Matrix product state as an unravelling of a completely positive map

Matrices $A^i$ define both the completely positive map $\mathcal{E}(X)={\sum }_i{A}^{i}X(A^i{)}^{†}$ and an MPS. Repeatedly retaining the Kraus label $i$ purifies the channel output into the physical MPS chain; tracing those labels recovers repeated application of $\mathcal{E}$.