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arXiv:1310.2269 · Physical Review A

Spin squeezing and entanglement for an arbitrary spin

Giuseppe Vitagliano, Iagoba Apellaniz, I. L. Egusquiza, Géza Tóth

Published Mar 7, 2014• 78 citations
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Abstract

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-$\frac{1}{2}$ 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 S\o{}rensen et al. [Nature (London) 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.

DOI
10.1103/physreva.89.032307

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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-12\frac{1}{2}21​ particles to entanglement conditions for spin-jjj 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 S\o{}rensen et al. [Nature (London) 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 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.
What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized spin squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular momentum components. They can be used for the experimental detection of entanglement in a system of spin‐12 particles in which the spins cannot be individually addressed. We also show how to obtain the complete set of entanglement conditions if the mean spin and the spin correlation matrix are known. Finally, we apply our criteria for entanglement detection in spin models, showing that they can be used to detect bound entanglement in the thermal state of these systems.
What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized-spin-squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular-momentum components Jk{J}_{k}Jk​. They can be used for the experimental detection of entanglement in a system of spin-12\frac{1}{2}21​ particles in which the spins cannot be individually addressed. We present various sets of inequalities that can detect all entangled states that can be detected based on the knowledge of (i) the mean values and variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (ii) the variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (iii) the mean values of Jk2{J}_{k}^{2}Jk2​ in three orthogonal directions, or (iv) the mean values and variances of Jk{J}_{k}Jk​ in arbitrary directions. We compare our inequalities to known spin-squeezing entanglement criteria and discuss to which extent spin squeezing is related to entanglement in the reduced two-qubit states. Finally, we apply our criteria for the detection of entanglement in spin models, showing that they can be used to detect bound entanglement in these systems.
Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective N -particle s u ( d ) operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the ( d 2 − 1 ) -dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of N qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding s u ( d ) -squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective N -particle s u ( d ) operators. We prove that, in the limit N ≫ 1 , such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the s u ( d ) singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the s u ( 3 ) -squeezing inequalities.
Particle number fluctuations, no matter how small, are present in experimental setups. One should rigorously take these fluctuations into account, especially for entanglement detection. In this context, we generalize the spin-squeezing inequalities introduced by G. T\'oth et al. [Phys. Rev. Lett. 99, 250405 (2007).]. These new inequalities are fulfilled by all separable states even when the number of particles is not constant and may present quantum fluctuations. These inequalities are useful for detecting entanglement in many-body systems when the superselection rule does not apply or when only a subspace of the total system Hilbert space is considered. We also define general dichotomic observables for which we obtain a coordinate-independent form of the generalized spin-squeezing inequalities. We give an example where our generalized coordinate-independent spin-squeezing inequalities present a clear advantage over the original ones.
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 spin-squeezing 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.