A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary ${\rm LGr}(N,2N)$

Abstract

Employing the fact that the geometry of the $N$-qubit ($N \geq 2$) Pauli group is embodied in the structure of the symplectic polar space $\mathcal{W}(2N-1,2)$ and using properties of the Lagrangian Grassmannian ${\rm LGr}(N,2N)$ 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 $2^{N-1}$-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases $N=3$ (also addressed recently by L\'evay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages, arXiv:1305.5689]) and $N=4$ are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space ${\rm PG}(2^N-1,2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where $G \equiv {\rm SL}(2,2)\times {\rm SL}(2,2)\times\cdots\times {\rm SL}(2,2)\rtimes S_N$, to decompose $\underline{\pi}({\rm LGr}(N,2N))$ into non-equivalent orbits. This leads to a partition of ${\rm LGr}(N,2N)$ into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.