| 1. |
- Appleby, D.M., et al.
(författare)
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Systems of Imprimitivity for the Clifford Group
- 2014
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Ingår i: Quantum information & computation. - : Rinton Press. - 1533-7146. ; 14:3-4, s. 339-360
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Tidskriftsartikel (refereegranskat)abstract
- It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).
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| 2. |
- Appleby, D. M., et al.
(författare)
-
The monomial representations of the Clifford group
- 2012
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Ingår i: Quantum information & computation. - : Rinton Press, Inc.. - 1533-7146. ; 12:5-6, s. 404-431
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Tidskriftsartikel (refereegranskat)abstract
- We show that the Clifford group-the normaliser of the Weyl-Heisenberg group-can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.
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| 3. |
- Björk, Gunnar, et al.
(författare)
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Stars of the quantum Universe : extremal constellations on the Poincare sphere
- 2015
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Ingår i: Physica Scripta. - : IOP Publishing. - 0031-8949 .- 1402-4896. ; 90:10
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Tidskriftsartikel (refereegranskat)abstract
- The characterization of the polarization properties of a quantum state requires the knowledge of the joint probability distribution of the Stokes variables. This amounts to assessing all the moments of these variables, which are aptly encoded in a multipole expansion of the density matrix. The cumulative distribution of these multipoles encapsulates in a handy manner the polarization content of the state. We work out the extremal states for that distribution, finding that SU(2) coherent states are maximal to any order, so they are the most polarized allowed by quantum theory. The converse case of pure states minimizing that distribution, which can be seen as the most quantum ones, is investigated for a diverse range of number of photons. Exploiting the Majorana representation, the problem appears to be closely related to distributing a number of points uniformly over the surface of the Poincare sphere.
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| 4. |
- Goldberg, Aaron Z., et al.
(författare)
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Quantum concepts in optical polarization
- 2021
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Ingår i: Advances in Optics and Photonics. - : The Optical Society. - 1943-8206. ; 13:1, s. 1-73
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Forskningsöversikt (refereegranskat)abstract
- We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincare sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincare space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.
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