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Comment at the
n-Category Cafe

Re: This Week’s Finds in Mathematical Physics (Week 251)

On Spekkens’ toy system and finite geometry

Background–

• In “Week 251” (May 5, 2007), John wrote:
  “Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”
• In the n-Category Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
  “It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”
• In the n-category Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
  “Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”

On finite geometry:

• In “Week 234” (June 12, 2006), John wrote:
  “For a pretty explanation of M₂₄… try this:
  … Steven H. Cullinane, Geometry of the 4 × 4 square,
  http://finitegeometry.org/sc/16/geometry.html”

The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.

Related material:

• Some links on finite geometry and quantum information theory, and
• The geometry of qubits.

See also arXiv:0707.0074v1 [quant-ph], June 30, 2007:

A fully epistemic model for a local hidden variable emulation of quantum dynamics,

by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
 

Skotiniotis et al. note that the group actions on the 4×4 square described in Spekkens' paper [1] may be viewed (as in Geometry of the 4×4 Square and Geometry of Logic) in the context of a hypercube, or tesseract, a structure in which adjacency is isomorphic to adjacency in the 4 × 4 square (on a torus).

Hypercube from the Skotiniotis paper:

Reference:

[1] Robert W. Spekkens, Phys. Rev. A 75, 032110 (2007),

Evidence for the epistemic view of quantum states: A toy theory,

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)

"There is such a thing
as a tesseract."
— A Wrinkle in Time