Daily Archives: February 5, 2008

The Generalized Pauli Group

I’ve been watching Perimeter Institute lectures again, and one that dips into the stuff I’m working came up. In addition to the lecture, there is an acrobat file that contains the slides and photos of the blackboard. This is 124 pages long and it is these pages that I will reference.

Applications of the generalized Pauli group in quantum information
Speaker(s): Thomas Durt – Vrije Universiteit Brussel
Abstract: It is known that finite fields with d elements exist only when d is a prime or a prime power. When the dimension d of a finite dimensional Hilbert space is a prime power, we can associate to each basis state of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group. Its elements can be expressed, in terms of the elements of a Galois field. This group presents numerous applications in Quantum Information Science e.g. tomography, dense coding, teleportation, error correction and so on. The aim of our talk is to give a general survey of these properties and to present recently obtained results in connection with three problems: -the so-called ”Mean King’s problem” in prime power dimension, -discrete Wigner distributions, -and quantum tomography . Finally we shall discuss a limitation of the possible dimensions in which the so-called epistemic interpretation can be consistently formulated, in relation with the existence of finite affine planes, Euler’s conjecture and the 36 officers problem.
Date: 10/10/2007 – 4:00 pm

This has been discussed a little over at Kea’s blog and at Matti’s blog as well, and I figure I should write something up here as well, since I am also applying “Mutually Unbiased Bases” (MUBs); what I call snuarks are the Pauli algebra example of a set of MUBs.

I’ve added an explicit calculation in density matrix formalism for a maximal MUB for the Dirac algebra, and I’ve converted the solution into spinor form. This is a wonderful illustration of how much more powerful the density matrix formalism is than the spinor formalism.
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