nLab graphical quantum channel

Under construction.

Idea

The idea here is to extend Coecke's graphical rules for completely positive maps in order to develop more sophisticated category-theoretic diagrams while maintaining the simplicity of the “object-arrow” representation (pdf). Two problems that could benefit from this analysis are the Birkhoff-von Neumann theorem and the study of extremal quantum channels.

Simple channels

Simple channel carrying quantum information

Consider a channel mapping a set of operators on a Hilbert space, $L(\mathcal{H}_{1})$, to another set of operators on a Hilbert space, $L(\mathcal{H}_{2})$. It can be represented by a simple digraph with an associated arrow diagram as follows:

$$\array{ Digraph && Arrow diagram \\ \\ \bullet && L(\mathcal{H}_{1}) \\ \downarrow && \downarrow \\ \bullet && L(\mathcal{H}_{2}) } \,.$$

Notice that this is a complete graph, $K_{2}$.

Simple channel carrying quantum and classical information

Consider a channel that takes as its input $L(\mathcal{H}_{1})\otimes C(X)$ where $C(X)$ is a set of continuous operators on some space $X$ and that represents classical information. Take the output of this channel to be $L(\mathcal{H}_{2})$. The associated representations are:

$$\array{ & Digraph && Arrow diagram & \\ \\ \bullet && \bullet && L(\mathcal{H}_{1}) && C(X) \\ \searrow && \swarrow && \searrow && \swarrow \\ & \bullet &&&& L(\mathcal{H}_{1}) \otimes C(X) & \\ & \downarrow &&&& \downarrow \\ & \bullet &&&& L(\mathcal{H}_{2}) } \,.$$

Copies of channels

Now consider two copies of a channel $\varepsilon: A \to R(A)$. Coecke’s model would look like:

and I’ll have to come back to this…