Genuine multipartite nonlocality in the one-dimensional ferromagnetic spin-1/2 chain

Genuine multipartite entanglement has been found in some spin chain systems. However, genuine multipartite nonlocality is much rarer than genuine multipartite entanglement. Here we present genuine multipartite nonlocality in a spin chain system. After introducing the definition of genuine multipartite nonlocality and a multipartite Bell-type inequality, we construct a group of joint measurements for the inequality in a one-dimensional ferromagnetic $N$-qubit chain with nearest-neighbor $XXZ$ interaction, and many violations to the inequality have been found. The violations do indicate that genuine multipartite nonlocality exists in this ferromagnetic spin-1/2 chain system. Last but not least, we also calculate genuine multipartite entanglement concurrence in the same spin chain to demonstrate the difference and relationship between genuine multipartite nonlocality and genuine multipartite entanglement.

Figure 1

The maximal violation of Eq. (4) is indicated by a solid line with changing $b/J_z$ and remaining $J_x/J_z$ unchanged. The value of the maximal violation of the 6,7,8-partite spin chain are respectively shown in (a)–(c). The maximum value of violation has steplike plateaus and a reflection symmetry structure. Different storey denotes different $|G_N^k\rangle$ in subfigures. From left to right, the GS is $|G_N^0\rangle ,|G_N^1\rangle ,\cdots ,|G_N^N\rangle$, respectively.

Figure 2

The solid line indicates the maximal violation of Eq. (4) with changing $J_x/J_z$ and remaining $b/J_z$ unchanged. The value of the maximal violation of the 6,7,8-partite spin chain are respectively shown in (a)–(c). In each subfigure, the GS is $|G_N^0\rangle ,|G_N^1\rangle ,\cdots ,|G_N^{\lfloor N/2\rfloor }\rangle$ respectively from left to right. The value of violation still varies gradiently at boundary values but decreases tardily with increasing of $J_x/J_z$.

Figure 3

Nodes indicate values of the maximal violation on $\langle G_N^1\left|\widehat{\mathcal{H}}\right|G_N^1\rangle$ with each $N$ from 4 to 43. The analog function of the solid line suggests that the result approaches a limit value greater than zero.

Figure 4

The solid line indicates GME concurrence with changing $b/J_z$ and remaining $J_x/J_z$ unchanged. The value of $C_{GME}$ of the 6,7,8-partite spin chain are respectively shown in (a)–(c). $C_{GME}$ has a steplike plateau and a reflection symmetry structure like Fig. 1. Different storey denotes different $|G_N^k\rangle$ in subfigures. From left to right, the GS is $|G_N^0\rangle ,|G_N^1\rangle ,\cdots ,|G_N^N\rangle$, respectively.

Figure 5

The solid line indicates GME concurrence with changing $J_x/J_z$ and remaining $b/J_z$ unchanged. The value of $C_{GME}$ of the 6,7,8-partite spin chain are respectively shown in (a)–(c). In each subfigure, the GS is $|G_N^0\rangle ,|G_N^1\rangle ,\cdots ,|G_N^{\lfloor N/2\rfloor }\rangle$ respectively from left to right. $C_{GME}$ varies gradiently at boundary values and decreases tardily with increasing $J_x/J_z$ in each subarea.