A Notable Relation between N-Qubit and 2(N-1)-Qubit Pauli Groups via Binary LGr(N, 2N)

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 (2 N - 1, 2) and using properties of the Lagrangian Grassmannian LGr(N, 2 N) 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 2 N 1 qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N = 3 (also addressed recently by Levay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and N = 4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2 N 1, 2) of the 2 N 1 - qubit Pauli group in terms of G - orbits, where G = SL(2, 2) Chi SL(2, 2) Chi . . . Chi SL(2, 2) Chi S-N, to decompose (pi)over bar (LGr(N, 2 N)) into non- equivalent orbits. This leads to a partition of LGr(N, 2 N) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.