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Under construction.
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.
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:
Notice that this is a complete graph, $K_{2}$.
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:
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…
Last revised on January 18, 2014 at 08:18:45. See the history of this page for a list of all contributions to it.