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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Zitieren: | 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. |
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Holweck, F. Saniga, M. Lévay, P. 2019-02-11T16:16:41Z 2019-02-11T16:16:41Z 2014 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 https://nasplib.isofts.kiev.ua/handle/123456789/146814 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. This work began while two of the authors (M.S. and P.L.) were fellows of the “Research in Pairs” Program of the Mathematisches Forschungsinstitut Oberwolfach (Oberwolfach, Germany), in the period from 24 February to 16 March, 2013. It was also partially supported by the PEPS ICQ 2013 project CoGIT of the CNRS (F.H. and M.S.), the VEGA Grant Agency, grant No. 2/0003/13 (M.S.) and by the MTA-BME Condensed Matter Physics Research Group, grant No. 04119 (P.L.). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) |
| spellingShingle |
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n) Holweck, F. Saniga, M. Lévay, P. |
| 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) |
| author |
Holweck, F. Saniga, M. Lévay, P. |
| author_facet |
Holweck, F. Saniga, M. Lévay, P. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.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. |
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