Operator networks, Schrodinger vs Heisenberg picture

Question

We can think of an asynchronous network of operators on a hilbert space to be something like a quantum algorithm. Wires represent the informatic objects, namely quantum states, and nodes represent operators on a hilbert space. This should be the Heisenberg picture. This has a dual, which is the Schrodinger picture and this should be Density operators at the nodes with wires representing causal, or informational relations between the density operators. For instance, the spectral order is a domain of density operators. These pictures are dual to each other in the same way that string diagrams and normal category theory diagrams are poincare dual. Is this correct? So the Heisenberg picture is poincare dual to the Schrodinger picture? The one problem is that density matrices are always square since we say that the most general form is $$\rho = \sum_i e_i | \psi_i \rangle \langle \psi_i |.$$ Perhaps their most general form is $$\rho = \sum_{i,j} e_{ij} | \psi_i \rangle \langle \psi_j |.$$ This would have to be true to have a complete duality. Is this correct?