Spin squeezing and entanglement for an arbitrary spin

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 Sørensen 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.

Figure 1

Different types of spin-squeezed states. (a) Almost fully polarized spin-squeezed states detected by ${\xi }_{\mathrm{s},j}^2,$ given in Eq. (5) and also by the new parameter ${\xi }_{\mathrm{os}}^2,$ defined in Eq. (6). (b) States close to symmetric Dicke states with $\langle J_z\rangle =0$ 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 ${\xi }_{\mathrm{os}}^2$ but are not detected by ${\xi }_{\mathrm{s},j}^2.$ (c) States close to many-body singlets with a small variance for all the three angular momentum components. Such states are detected by the criterion (9b). (d) 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).

Figure 2

(a) The polytope of separable states corresponding to Eq. (9) for $N=10$ spin-$j$ particles and for $\vec{J}=0.$ The completely mixed state defined in Eq. (39) corresponds to the origin of the coordinate axis, i.e., the point $(0,0,0)$ and it is inside the polytope. (b) The same polytope for $\vec{J}=(0,0,8)j.$ Note that this polytope is a subset of the polytope in (a). For the coordinates of the points $A_l$ and $B_l$ see Eq. (21).

Figure 3

Different types of spin-squeezed states violating the criterion based on the spin-squeezing parameter ${\xi }_{\mathrm{os}}^2$ [Eq. (6)] having $\vec{J}=0.$ (a) Mixture of two almost completely polarized spin-squeezed states pointing into opposite directions. (b) Mixture of several almost completely polarized spin-squeezed states. The original spin-squeezing inequality based on the spin-squeezing parameter and its generalization for arbitrary spin $j,$ ${\xi }_{\mathrm{s},j}^2$ [Eq. (5)] cannot detect these states since for these states the mean spin is zero.