bibkey: halliwellmawby2019finentimes
authors: J. J. Halliwell; C. Mawby
year: 2019
title: "Fine's theorem for Leggett-Garg tests with an arbitrary number of measurement times"
doi: 10.1103/PhysRevA.100.042103
url: https://arxiv.org/abs/1906.04865v2
claim: "For a dichotomic variable measured at n times with compatible nonnegative pairwise probabilities, the n-gon inequalities and all n-time Leggett-Garg inequalities are necessary for a joint probability matching the pairwise data; Section 4.B conjectures that the n-gon inequalities together with all three-time Leggett-Garg inequalities are sufficient, and proves this for n = 4 and n = 5 in the symmetric case."
strata_touched:
  - D5/S3/Quantum/Contextuality/LeggettGargSixTimeSufficiencyRefutation
license: citation-only
triage: anchor

Halliwell and Mawby, Fine’s theorem for Leggett-Garg tests with an arbitrary number of measurement times

J. J. Halliwell and C. Mawby, Phys. Rev. A 100, 042103 (2019); arXiv:1906.04865v2. Quotations are from the v2 TeX source.

A dichotomic variable $Q=\pm 1$ is measured at times $t_1,\dots ,t_n$, giving pairwise probabilities $p(s_i,s_j)$ that agree on shared times, with correlators $C_{ij}=⟨Q_i{Q}_j⟩$. Section 4.A derives the $n$-gon inequalities $n+2{\sum }_{i<j}{s}_i{s}_j{C}_{ij}\ge 1$ for odd $n$ and $\ge 0$ for even $n$. Section 4.B, “Sufficient Conditions: A Conjecture”, assumes all two-time inequalities and states:

A set of sufficient conditions for the existence of a joint probability matching all possible pairwise probabilities consists of the $n$-gon equalities Eq.(\ref{neweq}) together with all possible three-time LG inequalities of the form $$1+s_i{s}_j{C}_{ij}+s_i{s}_k{C}_{ik}+s_j{s}_k{C}_{jk}\ge 0,$$ where $i<j<k$.

The paper proves the conjecture for $n=4$ and $n=5$ in the symmetric case, where correlators of an odd number of variables vanish.

Verified locator

Source: https://arxiv.org/abs/1906.04865v2

• DOI: https://doi.org/10.1103/PhysRevA.100.042103 (Physical Review A 100(4), article 042103; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/1906.04865v2. The e-print member LGn_paper_revised.tex has 1092 lines and md5 9f186d04357875722ffef09fcdb26481. Locations:
  • compatibility of pairwise probabilities, l. 150–153; correlators, l. 218–220;
  • the $n$-gon inequalities, l. 494–512;
  • the conjecture and its proved cases, l. 548–566.