Theorem: Conditional semiclassical null and leakage allowance

Under Assumption ass:null, the complete accessible states are identical:

More generally, if a common state ${\rho }_{\ast }$ satisfies

then

Proof. In the identity-action case, tracing the message multiplies the accessible state by $\mathrm{Tr}{{\rho }_x}^A=1$. The stated factorization has the same consequence. Completeness excludes normalization by a message-dependent selected outcome. Thus the output is independent of $x$. Both differences in (eq:helstromscore) vanish. The approximate statement follows from Proposition prop:scorecontinuity, compared with the constant ensemble.