Definition: Operational admissibility

A strategy in Assumption ass:class is {operationally admissible at capacity $N$} if it satisfies the following requirements on every specified experiment whose entire execution remains within $N$ live qubits. Write $\mathsf{S}\in {\mathcal{A}}_N$.

{${\text{(C)}}_N$: composability.} For disjoint $S,T\subseteq L$, joint readout and both sequential orders have the same ordered joint law. Thus

on positive first branches, with value zero on a zero first branch; the corresponding equation holds in the reverse order.

{${\text{(NS)}}_N$: internal no-signaling.} For every bipartition, the terminal classical law of one local program is independent of the other local program and of the interleaving. Locally allocated qubits and returned quantum spectators are included. There is no exchange of names or messages.

{${\text{(N)}}_N$: nullity.}

{${\text{(K)}}_N$: continuity.} For each fixed $L,S$, the map

is continuous in total variation. No continuity of the unknown update is presumed.

{${\text{(R)}}_N$: typed records.} Let $L=S\bigsqcup B$, and let $u$ consume all of $B$, returning classical output and not using the first reported outcome. Compare:

1. read $S$, then run $u$ on the residual $B$, and report the ordered classical outputs;

2. allocate a fresh register $R$ of $\left|S\right|$ zero qubits, copy $S$ computationally to $R$ by bitwise CNOT, run the same $u$ on $B$, then read $R$, report the same ordered classical outputs, and return the original $S$.

Their classical observation laws must agree whenever both complete executions fit the budget.