A Notable Relation
Abstract
Employing the fact that the geometry of the Nqubit (N ≥ 2) Pauli group is embodied in the structure of the symplectic polar space W(2N1,2) 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 Nqubit Pauli group and a certain subset of elements of the 2^{N1}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(2^N1,2) of the 2^{N1}qubit Pauli group in terms of Gorbits, where G ≡ SL(2,2)× SL{2,2)×\cdots× SL(2,2)rtimes S_N, to decompose {π}({LGr}(N,2N)) into nonequivalent orbits. This leads to a partition of LGr(N,2N) into distinguished classes that can be labeled by elements of the abovementioned Pauli groups.
 Publication:

SIGMA
 Pub Date:
 April 2014
 DOI:
 10.3842/SIGMA.2014.041
 arXiv:
 arXiv:1311.2408
 Bibcode:
 2014SIGMA..10..041H
 Keywords:

 multiqubit Pauli groups; symplectic polar spaces W(2N1;
 2); Lagrangian Grassmannians LGr(N;
 2N) over the smallest Galois field;;
 Mathematical Physics;
 Mathematics  Combinatorics;
 Quantum Physics
 EPrint:
 SIGMA 10 (2014), 041, 16 pages