Möbius Pairs of Simplices and Commuting Pauli Operators

There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geomet-ric 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 com-muting and non-commuting group elements based on the existence of so-called Möbius pairs of n-simplices, i. e., pairs of n-simplices which are mu-tually inscribed and circumscribed to each other. For group elements repre-senting 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 Möbius pair of n-simplices ac-cording to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct ele-ments 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