Proposition: Finite full-output confidence bound

Under Specification spec:validation, choose $0<\alpha <1$ and let

Set

With probability at least $1-\alpha -\beta$, simultaneously for all tested tasks,

Consequently, acceptance only when $U_k\le {\epsilon }_k$ gives

The shot ledger is exactly

For a prescribed common radius $r>0$, taking every count equal to

gives that radius or smaller and an explicit total of $J_{\mathrm{val}}{n}_{k,\nu }$ attempted shots.

Proof. Hoeffding's inequality for scores in $[-1,1]$ gives

A conditional union bound followed by integration gives simultaneous sampling coverage with failure probability at most $\alpha$. Intersect with $G$. Then

where clipping at two is valid because both exact coefficients belong to $[-1,1]$.

The difference of the two normalized Choi states has Pauli expansion

Its identity coefficient is zero. Orthogonality and $\Vert X{\Vert }_1\le \sqrt{N_k}\Vert X{\Vert }_2$ give

For differences of channels with input dimension $d$,

To see the factor, a pure message-reference input can be written $(I\otimes B)|{\Omega }_d⟩$ with $\Vert B{\Vert }_{\infty }^2\le d$. Its output difference is the corresponding two-sided multiplication of ${\Delta }_k$, whose trace norm is at most $d\Vert {\Delta }_k{\Vert }_1$. The diamond norm of a channel difference is optimized on such referenced states Watrous (Chapter 3). Divide by two and add the transfer allowance to obtain (eq:validation-bound).

The joint failure probability is at most $\alpha +\beta$. Every false acceptance lies outside the simultaneous coverage event. The shot identities and prescribed-radius bound follow directly from the definitions.