# Can completely positive trace preserving maps be approximated with scalar costs?

In quantum physics, a quantum channel $$\mathcal{E}$$ is a completely positive trace-preserving map that sends density operators (hermitian, positive semi definite matrices with trace 1) to density operators. Call the space of density operators $$\mathcal{D}$$.

Frequently in quantum communications, we have a function $$F : \mathcal{D} \rightarrow \mathbb{R}$$, and such a function gives an indication of how "good" a state is with respect to an ideal state.

If you've got a network path, you can figure out the overall quality of transmission by evolving an initial state $$\rho \in \mathcal{D}$$ through each channel in the path, then applying $$F$$. For a particular path $$p$$, the overall cost is:

$$C(p) = F(\mathcal{E}_n \circ \mathcal{E}_{n-1} \circ \cdots \circ \mathcal{E}_1 (\rho))$$

If you've got a network with a small number of paths, you can find the optimal path with brute force, but in general this isn't feasible.

What I want to know is, is it possible to define a function $$\omega : \mathcal{D} \rightarrow \mathbb{R}^k$$ along with some partial ordering $$\preceq$$ on $$\mathbb{R}^k$$ such that for two paths $$p_1$$ $$p_2$$

$$C(p_1) \leq C(p_2) \leftrightarrow \sum_{\mathcal{E}\in p_1} \omega(\mathcal{E}) \preceq \sum_{\mathcal{E}\in p_2} \omega(\mathcal{E})$$

Essentially what I'm trying to do is approximate the linear maps with a vector of scalar costs to simplify pathfinding. Any thoughts (and especially recommended reading) would be greatly appreciated.

(Edit, made a correction on the last equation)