Proposition: Accessible expectations and residual separation

1. For channels $\Phi ,\Psi$ with common input and output spaces and a fixed output Hermitian $B$,

   $$\mathrm{Tr}(B\Phi (\rho ))=\mathrm{Tr}(B\Psi (\rho ))\text{for every input state }\rho$$

   if and only if

   $${\Phi }^{\ast }(B)={\Psi }^{\ast }(B).$$

   On a restricted state class, only the corresponding annihilator condition follows; operator equality need not follow.

2. Identical classical record channels do not imply identical channels on the record–residual output algebra.

Proof. Trace duality turns the first equality into

A Hermitian operator with zero expectation in every state is zero. Restricting the tested states restricts this conclusion to their real linear span.

For the second assertion, let ${\Delta }_Z(\rho )=P_0\rho P_0+P_1\rho P_1$ on a qubit and define

Their record marginals are the same constant record. For $\rho =|+⟩⟨+|$, their residual difference is $X/2$, so

The full residual algebra distinguishes them.