dorsal/arxiv
View SchemaSIC-POVMs and the Extended Clifford Group
| Authors | D. M. Appleby |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0412001 |
| URL | https://arxiv.org/abs/quant-ph/0412001 |
| DOI | 10.1063/1.1896384 |
| Journal | J. Math. Phys. 46, 052107 (2005) |
Abstract
We describe the structure of the extended Clifford Group (defined to be the group consisting of all operators, unitary and anti-unitary, 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 POVMs (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 conjuecture 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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"abstract": "We describe the structure of the extended Clifford Group (defined to be the\ngroup consisting of all operators, unitary and anti-unitary, which normalize\nthe generalized Pauli group (or Weyl-Heisenberg group as it is often called)).\nWe also obtain a number of results concerning the structure of the Clifford\nGroup proper (i.e. the group consisting just of the unitary operators which\nnormalize the generalized Pauli group). We then investigate the action of the\nextended Clifford group operators on symmetric informationally complete POVMs\n(or SIC-POVMs) covariant relative to the action of the generalized Pauli group.\nWe show that each of the fiducial vectors which has been constructed so far\n(including all the vectors constructed numerically by Renes et al) is an\neigenvector of one of a special class of order 3 Clifford unitaries. This\nsuggests a strengthening of a conjuecture of Zauner\u0027s. We give a complete\ncharacterization of the orbits and stability groups in dimensions 2-7. Finally,\nwe show that the problem of constructing fiducial vectors may be expected to\nsimplify in the infinite sequence of dimensions 7, 13, 19, 21, 31,... . We\nillustrate this point by constructing exact expressions for fiducial vectors in\ndimensions 7 and 19.",
"arxiv_id": "quant-ph/0412001",
"authors": [
"D. M. Appleby"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1896384",
"journal_ref": "J. Math. Phys. 46, 052107 (2005)",
"title": "SIC-POVMs and the Extended Clifford Group",
"url": "https://arxiv.org/abs/quant-ph/0412001"
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