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

SIGMA 10 (2014), 041, 16 pages      arXiv:1311.2408

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

Frédéric Holweck a, Metod Saniga b and Péter Lévay c
a) Laboratoire IRTES/M3M, Université de Technologie de Belfort-Montbéliard, F-90010 Belfort, France
b) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
c) Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Budafoki út. 8, H-1521, Budapest, Hungary

Received November 14, 2013, in final form April 02, 2014; Published online April 08, 2014

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 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 ${\rm PG}(2^N-1,2)$ 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.

Key words: multi-qubit Pauli groups; symplectic polar spaces $\mathcal{W}(2N-1,2)$; Lagrangian Grassmannians ${\rm LGr}(N,2N)$ over the smallest Galois field.

pdf (459 kb)   tex (27 kb)  Maple codes (40 kb)


  1. Alexeev B., Forbes M.A., Tsimerman J., Tensor rank: some lower and upper bounds, in 26th Annual IEEE Conference on Computational Complexity, IEEE Computer Soc., Los Alamitos, CA, 2011, 283-291, arXiv:1102.0072.
  2. Bremner M.R., Stavrou S.G., Canonical forms of $2\times 2\times 2$ and $2\times 2\times 2\times 2$ arrays over $\mathbb{F}_2$ and $\mathbb{F}_3$, Linear Multilinear Algebra 61 (2013), 986-997, arXiv:1112.0298.
  3. Cameron P.J., Projective and polar spaces, QMW Mathematics Notes, Vol. 13, Queen Mary and Westfield College, London, 1991, available at
  4. Carrillo-Pacheco J., Zaldivar F., On Lagrangian-Grassmannian codes, Des. Codes Cryptogr. 60 (2011), 291-298.
  5. Cox D., Little J., O'Shea D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007.
  6. Gel'fand I.M., Kapranov M.M., Zelevinsky A.V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994.
  7. Havlicek H., Odehnal B., Saniga M., Factor-group-generated polar spaces and (multi-)qudits, SIGMA 5 (2009), 096, 15 pages, arXiv:0903.5418.
  8. Havlicek H., Odehnal B., Saniga M., On invariant notions of Segre varieties in binary projective spaces, Des. Codes Cryptogr. 62 (2012), 343-356, arXiv:1006.4492.
  9. Holtz O., Schneider H., Open problems on GKK $\tau$-matrices, Linear Algebra Appl. 345 (2002), 263-267, math.RA/0109030.
  10. Holtz O., Sturmfels B., Hyperdeterminantal relations among symmetric principal minors, J. Algebra 316 (2007), 634-648, math.RA/0604374.
  11. Krutelevich S., Jordan algebras, exceptional groups, and Bhargava composition, J. Algebra 314 (2007), 924-977, math.NT/0411104.
  12. Landsberg J.M., Tensors: geometry and applications, Graduate Studies in Mathematics, Vol. 128, Amer. Math. Soc., Providence, RI, 2012.
  13. Lavrauw M., Sheekey J., Orbits of the stabiliser group of the Segre variety product of three projective lines, Finite Fields Appl. 26 (2014), 1-6, arXiv:1207.3972.
  14. Lévay P., Planat M., Saniga M., Grassmannian connection between three-and four-qubit observables, Mermin's contextuality and black holes, J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages, arXiv:1305.5689.
  15. Lin S., Sturmfels B., Polynomial relations among principal minors of a $4\times 4$-matrix, J. Algebra 322 (2009), 4121-4131, arXiv:0812.0601.
  16. Mermin N.D., Hidden variables and the two theorems of John Bell, Rev. Modern Phys. 65 (1993), 803-815.
  17. Oeding L., G-varieties and the principal minors of symmetric matrices, Ph.D.  Thesis, Texas A&M University, 2009.
  18. Oeding L., Set-theoretic defining equations of the tangential variety of the Segre variety, J. Pure Appl. Algebra 215 (2011), 1516-1527, arXiv:0911.5276.
  19. Oeding L., Set-theoretic defining equations of the variety of principal minors of symmetric matrices, Algebra Number Theory 5 (2011), 75-109, arXiv:0809.4236.
  20. Planat M., Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function?, J. Phys. A: Math. Theor. 44 (2011), 045301, 16 pages, arXiv:1009.3858.
  21. Planat M., Saniga M., On the Pauli graphs on $N$-qudits, Quantum Inf. Comput. 8 (2008), 127-146, quant-ph/0701211.
  22. Saniga M., Lévay P., Pracna P., Charting the real four-qubit Pauli group via ovoids of a hyperbolic quadric of ${\rm PG}(7,2)$, J. Phys. A: Math. Theor. 45 (2012), 295304, 16 pages, arXiv:1202.2973.
  23. Saniga M., Planat M., Multiple qubits as symplectic polar spaces of order two, Adv. Stud. Theor. Phys. 1 (2007), 1-4, quant-ph/0612179.
  24. Saniga M., Planat M., Prachna P., Projective curves over a ring that includes two-qubits, Theoret. and Math. Phys. 155 (2008), 905-913, quant-ph/0611063.
  25. Thas J.A., Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1981), 135-143.
  26. Thas K., The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs, Europhys. Lett. 86 (2009), 60005, 3 pages.
  27. Waegell M., Primitive nonclassical structures of the $N$-qubit Pauli group, arXiv:1310.3419.

Previous article  Next article   Contents of Volume 10 (2014)