A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n)
Employing the fact that the geometry of the n-qubit (n≥2) Pauli group is embodied in the structure of the symplectic polar space W(2n−1,2) and using properties of the Lagrangian Grassmannian LGr(n,2n) defined over the smallest Galois field, it is demonstrated that there exists a bijection between th...
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| Дата: | 2014 |
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| Мова: | English |
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Інститут математики НАН України
2014
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146814 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) / F. Holweck, M. Saniga, P. Lévay // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 27 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1468142019-02-12T01:24:51Z A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) Holweck, F. Saniga, M. Lévay, P. Employing the fact that the geometry of the n-qubit (n≥2) Pauli group is embodied in the structure of the symplectic polar space W(2n−1,2) and using properties of the Lagrangian Grassmannian LGr(n,2n) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the n-qubit Pauli group and a certain subset of elements of the 2ⁿ⁻¹-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases n=3 (also addressed recently by Lévay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and n=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2n−1,2) of the 2ⁿ⁻¹-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋯×SL(2,2)⋊Sn, to decompose π(LGr(n,2n)) into non-equivalent orbits. This leads to a partition of LGr(n,2n) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups. 2014 Article A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) / F. Holweck, M. Saniga, P. Lévay // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05B25; 51E20; 81P99 DOI:10.3842/SIGMA.2014.041 http://dspace.nbuv.gov.ua/handle/123456789/146814 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Employing the fact that the geometry of the n-qubit (n≥2) Pauli group is embodied in the structure of the symplectic polar space W(2n−1,2) and using properties of the Lagrangian Grassmannian LGr(n,2n) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the n-qubit Pauli group and a certain subset of elements of the 2ⁿ⁻¹-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases n=3 (also addressed recently by Lévay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and n=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2n−1,2) of the 2ⁿ⁻¹-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋯×SL(2,2)⋊Sn, to decompose π(LGr(n,2n)) into non-equivalent orbits. This leads to a partition of LGr(n,2n) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups. |
| format |
Article |
| author |
Holweck, F. Saniga, M. Lévay, P. |
| spellingShingle |
Holweck, F. Saniga, M. Lévay, P. A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Holweck, F. Saniga, M. Lévay, P. |
| author_sort |
Holweck, F. |
| title |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) |
| title_short |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) |
| title_full |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) |
| title_fullStr |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) |
| title_full_unstemmed |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) |
| title_sort |
notable relation between n-qubit and 2ⁿ⁻¹-qubit pauli groups via binary lgr(n,2n) |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146814 |
| citation_txt |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) / F. Holweck, M. Saniga, P. Lévay // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 27 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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