Proposition: Finite snapshot–containment equivalence

Let $A\subseteq M$ be a unital finite-dimensional $C^{\ast }$-subalgebra, and let $T:N\to M$ be UCP. The following are equivalent:

1. $T(N)\subseteq A$;

2. states with equal restrictions to $A$ have equal expectations on every $T(b)$, $b\in N$;

3. $T$ factors uniquely through a UCP map into $A$.

Proof. Containment gives state comparison. For the converse, if a Hermitian $T(b)\notin A$, its Hilbert–Schmidt orthogonal component $h_{\perp }$ is nonzero, Hermitian, and traceless because $I\in A$. For sufficiently small $\epsilon >0$, the density matrices

in a faithful matrix realization are positive. Their restrictions agree on $A$, but their expectations on $T(b)$ differ by $2\epsilon \Vert h_{\perp }{\Vert }_2^2$. This contradicts item two.

Under containment, restricting the codomain gives the unique factor. Matrix-level positivity is reflected by the subalgebra inclusion, so the factor is UCP.