Theorem: Finite-state reconstruction and coherent cutoff error

Under Assumption ass:trials, for $0<\beta <1$, define

With probability at least $1-\beta$,

In the oscillator embedding,

For the baseline, $p_{\mathbf{a}}\le {q_1}^{a_1+1}+{q_2}^{a_2+1}$.

Proof. A score in $[-1,1]$ has expectation error at most $2e_P$ under a record trace-distance error $e_P$. Lemma lem:concentration and a union bound over $m$ settings therefore imply, with the stated probability,

Pauli orthogonality gives

Euclidean projection onto the convex set of densities cannot increase distance from a member of that set. To see this, the minimizing projection satisfies $⟨{\tau }_{\mathrm{lin}}-\hat{\tau },\sigma -\hat{\tau }{⟩}_{\mathrm{HS}}\le 0$ for every density $\sigma$; expanding squares proves the asserted distance property. Thus

The second conclusion is Lemma lem:coherenttail and the triangle inequality.