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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 1 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-343/1/d/solution