Thursday, January 27, 2011

Theory Update 41

The dots and edges of the path table do not give the letters $X$ or $Y$ as arrows, suggesting an alternative view with the qubit sequence in terms of two dimensional arrows.
Note that the $4$ paths of the two qubit set are now given by arrow compositions. A qutrit triangle may be built from these arrows. By fixing the mismatch between the outer $XYZ$ and the inner $YZX$, we can still label the path simplex edges with single letters, as shown.

Ignoring the edge directions, we can form $24$ out of the $27$ length three qutrit paths with this two qutrit diagram. All paths of type $XYZ$ are closed triangles. When using these globule diagrams, the degenerate case of classical (one dimensional) objects is represented by basic strings of $1$-arrows, which is to say, ordinary paths. Qubits and qutrits are naturally higher dimensional objects. Observe now that the ordinary tetractys may be labeled by length two words along the edges, and these edges may be interpreted as globules.

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