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 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 socalled Möbius pairs of n-simplices, i. e., pairs of n-simplices which are 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 Möbius 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 threequbit generalised Pauli group serves as a non-trivial example to illustrate the theory for p = 2 and n = 5. Mathematics Subject Classification (2000): 51A50 – 81R05 – 20F99 PACS Numbers: 02.10.Ox, 02.40.Dr, 03.65.Ca Key-words: Möbius Pairs of Simplices – Factor Groups – Symplectic Polarity – Generalised Pauli Groups