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-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. Sorensen 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.
Highlights
One of the most rapidly developing areas in quantum physics is creating larger and larger entangled quantum systems with photons, trapped ions, and cold neutral atoms [1,2,3,4,5,6,7,8,9,10,11,12]
(ii) We present a generalization of the original spin squeezing parameter ξs2 defined in Eq (2) that can be used for entanglement detection even for particles with j
We present our spin-squeezing inequalities for particles with an arbitrary spin j and we examine the connection of these inequalities to the entanglement of the reduced two-particle state, and to the criterion based on the positivity of the partial transpose
Summary
One of the most rapidly developing areas in quantum physics is creating larger and larger entangled quantum systems with photons, trapped ions, and cold neutral atoms [1,2,3,4,5,6,7,8,9,10,11,12]. 1 2 case, which we will call optimal spin-squeezing inequalities for spin-j particles They are a complete set since, for large N, they detect all entangled states that can be detected knowing only the first moments and the modified second moments. Jz = 0 with a small variance for one of the angular momentum components and large variances in the two orthogonal directions Such states can be detected by ξo2s but are not detected by ξs2,j. Planar squeezed states with a small variance for two of the angular momentum components and a large variance in the orthogonal direction. Such states are detected by the criterion (9d). We present our spin-squeezing inequalities for particles with an arbitrary spin j and we examine the connection of these inequalities to the entanglement of the reduced two-particle state, and to the criterion based on the positivity of the partial transpose
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