Exact minimum measurement dependence for faithful local deterministic models of multipartite GHZ-Mermin correlations
Abstract
A local deterministic model can reproduce Bell-inequality violations only if the hidden state depends on the measurement settings. Hall found the minimum such dependence for the singlet state [Phys. Rev. Lett. 105, 250404 (2010)] and posed the corresponding problem for the Mermin state [Phys. Rev. A 84, 022102 (2011)]. This paper solves it, for the n-party GHZ-Mermin scenario at every n ≥ 3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is Fmin(n) = R/[2(R+1)], with R = 2⌊ (n−1)/2 ⌋ the Mermin violation ratio: 1/3, 2/5, 4/9, 8/17, …, always below 1/2. The lower bound follows from Mermin's inequality by a general overlap bound; the upper bound is an explicit model, uniform on the hidden-variable classes that violate the fewest constraints, with closed-form counts; a reduction theorem shows that for Mermin scenarios the marginals impose no additional dependence. The Mermin values are also certified by integer-arithmetic enumeration through n = 22. The formula was stated as a conjecture in the first version of this paper and was first proved, by a different route, by Wang, Yao and Chen; the proof here was obtained independently and the two agree. Every minimal contradiction of size k among stabilizer constraints has correlator-only minimum 1/(k−1). With the marginals included the minimum is the same on every core of the C5 ring cluster, on the size-4 and size-5 cores of C4 and on every four-core of C6, but larger on the size-6 and size-7 cores of C4 and on 112 of the 1,920 five-cores of C6, with exact certificates on both sides in every case.
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BibTeX
@misc{alai2026exact,
title = {{Exact minimum measurement dependence for faithful local deterministic models of multipartite GHZ-Mermin correlations}},
author = {Aaron Alai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.00124},
eprint = {2608.00124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00124},
}
RIS
TY - UNPB
TI - Exact minimum measurement dependence for faithful local deterministic models of multipartite GHZ-Mermin correlations
AU - Aaron Alai
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2608.00124
UR - https://arxiv.org/abs/2608.00124
AB - A local deterministic model can reproduce Bell-inequality violations only if the hidden state depends on the measurement settings. Hall found the minimum such dependence for the singlet state [Phys. Rev. Lett. 105, 250404 (2010)] and posed the corresponding problem for the Mermin state [Phys. Rev. A 84, 022102 (2011)]. This paper solves it, for the n-party GHZ-Mermin scenario at every n ≥ 3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is F_min(n) = R/[2(R+1)], with R = 2^⌊ (n−1)/2 ⌋ the Mermin violation ratio: 1/3, 2/5, 4/9, 8/17, …, always below 1/2. The lower bound follows from Mermin's inequality by a general overlap bound; the upper bound is an explicit model, uniform on the hidden-variable classes that violate the fewest constraints, with closed-form counts; a reduction theorem shows that for Mermin scenarios the marginals impose no additional dependence. The Mermin values are also certified by integer-arithmetic enumeration through n = 22. The formula was stated as a conjecture in the first version of this paper and was first proved, by a different route, by Wang, Yao and Chen; the proof here was obtained independently and the two agree. Every minimal contradiction of size k among stabilizer constraints has correlator-only minimum 1/(k−1). With the marginals included the minimum is the same on every core of the C_5 ring cluster, on the size-4 and size-5 cores of C_4 and on every four-core of C_6, but larger on the size-6 and size-7 cores of C_4 and on 112 of the 1,920 five-cores of C_6, with exact certificates on both sides in every case.
ER -