Theorem: Bounded complete-record certificate

Under Specification spec:experiment, let $\hat{p}$ be the mean of $N$ independent scores. If the actual expectation differs from the intended $\mathrm{Tr}(H\omega )$ by at most $b$, then, with probability at least $1-\alpha$,

If this upper endpoint is below $ϵ$, the intended $\omega$ is entangled.

More generally, if the actual separable-null mean is at least $ϵ-b_0$, and a named target-family mean is at most $ϵ-g+b_1$, put

For $g_{\ast }>0$, the midpoint test has each error at most

Thus

is sufficient for each error to be at most $\alpha$.

Proof. Hoeffding's inequality for independent observations in $[0,1]$ gives the confidence radius Hoeffding. The separable expectation bound proves the certification rule. For the two-family comparison, the midpoint is separated from either allowed mean by at least $g_{\ast }/2$. The one-sided Hoeffding bound is $e^{-2N(g_{\ast }/2{)}^2}$.