Theorem: General finite-record interface

Suppose a transfer on a named joint input class differs from the ideal $({\mathcal{B}}_N\otimes {\mathrm{id}}_S){\mathcal{I}}_h$ by at most $\eta$ in half trace norm, including finite references. Suppose a complete readout channel with the common alphabet $\mathcal{Y}$ has half-diamond error at most $\delta$. Set $e=\min \{1,\eta +\delta \}$.

For an input whose packet marginal is ${\rho }_h$, its actual full record satisfies

where ${\rho }_{hSR}={\mathrm{Tr}}_g{\rho }_{hgSR}$. Its decoded mean and law satisfy

The decoded record–spectator–reference state has the same second bound against ${\Omega }_{\infty ,SR}$.

Proof. Telescope through the ideal readout applied to the actual transferred state. Transfer contractivity and implementation complete-channel error give the full-record bound.

The reported observable

satisfies $0\le A_F\le NI$. For two states and such an observable,

because the positive and negative parts of $\rho -\sigma$ each have trace $d_{\mathrm{tr}}(\rho ,\sigma )$. Combine this with the exact ideal bias ${\nu }_h{q}_h^N$.

Classical decoding and reference discard are contractive. A triangle inequality through the saturated ideal record, followed by Proposition prop:recordtail, costs exactly ${q_h}^{N+1}$ for the unsaturated comparison.