Exploring the local orthogonality principle

Nonlocality is arguably one of the most fundamental and counterintuitive aspects of quantum theory. Nonlocal correlations could, however, be even more nonlocal than quantum theory allows, while still complying with basic physical principles such as no-signaling. So why is quantum mechanics not as nonlocal as it could be? Are there other physical or information-theoretic principles which prohibit this? So far, the proposed answers to this question have been only partially successful, partly because they are lacking genuinely multipartite formulations. In Fritz et al. [Nat. Commun. 4, 2263 (2013)], we introduced the principle of local orthogonality (LO), an intrinsically multipartite principle which is satisfied by quantum mechanics but is violated by nonphysical correlations. Here we further explore the LO principle, presenting additional results and explaining some of its subtleties. In particular, we show that the set of no-signaling boxes satisfying LO is closed under wirings, present a classification of all LO inequalities in certain scenarios, show that all extremal tripartite boxes with two binary measurements per party violate LO, and explain the connection between LO inequalities and unextendible product bases.

Figure 1

Schematic representation of the sets of no-signaling correlations (outer pentagon), quantum correlations (gray area), and classical correlations (striped area). The lines $B_1$ and $B_2$ separating the set of classical correlations from the nonlocal ones are examples of tight Bell inequalities. While $B_1$ is violated by some quantum correlations, $B_2$ is only violated by supraquantum boxes. In fact, $B_2$ is an example of a Bell inequality that is tight, as it corresponds to a facet of the set of classical correlations, nontrivial, because it is violated by no-signaling correlations, but with no quantum violation.

Figure 2

Orthogonality graph of the $(2,2,2)$ scenario. As mentioned in the text, each possible event corresponds to a node, while the edges connect locally orthogonal events.

Figure 3

$k$ copies of a PR box shared among $2k$ parties. Each party has access to one part of one box.

Figure 4

(a) Orthogonality graph of possible events for a PR box. It coincides with Fig. 2 in [36], where the authors study the CHSH inequality. (b) Nonorthogonality graph of the PR box, i.e., the complement of (a). In graph-theory terms, this is the circulant graph $C{i}_8(1,2)$ and also the 4-antiprism graph [37].

Figure 5

Three copies of a bipartite box wired between three groups of two parties each. The resulting wired box is tripartite, where each new party has access to one end of two boxes. To provide an example, consider a grouped party receiving input $y$. They choose to use first the box on their left-hand side, to which they input $f_1\left(y\right)$ and obtain an outcome $a_1$. Then, they use the box on the right-hand side by inputting $f_2(y,a_1)$ and obtaining an outcome $a_2$. Finally, they output $b=g(y,a_1,a_2)$ as the final outcome of the protocol. Here, $f_1$, $f_2$, and $g$ can be arbitrary functions.

Figure 6

(a) Optimal one-dimensional packing proving that $\alpha \left(G^{⊠1}\right)=2$ and that $\Theta \left(G\right)\ge 2$. (b) Optimal two-dimensional packing proving that $\alpha \left(G^{⊠2}\right)=5$ and that $\Theta \left(G\right)\ge \sqrt{5}$.