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

Original · EN

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, we first construct vector spaces over (p), p a prime, by factorising over appropriate normal subgroups. Then, by expressing (p) in terms of the commutator subgroup of, we construct alternating bilinear forms, which reflect whether or not two elements of 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 is ≤ 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. 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.