= Noisy GHZ parity constraints
{title2=$P_{\rm LR}(YYY=+1)\leq3\epsilon$}
If each of $XXY$, $XYX$ and $YXX$ has product $-1$ except with <probability> $\epsilon$, predetermined local X/Y values imply that a positive YYY product requires at least one of those constraints to fail. The <union bound> gives the displayed result without assuming independent errors. For the mixture of $(|000\rangle-i|111\rangle)/\sqrt2$ and its phase-flipped partner with weights $1-\epsilon,\epsilon$, quantum mechanics instead gives $P(YYY=+1)=1-\epsilon$. The violation survives for $\epsilon<1/4$. A noisy local outcome histogram is not uniquely determined by these parity statistics alone.
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