Let $A$ be a stochastic matrix living in a tensor product space $W=V \otimes V \otimes \cdots \otimes V$ of dimension $d^N$, where $dim(V)=d$. In particular, $A$ is a left stochastic matrix, a real square matrix with each column summing to $1$.
I'm trying to understand how to build a map from $A$ to $\tilde{A}$, such that:
- the resulting $\tilde{A}$ is a left stochastic matrix
- the trace is preserved $tr(A)=tr(\tilde{A})$
- $\tilde{A}$ lives in a tensor product space $\tilde{W}=V \otimes V \otimes \cdots \otimes V$ of dimension $d^\tilde{N}$, where $\tilde{N}\leq N$.
Question: How can I build such trace-preserving mapping?