Noisy GHZ parity constraints ($P_{\mathrm{LR}}(YYY=+1)\le 3ϵ$)

If each of $XXY$, $XYX$ and $YXX$ has product $-1$ except with probability $ϵ$, 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 $(|000⟩-i|111⟩)/\sqrt{2}$ and its phase-flipped partner with weights $1-ϵ,ϵ$, quantum mechanics instead gives $P(YYY=+1)=1-ϵ$. The violation survives for $ϵ<1/4$. A noisy local outcome histogram is not uniquely determined by these parity statistics alone.