Spin squeezing and 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 $J_k$. 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.

Figure 1

(Color online) (a) The polytope of separable states corresponding to Eq. (7) for $N=10$ and for $\vec{J}=\vec{0}$. The origin of the coordinate system corresponds to a many-body singlet state. (b) The same polytope for $\vec{J}=\left(0,0,4\right)$. Note that this polytope is a subset of the polytope in (a).

Figure 2

(Color online) The polytope of separable states corresponding to Eq. (7) for $N=10$ and for $\vec{J}=\vec{0}$. The points corresponding to random separable states fill the polytope.

Figure 3

(Color online) The polytope of separable states corresponding to Eq. (32) for the eigenvalues of $\mathfrak{X}$ for $N=10$ and for $\vec{J}=\vec{0}$. Compare with Fig. 1a.

Figure 4

(Color online) (a) Comparison of the optimal spin-squeezing inequalities and original spin squeezing for $\left(k,l,m\right)=\left(x,y,z\right)$, $N=10$, and $\vec{J}=\left(1,0,2\right)$. States detected by the latter are below the horizontal plane. (b) Optimal spin-squeezing inequalities and the inequality (40) for $N=10$ and $\vec{J}=\left(0,0,0\right).$ (c) Optimal spin squeezing and criterion (45) for $\vec{J}=\left(0,0,0\right).$

Figure 5

(Color online) Comparison of the critical temperatures of separability criteria for different site numbers in the Heisenberg-(left) and the $XY$ model (right). $T_c$ of the spin-squeezing inequality (7b) $(+)$ is higher than the critical temperatures of the PPT $(○)$ criterion [53] or the CCNR (◇) criterion [55].

Figure 6

(Color online) Schematic picture of the ${\text{Na}}_2{\text{V}}_3{\text{O}}_7$ system with coupling parameters $C_1$ and $C_2$ of the nine spin-$\frac{1}{2}$ Heisenberg ring model.