Quantum Bell inequalities from macroscopic locality

We propose a method to generate analytical quantum Bell inequalities based on the principle of macroscopic locality. By imposing locality over binary processings of virtual macroscopic intensities, we establish a correspondence between Bell inequalities and quantum Bell inequalities in bipartite scenarios with dichotomic observables. We discuss how to improve the latter approximation and how to extend our ideas to scenarios with more than two outcomes per setting.

Figure 1

A slice of the 3322 no-signaling polytope defined by the two extremal points $P_1$, $P_2$ ($I_{3322}=1$), and $P_n$ (random noise). In this slice, the bound for the local set $L$ and $Q$ are the same, which is $0$. The notation of the points is taken from [18] and the extremality of $P_1$ and $P_2$ follows from [20, 21].

Figure 2

(a) ML in the sense of binning into infinitely many parts. (b) Sign binning for scenarios with binary outcomes. (c) 3-binning induced by the probability $P(x)$. (d) Triangle binning for scenarios with three outcomes.

Figure 3

The line of random marginals defined by the random noise $P_n$ and the generalized PR box, $P_{\mathit{PR}}$ [23]. The bounds given by the local set $L$, macroscopic locality $Q^1$, and triangle binning $Q^{\mathrm{TB}}$ are 0.5, 0.75, and 0.73, respectively.