Proofs of Network Quantum Nonlocality in Continuous Families of Distributions

The study of nonlocality in scenarios that depart from the bipartite Einstein-Podolsky-Rosen setup is allowing one to uncover many fundamental features of quantum mechanics. Recently, an approach to building network-local models based on machine learning led to the conjecture that the family of quantum triangle distributions of [Renou et al., Phys. Rev. Lett. 123, 140401 (2019)] did not admit triangle-local models in a larger range than the original proof. We prove part of this conjecture in the affirmative. Our approach consists of reducing the family of original, four-outcome distributions to families of binary-outcome ones, and then using the inflation technique to prove that these families of binary-outcome distributions do not admit triangle-local models. This constitutes the first successful use of inflation in a proof of quantum nonlocality in networks for distributions whose nonlocality could not be proved with alternative methods. Moreover, we provide a method to extend proofs of network nonlocality in concrete distributions of a parametrized family to continuous ranges of the parameter. In the process, we produce a large collection of network Bell inequalities for the triangle scenario with binary outcomes, which are of independent interest.

Figure 1

(Top left) Realization of the family of quantum genuine triangle nonlocal distributions $P_u(a,b,c)$ of Ref. [16] [for completeness, also defined in Eq. (2)]. Each of the sources distributes a maximally entangled state $|{\psi }^{+}⟩\propto |01⟩+|10⟩$, and each of the parties performs the four-outcome measurement given by $\{|\overline{0}⟩,|{\overline{1}}_{+}⟩,|{\overline{1}}_{-}⟩,|\overline{2}⟩\}$ described in the main text. (Top right) Implications from the existence of a triangle-local model for $P_u(a,b,c)$, in the notation used in this Letter (see the Supplemental Material [17]). Reference [16] used only condition (i) for asserting the nonlocality of $P_u(a,b,c)$. In this Letter we analyze when condition (ii) is not true. (Bottom) The current standing of the proofs of nonlocality for the family.