Multipartite entanglement detection with nonsymmetric probing

We show that spin-squeezing criteria commonly used for entanglement detection can be erroneous if the probe is not symmetric. We then derive a lower bound on squeezing for separable states in spin systems probed asymmetrically. Using this we further develop a procedure that allows us to verify the degree of entanglement of a quantum state in the spin system. Finally, we apply our method for entanglement verification to existing experimental data, and use it to prove the existence of tripartite entanglement in a spin-squeezed atomic ensemble.

Figure 1

(a) Entanglement criteria for symmetric (dashed) and asymmetric (solid) probing for an atomic ensemble in a cylinder probed by a narrow Gaussian laser beam ($\nu =0.3$). The black, red, magenta and blue (listed from the top) curves correspond, respectively, to the minimum possible variance when allowing for separable, $n=2$, 5, and 20 particle-entangled states. The numbers between the lines therefore express the smallest possible group of entangled atoms in the corresponding regions. (b) Lower bounds for the normalized variance with respect to the average spin for a pair of atoms. The different curves correspond to different values of the coefficients ${\eta }_{2i-1}$ and ${\eta }_{2i}$, and were determined using a general wave function $|\psi \rangle =C_{\uparrow \uparrow }|\uparrow \uparrow \rangle +C_{\uparrow \downarrow }|\uparrow \downarrow \rangle +C_{\downarrow \uparrow }|\downarrow \uparrow \rangle +C_{\downarrow \downarrow }|\downarrow \downarrow \rangle$. Along with ${\eta }_{2i-1}$ and ${\eta }_{2i}, |\psi \rangle$ fully specify the state of the system; we can therefore use the complex numbers $C_{kl}$ ($k,l=\uparrow$ or $\downarrow$) as variables in the Lagrange $\mathrm{\Gamma }$ function, and obtain the minimal noise for each value of the mean spin, and ${\eta }_1$ and ${\eta }_2$. Letting the fraction ${\eta }_{2i-1}/{\eta }_{2i}$ vary continuously in the interval $[0,1]$, we find that the minimal noise is obtained for ${\eta }_1={\eta }_2$.

Figure 2

Entanglement and three-particle entanglement criteria for the experiment described in Ref. [27]. The solid curves give bounds on the variance assuming a separable state, while the dashed curves give the bound when allowing for bipartite entanglement. The colors blue (top), black (middle), and red (bottom) correspond respectively to symmetric probing, asymmetric probing with no displacement, and asymmetric probing with displacement between the FORT and the probes. Parameters used are $T=50\mu \mathrm{K}, {\omega }_t=50\mu \mathrm{m}, {\lambda }_t=1032\mathrm{nm}, {\omega }_p=27\mu \mathrm{m}, {\lambda }_p=852\mathrm{nm}$, and $V_0/k_B=1.73\times {10}^{-4}\mathrm{K}$.