# Contents

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

$\begin{array}{ccc}\mathrm{Digraph}& & \mathrm{Arrow}\mathrm{diagram}\\ \\ •& & L\left({ℋ}_{1}\right)\\ ↓& & ↓\\ •& & L\left({ℋ}_{2}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $L\left({ℋ}_{1}\right)\otimes C\left(X\right)$ where $C\left(X\right)$ is a set of continuous operators on some space $X$ and that represents classical information. Take the output of this channel to be $L\left({ℋ}_{2}\right)$. The associated representations are:

$\begin{array}{ccccc}& \mathrm{Digraph}& & \mathrm{Arrow}\mathrm{diagram}& \\ \\ •& & •& & L\left({ℋ}_{1}\right)& & C\left(X\right)\\ ↘& & ↙& & ↘& & ↙\\ & •& & & & L\left({ℋ}_{1}\right)\otimes C\left(X\right)& \\ & ↓& & & & ↓\\ & •& & & & L\left({ℋ}_{2}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\epsilon :A\to R\left(A\right)$. Coecke’s model would look like:

and I’ll have to come back to this…

Revised on July 4, 2010 13:42:31 by Eric Forgy (119.247.164.98)