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)⋊Sn, to decompose π(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.