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

The Veldkamp space of multiple qubits

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.Comment: 15 pages, 1 figure; added references, corrected typos; minor change