Moebius Pairs of Simplices and Commuting Pauli Operators
| Autor: | Havlicek, Hans, Odehnal, Boris, Saniga, Metod |
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| Přispěvatelé: | Institut fuer Diskrete Mathematik und Geometrie (TUW), Vienna University of Technology (TU Wien), Astronomical Institute of the Slovak Academy of Sciences, Slovak Academy of Science [Bratislava] (SAS) |
| Jazyk: | angličtina |
| Rok vydání: | 2009 |
| Předmět: |
Quantum Physics
[PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Symplectic Polarity PACS Numbers: 02.10.Ox 02.40.Dr 03.65.Ca FOS: Physical sciences Factor Groups Mathematical Physics (math-ph) Quantum Physics (quant-ph) Generalised Pauli Groups Mathematical Physics Moebius Pairs of Simplices |
| Zdroj: | Mathematica Pannonica Mathematica Pannonica, 2010, 21, pp.115-128 |
| Popis: | There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG($n, p$), $n$ being odd and $p$ a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of $n$-simplices, i. e., pairs of $n$-simplices which are \emph{mutually inscribed and circumscribed} to each other. For group elements representing an $n$-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension $n$ of the associated polar space in group theoretic terms. Any Moebius pair of $n$-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for $p=2$ and $n=5$. 13 pages, 1 example; Version 2 - slightly polished and updated |
| Databáze: | OpenAIRE |
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