A 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 (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)×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.