Matrix product state as an unravelling of a completely positive map
= Matrix product state as an unravelling of a completely positive map
Matrices $A^i$ define both the completely positive map $\mathcal E(X)=\sum_iA^iX(A^i)^\dagger$ 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 $\mathcal E$.