Proposition: Orders and adequacy

For every channel $\Lambda$ and output effect $M$,

Post-processing can only increase order and decrease each weight. The state order is one whenever two supported branches coexist, and is infinite otherwise.

The question order is at least $m$ if and only if all dephasings of fewer than $m$ records leave its probability unchanged. The corresponding statement holds for interface order with agreement on every output effect.

Proof. If $Z_J=0$, then $\Lambda (Z_J)=0$. If $\Lambda (Z_J)=0$, then $\mathrm{tr}(M\Lambda (Z_J))=0$. Contractivity proves the post-processing statement.

Any two distinct labels differ on some $j$, and Proposition prop:statecomplex then gives $Z_{\{j\}}\ne 0$. With one supported branch all nonempty elements vanish.

Finally,

Vanishing below size $m$ implies adequacy below size $m$. Conversely, induction on $|J|$ recovers the vanishing of the coefficients from adequacy. The operator argument is identical after applying $\Lambda$.