Equation involving a partial trace Is there, in general, a solution to the following equation?
$\text{Tr}_{V_1}(A(X\otimes I_{V_2})) = B$
where 


*

*A is an operator on $V_1\otimes V_2$, 

*$B$ is an operator on $V_2$,

*$I_{V_2}$ is the identity on $V_2$.

*$X$ is an unknown operator on $V_1$.
Assuming that $V_1\cong V_2$ and reasonable invertibility constraints, can we solve for $X$?
Intuitively, I am wondering if composing and partially tracing on one component loses information.

If $V_1,V_2$ are finite dimensional, we can write $A=\sum_{i=1}^nA_i\otimes B_i$, where $A_1,\ldots,A_n$ are linear independent operators acting on $V_1$ and $B_1,\ldots,B_n$ are linear independent operator acting on $V_2$.
Your equation is equivalent to  $\sum_{i=1}^ntr(A_iX)B_i=B$.
You can solve this equation iff $B\in \text{span}\{B_1,\ldots,B_n\}$.
