graphical quantum channel



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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.

Simple channels

Simple channel carrying quantum information

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

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

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

Simple channel carrying quantum and classical information

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

Digraph Arrowdiagram L( 1) C(X) L( 1)C(X) L( 2). \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 ε:AR(A)\varepsilon: A \to R(A). Coecke’s model would look like:

<!-- Created with SVG-edit - --> Layer 1 A A R(A) R(A)

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.