Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

Abstract

In the present paper we show that it is possible to obtain the well known Pauli group P = 〈X,Y,Z | X2 = Y2 = Z2 = 1,(Y Z)4 = (ZX)4 = (XY )4 = 1〉 of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere S3. The first of these spaces of orbits is realized via an action of the quaternion group Q8 on S3; the second one via an action of the cyclic group of order four \(\mathbb {Z}(4)\) on S3. We deduce a result of decomposition of P of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.

References

  1. 1.

    Bagarello, F., Russo, F. G.: A description of pseudo-bosons in terms of nilpotent Lie algebras. J. Geom. Phy. 125, 1–11 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Bagarello, F., Russo, F. G.: On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank. J. Geom. Phy. 125, 124–131 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Bagarello, F., Russo, F. G.: Realization of Lie Algebras of High Dimension via Pseudo-Bosonic Operators. J. Lie Theory 30, 925–938 (2020)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bagarello, F.: Weak pseudo-bosons. J. Phys. A 53, 135201 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Bagarello, F., et al.: Deformed canonical (anti-)commutation relations and non hermitian Hamiltonians. In: aspects, Mathematical, Znojil, M. (eds.) Non-selfadjoint operators in quantum physics, pp 121–188. Wiley, Hoboken (2015)

  6. 6.

    Bagarello, F.: Linear pseudo-fermions. J. Phys. A 45, 444002 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Bagarello, F.: Damping and Pseudo-fermions. J. Math. Phys. 54, 023509 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Bagarello, F., Gargano, F.: Model pseudofermionic systems: connections with exceptional points. Phys. Rev. A 89, 032113 (2014)

    ADS  Article  Google Scholar 

  9. 9.

    Bender, C. M.: PT Symmetry in Quantum and Classical Physics. World Scientific, Singapore (2019)

    Google Scholar 

  10. 10.

    Borzellino, J. E.: Riemannian geometry of orbifolds, UCLA PhD thesis (1992)

  11. 11.

    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, AMS Providence (2001)

  12. 12.

    Chepilko, N.M., Romanenko, A.V.: Quantum mechanics on Riemannian Manifold in Schwinger’s Quantization Approach I, II, III, IV, see respectively: arXiv:hep-th/0102139, arXiv:hep-th/0102115, arXiv:hep-th/0102117, arXiv:hep-th/0201074

  13. 13.

    Cherbal, O., Drir, M., Maamache, M., Trifonov, D. A.: Fermionic coherent states for pseudo-Hermitian two-level systems. J. Phys. A 40, 1835–1844 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Doerk, K., Hawkes, T.: Finite Soluble Groups. W. de Gruyter, Berlin (1994)

    MATH  Google Scholar 

  15. 15.

    Gorenstein, D.: Finite Groups. Chelsea Publishing Company, New York (1980)

    MATH  Google Scholar 

  16. 16.

    Gottesman, D.: Stabilizer codes and quantum error correction, preprint, arXiv:quant-ph/9705052 (1997)

  17. 17.

    Gottesman, D.: Fault-Tolerant Quantum Computation with Higher-Dimensional Systems. In: Williams, C.P. (ed.) Quantum Computing and Quantum Communications. LNCS 1509, pp. 302–313. Springer, Berlin (1999)

  18. 18.

    Greenberg, O. W.: Particles with small violations of Fermi or Bose statistics. Phys. Rev. D 43, 4111–4120 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  20. 20.

    Hofmann, K.H., Morris, S.: The Structure of Compact Groups. de Gruyter, Berlin (2006)

    Book  Google Scholar 

  21. 21.

    Kibler, M. R.: Variations on a theme of Heisenberg, Pauli and Weyl. J. Phys. A 41(37) (2008)

  22. 22.

    Knill, E.: Non-binary unitary error bases and quantum codes, preprint, arXiv:9608048 (1996)

  23. 23.

    Kosniowski, C.: Introduction to Algebraic Topology. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  24. 24.

    Krüger, O., Werner, R.F.: Open problems in quantum information, preprint, arXiv:quant-ph/0504166 (2005)

  25. 25.

    Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Dover Publications, New York (1976)

    MATH  Google Scholar 

  26. 26.

    Maleki, Y.: Para-Grassmannian coherent and squeezed states for Pseudo-Hermitian q-oscillator and their entanglement. SIGMA 7, 084 (2011)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Messiah, A.: Quantum Mechanics, vol. 1. North Holland Publishing Company, Amsterdam (1967)

    MATH  Google Scholar 

  28. 28.

    Mohapatra, R. N.: Infinite statistics and a possible small violation of the Pauli principle. Phys. Lett. B. 242, 407–411 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Pauli, W.: Zur Quantenmechanik des magnetischen Elektrons. Z. Phys. 43, 601–623 (1927)

    ADS  Article  Google Scholar 

  30. 30.

    Provost, J. P., Vallee, G.: Riemannian structure on manifolds of quantum states. Commun. Math. Phys. 76, 289–301 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Reni, M., Zimmermann, B. P.: Hyperelliptic involutions of hyperbolic 3-manifolds. Math. Ann. 321, 295–317 (2001)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Rocchetto, A., Russo, F. G.: Decomposition of Pauli groups via weak central products, preprint arXiv:1911.10158 (2019)

  33. 33.

    Zimmermann, B. P.: On finite groups acting on spheres and finite subgroups of orthogonal groups. Sib. Elektron. Mat. Izv. 9, 1–12 (2012)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Zimmermann, B. P.: On topological actions of finite groups on S3. Topology Appl. 236, 59–63 (2018)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Zimmermann, B. P.: On hyperbolic knots with homeomorphic cyclic branched coverings. Math. Ann. 311, 665–673 (1998)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Zimmermann, B. P.: On topological actions of finite non-standard groups on spheres. Monatsh. Math. 183, 219–223 (2017)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

F.B. acknowledges support from Palermo University and from the Gruppo Nazionale di Fisica Matematica of the I.N.d.A.M. The other two authors (Y.B. and F.G.R.) thank Shuttleworth Postgraduate Scholarship Programme 2019 and NRF for grants no. 118517 and 113144.

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Appendix

Appendix

The present appendix is meant to make the paper self-contained, by giving some essential definitions and results on pseudo-fermions. We consider two operators a and b, acting on the Hilbert space \({\mathscr{H}}=\Bbb C^{2}\), which satisfy the following rules, [6]:

(A.1)

where {x,y} = xy + yx is the anti-commutator between x and y. We first observe that a non zero vector φ0 exists in \({\mathscr{H}}\) such that aφ0 = 0. Similarly, a non zero vector Ψ0 exists in \({\mathscr{H}}\) such that \(b^{\dagger } {\Psi }_{0}=0\). This is because the kernels of a and b are non-trivial.

It is now possible to deduce the following results. We first introduce the following non zero vectors

$$ {\varphi}_1:=b {\varphi}_0,\quad {\Psi}_{1}=a^{\dagger} {\Psi}_{0}, $$
(A.2)

as well as the non self-adjoint operators

$$ N=ba,\quad N^{\dagger}=a^{\dagger} b^{\dagger}. $$
(A.3)

Of course, it makes no sense to consider \(b^{n} {\varphi }_{0}\) or \({\left (a^{\dag }\right )}^n {\Psi }_{0}\) for n ≥ 2, since all these vectors are automatically zero. This is analogous to what happens for ordinary fermions. Let now introduce the self-adjoint operators Sφ and SΨ via their action on a generic \(f\in {\mathscr{H}}\):

$$ S_{\varphi} f=\sum\limits_{n=0}^1\left<{\varphi}_{n},f\right> {\varphi}_{n}, \quad S_{\Psi} f=\sum\limits_{n=0}^{1}\left<{\Psi}_{n},f\right> {\Psi}_{n}. $$
(A.4)

The following results can be easily proved:

  • $$ a\varphi_{1}=\varphi_{0},\quad b^{\dagger}{\Psi}_{1}={\Psi}_0. $$
    (A.5)
  • $$ N{\varphi}_{n}=n{\varphi}_{n},\quad N^{\dagger} {\Psi}_{n}=n{\Psi}_{n}, \ \text{for} \ n=0,1. $$
    (A.6)
  • If the normalizations of φ0 and Ψ0 are chosen in such a way that \(\left <\varphi _{0},{\Psi }_{0}\right >=1\), then

    $$ \left<\varphi_k,{\Psi}_{n}\right>=\delta_{k,n}, \ \text{for} \ k,n=0,1. $$
    (A.7)
  • Sφ and SΨ are bounded, strictly positive, self-adjoint, and invertible. They satisfy

    $$ \|S_{\varphi}\|\leqslant\|\varphi_0\|^2+\|\varphi_1\|^2, \quad \|S_{\Psi}\|\leqslant\|{\Psi}_0\|^2+\|{\Psi}_{1}\|^{2}, $$
    (A.8)
    $$ S_{\varphi} {\Psi}_{n}={\varphi}_{n},\qquad S_{\Psi} {\varphi}_{n}={\Psi}_{n}, $$
    (A.9)

    for n = 0, 1, as well as \(S_{\varphi }=S_{\Psi }^{-1}\) and the following intertwining relations

    $$ S_{\Psi} N=N^{\dagger} S_{\Psi},\qquad S_{\varphi} N^{\dagger} =N S_{\varphi}. $$
    (A.10)

Notice that, being biorthogonal, the vectors of both \(\mathcal {F}_{\varphi }\) and \(\mathcal {F}_{\Psi }\) are linearly independent. Hence φ0 and φ1 are two linearly independent vectors in a two-dimensional Hilbert space, so that \(\mathcal {F}_{\varphi }\) is a basis for \({\mathscr{H}}\). The same argument obviously can be used for \(\mathcal {F}_{\Psi }\). More than this: both these sets are also Riesz bases. We refer to [5] for more details.

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Bagarello, F., Bavuma, Y. & Russo, F.G. Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems. Math Phys Anal Geom 24, 16 (2021). https://doi.org/10.1007/s11040-021-09387-1

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Keywords

  • Pauli groups
  • Actions of groups
  • Hamiltonians
  • Pseudo-fermions
  • Central products

Mathematics Subject Classification (2010)

  • Primary: 57M07
  • 57M60; Secondary: 81R05
  • 22E70