A Notable Relation Between N-Qubit and 2^{N-1}-Qubit Pauli Groups via Binary LGr(N,2N) - Archive ouverte HAL Access content directly
Journal Articles Symmetry, Integrability and Geometry : Methods and Applications Year : 2014

A Notable Relation Between N-Qubit and 2^{N-1}-Qubit Pauli Groups via Binary 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 $LGr(N,\,2N)$ 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 (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 $PG(2^N-1,2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where $G \equiv SL(2,2)\times SL(2,2)\times\dots\times SL(2,2)\rtimes S_N$, to decompose $\underline{\pi}(LGr(N,2N))$ 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.
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Dates and versions

hal-00903272 , version 1 (11-11-2013)

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Frédéric Holweck, Metod Saniga, Peter Levay. A Notable Relation Between N-Qubit and 2^{N-1}-Qubit Pauli Groups via Binary LGr(N,2N). Symmetry, Integrability and Geometry : Methods and Applications, 2014, 10, pp.041. ⟨10.3842/SIGMA.2014.041⟩. ⟨hal-00903272⟩
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