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Fundamentals of Continuous Variables

  • Gianfranco CariolaroEmail author
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

The fundamentals introduced in Chap.  3, based on the four postulates of Quantum Mechanics, are concerned with discrete variables, with operators and quantum measurements specified by enumerable sets. But Quantum Information makes use of both discrete and continuous variables. The extension of fundamentals to continuous variables is done starting from the quantum harmonic oscillator, which is the basis of the theory of the electromagnetic field, in which the electromagnetic radiation is represented as a combination of harmonic oscillators. In this context, position and momentum (canonical variables), and annihilator and the creation operator (bosonic variables) are the fundamental operators for the development of the theory of continuous variables. The environment is an infinite dimensional Hilbert space, but an alternative environment is given by the phase space, a two-dimensional real space where quantum states of infinite dimensions are simply represented (in the single mode) by two functions of two real variables, the Wigner and the characteristic functions. These functions allow for the introduction of Gaussian states and Gaussian transformations. In addition to offering an easy description in terms of Gaussian functions, Gaussian states and transformations are of great practical relevance and represent the main tool of Quantum Information processing, with applications to quantum computation, quantum cryptography, and quantum communications. Coherent states are notable examples of Gaussian states, but in this chapter several other examples of Gaussian states are also considered. The most interesting applications, in particular the ones based on the entanglement, are concerned with continuous variables in the multimode, where the Hilbert space is given by \(N\) replicas (tensor product) of the space of the single mode and the phase space becomes \(2N\)-dimensional. The extension to multimode continuous variables represents the main difficulty of the chapter.

References

  1. 1.
    C. Weedbrook, S. Pirandola, R. García-Patrón, N.J. Cerf, T.C. Ralph, J.H. Shapiro, S. Lloyd, Gaussian quantum information. Rev. Mod. Phys. 84, 621–669 (2012)CrossRefGoogle Scholar
  2. 2.
    P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1958)zbMATHGoogle Scholar
  3. 3.
    S.L. Braunstein, P. van Loock, Quantum information with continuous variables. Rev. Mod. Phys. 77, 513–577 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    U. Andersen, G. Leuchs, C. Silberhorn, Continuous-variable quantum information processing. Laser Photon. Rev. 4(3), 337–354 (2010)CrossRefGoogle Scholar
  5. 5.
    S.L. Braunstein, A.K. Pati, Quantum Information with Continuous Variables (Kluwer Academic Publishers, Dordrecht, 2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    N. Cerf, G. Leuchs, E. Polzik, Quantum Information with Continuous Variables of Atoms and Light (Imperial College Press, London, 2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    X.B. Wang, T. Hiroshima, A. Tomita, M. Hayashi, Quantum information with Gaussian states. Phys. Rep. 448(1–4), 1–111 (2007)CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. Ferraro, S. Olivares, M. Paris, Gaussian States in Continuous Variable Quantum Information, Napoli Series on Physics and Astrophysics (2005). (ed. Bibliopolis, Napoli, 2005)Google Scholar
  9. 9.
    S. Olivares, Quantum optics in the phase space. Eur. Phys. J. Spec. Top. 203(1), 3–24 (2012)CrossRefGoogle Scholar
  10. 10.
    J. Eisert, M.B. Plenio, Introduction to the basics of entanglement theory in continuous-variable systems. Int. J. Quantum Inf. 01(04), 479–506 (2003)CrossRefGoogle Scholar
  11. 11.
    W.H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964)Google Scholar
  12. 12.
    A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965)Google Scholar
  13. 13.
    R.J. Glauber, The quantum theory of optical coherence. Phys. Rev. 130, 2529–2539 (1963)CrossRefMathSciNetGoogle Scholar
  14. 14.
    K.E. Cahill, R.J. Glauber, Ordered expansions in Boson amplitude operators. Phys. Rev. 177, 1857–1881 (1969)Google Scholar
  15. 15.
    R. Simon, N. Mukunda, B. Dutta, Quantum-noise matrix for multimode systems: u(n) invariance, squeezing, and normal forms. Phys. Rev. A 49, 1567–1583 (1994)CrossRefMathSciNetGoogle Scholar
  16. 16.
    W. Rudin, Fourier Analysis on Groups (Interscience Publishers, New York, 1962)zbMATHGoogle Scholar
  17. 17.
    G. Cariolaro, Unified Signal Theory (Springer, London, 2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd edn. (Addison-Wesley, Redwood, 1991)Google Scholar
  19. 19.
    E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)CrossRefGoogle Scholar
  20. 20.
    K.E. Cahill, R.J. Glauber, Density operators and quasiprobability distributions. Phys. Rev. 177, 1882–1902 (1969)Google Scholar
  21. 21.
    M.G.A. Paris, The modern tools of quantum mechanics. Eur. Phys. J. Spec. Top. 203(1), 61–86 (2012)CrossRefGoogle Scholar
  22. 22.
    R. Simon, E.C.G. Sudarshan, N. Mukunda, Gaussian-Wigner distributions in quantum mechanics and optics. Phys. Rev. A 36, 3868–3880 (1987)CrossRefMathSciNetGoogle Scholar
  23. 23.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)Google Scholar
  24. 24.
    G. Cariolaro, G. Pierobon, The Weyl operator as an eigenoperator of the symplectic Fourier transform, to be publishedGoogle Scholar
  25. 25.
    I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, 7th edn. (Elsevier, Amsterdam, 2007)Google Scholar
  26. 26.
    X. Ma, W. Rhodes, Multimode squeeze operators and squeezed states. Phys. Rev. A 41, 4625–4631 (1990)CrossRefGoogle Scholar
  27. 27.
    D.F. Walls, G.J. Milburn, Quantum Optics (Springer, Berlin, 2008)CrossRefzbMATHGoogle Scholar
  28. 28.
    C.M. Caves, C. Zhu, G.J. Milburn, W. Schleich, Photon statistics of two-mode squeezed states and interference in four-dimensional phase space. Phys. Rev. A 43, 3854–3861 (1991)CrossRefGoogle Scholar
  29. 29.
    B.L. Schumaker, Quantum mechanical pure states with Gaussian wave functions. Phys. Rep. 135(6), 317–408 (1986)CrossRefMathSciNetGoogle Scholar
  30. 30.
    X. Ma, Time evolution of stable squeezed states. J. Mod. Opt. 36(8), 1059–1064 (1989)CrossRefzbMATHGoogle Scholar
  31. 31.
    H.P. Yuen, R. Kennedy, M. Lax, Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inf. Theory 21(2), 125–134 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    H.P. Yuen, Two-photon coherent states of the radiation field. Phys. Rev. A 13, 2226–2243 (1976)CrossRefGoogle Scholar
  33. 33.
    N.L. Johnson, S. Kotz, N. Balakrishnan, Discrete Multivariate Distributions (Wiley, New York, 1997)zbMATHGoogle Scholar
  34. 34.
    C.M. Caves, B.L. Schumaker, New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states. Phys. Rev. A 31, 3068–3092 (1985)CrossRefMathSciNetGoogle Scholar
  35. 35.
    M.S. Kim, W. Son, V. Bužek, P.L. Knight, Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement. Phys. Rev. A 65, paper no. 032323 (2002). http://link.aps.org/doi/10.1103/PhysRevA.65.032323
  36. 36.
    M.M. Wolf, J. Eisert, M.B. Plenio, Entangling power of passive optical elements. Phys. Rev. Lett. 90, paper no. 047904 (2003)Google Scholar
  37. 37.
    G. Cariolaro, R. Corvaja, G. Pierobon, Gaussian states and geometrically uniform symmetry. Phys. Rev. A 90(4) (2014)Google Scholar
  38. 38.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1998)Google Scholar
  39. 39.
    M.S. Kim, F.A.M. de Oliveira, P.L. Knight, Properties of squeezed number states and squeezed thermal states. Phys. Rev. A 40, 2494–2503 (1989)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Information EngineeringUniversity of PadovaPadovaItaly

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