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The Stinespring dilation isometry isApplying it successively to fresh environment systems givesContracting 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 istogether 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.

Cutting an open-boundary matrix product state crosses one virtual bond of dimension $\chi$. Its Schmidt rank is at most $\chi$, soA periodic interval crosses two bonds and obeys $S(A)\le 2\log \chi$. The one-dimensional boundary has a constant number of points, so this is an area law.

For a projected entangled pair state, cut every virtual bond crossing the boundary of a region $A$. If $n_{\partial }$ bonds are cut, the Schmidt rank is at most ${\chi }^{n_{\partial }}$, and thereforeSince $n_{\partial }$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.

Solved by gpt-5.6-sol high.