Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
In: ISSN: 2227-7390 ; Mathematics ; https://hal.science/hal-03551957 ; Mathematics , 2021, 2021
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Zugriff:
International audience ; We study certain physically-relevant subgeometries of binary symplectic polar spaces W(2N−1,2) of small rank N, when the points of these spaces canonically encode N-qubit observables. Key characteristics of a subspace of such a space W(2N−1,2) are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of W(2N−1,2) and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of W(2N−1,2) whose rank is N−1. W(3,2) features three negative lines of the same type and its W(1,2)’s are of five different types. W(5,2) is endowed with 90 negative lines of two types and its W(3,2)’s split into 13 types. A total of 279 out of 480 W(3,2)’s with three negative lines are composite, i.e., they all originate from the two-qubit W(3,2). Given a three-qubit W(3,2) and any of its geometric hyperplanes, there are three other W(3,2)’s possessing the same hyperplane. The same holds if a geometric hyperplane is replaced by a ‘planar’ tricentric triad. A hyperbolic quadric of W(5,2) is found to host particular sets of seven W(3,2)’s, each of them being uniquely tied to a Conwell heptad with respect to the quadric. There is also a particular type of W(3,2)’s, a representative of which features a point each line through which is negative. Finally, W(7,2) is found to possess 1908 negative lines of five types and its W(5,2)’s fall into as many as 29 types. A total of 1524 out of 1560 W(5,2)’s with 90 negative lines originate from the three-qubit W(5,2). Rem arkably, the difference in the number of negative lines for any two distinct types of four-qubit W(5,2)’s is a multiple of four.
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Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
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Autor/in / Beteiligte Person: | Saniga, Metod ; de Boutray, Henri ; Holweck, Frédéric ; Giorgetti, Alain ; Slovak Academy of Sciences (SAS) ; Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174) (FEMTO-ST) ; Université de Technologie de Belfort-Montbeliard (UTBM)-Ecole Nationale Supérieure de Mécanique et des Microtechniques (ENSMM)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC) ; Université Bourgogne Franche-Comté COMUE (UBFC)-Université Bourgogne Franche-Comté COMUE (UBFC) ; Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB) ; Université de Technologie de Belfort-Montbeliard (UTBM)-Université de Bourgogne (UB)-Université Bourgogne Franche-Comté COMUE (UBFC)-Centre National de la Recherche Scientifique (CNRS) ; ANR-17-EURE-0002,EIPHI,Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité(2017) |
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Zeitschrift: | ISSN: 2227-7390 ; Mathematics ; https://hal.science/hal-03551957 ; Mathematics , 2021, 2021 |
Veröffentlichung: | HAL CCSD ; MDPI, 2021 |
Medientyp: | academicJournal |
DOI: | 10.3390/math9182272 |
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