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

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 $\le 2$. 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.

20 pages, 6 figures, 1 table; Version 2 - slightly polished, updated references; Version 3 - published version in SIGMA

References in corpus (8)

• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• Black Hole Entropy and Finite Geometry
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line of an Arbitrary Single Qudit
• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of a Specific Qudit
• On the Pauli graphs of N-qudits
• Two-qubit analogs of octonions and projective planes