Theorem: Single-product-effect read-chain bias

For the $c$-member GHZ read-chain at full access, let

Then

Proof. Here

so a product effect has bias

For a two-dimensional effect, the eigenvalue gap is at most one. Writing it as $\left(\begin{array}{cc}a& z\\ \bar{z}& b\end{array}\right)$, this gap is $\sqrt{(a-b{)}^2+4|z{|}^2}$, hence $|z|\le 1/2$. The bias is therefore at most $2^{-c}$. Taking every $M_q=|+⟩⟨+|$ attains it. The unrestricted value follows from the two eigenvalues $\pm 1/2$ of $X_J$.