V = \sum_{α} K_{α} \otimes ∣ α ⟩_{E} .

Applying it successively to fresh environment systems gives

∣ Ψ_{N} ⟩ = \sum_{α_{1}, \dots, α_{N}} K_{α_{N}} \dots K_{α_{1}} ∣ ϕ ⟩ \otimes ∣ α_{1} \dots α_{N} ⟩ .

(2)

Contracting the final system with a boundary vector turns every amplitude into a boundary contraction of the matrices

K_{α}

. This is a matrix product state with bond dimension at most the system dimension.

For an injective MPS, the fundamental gauge freedom is

K_{α} \mapsto X K_{α} X^{- 1},

(3)

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.