2026-10-19 · arXiv Quantum Physics
arXiv:2608.00124v2 Announce Type: replace
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 $n \geq 3$, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is $F_{\min}(n) = R/[2(R+1)]$, with $R = 2^{\lfloor (n-1)/2 \rfloor}$ the Mermin violation ratio: $1/3, 2/5, 4/9, 8/17, \ldots$, 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 $n = 22$. 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.
原文链接：https://arxiv.org/abs/2608.00124
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