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

Abstract. 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) and using properties of the Lagrangian Grassmannian LGr(N, 2N) 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 [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(2N − 1, 2) of the 2N−1-qubit Pauli group in terms of G-orbits, where G ≡ SL(2, 2)×SL(2, 2)× · · ·×SL(2, 2)oSN, to decompose pi(LGr(N, 2N)) 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