### 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 2^{N - 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(2^{N} - 1, 2) of the 2^{N - 1}-qubit Pauli group in terms of G-orbits, where G ≡ SL(2, 2) × SL(2, 2) × · · · × SL(2, 2) ⋊ S_{N}, 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.

Original language | English |
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Article number | 041 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 10 |

DOIs | |

Publication status | Published - Apr 8 2014 |

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### Keywords

- Lagrangian Grassmannians LGr(N,2N) over the smallest Galois field
- Multi-qubit Pauli groups
- Symplectic polar spaces W(2N - 1,2)

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Geometry and Topology