Generating nonclassical correlations without fully aligning measurements

We investigate the scenario where spatially separated parties perform measurements in randomly chosen bases on an $N$-partite Greenberger-Horne-Zeilinger state. We show that without any alignment of the measurements, the observers will obtain correlations that violate a Bell inequality with a probability that rapidly approaches 1 as $N$ increases and that this probability is robust against noise. We also prove that restricting these randomly chosen measurements to a plane perpendicular to a common direction will always generate correlations that violate some Bell inequality. Specifically, if each observer chooses their two measurements to be locally orthogonal, then the $N$ observers will violate one of two Bell inequalities by an amount that increases exponentially with $N$. These results are also robust against noise and perturbations of each observer’s reference direction from the common direction.

Figure 1

(a) Contour plot of the probability of violating ${\mathcal{S}}_1^N$ or ${\mathcal{S}}_2^N$ for $N=6$ when sampling measurements via PROM with a reference direction as in Eq. (38). (b) Contour plot of the probability of violating one of the WWZB inequalities for $N=6$ when sampling measurements via PROM with the reference direction as defined in Eq. (38). In both contour plots, $\lambda =0^{°}$ in the center and increases radially to a maximum of ${90}^{°}$, and $p_{\mathcal{I}}=1$ for smaller $\lambda$ and generally decreases as $\lambda$ increases along a line of fixed $\alpha$ (i.e., along a radial line).

Figure 2

Probability of violation $p_{\mathrm{NL}}$, sampled using RIM for $N=2,\dots ,6$ observers, as a function of the noise parameter $\nu$ for (a) depolarizing noise and (b) dephasing noise.

Figure 3

Probability of violation $p_{\mathrm{NL}}$, sampled using ROM for $N=2,\dots ,6$ observers, as a function of the noise parameter $\nu$ for (a) depolarizing noise and (b) dephasing noise.

Figure 4

Representation of the normal (${\mathrm{n⃗}}_k$) and measurement directions (${\Omega }_{s_0}^k$ and ${\Omega }_{s_0}^k$) for several parties.

Figure 5

(a) Probability of violation $p_{\mathrm{MABK}}$ when measurements are sampled using orthogonal measurements in the plane perpendicular to the normals ${\mathrm{n⃗}}_k$ that are distributed according to a pseudo-Gaussian distribution as a function of the standard deviation of the distribution of polar angles ${\lambda }_{\mathrm{std}}$ for $N=2,\dots ,6$ observers. (b) Increase in the probability of observers finding that their correlation functions do not correspond to a locally causal model when they check against the the full set of $2^N$ MABK inequalities, rather than just ${\mathcal{S}}_1^N$ and ${\mathcal{S}}_2^N$.