= Solution
An open-boundary <matrix product state> with physical dimension $q$ and bond dimension $\chi$ is
$$
|\Psi_N\rangle
=\sum_{i_1,\ldots,i_N=1}^q
\langle \ell|A^{i_1}A^{i_2}\cdots A^{i_N}|r\rangle
|i_1i_2\cdots i_N\rangle,
$$
where the $A^i$ are $\chi\times\chi$ matrices and $|\ell\rangle,|r\rangle$ are boundary vectors. For periodic boundary conditions, replace the boundary contraction by the <matrix trace> $\operatorname{Tr}(A^{i_1}\cdots A^{i_N})$.
The same matrices define the <matrix product state transfer map>
$$
\mathcal E(X)=\sum_iA^iX(A^i)^\dagger.
$$
When $\sum_i(A^i)^\dagger A^i=I$, the <Stinespring dilation>
$$
V|\psi\rangle=\sum_i|i\rangle\otimes A^i|\psi\rangle
$$
is an <isometry>. Repeatedly applying $V$ stores each Kraus label in a fresh physical register:
$$
V_N\cdots V_1|r\rangle
=\sum_{i_1,\ldots,i_N}|i_1\cdots i_N\rangle
\otimes A^{i_N}\cdots A^{i_1}|r\rangle.
$$
Contracting the remaining virtual system with $\langle\ell|$ gives an MPS, while tracing over all recorded labels gives repeated application of the <completely positive map> $\mathcal E$. The MPS is therefore a coherent <matrix product state as an unravelling of a completely positive map>[unravelling] of the channel. Equivalently, retaining the Kraus-label registers realizes a <purification of a density operator>[purification] of its output. A non-normalized MPS tensor gives the same construction with a general completely positive map; an appropriate canonical gauge normalizes the transfer map on its support.
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