Theorem: Functions supplied by a recovery certificate

Suppose

Then:

1. Referenced payload return. For every state ${\rho }_{QL}$,

   $$d_{\mathrm{tr}}(({\mathrm{id}}_Q\otimes \Lambda )({\rho }_{QL}),{\rho }_{QL})\le e.$$

2. Entanglement transfer. For a Bell input, the decoded Bell overlap satisfies

   $$⟨{\Phi }_d|({\mathrm{id}}_Q\otimes \Lambda )(|{\Phi }_d⟩⟨{\Phi }_d|)|{\Phi }_d⟩\ge 1-e.$$

3. Classical message transmission. Encoding the $d$ orthonormal basis messages and measuring the decoded output in that basis has error at most $e$ for every message.

4. Logical processing after return. For every declared CPTP task $\mathcal{A}:L\to T$,

   $$d_{\diamond }(\mathcal{A}\Lambda ,\mathcal{A})\le e.$$

   In particular, any specified logical unitary may be applied after return without increasing this allowance.

5. Measurement-task transfer. Every effect $0\le F\le I$, including a joint reference–output effect, changes probability by at most $e$. A bounded Hermitian observable $O$ changes expectation by at most $2e{\Vert O\Vert }_{\infty }$.

Proof. The first assertion is the definition of diamond control. The Bell projector and each correct-message projector are effects, which proves the next two assertions. CPTP postprocessing contracts the diamond norm. The final claims follow from the operational characterization of trace distance and trace/operator-norm duality.