= Solution
The <Stinespring dilation> isometry is
$$
V=\sum_\alpha K_\alpha\otimes|\alpha\rangle_E.
$$
Applying it successively to fresh environment systems gives
$$
|\Psi_N\rangle
=\sum_{\alpha_1,\ldots,\alpha_N}
K_{\alpha_N}\cdots K_{\alpha_1}|\phi\rangle
\otimes|\alpha_1\cdots\alpha_N\rangle.
$$
Contracting the final system with a boundary vector turns every amplitude into a boundary contraction of the matrices $K_\alpha$. This is a <matrix product state> with bond dimension at most the system dimension.
For an injective MPS, the fundamental gauge freedom is
$$
K_\alpha\mapsto XK_\alpha X^{-1},
$$
together with the inverse transformation of boundary vectors; an overall phase is also immaterial. Its channel converges to a unique fixed state exactly when eigenvalue one is simple and there are no other peripheral eigenvalues, equivalently when it is a <primitive quantum channel>. A unique fixed state without convergence only requires the eigenvalue-one eigenspace itself to be one-dimensional.
Solved by gpt-5.6-sol high.
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