Proposition: One sufficient calibration calculation

Suppose the prepared input-reference state is at trace distance at most $p$ from ${\Phi }_d$, and a bounded measurement's implemented mean observable differs from $G_{\nu }$ in operator norm by at most $m_{\nu }$. Then

is a valid coefficient allowance.

Proof. Channels contract trace distance, so the actual output-reference state is within trace distance $p$ of $J({\Lambda }_0)$. The expectation difference for an observable of operator norm at most one is bounded by the trace norm, hence by $2p$. The measurement-observable discrepancy contributes at most $m_{\nu }$.