See also Spekkens in this journal.
Thursday, January 10, 2019
Wednesday, January 9, 2019
Valid States of Maximal Knowledge
"Few scripts would have the audacity
to have the deus ex machina be
a Captain Midnight decoder ring."
— Review of "The House with
a Clock in Its Walls" (2018 film)
Related mathematics (click to enlarge) . . .
The "uwa.edu.au" above is for the University of Western Australia.
See the black swan in its coat of arms (and in the above film).
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Wednesday, June 10, 2015
Epistemic States
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Epistemic* Tetrads
"Those that can be obtained…." —
Related music video: Waterloo.
* "In defense of the epistemic view of quantum states:
a toy theory," by Robert W. Spekkens, Perimeter Institute
for Theoretical Physics, Waterloo, Canada
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Thursday, January 5, 2012
Precisely
From a review of Truth and Other Enigmas , a book by the late Michael Dummett—
"… two issues stand out as central, recurring as they do in many of the
essays. One issue is the set of debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent of the way we
represent that aspect to ourselves. For example, is there a realm of mathematical
entities that exists fully formed independently of our mathematical activity? Are
there facts about the past that our use of the past tense aims to capture? The other
issue is the view— which Dummett learns primarily from the later Wittgenstein—
that the meaning of an expression is fully determined by its use, by the way it
is employed by speakers. Much of his work consists in attempts to argue for this
thesis, to clarify its content and to work out its consequences. For Dummett one
of the most important consequences of the thesis concerns the realism debate and
for many other philosophers the prime importance of his work precisely consists
in this perception of a link between these two issues."
— Bernhard Weiss, pp. 104-125 in Central Works of Philosophy , Vol. 5,
ed. by John Shand, McGill-Queen's University Press, June 12, 2006
The above publication date (June 12, 2006) suggests a review of other
philosophical remarks related to that date. See …
- a June 12, 2006 link (in a 2007 post) to Geometry of the 4×4 Square,
- some related remarks on Dummett and Universals in this journal, and
- the following excerpt from Spekkens's toy model—
"Every partitioning of the set of sixteen ontic states
into four disjoint pure epistemic states
yields a maximally informative measurement."—
For some more-personal remarks on Dummett, see yesterday afternoon's
"The Stone" weblog in The New York Times.
I caught the sudden look of some dead master….
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Thursday, February 28, 2008
Thursday February 28, 2008
Popularity of MUB’s
From an entry today at the weblog of Lieven Le Bruyn (U. of Antwerp):
“MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias….”
The link to Kea (Marni Dee Sheppeard (pdf) of New Zealand) and a link in her Mutual Unbias III (Feb. 13) lead to the following illustration, from a talk, “Discrete phase space based on finite fields,” by William Wootters at the Perimeter Institute in 2005:
This illustration makes clear the
close relationship of MUB’s to the
finite geometry of the 4×4 square.
See also “Reflections on Symmetry,”
“Quantum Information Theory Related to Finite Geometry,”
and a comment at The n-Category Cafe,
On Spekkens’ toy system and finite geometry.
The Wootters talk was on July 20, 2005. For related material from that July which some will find more entertaining, see “Steven Cullinane is a Crank,” conveniently reproduced as a five-page thread in the Mathematics Forum at groupsrv.com.
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Sunday, September 2, 2007
Sunday September 2, 2007
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:
Update of
Sept. 5, 2007
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 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,
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.)
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