208 research outputs found
MUBs, polytopes, and finite geometries
A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces
of dimension N = p^k, where p is a prime number. They mesh naturally with
finite affine planes of order N, that exist when N = p^k. The existence of MUBs
for other values of N is an open question, and the same is true for finite
affine planes. I explore the question whether the existence of complete sets of
MUBs is directly related to the existence of finite affine planes. Both
questions can be shown to be geometrical questions about a convex polytope, but
not in any obvious way the same question.Comment: 15 pages; talk at the Vaxjo conference on probability and physic
Geometrical Statistics--Classical and Quantum
This is a review of the ideas behind the Fisher--Rao metric on classical
probability distributions, and how they generalize to metrics on density
matrices. As is well known, the unique Fisher--Rao metric then becomes a large
family of monotone metrics. Finally I focus on the Bures--Uhlmann metric, and
discuss a recent result that connects the geometric operator mean to a geodesic
billiard on the set of density matrices.Comment: Talk at the third Vaxjo conference on Quantum Theory: Reconsideration
of foundation
How much complementarity?
Bohr placed complementary bases at the mathematical centre point of his view
of quantum mechanics. On the technical side then my question translates into
that of classifying complex Hadamard matrices. Recent work (with Barros e Sa)
shows that the answer depends heavily on the prime number decomposition of the
Hilbert space. By implication so does the geometry of quantum state space.Comment: 6 pages; talk at the Vaxjo conference on Foundations of Probability
and Physics, 201
Three ways to look at mutually unbiased bases
This is a review of the problem of Mutually Unbiased Bases in finite
dimensional Hilbert spaces, real and complex. Also a geometric measure of
"mubness" is introduced, and applied to some recent calculations in six
dimensions (partly done by Bjorck and by Grassl). Although this does not yet
solve any problem, some appealing structures emerge.Comment: 18 pages. Talk at the Vaxjo Conference on Foundations of Probability
and Physics, June 200
2+1 gravity, chaos and time machines
2+1 gravity for spacetimes with topology RxT^2 has been much studied. We add
a description of how to extend these spacetimes across a Cauchy horizon into a
region where the torus becomes Lorentzian. The result is a one parameter family
of tori given by a geodesic in the "Teichmueller space" of Lorentzian tori. We
describe this in detail. We also point out that if the modular group is
regarded as part of the gauge group then these spacetimes offer a nice toy
model for the dynamics of Bianchi IX models; in the region where the tori are
spacelike the dynamics is described exactly by a hyperbolic billiard. On the
other hand the modular group acts ergodically on the Teichmueller space of
Lorentzian tori
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