Many-box locality

There is an ongoing search for a physical or operational definition for quantum mechanics. Several informational principles have been proposed which are satisfied by a theory less restrictive than quantum mechanics. Here, we introduce the principle of “many-box locality,” which is a refined version of the previously proposed “macroscopic locality.” These principles are based on coarse graining the statistics of several copies of a given box. The set of behaviors satisfying many-box locality for $N$ boxes is denoted $L_N^{MB}$. We study these sets in the bipartite scenario with two binary measurements, in relation with the sets $\mathcal{Q}$ and ${\mathcal{Q}}_{1+AB}$ of quantum and “almost quantum” correlations, respectively. We find that the $L_N^{MB}$ sets are, in general, not convex. For unbiased marginals, by working in the Fourier space we can prove analytically that $L_N^{MB}⊈\mathcal{Q}$ for any finite $N$, while $L_{\infty }^{MB}=\mathcal{Q}$. Then, with suitably developed numerical tools, we find an example of a point that belongs to $L_{16}^{MB}$ but not to ${\mathcal{Q}}_{1+AB}$. Among the problems that remain open is whether $\mathcal{Q}\subset L_{\infty }^{MB}$.

Figure 1

Comparison between the usual Bell scenario and the many-box scenario. (a) The usual Bell scenario. We called this a “box.” (b) The many-box scenario. An “$N$-box” is formed by combining $N$ boxes together in a particular way: All individual boxes are identical and independent, the same setting is provided to each box, and the outcome of each box is cumulated to produce the outcome of the many box; i.e., $A={\sum }_i{a}_i$, $B={\sum }_i{b}_i$.

Figure 2

The $L_1^{MB}$, $L_2^{MB}$, and $L_3^{MB}$ in the no-signaling set. The region inside the dashed brown lines is the set $L_1^{MB}$, which is the local set. Here, the thick black, the blue with star markers, and the red lines are the boundary of $Q$, $L_2^{MB}$, and $L_3^{MB}$, respectively.

Figure 3

The boundary of the $L_N^{MB}$ sets in the plane $\gamma =0$ for $N$ from 2 to 10. Here, solid lines from top to bottom are $NS$, $Q$, $L_{10}^{MB}$, $L_8^{MB}$, $L_6^{MB}$, $L_4^{MB}$, $L_2^{MB}$, respectively, and dashed lines from top to bottom are $L_9^{MB}$, $L_7^{MB}$, $L_5^{MB}$, $L_3^{MB}$, $L_1^{MB}$, respectively. Note that the shaded is the set $MB{L}_1$, i.e., the local set. In particular, we magnify the region inside the green rectangle box to show more details.

Figure 4

The boundary of the $L_N^{MB}$ sets in the plane $\beta =0$ for $N$ from 2 to 10. Here, solid lines, from top to bottom, are $NS$, $Q$, $L_{10}^{MB}$, $L_8^{MB}$, $L_6^{MB}$, $L_4^{MB}$, $L_2^{MB}$, respectively, and dashed lines from top to bottom are $L_9^{MB}$, $L_7^{MB}$, $L_5^{MB}$, $L_3^{MB}$, $L_1^{MB}$, respectively. Note that the shaded area is the set $L_1^{MB}$, i.e., the local set.