Proposition: Finite snapshot–support equivalence

Let $\Phi$ be a channel on a finite tensor-product system. Let $\mathcal{A}(I)$ and $\mathcal{A}(Y)$ be the input and output local algebras. The following are equivalent:

1. the output reduction on $Y$ factors through input reduction on $I$ by a CPTP map;

2. equal input reductions on $I$ give equal output reductions on $Y$;

3. ${\Phi }^{\ast }(\mathcal{A}(Y))\subseteq \mathcal{A}(I)$.

The resulting equality of channels preserves external references.

Proof. Factorization gives equal-snapshot dependence and the adjoint containment.

For the converse from equal snapshots, let $A=A^{†}$ be a pulled-back output observable. If $A\notin \mathcal{A}(I)$, orthogonal projection in the Hilbert–Schmidt inner product gives a nonzero Hermitian component $F$ orthogonal to $\mathcal{A}(I)$. Then ${\mathrm{Tr}}_{I^c}F=0$ and $\mathrm{Tr}F=0$. For sufficiently small $t>0$, the states $I/\dim \mathcal{H}\pm tF$ are positive, have equal $I$-reductions, and distinguish $A$, a contradiction.

Under containment, define the adjoint local map by restricting ${\Phi }^{\ast }$ to the embedded output algebra. It is unital and completely positive because the inclusions preserve matrix positivity. Its dual is CPTP and yields factorization. Equality of linear maps is stable under reference extension.