Exploring Bell Nonlocality with Extremal Non-Signaling Boxes

Abstract

Extremal non-signaling (ENS) boxes are correlations that correspond to vertices of the non-signaling polytope of a Bell scenario. Neither quantum theory nor any theory for ideal measurements allows for ENS boxes. That is, according to quantum theory, ENS boxes are nonphysical. Still, ENS boxes are crucial for addressing a number of problems in Bell nonlocality. Here, we obtain ENS boxes in arbitrary bipartite Bell scenarios and present the complete list of ENS boxes for several unexplored scenarios. Equipped with the boxes, we revisit several foundational questions. We find that already two copies of any ENS box violate the exclusivity (or local orthogonality) and Specker's principles. We provide the minimal decomposition of the magic square correlation - the simplest known perfect correlation in nature - in terms of ENS boxes. We identify the minimal scenario in which a dit of communication (with d < 6) is insufficient to simulate ENS boxes. Our results show that the ENS boxes approach leads to new results and opens new avenues for research.

Figures (2)

• Figure 1: Geometric depiction of our main findings described with respect to different correlation sets: local ($\mathcal{L}$, inner set, white), quantum ($\mathcal{Q}$, light gray), and non-signaling ($\mathcal{NS}$, dark gray). Note that $\mathrm{LO}^{(1)}=\mathcal{NS}$, whereas $\mathrm{LO}^{(2)}$ is drawn with blue dashed lines. While local vertices are indicated by green disks, nonlocal non-signaling vertices are distinguished by three different green markers (square, pentagon, and triangle, illustrating the different amounts of classical communication needed to simulate these boxes. The red star represents magic square correlations, which we show can be decomposed into a convex combination of two ENS boxes.
• Figure 2: Illustration showing the bipartite Bell scenarios for which the complete list of ENS boxes is now known. Dark gray cells indicate scenarios solved in this work, while light gray ones correspond to scenarios for which we only conjecture our list of extreme points to be complete. The scenarios $(2,2,d,d)$ and $(d,d,2,2)$ were respectively solved in Barrett2005Masanes2014 and are shown with hatch patterns.