Real-time simulation of H–P noisy Schrödinger equation and Belavkin filter

  • Naman Garg
  • Harish Parthasarathy
  • D. K. Upadhyay
Article

Abstract

The Hudson–Parthasarathy noisy Schrödinger equation is an infinite-dimensional differential equation where the noise operators—Creation, Annihilation and Conservation processes—take values in Boson Fock space. We choose a finite truncated basis of exponential vectors for the Boson Fock space and obtained the unitary evolution in a truncated orthonormal basis using the Gram–Schmidt orthonormalization process to the exponential vectors. Then, this unitary evolution is used to obtained the approximate evolution of the system state by tracing out over the bath space. This approximate evolution is compared to the exact Gorini–Kossakowski–Sudarshan–Lindblad equation for the system state. We also perform a computation of the rate of change of the Von Neumann entropy for the system assuming vacuum noise state and derive condition for entropy increase. Finally, by taking non-demolition measurement in the sense of Belavkin, we simulate the Belavkin quantum filter and show that the Frobenius norm of the error observables \(j_t(X)-\pi _t(X)\) becomes smaller with time for a class of observable X. Here \(j_t(X)\) is the H–P equation observable and \(\pi _t(X)\) is the Belavkin filter output observable. In last, we have derived an approximate expression for the filtered density and entropy of the system after filtering.

Keywords

H–P quantum Itô formula Noisy Schrödinger equation Gram–Schmidt orthonormalization GKSL equation Non-demolition measurements Kushner–Belavkin quantum filter Von Neumann entropy 

References

  1. 1.
    Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Birkhauser, Berlin (1992)MATHCrossRefGoogle Scholar
  2. 2.
    Hudson, R.L., Parhasarathy, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984)ADSMathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Varadarajan, V.S.: Lie Groups, Lie Algebras, and Their Representations (Graduate text in Mathematics). Springer, Berlin (1984)MATHCrossRefGoogle Scholar
  4. 4.
    Hayasi, M.: Quantum Information Theory: An Introduction. Springer, Berlin (2006)Google Scholar
  5. 5.
    Gough, J., Kostler, C.: Quantum filtering in coherent states. Commun. Stoch. Anal. 4(4), 505–521 (2010)MathSciNetMATHGoogle Scholar
  6. 6.
    Kushner, H.J.: Jump-diffusion approximations for ordinary differential equations with wide-band random right hand sides. SIAM J. Control Optim. 17, 729–744 (1979)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kushner, H.J.: Diffusion approximations to output processes of nonlinear systems with wide-band inputs and applications. IEEE Trans. Inf. Theory 26, 715–725 (1990)ADSMathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Geb. 11, 230–243 (1969)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Davis, M.H.A., Marcus, S.I.: An introduction to nonlinear filtering. Stoch. Syst. Math. Filter. Identif. Appl. 78, 53–75 (1981)MATHGoogle Scholar
  10. 10.
    Belavkin, V.P.: Quantum filtering of Markov signals with white quantum noise. Radiotech. Electron. 25, 1445–1453 (1980)ADSGoogle Scholar
  11. 11.
    Belavkin, V.P.: Quantum continual measurements and a posteriori collapse on CCR. Commun. Math. Phys. 146, 611–635 (1992)ADSMathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Belavkin, V.P.: Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42, 171–201 (1992)ADSMathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Belavkin, V.P.: Quantum stochastic positive evolutions: characterization, construction, dilationcalculus and quantum nonlinear filtering. Commun. Math. Phys. 184, 533–566 (1997)ADSMATHCrossRefGoogle Scholar
  14. 14.
    Belavkin, V.P.: Quantum quasi-Markov processes in eventum mechanics dynamics, observation, filtering and control. Quantum Inf. Process. 12, 1539–1626 (2013)ADSMathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gough, J., Belavkin, V.P.: Quantum control and information processing. Quantum Inf. Process. 12, 1397–1415 (2013)ADSMathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Statistical Signal Processing Lab, Room No. 127, ECE Division, Netaji Subhas Institute of TechnologyUniversity of DelhiNew DelhiIndia

Personalised recommendations