Nonlocality and entanglement in the $\mathit{XY}$ model

Nonlocality and quantum entanglement constitute two special features of quantum systems of paramount importance in quantum-information theory (QIT). Essentially regarded as identical or equivalent for many years, they constitute different concepts. Describing nonlocality by means of the maximal violation of two Bell inequalities, we study both entanglement and nonlocality for two and three spins in the $\mathit{XY}$ model. Our results shed light on the description of nonlocality and the possible information-theoretic task limitations of entanglement in an infinite quantum system.

Figure 1

The oscillating curve depicts $B_{\mathrm{CHSM}}^{max}$ [Eq. (8)] vs time (in s) for $h_0=0.5,h_f=0$, $\gamma =0.5$, and $R=1$ for the evolved state ${\rho }_{\mathit{ij}}^{(R)}(t)$ [Eq. (3)] at $T=0$. Final and equilibrium values do not coincide (nonergodic). Inset (a) depicts $B_{\mathrm{CHSM}}^{max}$ vs $h$ for $\gamma =0.5$ and different distances between spins $R=1,2,3$, and $\infty$ (dotted line). Though no Bell violation is observed, $B_{\mathrm{CHSM}}^{max}$ exhibits a long-range order. Inset (b) depicts equilibrium nonlocality (upper curves) and concurrence (lower curves) measures vs $h$ for $\gamma \in [0,1]$ and $R=1$. The isotropic case $\gamma =0$ collapses to 2 for $h⩾1$. See the text for details.

Figure 2

(a) Value of ${\mathcal{M}}^{max}$ vs $h$ (10) for several $\gamma$ values for the $a=b=1$ block configuration. (b) Similar curves for $a=1,b=2$ and $a=2,b=2$, with $\gamma =0.5$. (c) Evolution of multipartite entanglement $S_{\mathit{vN}}$ ($a=b=1$) vs $h$ for several values of the anisotropy. (d) ${\mathcal{M}}^{max}$ vs $S_{\mathit{vN}}$ plots for $\gamma =0.5$ and all previous configurations, $a=b=1$, $a=1,b=2$, and $a=b=2$. Notice the apparent monotonic decreasing evolution. See the text for details.