bibkey: abbottwechscostabranciard2017gmnc
authors: Alastair A. Abbott, Julian Wechs, Fabio Costa and Cyril Branciard
year: 2017
title: "Genuinely multipartite noncausality"
doi: 10.22331/q-2017-12-14-39
url: https://arxiv.org/abs/1708.07663v2
claim: "In the lazy scenario, the inequality J₁(N) ≥ 0 is valid for every 2-causal N-partite correlation; the paper reports that it is a facet of the 2-causal polytope for N = 2, 3, 4 and conjectures that it is a facet for all N."
strata_touched:
  - D5/S3/Quantum/Contextuality/LazyTwoCausalJOneFacet
license: citation-only
triage: anchor

Abbott–Wechs–Costa–Branciard, Genuinely multipartite noncausality

A. A. Abbott, J. Wechs, F. Costa and C. Branciard, Quantum 1, 39 (2017); arXiv:1708.07663v2. Quotations are from the v2 TeX source.

An $N$-partite correlation $P(\vec{a}|\vec{x})$ is 2-causal if it is a convex mixture, over nonempty proper groups $\mathcal{K}$, of correlations in which the parties of $\mathcal{K}$ act first and the remaining parties may depend on their inputs and outputs (Definition 2); the 2-causal correlations form a polytope whose vertices are the deterministic 2-causal correlations. In the lazy scenario each party has a binary input and outputs $0$ on input $0$. The inequality $J_1(N)={\sum }_{\varnothing ⊊\mathcal{K}⊊\mathcal{N}}P({\vec{a}}_{\mathcal{K}}=\vec{1}|{\vec{x}}_{\mathcal{K}}=\vec{1},{\vec{x}}_{\mathcal{N}\setminus \mathcal{K}}=\vec{0})-P(\vec{a}=\vec{1}|\vec{x}=\vec{1})\ge 0$ holds for all 2-causal correlations (Eq. (10)). The paper states:

It is interesting to ask if this inequality is tight (i.e., facet inducing) for all $N$. For $N=2$ it reduces to the LGYNI inequality which is indeed tight, and for $N=3$ it was also found to be a facet. By explicitly enumerating the vertices of the 2-causal polytope for $N=4$ (of which there are 136 818 592) we were able to verify that $J_1(4)\ge 0$ is indeed also a facet, and we conjecture that this is true for all $N$.

Verified locator

Source: https://arxiv.org/abs/1708.07663v2

• DOI: https://doi.org/10.22331/q-2017-12-14-39 (Quantum 1, article 39; title, authors, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/1708.07663v2. The e-print member genuine_multi_noncausal.tex has 900 lines and md5 03974e3e48407d2a35b64870af0654f3. Locations:
  • Definition 2, l. 146–154; the vertices of the 2-causal polytope, l. 222–227; the lazy scenario, l. 213–214;
  • Eq. (10), l. 333–337; the conjecture, l. 343–344.