nLab
graphical quantum channel

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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( 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 - http://svg-edit.googlecode.com/ --> Layer 1 A A R(A) R(A)

and I’ll have to come back to this…

Revised on January 18, 2014 08:18:45 by Urs Schreiber (82.113.106.29)