Matrix product state as an unravelling of a completely positive map

Matrices $A^i$ acting on a finite-dimensional virtual space define a completely positive mapThe same matrices form the local tensor of a matrix product state. To interpret $\mathcal{E}$ as a quantum channel, require the additional trace normalization ${\sum }_i(A^i{)}^{†}{A}^i=I$.

Under this condition,is an isometry, because $W^{†}W={\sum }_i(A^i{)}^{†}{A}^i=I$. Keeping the physical label $i$ preserves a coherent record of each Kraus operator. Tracing it out gives ${\mathrm{Tr}}_{\mathrm{phys}}(WX{W}^{†})=\mathcal{E}(X)$. This is the finite-dimensional Stinespring representation of the quantum channel. ^[ref]

Starting from $|L⟩$ and applying this isometry repeatedly produces amplitudes $A^{i_n}\cdots A^{i_1}|L⟩$ for the physical chain and remaining virtual system. A final boundary contraction with $⟨R|$ gives the matrix product state amplitudes $⟨R|A^{i_n}\cdots A^{i_1}|L⟩$. Tracing all physical labels instead gives ${\mathcal{E}}^n(|L⟩⟨L|)$.

Without the stated normalization, the matrices still define a completely positive map and a matrix product state tensor. They do not automatically describe a deterministic quantum channel; normalization or conditioning must be specified.