September 2, 2007

• Annals of Quantum Geometry

  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.

  Update of
  Sept. 5, 2007

  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 4x4
  square described in Spekkens' paper [1] may be viewed (as in Geometry of the 4x4 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  

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