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, $L({\mathscr{H}}_1)$, to another set of operators on a Hilbert space, $L({\mathscr{H}}_2)$. It can be represented by a simple digraph with an associated arrow diagram as follows:

$$\begin{array}{ccc}\mathrm{Digraph}& & \mathrm{Arrow}\mathrm{diagram}\\ \\ •& & L({\mathscr{H}}_1)\\ \downarrow & & \downarrow \\ •& & L({\mathscr{H}}_2)\end{array}.$$\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 $L({\mathscr{H}}_1)\otimes C(X)$ 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 $L({\mathscr{H}}_2)$. The associated representations are:

$$\begin{array}{ccccc}& \mathrm{Digraph}& & \mathrm{Arrow}\mathrm{diagram}& \\ \\ •& & •& & L({\mathscr{H}}_1)& & C(X)\\ \searrow & & \swarrow & & \searrow & & \swarrow \\ & •& & & & L({\mathscr{H}}_1)\otimes C(X)& \\ & \downarrow & & & & \downarrow \\ & •& & & & L({\mathscr{H}}_2)\end{array}.$$\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 $\epsilon :A\to R(A)$. Coecke’s model would look like:

and I’ll have to come back to this…