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arXiv:0806.1048 · Physical Review A

Spin squeezing and entanglement

Géza Tóth, Christian Knapp, Otfried Gühne, Hans J. Briegel

Published Apr 27, 2009• 222 citations
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Abstract

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 ${J}_{k}$. They can be used for the experimental detection of entanglement in a system of spin-$\frac{1}{2}$ 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 ${J}_{k}$ in three orthogonal directions, or (ii) the variances of ${J}_{k}$ in three orthogonal directions, or (iii) the mean values of ${J}_{k}^{2}$ in three orthogonal directions, or (iv) the mean values and variances of ${J}_{k}$ 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.

DOI
10.1103/physreva.79.042334

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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.
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.
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 present detailed derivations, various improvements and application to concrete experimental data of spin squeezing inequalities formulated recently by some of us [Phys. Rev. Lett. 95, 120502 (2005)]. These inequalities generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine two-, or three-qubit entanglement for symmetric states, and sufficient condition for general NNN-qubit states. We apply our method to theoretical study of Dicke states, and, in particular, to W\mathrm{W}W states of NNN qubits. Then, we analyze the recently experimentally generated seven- and eight-ion W\mathrm{W}W states [Nature 438, 643 (2005)]. we also present some details concerning this experiment. Finally, we improve criteria for detection of genuine tripartite entanglement based on entanglement witnesses.
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.