Analytic proof of the optimal MABK robustness bound for the remaining cases

Prove that the positivity quantity \(\lambda_{\vec a}\) is nonnegative on its full domain for the cases \(h=1\) and \(h=2\) for every integer \(n\geq 6\), thereby establishing analytically the conjectured optimal robustness parameters \(\bar{s}_n\) and \(\bar{\mu}\) for the \(n\)-qubit MABK self-testing bound.

Background

The paper derives an analytic, size-independent robustness bound for self-testing the $n$-qubit GHZ state from the MABK inequality. It also identifies a conjectured optimal linear bound with parameters $\bar{s}_n$ and $\bar{\mu}$.

The proof of optimality reduces to verifying nonnegativity of a positivity function $\lambda_{\vec a}$. Analytic arguments resolve the cases $h\geq 3$ and $h=0$, while numerical grid searches and local optimization support nonnegativity for the remaining cases $h=1$ and $h=2$, including computations up to $n=100$. A general analytic proof for these cases, and hence for all $n\geq 6$, remains unresolved.