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

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 $\mathbf G$, we first construct vector spaces over $\mathrm{GF}(p)$, $p$ a prime, by factorising $\mathbf G$ over appropriate normal subgroups. Then, by expressing $\mathrm{GF}(p)$ in terms of the commutator subgroup of $\mathbf G$, we construct alternating bilinear forms, which reflect whether or not two elements of $\mathbf G$ 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 $\mathbf G$ 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 $\mathbf G$. 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.