SICs and the Triangle Group (3, 3, 3)
The problem of the existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action....
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212263 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | SICs and the Triangle Group (3, 3, 3). Danylo Yakymenko. SIGMA 20 (2024), 044, 12 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862528937250258944 |
|---|---|
| author | Yakymenko, Danylo |
| author_facet | Yakymenko, Danylo |
| citation_txt | SICs and the Triangle Group (3, 3, 3). Danylo Yakymenko. SIGMA 20 (2024), 044, 12 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The problem of the existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group (3, 3, 3). We give a full description of such representations and show how they can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension > 3 is prime.
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| first_indexed | 2026-03-12T12:06:35Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212263 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T12:06:35Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Yakymenko, Danylo 2026-02-03T07:57:12Z 2024 SICs and the Triangle Group (3, 3, 3). Danylo Yakymenko. SIGMA 20 (2024), 044, 12 pages 1815-0659 2020 Mathematics Subject Classification: 20C35; 81P15; 81R05 arXiv:2312.13400 https://nasplib.isofts.kiev.ua/handle/123456789/212263 https://doi.org/10.3842/SIGMA.2024.044 The problem of the existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group (3, 3, 3). We give a full description of such representations and show how they can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension > 3 is prime. The author is grateful to Ingemar Bengtsson for valuable discussions and to the referees for important suggestions. The author has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 873071. This work was also supported by a grant from the Simons Foundation (1290607, DY). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications SICs and the Triangle Group (3, 3, 3) Article published earlier |
| spellingShingle | SICs and the Triangle Group (3, 3, 3) Yakymenko, Danylo |
| title | SICs and the Triangle Group (3, 3, 3) |
| title_full | SICs and the Triangle Group (3, 3, 3) |
| title_fullStr | SICs and the Triangle Group (3, 3, 3) |
| title_full_unstemmed | SICs and the Triangle Group (3, 3, 3) |
| title_short | SICs and the Triangle Group (3, 3, 3) |
| title_sort | sics and the triangle group (3, 3, 3) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212263 |
| work_keys_str_mv | AT yakymenkodanylo sicsandthetrianglegroup333 |