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A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary $\mathrm{LGr}(N,2N)$
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\mathrm{LGr}(N,2N)$ 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 $\mathrm{PG}(2^N-1,2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where $G \equiv \mathrm{SL}(2,2)\times \mathrm{SL}(2,2)\times\cdots\times \mathrm{SL}(2,2)\rtimes S_N$, to decompose $\underline{\pi}(\mathrm{LGr}(N,2N))$ into non-equivalent orbits. This leads to a partition of $\mathrm{LGr}(N,2N)$ into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.
Keywords: multi-qubit Pauli groups; symplectic polar spaces $\mathcal{W}(2N-1,2)$; Lagrangian Grassmannians $\mathrm{LGr}(N,2N)$ over the smallest Galois field. 
@article{SIGMA_2014_10_a40,
     author = {Fr\'ed\'eric Holweck and Metod Saniga and P\'eter L\'evay},
     title = {A {Notable} {Relation} between $N${-Qubit} and $2^{N-1}${-Qubit} {Pauli} {Groups} via {Binary} $\mathrm{LGr}(N,2N)$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a40/}
}
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Frédéric Holweck; Metod Saniga; Péter Lévay. A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary $\mathrm{LGr}(N,2N)$. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a40/

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