Advertisement

Formalization of Infinite Dimension Linear Spaces with Application to Quantum Theory

  • Mohamed Yousri Mahmoud
  • Vincent Aravantinos
  • Sofiène Tahar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7871)

Abstract

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

Keywords

Linear Space Linear Algebra Beam Splitter Scalar Multiplication Computer Algebra System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aladev, V.Z.: Computer Algebra Systems: A New Software Toolbox For Maple. Computer Mathematics Series. Fultus Books (2004)Google Scholar
  2. 2.
    Bakshi, U.A., Bakshi, V.: Modern Control Theory. Technical Publications (2009)Google Scholar
  3. 3.
    Chandrasekaran, S., Manjunath, B.S., Wang, Y.F., Winkeler, J., Zhang, H.: An Eigenspace Update Algorithm for Image Analysis. Graphical Models and Image Processing 59(5), 321–332 (1997)CrossRefGoogle Scholar
  4. 4.
    Dettman, J.W.: Introduction to Linear Algebra. Dover Books on Mathematics Series. Dover (1974)Google Scholar
  5. 5.
    Fox, M.: Quantum Optics: An Introduction. Oxford Master Series in Physics. Oxford University Press (2006)Google Scholar
  6. 6.
    Griffiths, D.J.: Introduction to Quantum Mechanics. Pearson Prentice Hall (2005)Google Scholar
  7. 7.
    Harrison, J.: HOL Light: A Tutorial Introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  8. 8.
    Harrison, J.: A HOL Theory of Euclidean Space. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 114–129. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Herencia-Zapana, H., Jobredeaux, R., Owre, S., Garoche, P.-L., Feron, E., Perez, G., Ascariz, P.: PVS linear algebra libraries for verification of control software algorithms in C/ACSL. In: Goodloe, A.E., Person, S. (eds.) NFM 2012. LNCS, vol. 7226, pp. 147–161. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Hirvensalo, M.: Quantum Computing. Natural Computing Series. Springer (2004)Google Scholar
  11. 11.
  12. 12.
  13. 13.
    Leonhardt, U.: Quantum Physics of Simple Optical Instruments. Reports on Progress in Physics 66(7), 1207 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Leonhardt, U.: Essential Quantum Optics: From Quantum Measurements to Black Holes. Cambridge University Press (2010)Google Scholar
  15. 15.
  16. 16.
    The Coq development team: The Coq Proof Assistant Reference Manual. LogiCal Project, Version 8.0 (2004)Google Scholar
  17. 17.
    Ralph, T.C., Gilchrist, A., Milburn, G.J., Munro, W.J., Glancy, S.: Quantum Computation with Optical Coherent States. Physical Review A 68, 042319 (2003)Google Scholar
  18. 18.
    Strang, G.: Introduction to Linear Algebra. Wellsley-Cambrige Press (2003)Google Scholar
  19. 19.
    Vinga, S., Almeida, J.: Alignment-Free Sequence Comparison – A Review. Bioinformatics 19(4), 513–523 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mohamed Yousri Mahmoud
    • 1
  • Vincent Aravantinos
    • 1
  • Sofiène Tahar
    • 1
  1. 1.Electrical and Computer Engineering Dept.Concordia UniversityMontrealCanada

Personalised recommendations