Nonlocal correlations in a macroscopic measurement scenario

Nonlocality is one of the main characteristic features of quantum systems involving more than one spatially separated subsystem. It is manifested theoretically as well as experimentally through violation of some local realistic inequality. On the other hand, classical behavior of all physical phenomena in the macroscopic limit gives a general intuition that any physical theory for describing microscopic phenomena should resemble classical physics in the macroscopic regime, the so-called macrorealism. In the 2-2-2 scenario (two parties, with each performing two measurements and each measurement having two outcomes), contemplating all the no-signaling correlations, we characterize which of them would exhibit classical (local realistic) behavior in the macroscopic limit. Interestingly, we find correlations which at the single-copy level violate the Bell-Clauser-Horne-Shimony-Holt inequality by an amount less than the optimal quantum violation (i.e., Cirel'son bound $2\sqrt{2}$), but in the macroscopic limit gives rise to a value which is higher than $2\sqrt{2}$. Such correlations are therefore not considered physical. Our study thus provides a sufficient criterion to identify some of unphysical correlations.

Figure 1

Single-pair setup: $X,Y\in \{0,1\}$ are Alice's and Bob's measurements, respectively. After the measurement interaction, going through the paths $a=0$ and $a=1$, particles are collected at Alice's detectors $D_0\left(A\right)$ and $D_1\left(A\right)$, respectively, and similarly on Bob's end at the detectors $D_0\left(B\right)$ and $D_1\left(B\right)$.

Figure 2

Multipair setup: Source produces $M$ independent pairs of particles. Since information about ordering between Alice's and Bob's particles is lost during their transmission, they address the beams of particles as a whole. A particle gets in the detector $D_s\left(\kappa \right)$ if $s=a\in \{0,1\}$ ($s=b\in \{0,1\}$) is the outcome in the measurement $X\in \{0,1\}$ ($Y\in \{0,1\}$) on $\kappa =A$, i.e., Alice's particle ($\kappa =B$, i.e., Bob's particle). A few particles ($n_0$ in number) are collected at the detector $D_0\left(\kappa \right)$, and the rest are collected at the detector $D_1\left(\kappa \right)$, where $\kappa =A,B$.

Figure 3

$A_{\mathbf{XY}}^{\left(M\right)}$ for the probability distribution $\mathcal{P}:=\left\{P\right(00\left|XY\right),P\left(01\right|XY),P(10\left|XY\right),P\left(11\right|XY\left)\right\}=(0,\beta ,\delta ,0)$. Solid red curve: $\beta =0.5$; dotted black curve: $\beta =0.4$; dashed blue curve: $\beta =0.8$.

Figure 4

$A_{\mathbf{XY}}^{\left(M\right)}$ for the probability distribution $\mathcal{P}=(\alpha ,\beta ,\delta ,0).\beta =\delta$ and $\alpha =0.4$ (dotted black curve), $\alpha =0.5$ (solid red curve), and $\alpha =0.6$ (dashed blue curve).

Figure 5

$A_{\mathbf{XY}}^{\left(M\right)}$ for the probability distribution $\mathcal{P}=(0,\beta ,\delta ,\gamma ).\beta =\delta$ and $\gamma =0.6$ (dotted black curve), $\gamma =0.5$ (solid red curve), and $\gamma =0.4$ (dashed blue curve).

Figure 6

$A_{\mathbf{XY}}^{\left(M\right)}$ for the probability distribution $\mathcal{P}=(\alpha ,0,\delta ,\gamma ).\alpha =\gamma$ and $\alpha =0.2$ (dotted black curve), $\alpha =0.25$ (solid red curve), and $\alpha =0.3$ (dashed blue curve).

Figure 7

$A_{\mathbf{XY}}^{\left(M\right)}$ for the probability distribution $\mathcal{P}=(\alpha ,\beta ,\delta ,\gamma ).\alpha =\gamma ,\beta =\delta$ and $\alpha =0.2$ (dashed blue curve), $\alpha =0.25$ (solid red curve), and $\alpha =0.3$ (dotted black curve).

Figure 8

The red surface represents the function $F(p_1,y)$ in Eq. (30). The green surface represents an $F=0$ surface. The points $(p_1,y)$ on the right side of the green surface satisfy the necessary condition of IC.