[go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

Entanglement entropy distinguishes PT-symmetry and topological phases in a class of non-unitary quantum walks

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We calculate the hybrid entanglement entropy between coin and walker degrees of freedom in a class of two-step non-unitary quantum walks. The model possesses a joint parity and time-reversal symmetry or PT-symmetry and supports topological phases when this symmetry is unbroken by its eigenstates. An asymptotic analysis at long times reveals that the quantum walk can indefinitely sustain hybrid entanglement in the unbroken symmetry phase even when gain and loss mechanisms are present. However, when the gain-loss strength is too large, the PT-symmetry of the model is spontaneously broken and entanglement vanishes. The entanglement entropy is therefore an effective and robust parameter for constructing PT-symmetry and topological phase diagrams in this non-unitary dynamical system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Nayak, A., Vishwanath, A.: Quantum walk on the line. arXiv:quant-ph/0010117 (2000)

  2. Yang, Y.-G., Pan, Q.-X., Sun, S.-J., Xu, P.: Novel image encryption based on quantum walks. Sci. Rep. 5, 7784 (2015). https://doi.org/10.1038/srep07784

    Article  Google Scholar 

  3. Abd El-Latif, A.A., Abd-El-Atty, B., Amin, M., Iliyasu, A.M.: Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications. Sci. Rep. 10, 1930 (2020). https://doi.org/10.1038/s41598-020-58636-w

    Article  ADS  Google Scholar 

  4. Aaronson, S., Ambainis, A.: Quantum search of spatial regions. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 200–209. IEEE Computer Society, Los Alamitos, CA (2003). https://doi.org/10.1109/SFCS.2003.1238194

  5. Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inform. 1(4), 507–518 (2003). https://doi.org/10.1142/S0219749903000383

    Article  MATH  Google Scholar 

  6. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1099–1108. SIAM, Philadelphia, PA (2005). e-print https://doi.org/10.48550/arXiv.quant-ph/0402107

  7. Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18), 180501 (2009). https://doi.org/10.1103/PhysRevLett.102.180501

    Article  ADS  MathSciNet  Google Scholar 

  8. Lovett, N.B., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81(4), 042330 (2010). https://doi.org/10.1103/PhysRevA.81.042330

    Article  ADS  MathSciNet  Google Scholar 

  9. Kurzyński, P., Wójcik, A.: Discrete-time quantum walk approach to state transfer. Phys. Rev. A 83(6), 062315 (2011). https://doi.org/10.1103/PhysRevA.83.062315

    Article  ADS  Google Scholar 

  10. Shang, Y., Wang, Y., Li, M., Lu, R.: Quantum communication protocols by quantum walks with two coins. Europhys. Lett. 124(6), 60009 (2018). https://doi.org/10.1209/0295-5075/124/60009

    Article  Google Scholar 

  11. Perets, H.B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y.: Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100(17), 170506 (2008). https://doi.org/10.1103/PhysRevLett.100.170506

    Article  ADS  Google Scholar 

  12. Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Mosley, P.J., Andersson, E., Jex, I., Silberhorn, C.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104(5), 050502 (2010). https://doi.org/10.1103/PhysRevLett.104.050502

    Article  ADS  Google Scholar 

  13. Karski, M., Förster, L., Choi, J.-M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325(5937), 174–177 (2009). https://doi.org/10.1126/science.1174436

    Article  ADS  Google Scholar 

  14. Preiss, P.M., Ma, R., Eric Tai, M., Lukin, A., Rispoli, M., Zupancic, P., Lahini, Y., Islam, R., Greiner, M.: Strongly correlated quantum walks in optical lattices. Science 347(6227), 1229–1233 (2015). https://doi.org/10.1126/science.1260364

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Manouchehri, K., Wang, J.: Physical implementation of quantum walks, 1st edn. Springer, Berlin (2014). https://doi.org/10.1007/978-3-642-36014-5

  16. De Nicola, F., Sansoni, L., Crespi, A., Ramponi, R., Osellame, R., Giovannetti, V., Fazio, R., Mataloni, P., Sciarrino, F.: Quantum simulation of bosonic-fermionic noninteracting particles in disordered systems via a quantum walk. Phys. Rev. A 89(3), 032322 (2014). https://doi.org/10.1103/PhysRevA.89.032322

    Article  ADS  Google Scholar 

  17. Regensburger, A., Bersch, C., Miri, M.-A., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity-time synthetic photonic lattices. Nature 488(7410), 167–171 (2012). https://doi.org/10.1038/nature11298

    Article  ADS  Google Scholar 

  18. Xiao, L., Zhan, X., Bian, Z., Wang, K., Zhang, X., Wang, X., Li, J., Mochizuki, K., Kim, D., Kawakami, N., Yi, W., Obuse, H., Sanders, B.C., Xue, P.: Observation of topological edge states in parity-time-symmetric quantum walks. Nat. Phys. 13(11), 1117–1123 (2017). https://doi.org/10.1038/nphys4204

    Article  Google Scholar 

  19. Zhan, X., Xiao, L., Bian, Z., Wang, K., Qiu, X., Sanders, B.C., Yi, W., Xue, P.: Detecting topological invariants in nonunitary discrete-time quantum walks. Phys. Rev. Lett. 119(13), 130501 (2017). https://doi.org/10.1103/PhysRevLett.119.130501

    Article  ADS  MathSciNet  Google Scholar 

  20. Özdemir, ŞK., Rotter, S., Nori, F., Yang, L.: Parity-time symmetry and exceptional points in photonics. Nat. Mater. 18(8), 783–798 (2019). https://doi.org/10.1038/s41563-019-0304-9

    Article  ADS  Google Scholar 

  21. Hatano, N., Obuse, H.: Delocalization of a non-Hermitian quantum walk on random media in one dimension. Ann. Phys. 435, 168615 (2021). https://doi.org/10.1016/j.aop.2021.168615

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin, Q., Li, T., Xiao, L., Wang, K., Yi, W., Xue, P.: Observation of non-Hermitian topological Anderson insulator in quantum dynamics. Nat. Commun. 13(1), 3229 (2022). https://doi.org/10.1038/s41467-022-30938-9

    Article  ADS  Google Scholar 

  23. Carneiro, I., Loo, M., Xu, X., Girerd, M., Kendon, V., Knight, P.L.: Entanglement in coined quantum walks on regular graphs. New J. Phys. 7(1), 156 (2005). https://doi.org/10.1088/1367-2630/7/1/156

    Article  ADS  Google Scholar 

  24. Abal, G., Siri, R., Romanelli, A., Donangelo, R.: Quantum walk on the line: Entanglement and nonlocal initial conditions. Phys. Rev. A 73(4), 042302 (2006). https://doi.org/10.1103/PhysRevA.73.042302

    Article  ADS  MathSciNet  Google Scholar 

  25. Ide, Y., Konno, N., Machida, T.: Entanglement for discrete-time quantum walks on the line. Quantum Inf. Comput. 11(9–10), 855–866 (2011). https://doi.org/10.48550/arXiv.1012.4164

    Article  MathSciNet  MATH  Google Scholar 

  26. Neves, L., Lima, G., Delgado, A., Saavedra, C.: Hybrid photonic entanglement: realization, characterization, and applications. Phys. Rev. A 80(4), 042322 (2009). https://doi.org/10.1103/PhysRevA.80.042322

    Article  ADS  Google Scholar 

  27. Li, Y., Gessner, M., Li, W., Smerzi, A.: Hyper- and hybrid nonlocality. Phys. Rev. Lett. 120, 050404 (2018). https://doi.org/10.1103/PhysRevLett.120.050404

    Article  ADS  Google Scholar 

  28. Flamini, F., Spagnolo, N., Sciarrino, F.: Photonic quantum information processing: a review. Rep. Prog. Phys. 82(1), 016001 (2019). https://doi.org/10.1088/1361-6633/aad5b2

    Article  ADS  Google Scholar 

  29. Gratsea, A., Metz, F., Busch, T.: Universal and optimal coin sequences for high entanglement generation in 1D discrete time quantum walks. J. Phys. A: Math. Theor. 53(44), 445306 (2020). https://doi.org/10.1088/1751-8121/abb54d

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Gratsea, A., Lewenstein, M., Dauphin, A.: Generation of hybrid maximally entangled states in a one-dimensional quantum walk. Quantum Sci. Technol. 5(2), 025002 (2020). https://doi.org/10.1088/2058-9565/ab6ce6

    Article  ADS  Google Scholar 

  31. Vieira, R., Amorim, E.P.M., Rigolin, G.: Dynamically disordered quantum walk as a maximal entanglement generator. Phys. Rev. Lett. 111(18), 180503 (2013). https://doi.org/10.1103/PhysRevLett.111.180503

    Article  ADS  Google Scholar 

  32. Maloyer, O., Kendon, V.: Decoherence versus entanglement in coined quantum walks. New J. Phys. 9(4), 87 (2007). https://doi.org/10.1088/1367-2630/9/4/087

    Article  ADS  Google Scholar 

  33. Dey, S., Raj, A., Goyal, S.K.: Controlling decoherence via PT-symmetric non-Hermitian open quantum systems. Phys. Lett. A 383(30), 125931 (2019). https://doi.org/10.1016/j.physleta.2019.125931

    Article  MathSciNet  MATH  Google Scholar 

  34. Fring, A., Frith, T.: Eternal life of entropy in non-Hermitian quantum systems. Phys. Rev. A 100(1), 010102 (2019). https://doi.org/10.1103/PhysRevA.100.010102

    Article  ADS  MathSciNet  Google Scholar 

  35. Chakraborty, S., Sarma, A.K.: Delayed sudden death of entanglement at exceptional points. Phys. Rev. A 100(6), 063846 (2019). https://doi.org/10.1103/PhysRevA.100.063846

    Article  ADS  MathSciNet  Google Scholar 

  36. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \({\cal{P} }{\cal{T} }\) symmetry. Phys. Rev. Lett. 80(24), 5243–5246 (1998). https://doi.org/10.1103/PhysRevLett.80.5243

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Bender, C.M., Boettcher, S., Meisinger, P.N.: \({\cal{P} }{\cal{T} }\)-symmetric quantum mechanics. J. Math. Phys. 40(5), 2201–2229 (1999). https://doi.org/10.1063/1.532860

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Mochizuki, K., Kim, D., Obuse, H.: Explicit definition of \(\mathscr{P}\mathscr{T}\) symmetry for nonunitary quantum walks with gain and loss. Phys. Rev. A 93(6), 062116 (2016). https://doi.org/10.1103/PhysRevA.93.062116

    Article  ADS  Google Scholar 

  39. Lambert, N., Emary, C., Brandes, T.: Entanglement and entropy in a spin-boson quantum phase transition. Phys. Rev. A 71(5), 053804 (2005). https://doi.org/10.1103/PhysRevA.71.053804

    Article  ADS  Google Scholar 

  40. De Chiara, G., Lepori, L., Lewenstein, M., Sanpera, A.: Entanglement spectrum, critical exponents, and order parameters in quantum spin chains. Phys. Rev. Lett. 109(23), 237208 (2012). https://doi.org/10.1103/PhysRevLett.109.237208

    Article  ADS  Google Scholar 

  41. Wang, Q.-Q., Xu, X.-Y., Pan, W.-W., Tao, S.-J., Chen, Z., Zhan, Y.-T., Sun, K., Xu, J.-S., Chen, G., Han, Y.-J., Li, C.-F., Guo, G.-C.: Robustness of entanglement as an indicator of topological phases in quantum walks. Optica 7(1), 53–58 (2020). https://doi.org/10.1364/OPTICA.375388

    Article  ADS  Google Scholar 

  42. Mochizuki, K., Kim, D., Kawakami, N., Obuse, H.: Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry. Phys. Rev. A 102(6), 062202 (2020). https://doi.org/10.1103/PhysRevA.102.062202

    Article  ADS  MathSciNet  Google Scholar 

  43. Wang, Q., Li, Z.-J.: Topological invariants of nonunitary quantum walk with chiral symmetry. Results Phys. 34, 105279 (2022). https://doi.org/10.1016/j.rinp.2022.105279

    Article  Google Scholar 

  44. Asbóth, J.K., Obuse, H.: Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88(12), 121406 (2013). https://doi.org/10.1103/PhysRevB.88.121406

    Article  ADS  Google Scholar 

  45. Mostafazadeh, A.: Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102(22), 220402 (2009). https://doi.org/10.1103/PhysRevLett.102.220402

    Article  ADS  Google Scholar 

  46. Sergi, A., Zloshchastiev, K.G.: Quantum entropy of systems described by non-Hermitian Hamiltonians. J. Stat. Mech. Theor. Exp. 2016(3), 033102 (2016). https://doi.org/10.1088/1742-5468/2016/03/033102

    Article  MathSciNet  MATH  Google Scholar 

  47. Wen, J., Zheng, C., Ye, Z., Xin, T., Long, G.: Stable states with nonzero entropy under broken \(\cal{PT} \) symmetry. Phys. Rev. Research 3(1), 013256 (2021). https://doi.org/10.1103/PhysRevResearch.3.013256

    Article  ADS  Google Scholar 

  48. Herviou, L., Regnault, N., Bardarson, J.H.: Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models. SciPost Phys. 7, 069 (2019). https://doi.org/10.21468/SciPostPhys.7.5.069

    Article  ADS  MathSciNet  Google Scholar 

  49. Xiao, L., Deng, T., Wang, K., Zhu, G., Wang, Z., Yi, W., Xue, P.: Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nature Phys. 16(7), 761–766 (2020). https://doi.org/10.1038/s41567-020-0836-6

    Article  ADS  Google Scholar 

  50. Ju, C.-Y., Miranowicz, A., Chen, G.-Y., Nori, F.: Non-Hermitian Hamiltonians and no-go theorems in quantum information. Phys. Rev. A 100(6), 062118 (2019). https://doi.org/10.1103/PhysRevA.100.062118

    Article  ADS  MathSciNet  Google Scholar 

  51. Badhani, H., Banerjee, S., Chandrashekar, C.: Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction. arXiv:2109.10682 (2021)

  52. Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 73(8), 1060–1062 (1994). https://doi.org/10.1103/PhysRevLett.73.1060

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Chen, L.-M., Chen, S.A., Ye, P.: Entanglement, non-Hermiticity, and duality. SciPost Phys. 11, 003 (2021). https://doi.org/10.21468/SciPostPhys.11.1.003

    Article  ADS  MathSciNet  Google Scholar 

  54. Hatsugai, Y.: Quantized Berry phases as a local order parameter of a quantum liquid. J. Phys. Soc. Jpn. 75(12), 123601 (2006). https://doi.org/10.1143/JPSJ.75.123601

    Article  ADS  MATH  Google Scholar 

  55. Liang, S.-D., Huang, G.-Y.: Topological invariance and global Berry phase in non-Hermitian systems. Phys. Rev. A 87(1), 012118 (2013). https://doi.org/10.1103/PhysRevA.87.012118

    Article  ADS  Google Scholar 

  56. Zak, J.: Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989). https://doi.org/10.1103/PhysRevLett.62.2747

    Article  ADS  Google Scholar 

  57. Cardano, F., D’Errico, A., Dauphin, A., Maffei, M., Piccirillo, B., de Lisio, C., De Filippis, G., Cataudella, V., Santamato, E., Marrucci, L., Lewenstein, M., Massignan, P.: Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Commun. 8(1), 15516 (2017). https://doi.org/10.1038/ncomms15516

    Article  ADS  Google Scholar 

  58. Shen, H., Zhen, B., Fu, L.: Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120(14), 146402 (2018). https://doi.org/10.1103/PhysRevLett.120.146402

    Article  ADS  MathSciNet  Google Scholar 

  59. Zhang, X.Z., Song, Z.: Partial topological Zak phase and dynamical confinement in a non-Hermitian bipartite system. Phys. Rev. A 99, 012113 (2019). https://doi.org/10.1103/PhysRevA.99.012113

    Article  ADS  Google Scholar 

  60. Brody, D.C.: Biorthogonal quantum mechanics. J. Phys. A Math. Theor. 47(3), 035305 (2013). https://doi.org/10.1088/1751-8113/47/3/035305

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. Song, H.F., Rachel, S., Flindt, C., Klich, I., Laflorencie, N., Le Hur, K.: Bipartite fluctuations as a probe of many-body entanglement. Phys. Rev. B 85(3), 035409 (2012). https://doi.org/10.1103/PhysRevB.85.035409

    Article  ADS  Google Scholar 

  62. Acharya, A.P., Chakrabarty, A., Sahu, D.K., Datta, S.: Localization, \(\cal{PT} \) symmetry breaking, and topological transitions in non-Hermitian quasicrystals. Phys. Rev. B 105(1), 014202 (2022). https://doi.org/10.1103/PhysRevB.105.014202

    Article  ADS  Google Scholar 

  63. Ashida, Y., Furukawa, S., Ueda, M.: Parity-time-symmetric quantum critical phenomena. Nat. Commun. 8, 15791 (2017). https://doi.org/10.1038/ncomms15791

    Article  ADS  Google Scholar 

  64. Kawabata, K., Bessho, T., Sato, M.: Classification of exceptional points and non-Hermitian topological semimetals. Phys. Rev. Lett. 123(6), 066405 (2019). https://doi.org/10.1103/PhysRevLett.123.066405

    Article  ADS  MathSciNet  Google Scholar 

  65. Hanai, R., Littlewood, P.B.: Critical fluctuations at a many-body exceptional point. Phys. Rev. Research 2(3), 033018 (2020). https://doi.org/10.1103/PhysRevResearch.2.033018

    Article  ADS  Google Scholar 

Download references

Acknowledgements

G. M. M. Itable acknowledges support from the Department of Science and Technology–Science Education Institute through its Accelerated Science and Technology Human Resource Development Program. The authors are grateful for the guidance, mentorship, and valuable advice provided by J. P. H. Esguerra.

Author information

Authors and Affiliations

  1. National Institute of Physics, University of the Philippines Diliman, 1101, Quezon City, Philippines

    Gene M. M. Itable & Francis N. C. Paraan

Authors
  1. Gene M. M. Itable
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francis N. C. Paraan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gene M. M. Itable.

Ethics declarations

Note added in proof

In the thermodynamic limit of a non-Hermitian Su-Schrieffer-Heeger model, the exceptional point is characterized by a non-analyticity in the sublattice entanglement entropy where a transition in volume-law scaling occurs [Le Gal, Y., Turkeshi, X., Schiró, M.: arXiv:2210.11937 (2022)].

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Itable, G.M.M., Paraan, F.N.C. Entanglement entropy distinguishes PT-symmetry and topological phases in a class of non-unitary quantum walks. Quantum Inf Process 22, 106 (2023). https://doi.org/10.1007/s11128-023-03848-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-023-03848-y

Keywords