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Symmetric informationally complete–positive operator valued measures and the extended Clifford group

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Appleby, D. M.·2005·Journal of Mathematical Physics
DOI 10.1063/1.1896384
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Abstract

We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7,13,19,21,31,… . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.

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  • In dimension 4, there is only one known SIC POM, and all other known SIC POMs are unitarily equivalent to it.

    “In dimension 4, there is only one known SIC POM, and all the other known SIC POMs are unitarily equivalent to it [6].”

    From Experimental proposal for symmetric minimal two-qubit state tomography (2012) · cites this paper as [6]

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Research works printed but not linked: 20

  1. [1][1] Davies, E.B., Quantum Theory of Open Systems (Academic Press, London, 1976). -
  2. [2][2] Busch, P., Grabowski, M., and Lahti, P.J., Operational Quantum Physics (Springer, Berlin, 1995). -
  3. [8][8] Flammia, S.T., Silberfarb, A., and Caves, C.M., 'Minimal informationally complete measurements for pure states', quant-ph/0404137. - · arXiv
  4. [9][9] Finkelstein J., 'On PSI-complete and PSIR-complete measurements', quant-ph/0407078. - · arXiv
  5. [13][13] Fuchs, C.A., 'Quantum mechanics as quantum information (and only a little more)', quant-ph/0205039. - · arXiv
  6. [14][14] Hardy, L., 'Quantum theory from five reasonable axioms', quant-ph/0101012. - · arXiv
  7. [15][15] Hardy, L., 'Why quantum theory?', in J. Butterfield and T. Placek (eds), Proceedings of the NATO Advanced Research Workshop on Modality, Probability and Bell's Theorem (IOS Press, Amsterdam, 2002). Available as e-print quant-ph/0111068. - · arXiv
  8. [16][16] Zauner, G., 'Quantum designs-foundations of a non-commutative theory of designs' (in German), Ph.D. thesis, University of Vienna, 1999. Available online at http://www.mat.univie.ac.at/˜neum/papers/physpapers.html. -
  9. [18][18] Wootters, W.K., 'Quantum measurements and finite geometry', quant-ph/0406032. - · arXiv
  10. [19][19] Bengtsson, I., 'MUBs, polytopes, and finite geometries' quant-ph/0406174. - · arXiv
  11. [21][21] Grassl, M., 'On SIC-POVMs and MUBs in dimension 6', quant-ph/0406175. - · arXiv
  12. [23][23] Gibbons, K.S., Hoffman, M. J., and Wootters, W.K., 'Discrete phase space based on finite fields', quant-ph/0401155. - · arXiv
  13. [26][26] Gottesman, D., 'Fault-tolerant quantum computation with higher-dimensional systems', in C. Williams (ed.) Quantum Computing and Quantum Communications, Proceedings of the First NASA International Conference on Quantum Computing and Quantum Communications(QCQC), Palm Springs, California , 302-313. Available as e-print quant-ph/9802007. - · arXiv
  14. [27][27] Gottesman, D., 'The Heisenberg representation of quantum computers', quant-ph/9807006. - · arXiv
  15. [29][29] Hostens, E., Dehaene, J., and De Moor, B., 'Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic', quant-ph/0408190. - · arXiv
  16. [30][30] van den Nest, M., Dehaene, J., and de Moor, B., 'The invariants of the local Clifford group', quant-ph/0410035. - · arXiv
  17. [31][31] van den Nest, M., Dehaene, J., and de Moor, B., 'Finite set of invariants to characterize local Clifford equivalence of stabilizer states', quant-ph/0410165. - · arXiv
  18. [32][32] Berndt, R., and Schmidt, R., Elements of the Representation Theory of the Jacobi Group , Progress in Mathematics vol. 163 (Birkha¨ user Verlag, Basel, 1998). -
  19. [33][33] Nathanson, M.B., Elementary Methods in Number Theory , Graduate Texts in Mathematics no. 195 (Springer, New York, 2000). -
  20. [34][34] Rose, H.E., A Course in Number Theory , 2 nd edition (Clarendon Press, Oxford, 1994).

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Authors
Appleby, D. M.
Affiliations
Queen Mary University of London (GB)
Corresponding author
Appleby, D. M.
Venue
Journal of Mathematical Physics
Publisher
AIP Publishing
Published
2005-04-19
Volume
46
Issue
5
ISSN
  • 0022-2488 (ISSN-L)
  • 1089-7658
DOI
10.1063/1.1896384
OpenAlex
W1977809994
Cited by
304
References
2
Citations by year
2026: 11 · 2025: 14 · 2024: 24 · 2023: 6 · 2022: 27 · 2021: 11 · 2020: 15 · 2019: 14 · 2018: 12 · 2017: 20 · 2016: 13 · 2015: 16 · 2014: 16 · 2013: 13 · 2012: 10
FWCI
14.45
Type
journal article
Language
en
Open access
Locations
  • Journal of Mathematical Physics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
symmetric informationally complete POVM, SIC-POVM, Clifford group, Weyl-Heisenberg group, finite-dimensional quantum systems

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