Entanglement depth for symmetric states

Entanglement depth characterizes the minimal number of particles in a system that are mutually entangled. For symmetric states, there is a dichotomy for entanglement depth: An $N$-particle symmetric state is either fully separable or fully entangled—the entanglement depth is either 1 or $N$. We show that this dichotomy property for entangled symmetric states is even stable under nonsymmetric noise. We propose an experimentally accessible method to detect entanglement depth in atomic ensembles based on a bound on the particle number population of Dicke states, and demonstrate that the entanglement depth of some Dicke states, for example the twin Fock state, is very stable even under a large arbitrary noise. Our observation can be applied to atomic Bose-Einstein condensates to infer that these systems can be highly entangled with the entanglement depth that is on the order of the system size (i.e., several thousands of atoms).

Figure 1

(a) $p_{N,r}$ with different $N$ and $r$ for Dicke states $|D_{N,r}\rangle$. The vertical axis is $p_{N,r}$, and the horizontal axis is $r/N$. From top to bottom, the number of particles $N$ increases as $N=10–60$. With increasing $N$, the data collapse onto the same line. And the $N=60$ line is already very close to the limit case of $N\to \infty$. (b) $p_{N,N/2}$ for twin Fock state $|D_{N,N/2}\rangle$ with different $N$ and its finite size extrapolation.

Figure 2

(a) $p_{N,r}^{\prime }$ with different $N$ and $r$ for Dicke states $|D_{N,r}\rangle$. The vertical axis is $p_{N,r}^{\prime }$, and the horizontal axis is $r/N$. From bottom to top, the number of particles $N$ increases as $N=50,100,5\times {10}^2,1\times {10}^3,5\times {10}^3,1\times {10}^4$. The top line with $N=1\times {10}^4$ is close to the $N\to \infty$ limit case with $p_{N,r}^{\prime }=1$. (b) $p_{N,N/2}^{\prime }$ with different $N$'s for twin Fock state $|D_{N,N/2}\rangle$. For clarity, the horizontal axis is $\ln N$.