Theorem: Effect pullback and the full residual criterion

The record marginals of $\mathcal{F}$ and $\mathcal{G}$ agree for every input state if and only if

For an observation space $\mathcal{O}\subseteq \mathcal{B}(K)$, agreement on every observable ${\sum }_w|w⟩⟨w{|}_R\otimes B_w$, $B_w\in \mathcal{O}$, is equivalent to

Both are equalities of linear maps on the declared observation spaces and remain valid after tensoring with any untouched reference identity.

Proof. The fine aggregated probability is

while the coarse probability is $\mathrm{Tr}(\rho T^{\ast }(E_w))$. Equality against all states is equality of Hermitian effects.

The two $w$-conditioned expectations of $B$ pull back to the two sides of (eq:residual-pullback). States separate Hermitian matrices, and complex linearity gives the general equality. Tensoring an equality of linear maps with a reference identity preserves it.