Advertisement

Kraus operator formalism for quantum multiplexer operations for arbitrary two-qubit mixed states

  • Pavithra Udayakumar
  • Preethika Kumar-EslamiEmail author
Article
  • 59 Downloads

Abstract

The dynamics of open quantum systems are described using a set of operators called Kraus operators. In this paper, we show how to find system parameters for a closed system of two qubits undergoing quantum multiplexer operations such that Kraus operators can be written for a single open qubit system (the target qubit) when the initial density matrix of the joint system is in any arbitrary mixed state. The strategy used is to extend the single-qubit open system to a larger two-qubit closed system, which can evolve using unitary dynamics. The constructed two-qubit system is evolved using quantum multiplexer operations, and system parameters are derived to implement the operation. To derive the parameters, we use a reduced Hamiltonian technique wherein the qubit of interest (the target) evolves only in subspaces of the second qubit (the control). The main advantage of our scheme is that it is not restricted to separable product states and/or local unitary evolution of the joint system.

Keywords

Quantum Density matrix Quantum multiplexer Quantum switch QMUX Kraus Open quantum system Ising 

Notes

References

  1. 1.
    Nielson, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2001)Google Scholar
  2. 2.
    Makhlin, Y., Schon, G., Shnirman, A.: Quantum state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357 (2001)ADSCrossRefGoogle Scholar
  3. 3.
    Gagnebin, P.K., Skinner, S.R., Behrman, E.C., Steck, J.E., Han, S., Zhou, Z.: Quantum gates using a pulsed bias scheme. Phys. Rev. A 72, 042311 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    Kane, B.E.: A silicon-based nuclear spin quantum computer. Nature 393, 133 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    Pachos, J.K., Knight, P.L.: Quantum computation with a one-dimensional optical lattice. Phys. Rev. Lett. 91, 107902 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    van der Ploeg, S.H.W., Izmalkov, A., van den Brink, A.M., Hübner, U., Grajcar, M., Il’ichev, E., Meyer, H.-G., Zagoskin, A.M.: Controllable coupling of superconducting flux qubits. Phys. Rev. Lett. 98, 057004 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    Stock, R., James, D.F.V.: A scalable, high-speed measurement-based quantum computer using trapped ions. Rev. Lett. 102, 170501 (2009)ADSCrossRefGoogle Scholar
  8. 8.
    Gardiner, C.W.: Quantum Noise. Springer, Berlin (1991)CrossRefGoogle Scholar
  9. 9.
    Weiss, U.: Quantum Dissipative Systems, 3rd edn. World Scientific, Singapore (2008)CrossRefGoogle Scholar
  10. 10.
    Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, New York (2002)zbMATHGoogle Scholar
  11. 11.
    Arrigoni, E., Knap, M., von der Linden, W.: Non-equilibrium dynamical mean field theory: an auxiliary quantum master equation approach. Phys. Rev. Lett. 110, 086403 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Dyre, J.C.: Master-equation approach to the glass transition. Phys. Rev. Lett. 58, 792 (1987)ADSCrossRefGoogle Scholar
  13. 13.
    Giovannetti, V., Palma, G.M.: Master equations for correlated quantum channels. Phys. Rev. Lett. 108, 040401 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Wu, L.-A., Kurizki, G., Brumer, P.: Master equation and control of an open quantum system with leakage. Phys. Rev. Lett. 102, 080405 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    Ferialdi, L.: Exact closed master equation for Gaussian non-Markovian dynamics. Phys. Rev. Lett. 116, 120402 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    Hall, M.J.W., Cresser, C.D., Li, L., Andersson, E.: Canonical form of master equations and characterization of non-Markovianity. Phys. Rev. A 89, 042120 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Maldunado-Mundo, D., Öhberg, P., Lovett, B.W., Andersson, E.: Investigating the generality of time-local master equations. Phys. Rev. A 86, 042107 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    Joshi, P., Öhberg, C., Cresser, J.D., Andersson, E.: Markovian evolution of strongly coupled harmonic oscillators. Phys. Rev. A 90, 063815 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    Vacchini, B.: Generalized master equations using to completely positive dynamics. Phys. Rev. Lett. 117, 230401 (2016)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Diósi, L., Ferialdi, L.: General non-Markovian structure of Gaussian master and stochastic Schrödinger equations. Phys. Rev. Lett. 113, 200403 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    Lindblad, G.: Brownian motion of a quantum harmonic oscillator. Rep. Math. Phys. 10, 393 (1976)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Rivas, A., Huelga, S.F.: Open quantum systems. An introduction (2012). arXiv:1104.5242v2 CrossRefGoogle Scholar
  23. 23.
    de Vega, I., Alonso, D.: Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys. 89, 15001 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kraus, K.: States. Effects and Operations. Spring, Berlin (1983)zbMATHGoogle Scholar
  25. 25.
    Romero, L.D., Paz, J.P.: Decoherence and initial correlations in quantum Brownian motion. Phys. Rev. A 55, 4070 (1997)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Bouda, J., Bužek, V.: Purification and correlated measurements of bipartite mixed states. Phys. Rev. A 65, 034304 (2003)ADSCrossRefGoogle Scholar
  27. 27.
    Pechukas, P.: Reduced dynamics needs not be completely positive. Phys. Rev. Lett. 73, 1060 (1994)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Kimura, G.: Restriction on relaxation times derived from the Lindblad-type master equations of two-level systems. Phys. Rev. A 66, 062113 (2002)ADSCrossRefGoogle Scholar
  29. 29.
    Štelmachovič, P., Bužek, V.: Dynamics of open quantum systems initially entangled with the environment: beyond the Kraus representation. Phys. Rev. A 64, 062106 (2001)ADSCrossRefGoogle Scholar
  30. 30.
    Salgado, D., S´anchez-G´omez, J. L.: arXiv:quant-ph/0211164 (2002)
  31. 31.
    Hayashi, H., Kimura, G., Ota, Y.: Kraus representation in the presence of initial correlations. Phys. Rev. A 67, 062109 (2003)ADSCrossRefGoogle Scholar
  32. 32.
    Tong, D.M., Chen, J., Kwek, L.C., Oh, C.H.: Kraus representation for density operator of arbitrary open qubit system. Laser Phys. 16(11), 1512 (2006)ADSCrossRefGoogle Scholar
  33. 33.
    Arsenijevic, M., Jeknic-Dugic, J., Dugic, M.: Kraus operators for a pair of interacting qubits: a case study. Braz. J. Phys. 48(3), 242 (2018)ADSCrossRefGoogle Scholar
  34. 34.
    Garigipati, R., Kumar, P.: Mirror inverse operations in linear nearest neighbors using dynamic learning algorithm. IEEE Trans. Neural Netw. Learn. Syst. 27(1), 202 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer ScienceWichita State UniversityWichitaUSA

Personalised recommendations