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)



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