Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups

Abstract

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type $\widetilde{D}_n$, $4 \leq n \leq 8$, it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG$(3,2)$. Proper labelling of the vertices of the diagram (for $4 \leq n \leq 7$) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG$(3,2)$. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space $W(3,2)$ of the PG$(3,2)$ whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin-Peres magic square. In the case of $n=8$, a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG$(5,2)$, the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG$(3,2)$ are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin-Peres square. Moreover, save for $n=4$, each Veldkamp space is also endowed with some `exceptional' point(s). Interestingly, two such points in the $n=8$ case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix $Y$.