Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems

We develop criteria to detect three classes of nonlocality that have been shown by Wiseman et al. [Phys. Rev. Lett. 98, 140402 (2007)] to be nonequivalent: entanglement, EPR steering, and the failure of local hidden-variable theories. We use the approach of Cavalcanti et al. [Phys. Rev. Lett. 99, 210405 (2007)] for continuous variables to develop the nonlocality criteria for arbitrary spin observables defined on a discrete Hilbert space. The criteria thus apply to multisite qudits, i.e., systems of fixed dimension $d$, and take the form of inequalities. We find that the spin moment inequalities that test local hidden variables (Bell inequalities) can be violated for arbitrary $d$ by optimized highly correlated nonmaximally entangled states provided the number of sites $N$ is high enough. On the other hand, the spin inequalities for entanglement are violated and thus detect entanglement for such states, for arbitrary $d$ and $N$, and with a violation that increases with $N$. We show that one of the moment entanglement inequalities can detect the entanglement of an arbitrary generalized multipartite Greenberger-Horne-Zeilinger state. Because they involve the natural observables for atomic systems, the relevant spin-operator correlations should be readily observable in trapped ultracold atomic gases and ion traps.

Figure 1

Three famous types of quantum nonlocality: Bell nonlocality is a stronger result than EPR steering, which is stronger than entanglement.

Figure 2

Hybrid model of Ref. [11] involves different local hidden states, either quantum (LQS) or local hidden variable (LHV), at each spatially separated site $1$ and $2$. The asymmetric use of quantum uncertainty relations that results because of this gives rise to criteria for steering and the EPR paradox (EPR steering) [12].

Figure 3

Spin $1/2$: Entanglement (dotted), EPR steering ($T=1$) (dashed), and Bell nonlocalities (solid) can be detected for GHZ states of Eq. (49) using $C_J$ criteria in Eqs. (47) and (48) and Bell inequalities (43) when $B_{\mathrm{ent}},B_{\mathrm{EPR}},B_{\mathrm{Bell}}>1$. $B_{\mathrm{ent}}$ for the generalized HZ entanglement criterion (25), i.e., Eq. (44), becomes infinite in this case.

Figure 4

Spin-$1$ case for the maximally entangled state (37). Nonlocality detected by the Bell inequality (31) and the $C_J$ entanglement and EPR steering criteria (34) and (35). The nonlocality is detected when the appropriate $B>1$. The entanglement measured by the generalized HZ entanglement criterion (25) is plotted for comparison.

Figure 5

Spin-$1$ case for the nonmaximally entangled state (42) with optimal $r.$ The amount of nonlocality $B_{\mathrm{Bell}}$ (solid), $B_{\mathrm{EPR}}\hspace{0.5em}(T=1)$ (dashed), and $B_{\mathrm{ent}}$ (dotted) detected by the Bell inequality (31) and $C_J$ criteria (35) and (34) is larger than for state (37) shown in Fig. 4. Here, $B_{\mathrm{Bell}},B_{\mathrm{EPR}},B_{\mathrm{ent}}>1$ implies the detection of the respective nonlocality. The entanglement detected by the generalized HZ entanglement criterion (25) with optimal $r$ is shown in here for comparison.

Figure 6

Violation of the Bell inequality (31) as measured by $B_{\mathrm{Bell}}$ vs $N$ for the symmetric nonmaximally entangled state (41): (a) $J=1,2,3$ and (b) $J=1/2,3/2,5/2$. Bell nonlocality is confirmed when $B_{\mathrm{Bell}}>1$.

Figure 7

Minimum number of sites $N$ needed for a given $d$ ($d=2J+1$) in order to violate the Bell inequality (31) for a nonmaximally entangled state (41): even $d$ (dashed curve), odd $d$ (solid curve).

Figure 8

Entanglement as measured by the $C_J$ criterion (34): $B_{\mathrm{ent}}$ vs $N$ for (a) the maximally entangled state (37) and (b) the optimal nonmaximally entangled state (41). Entanglement is confirmed when $B_{\mathrm{ent}}>1$.

Figure 9

Entanglement as measured by the generalized HZ entanglement criterion (25): $B_{\mathrm{ent}}$ vs $N$ for (a) the maximally entangled state (37) and (b) the optimal nonmaximally entangled state (41). Entanglement is confirmed when $B_{\mathrm{ent}}>1$.

Figure 10

Bipartite case: Entanglement as measured by the generalized HZ criterion (71) (dashed curve) and the $C_J$ criterion (solid curve): $B_{\mathrm{ent}}$ vs $d=2J+1$ for (a) the bipartite maximally entangled state (37) and (b) the nonmaximally entangled state (41) with optimal $r_m$. Entanglement is confirmed when $B_{\mathrm{ent}}>1$.

Figure 11

Effect of increasing the number of sites $N$ for fixed $d$ on the strength $B$ of the nonlocality as measured by the Bell inequality (31) and the $C_J$ criteria (34), (35), and (36): (a) $d=6$ ($J=5/2$) and (b) $d=21$ ($J=10$). Results are for the nonmaximally entangled states (41): entanglement (dotted) can be detected when $B_{\mathrm{ent}}>1$; Bell nonlocalities (solid) can be detected when $B_{\mathrm{Bell}}>1$. The $C_J$ criterion (36) for the more general model of $T$ quantum sites can be verified when $B_T>1$.

Figure 12

Entanglement, EPR steering, and Bell nonlocalities vs $d$ (i.e., $d=2J+1$) as measured by the $C_J$ criteria (34), (35) and the Bell inequality (31) for the nonmaximally entangled states (41): $N=10$. The nonlocality is confirmed when $B>1$.