Papers
Topics
Authors
Recent
Search
2000 character limit reached

A geometrical description of non-Hermitian dynamics: speed limits in finite rank density operators

Published 22 May 2024 in quant-ph | (2405.13913v2)

Abstract: Non-Hermitian dynamics in quantum systems preserves the rank of the state density operator. Using this insight, we develop a geometric framework to describe its time evolution. In particular, we identify mutually orthogonal coherent and incoherent directions and provide their physical interpretation. This understanding enables us to optimize the success rate of non-Hermitian driving along prescribed trajectories, with direct relevance to shortcuts to adiabaticity. Next, we explore the geometric interpretation of a speed limit for non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit expression that saturates this bound and illustrate our results with a minimal example of a dissipative qubit.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (60)
  1. J. Muga, J. Palao, B. Navarro, and I. Egusquiza, “Complex absorbing potentials,” Phys. Rep., vol. 395, no. 6, pp. 357, 2004.
  2. Y. Ashida, Z. Gong, and M. Ueda, “Non-Hermitian physics,” Adv. in Phys., vol. 69, no. 3, pp. 249, 2020.
  3. H. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields. Theoretical and Mathematical Physics, Springer Berlin Heidelberg, 2007.
  4. K. Jacobs, Quantum Measurement Theory. Cambridge University Press, 2014.
  5. S. Alipour, A. Chenu, A. T. Rezakhani, and A. del Campo, “Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution,” Quantum, vol. 4, p. 336, 2020.
  6. S. Amari, Information geometry and its applications. Springer, 2016.
  7. S. Amari and H. Nagaoka, “Methods of Information Geometry,” 2007. American Mathematical Society Series: Transl. of Math. Monographs vol.: 191.
  8. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Roy. London A, vol. 392, 1984.
  9. F. Wilczek and A. Shapere, Geometric Phases in Physics. Advanced series in Math. Phys., World Scientific, 1989.
  10. D. Chruscinski and A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics. Progress in Math. Phys., Birkhäuser Boston, 2004.
  11. M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor., vol. 42, p. 365303, 2009.
  12. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, “Shortcuts to adiabaticity: Concepts, methods, and applications,” Rev. of Modern Phys., vol. 91, p. 045001, 2019.
  13. S. Deffner and S. Campbell, “Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control,” J. of Phys. A: Math. and Theor., vol. 50, p. 453001, 2017.
  14. Z. Gong and R. Hamazaki, “Bounds in nonequilibrium quantum dynamics,” Int. J. Mod. Phys. B, vol. 36, 2022.
  15. M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A, vol. 107, no. 46, p. 9937, 2003.
  16. M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B, vol. 109, no. 14, p. 6838, 2005.
  17. M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys., vol. 129, no. 15, p. 154111, 2008.
  18. G. Vacanti, R. Fazio, S. Montangero, G. M. Palma, M. Paternostro, and V. Vedral, “Transitionless quantum driving in open quantum systems,” New J. of Phys., vol. 16, p. 053017, 2014.
  19. S. Alipour, A. T. Rezakhani, A. Chenu, A. del Campo, and T. Ala-Nissila, “Entropy-based formulation of thermodynamics in arbitrary quantum evolution,” Phys. Rev. A, vol. 105, p. L040201, 2022.
  20. W. F. Stinespring, “Positive Functions on C*-Algebras,” Proc. of the American Math. Society, vol. 6, no. 2, pp. 211, 1955.
  21. M. Hayashi, Quantum Information Theory: Mathematical Foundation. Graduate Texts in Physics, Springer Berlin Heidelberg, 2016.
  22. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. 2010.
  23. H. M. Wiseman and G. J. Milburn, Quantum measurement and control. Cambridge University Press, 2009.
  24. J. Audretsch, “Mixed States and the Density Operator,” in Entangled Systems, pp. 73, John Wiley & Sons, Ltd, 2007.
  25. C. A. Fuchs and K. Jacobs, “Information-tradeoff relations for finite-strength quantum measurements,” Phys. Rev. A, vol. 63, p. 062305, 2001.
  26. M. A. Nielsen, “Characterizing mixing and measurement in quantum mechanics,” Physical Review A, vol. 63, p. 022114, 2001.
  27. J. Grabowski, M. Kuś, and G. Marmo, “Geometry of quantum systems: density states and entanglement,” J. of Phys. A: Math. and General, vol. 38, p. 10217, 2005.
  28. F. M. Ciaglia, “Quantum states, groups and monotone metric tensors,” The European Phys. J. Plus, vol. 135, p. 530, 2020.
  29. D. C. Brody and E.-M. Graefe, “Mixed-state evolution in the presence of gain and loss,” Phys. Rev. Lett., vol. 109, p. 230405, 2012.
  30. J. Cornelius, Z. Xu, A. Saxena, A. Chenu, and A. del Campo, “Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos,” Phys. Rev. Lett., vol. 128, p. 190402, 2022.
  31. A. S. Matsoukas-Roubeas, F. Roccati, J. Cornelius, Z. Xu, A. Chenu, and A. del Campo, “Non-Hermitian Hamiltonian deformations in quantum mechanics,” J. High Energy Phys., vol. 2023, 2023.
  32. A. S. Matsoukas-Roubeas, M. Beau, L. F. Santos, and A. del Campo, “Unitarity breaking in self-averaging spectral form factors,” Phys. Rev. A, vol. 108, p. 062201, 2023.
  33. S. Das and J. R. Green, “Density matrix formulation of dynamical systems,” Physical Review E, vol. 106, p. 054135, 2022.
  34. M. Hübner, “Computation of Uhlmann’s parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space,” Phys. Lett. A, vol. 179, pp. 226, 1993.
  35. H. Hasegawa, “α𝛼\alphaitalic_α-divergence of the non-commutative information geometry,” Rep. on Math. Phys., vol. 33, pp. 87, 1993.
  36. D. Petz and H. Hasegawa, “On the Riemannian metric of α𝛼\alphaitalic_α-entropies of density matrices,” Lett. in Math. Phys., vol. 38, pp. 221, 1996.
  37. J. Dittmann and A. Uhlmann, “Connections and metrics respecting purification of quantum states,” J. of Math. Phys., vol. 40, pp. 3246, 1999.
  38. O. Andersson, “Holonomy in Quantum Information Geometry,” 2019. arXiv:1910.08140.
  39. D. Girolami, “How Difficult is it to Prepare a Quantum State?,” Phys. Rev. Lett., vol. 122, p. 010505, 2019.
  40. K. Funo, N. Shiraishi, and K. Saito, “Speed limit for open quantum systems,” New J. of Phys., vol. 21, p. 013006, 2019.
  41. L. P. García-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, “Unifying Quantum and Classical Speed Limits on Observables,” Phys. Rev. X, vol. 12, p. 011038, 2022.
  42. S. Alipour, A. T. Rezakhani, A. Chenu, A. del Campo, and T. Ala-Nissila, “Entropy-based formulation of thermodynamics in arbitrary quantum evolution,” Phys. Rev. A, vol. 105, 2022.
  43. I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, 2006.
  44. E. A. Morozova and N. N. Čencov, “Markov invariant geometry on manifolds of states,” J Math Sci, vol. 56, 1991.
  45. D. Petz, “Monotone metrics on matrix spaces,” Linear Algebra and its Applications, vol. 244, pp. 81–96, 1996.
  46. J. Dittmann, “On the Riemannian metric on the space of density matrices,” Rep. on Math. Phys., vol. 36, pp. 309, 1995.
  47. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett., vol. 72, pp. 3439, 1994.
  48. T. Kato, “On the adiabatic theorem of quantum mechanics,” J. Phys. Soc. Jpn., vol. 5, pp. 435, 1950.
  49. S. Jansen, M.-B. Ruskai, and R. Seiler, “Bounds for the adiabatic approximation with applications to quantum computation,” J. of Math. Phys., vol. 48, p. 102111, 2007.
  50. N. Hörnedal, D. Allan, and O. Sönnerborn, “Extensions of the Mandelstam-Tamm quantum speed limit to systems in mixed states,” New J. of Phys., vol. 24, p. 055004, 2022.
  51. Boca Raton: CRC Press, 2 ed., 2017.
  52. A. Uhlmann, “The ‘transition probability’ in the state space of a *-algebra,” Rep. on Math. Phys., vol. 9, pp. 273, 1976.
  53. A. Uhlmann, “An energy dispersion estimate,” Phys. Lett. A, vol. 161, pp. 329, 1992.
  54. F. Fröwis, “Kind of entanglement that speeds up quantum evolution,” Phys. Rev. A, vol. 85, p. 052127, 2012.
  55. M. M. Taddei, B. M. Escher, L. Davidovich, R. L. de Matos Filho, “Quantum Speed Limit for Physical Processes”, Phys. Rev. Lett, vol. 110, p. 050402, 2013.
  56. D. Petz, “State Estimation,” in Quantum Information Theory and Quantum Statistics (D. Petz, ed.), pp. 143, Springer Berlin Heidelberg, 2008.
  57. D. Thakuria, A. Srivastav, B. Mohan, A. Kumari, and A. K. Pati, “Generalised quantum speed limit for arbitrary time-continuous evolution,” J. of Phys. A: Math. and Theor., vol. 57, p. 025302, 2023.  
  58. Å. Ericsson, “Geodesics and the best measurement for distinguishing quantum states,” J. of Phys. A: Math. and General, vol. 38, p. L725, 2005.
  59. D. Spehner, “Bures geodesics and quantum metrology,” 2023. arXiv:2308.08706.
  60. F. D’Andrea, D. Franco, “On the pseudo-manifold of quantum states,” ScienceDirect, vol. 78, p. 101800, 2021.
Citations (2)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free