin components. The class of states detected goes beyond traditional spin-squeezed states by including Dicke states and other unpolarized states. The criteria derived are easy to evaluate numerically even for systems of very many particles and outperform past approaches, especially in practical situations where noise is present. We also derive analytic lower bounds based on the linearization of our criteria, which make it possible to define spinsqueezing parameters for Dicke states. In addition, we obtain spin squeezing parameters also from the condition derived in (Sørensen and Mølmer 2001 Phys. Rev. Lett. 86 4431). We also extend our results to systems with fluctuating number of particles. consider a subgroup of  k N particles and define its total spin asWe also need to define a function F J via a minimization over quantum states of such a group aswhere L l are the spin components of the group. In practice, the minimum will be the same if we carry out the minimization over states of a single particle with a spin J [19]. Then, for all pure states with an entanglement depth of at most kholds. It is easy to see that (4) is valid even for mixed states with an entanglement depth of at most k since the variance is concave in the state and F J (X) is convex 5 . Thus, every state that violates (4) must have a depth of entanglement of ( ) + k 1 or larger. It is important to stress that the criterion (4) provides a tight lower bound on ( ) DJ x 2 based on á ñ J .z Spin squeezing has been demonstrated in many experiments, from cold atoms [7, 20-26] to trapped ions [27], magnetic systems [28] and photons [29], and in many of these experiments even multipartite entanglement has been detected using the condition (4) [7,[23][24][25][26]29].Recently, the concept of spin squeezing has been extended to unpolarized states [30][31][32][33][34]. In particular, Dicke states are attracting increasing attention, since their multipartite entanglement is robust against particle loss, and they can be used for high precision quantum metrology [8]. Dicke states are produced in experiments with photons [35,36] and Bose-Einstein condensates [8,37,38]. Suitable criteria to detect the depth of entanglement of Dicke states have also been derived. However, either they are limited to spin-1/2 particles [8,39] or they do not give a tight lower bound on ( ) DJ x 2 based on the expectation value measured for the criterion, concretely,with F J (X) defined as in equation (3) and J = kj as in (2). Our approach is motivated by the fact that equation (4) fails to be a good criterion for mixed states with a low polarization  á ñ + á ñ J J N j y z 2 2 22 . Thus, we consider the second moments á + ñ J J y z 2 2 instead, which are still large for many useful unpolarized quantum states, such as Dicke states. Using the second moments is advantageous even for states with a large spin polarization since c...