Accessible output algebras

This criterion explains why a later coherence-sensitive continuation must declare a larger boundary if the earlier interface retained only populations CheungVIIE.

restricted classical population boundary. Consider the specifically declared zero-Hamiltonian generator

Directly,

Diagonal input states remain diagonal. Instruments whose admitted population updates are substochastic classical matrices, with complete sums, then give exactly the same complete history law as the corresponding classical model, by branch induction. For this generator there is no coherence-to-population term between interventions. Introducing a Hamiltonian or a coherence-to-population intervention changes that statement. This is the restricted comparison recorded in Psy03-A CheungPsy03A, not a theorem about every zero-Hamiltonian quantum model. A classical geometric label or diagonal record does not certify a quantum residual.

observation-law allowances. If actual laws satisfy uniformly in $\rho$

the triangle inequality gives

These allowances must be certified; the exact pullback identity does not provide them.

More generally, if $d_{\mathrm{TV}}(P,Q)\le e$, every score $0\le f\le 1$ satisfies $|E_{P}f-E_{Q}f|\le e$. Indeed, $f={{\int }_0}^1{1}_{\{f>u\}}\mathrm{d}u$. Taking suprema over one fixed test family preserves this allowance. It applies to accuracy when the joint labelled observation law is the compared law, as in the leakage interface of CheungPsy03D.

separate assisted channel distance. If channel comparison with a reference is needed, define separately

Here the diamond norm includes reference-assisted optimization. For fixed CPTP pre- and postprocessing,

This follows from the induced trace-norm definition and $\Vert \Omega {\Vert }_{\diamond }=\Vert \Lambda {\Vert }_{\diamond }=1$ Watrous2018. It uses the half-diamond convention compared in Em05-A CheungEm05A, and does not replace eq:delta-exact.