= Solution
Write $\epsilon=0.08$ and $v=1-2\epsilon=0.84$. The ensemble is the mixture $(1-\epsilon)|\mathrm{GHZ}\rangle\langle\mathrm{GHZ}|+\epsilon|\mathrm{GHZ}'\rangle\langle\mathrm{GHZ}'|$, rather than a <quantum superposition>[coherent superposition] of the two phases. The ZZZ <probabilities> stay unchanged. Reversing the <relative phase> interchanges the allowed parity sector in every equatorial basis, so for XXY, XYX and YXX,
$$
\boxed{P(a,b,c)=\frac{1-vabc}{8}
=\begin{cases}0.23,&abc=-1,\\0.02,&abc=+1.\end{cases}}
$$
In YYY the quantum prediction is the opposite:
$$
\boxed{P_{\rm QM}^{YYY}(a,b,c)=\frac{1+vabc}{8}
=\begin{cases}0.23,&abc=+1,\\0.02,&abc=-1.\end{cases}}
$$
For a concrete continuation of the ideal local analysis, apply the same <phase flip>[phase-flip] event to a predetermined local assignment: it reverses X and Y at the affected <photon>. All three mixed-basis products and the YYY product then reverse together. With unbiased marginals in each parity sector this shared-error local model predicts negative-product YYY outcomes at $0.23$ each and positive-product outcomes at $0.02$ each. The histograms compare this specified model with the quantum result.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019-ghz-yyy-histograms.png]
{title=YYY outcome histograms for ideal and phase-flipped GHZ correlations}
{description=The ideal local prediction has negative parity, while quantum mechanics has positive parity. The noisy local bars use the explicitly stated shared phase-flip model; general noisy local models are constrained by a bound rather than by a unique histogram.}
Imperfect mixed-basis statistics alone do not force that particular local histogram. A model-independent constraint follows because a positive YYY product requires at least one of the three ideal mixed-product constraints to fail. Each fails with <probability> $\epsilon$, so the <union bound> gives
$$
\boxed{P_{\rm LR}(Y_1Y_2Y_3=+1)\leq3\epsilon=0.24,
\qquad P_{\rm QM}(Y_1Y_2Y_3=+1)=0.92.}
$$
Equivalently, local assignments satisfy $|\langle XXY\rangle+\langle XYX\rangle+\langle YXX\rangle-\langle YYY\rangle|\leq2$, whereas this noisy <quantum state> gives $4v=3.36$. Thus the violation survives the specified phase error without assuming a unique noisy local histogram or independent errors in the three <quantum measurement>[measurement] contexts.
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