Types of quantum field thories
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, , to another set of operators on a Hilbert space, . It can be represented by a simple digraph with an associated arrow diagram as follows:
Notice that this is a complete graph, .
Consider a channel that takes as its input where is a set of continuous operators on some space and that represents classical information. Take the output of this channel to be . The associated representations are:
Now consider two copies of a channel . Coecke’s model would look like:
and I’ll have to come back to this…