Factor-Group-Generated Polar Spaces and (Multi-)Qudits

TL;DR

This paper develops a unifying geometric framework linking generalized Pauli groups to finite polar spaces, revealing new insights into the structure of multi-qudit systems and their algebraic properties.

Contribution

It introduces a comprehensive method to relate group factorization to polar spaces, unifying various known relations and enabling detailed analysis of quantum systems.

Findings

Unified geometric framework for Pauli groups and finite geometries

Explicit construction of vector spaces from group factorization

Application to multi-qubit Pauli groups illustrating the formalism

Abstract

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group $\vG$, we first construct vector spaces over $\GF(p)$, $p$ a prime, by factorising $\vG$ over appropriate normal subgroups. Then, by expressing $\GF(p)$ in terms of the commutator subgroup of $\vG$, we construct alternating bilinear forms, which reflect whether or not two elements of $\vG$ commute. Restricting to $p=2$, we search for ``refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of $\vG$ is $\leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation'' of several distinct elements of $\vG$. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.