# Moebius Pairs of Simplices and Commuting Pauli Operators - Quantum Physics

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Abstract: There exists a large class of groups of operators acting on Hilbert spaces,where commutativity of group elements can be expressed in the geometriclanguage 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 commutingand non-commuting group elements based on the existence of so-called Moebiuspairs of $n$-simplices, i. e., pairs of $n$-simplices which are \emph{mutuallyinscribed and circumscribed} to each other. For group elements representing an$n$-simplex there is no element outside the centre which commutes with all ofthem. This allows to express the dimension $n$ of the associated polar space ingroup theoretic terms. Any Moebius pair of $n$-simplices according to ourconstruction corresponds to two disjoint families of group elements operatorswith the following properties: i Any two distinct elements of the same familydo not commute. ii Each element of one family commutes with all but one ofthe elements from the other family. A three-qubit generalised Pauli groupserves as a non-trivial example to illustrate the theory for $p=2$ and $n=5$.

Author: ** Hans Havlicek TUW, Boris Odehnal TUW, Metod Saniga ASTRINSTSAV**

Source: https://arxiv.org/