Moebius Pairs of Simplices and Commuting Pauli Operators

TL;DR

This paper explores the geometric and algebraic structure of commuting and non-commuting operators in groups acting on Hilbert spaces, using Moebius pairs of simplices and symplectic polar spaces to characterize their relationships.

Contribution

It introduces a novel geometric framework using Moebius pairs of simplices to analyze commuting properties of operators in symplectic polar spaces, linking group theory and geometry.

Findings

Moebius pairs of simplices correspond to two disjoint families of operators with specific commutation relations

The dimension of the associated polar space can be expressed in group-theoretic terms

A three-qubit Pauli group exemplifies the theory for p=2 and n=5

Abstract

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 $n=5$.