Description:
Employing the fact that the geometry of the N-qubit (N>=2) Pauli group is embodied in the structure of the symplectic polar space W(2N-1, 2) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N-qubit Pauli group and a certain subset of elements of the 2N-1-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N=3 (also addressed recently by Lévay, Planat and Saniga (JHEP 09 (2013) 037)) and N=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2N-1, 2) of the 2N-1-qubit Pauli group in terms of G-orbits, where G=SL (2,2) x SL (2,2) x ... x SL (2,2) x|Sn, to decompose Pi(LGr(N,"N)) into non-equivalent orbits. This leads to a partition of LGr(N, 2N) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.