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SICs and the Triangle Group $(3,3,3)$
Danylo Yakymenko Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine
Аннотация:
The problem of 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 it 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 $d>3$ is prime.
Ключевые слова:
quantum design, SIC-POVM, equiangular tight frame, Zauner's conjecture, Weyl–Heisenberg group, triangle group, projective representation.
Поступила: 2 января 2024 г.; в окончательном варианте 10 мая 2024 г.; опубликована 29 мая 2024 г.
Образец цитирования:
Danylo Yakymenko, “SICs and the Triangle Group $(3,3,3)$”, SIGMA, 20 (2024), 044, 12 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma2046 https://www.mathnet.ru/rus/sigma/v20/p44
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