dorsal/arxiv
View SchemaOn SIC-POVMs in Prime Dimensions
| Authors | Steven T. Flammia |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605050 |
| URL | https://arxiv.org/abs/quant-ph/0605050 |
| DOI | 10.1088/0305-4470/39/43/007 |
| Journal | J. Phys. A: Math. Gen. 39 (2006) 13483-13493 |
Abstract
The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a "canonical" unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and give a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with computer computations in all dimensions < 48.
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"abstract": "The generalized Pauli group and its normalizer, the Clifford group, have a\nrich mathematical structure which is relevant to the problem of constructing\nsymmetric informationally complete POVMs (SIC-POVMs). To date, almost every\nknown SIC-POVM fiducial vector is an eigenstate of a \"canonical\" unitary in the\nClifford group. I show that every canonical unitary in prime dimensions p \u003e 3\nlies in the same conjugacy class of the Clifford group and give a class\nrepresentative for all such dimensions. It follows that if even one such\nSIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them\nare (for a given such dimension). I also conjecture that in all dimensions d,\nthe number of conjugacy classes is bounded above by 3 and depends only on d mod\n9, and I support this claim with computer computations in all dimensions \u003c 48.",
"arxiv_id": "quant-ph/0605050",
"authors": [
"Steven T. Flammia"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/39/43/007",
"journal_ref": "J. Phys. A: Math. Gen. 39 (2006) 13483-13493",
"title": "On SIC-POVMs in Prime Dimensions",
"url": "https://arxiv.org/abs/quant-ph/0605050"
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