Gate errors, repeated records, and exact causality

This proves a tolerance statement only after the comparison interfaces are fixed. It is the local counterpart of the resource-controlled complete-channel allowance used in En08-B CheungEn08B.

For $n$ independent repetitions, the complete word-list channel has the separate bound

This is not the same ledger as the per-shot mean-bias allowance $\delta /b$ in (eq:robust).

Nor does a small allowance prove exact non-signalling or exact geometric containment. For example, let $\mathcal{T}$ be the constant qubit-output channel and $\mathcal{Z}$ the channel that measures the input in the computational basis and prepares the corresponding basis state. Then

but for every $\eta >0$ the outputs on inputs $0$ and $1$ can differ. Approximate channel agreement is not exact independence from an input.

A hardware failure alphabet, if needed, must be added to a common comparison interface with all attempted outcomes retained. Discarding failed trials without an explicit statistical adjustment does not satisfy Assumption ass:statistics.