Log in

Classical optical analogue of quantum discord

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

A Publisher Correction to this article was published on 10 October 2023

This article has been updated

Abstract

Quantum discord has been shown to be a resource for quantum advantage in addition to quantum entanglement. While many experiments have demonstrated classical analogies of entanglement, none have done so for discord. We present a proof-of-concept demonstration for creating a classical analogue of quantum discord using classical light that takes advantage of the analogy between the state of two qubits and the spatial modes of a Laguerre-Gauss beam. We demonstrate the validity of this approach by comparing the intensity profiles of theoretical simulations to experimental results for different values of discord. Such a classical analogue of quantum discord may provide further insight in understanding and development of quantum information technologies that make use of discord.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /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 includes VAT (Thailand)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Change history

References

  1. F. Arute, K. Arya, R. Babbush, D. Bacon, J. Bardin, R. Barends, R. Biswas, S. Boixo, F. Brandao, D. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, J. Martinis, Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5

    Article  ADS  CAS  PubMed  Google Scholar 

  2. C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing. Proc. IEEE Int. Conf. Comput. Syst. Signal Process. 175, 8 (1984)

    Google Scholar 

  3. C.H. Bennett, F. Bessette, L. Brassard, Gilles, J. Salvail, Smolin, Experimental quantum cryptography. J. Cryptol. (1992). https://doi.org/10.1007/BF00191318

    Article  Google Scholar 

  4. R.C. Pooser, N. Savino, E. Batson, J.L. Beckey, J. Garcia, B.J. Lawrie, Truncated nonlinear interferometry for quantum-enhanced atomic force microscopy. Phys. Rev. Lett. 124, 230504 (2020). https://doi.org/10.1103/PhysRevLett.124.230504

    Article  ADS  CAS  PubMed  Google Scholar 

  5. V. Giovannetti, S. Lloyd, L. Maccone, Quantum-enhanced measurements: beating the standard quantum limit. Science (New York, N.Y.) 306, 1330–6 (2004). https://doi.org/10.1126/science.1104149

    Article  ADS  CAS  PubMed  Google Scholar 

  6. F. Hudelist, J. Kong, C. Liu, J. **g, Z.Y. Ou, W. Zhang, Quantum metrology with parametric amplifier-based photon correlation interferometers. Nat. Commun. 5, 3049 (2014). https://doi.org/10.1038/ncomms4049

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Y. Ma, H. Miao, B. Pang, M. Evans, C. Zhao, J. Harms, R. Schnabel, Y. Chen, Proposal for gravitational-wave detection beyond the standard quantum limit via epr entanglement. Nat. Phys. (2016). https://doi.org/10.1038/nphys4118

    Article  Google Scholar 

  8. R.P. Feynman, Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  9. E. Knill, R. Laflamme, Power of one bit of quantum information. Phys. Rev. Lett. 81(25), 5672–5675 (1998). https://doi.org/10.1103/PhysRevLett.81.5672

    Article  ADS  CAS  Google Scholar 

  10. A. Datta, A. Shaji, C.M. Caves, Quantum discord and the power of one qubit. Phys. Rev. Lett. 100(5), 050502 (2008). https://doi.org/10.1103/PhysRevLett.100.050502

    Article  ADS  CAS  PubMed  Google Scholar 

  11. B.P. Lanyon, M. Barbieri, M.P. Almeida, A.G. White, Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008). https://doi.org/10.1103/PhysRevLett.101.200501

    Article  ADS  CAS  PubMed  Google Scholar 

  12. L. Chen, Quantum discord of thermal two-photon orbital angular momentum state: mimicking teleportation to transmit an image. Light: Sci. Appl. 10(1), 148 (2021). https://doi.org/10.1038/s41377-021-00585-8. (Accessed 2022-04-18)

    Article  ADS  CAS  PubMed  Google Scholar 

  13. H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88(1), 017901 (2001). https://doi.org/10.1103/PhysRevLett.88.017901. (Accessed 2021-07-22)

    Article  ADS  CAS  PubMed  Google Scholar 

  14. S. Luo, Quantum discord for two-qubit systems. Phys. Rev. A 77(4), 042303 (2008). https://doi.org/10.1103/PhysRevA.77.042303. (Accessed 2021-09-21)

    Article  ADS  CAS  Google Scholar 

  15. V. Chille, N. Quinn, C. Peuntinger, C. Croal, L. Mišta, C. Marquardt, G. Leuchs, N. Korolkova, Quantum nature of gaussian discord: experimental evidence and role of system-environment correlations. Phys. Rev. A 91, 050301 (2015). https://doi.org/10.1103/PhysRevA.91.050301

    Article  ADS  CAS  Google Scholar 

  16. P.A.M. Dirac, On the analogy between classical and quantum mechanics. Rev. Mod. Phys. 17, 195–199 (1945). https://doi.org/10.1103/RevModPhys.17.195

    Article  ADS  MathSciNet  Google Scholar 

  17. A. Enders, G. Nimtz, Evanescent-mode propagation and quantum tunneling. Phys. Rev. E 48, 632–634 (1993). https://doi.org/10.1103/PhysRevE.48.632

    Article  ADS  CAS  Google Scholar 

  18. J.J. Hupert, Evanescent fields in physics and their interpretations in terms of flowgraphs. Appl. Phys. 6(2), 131–149 (1975). https://doi.org/10.1007/BF00883744

    Article  ADS  Google Scholar 

  19. X.-F. Qian, B. Little, J.C. Howell, J.H. Eberly, Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields. Optica 2(7), 611–615 (2015). https://doi.org/10.1364/OPTICA.2.000611. (Publisher: Optical Society of America. Accessed 2022-01-20)

    Article  ADS  Google Scholar 

  20. D. Dragoman, M. Dragoman, Quantum-classical analogies. The frontiers collection (Springer, Berlin, 2004). https://doi.org/10.1007/978-3-662-09647-5_1

    Book  Google Scholar 

  21. J.H. Eberly, X.-F. Qian, A.A. Qasimi, H. Ali, M.A. Alonso, R. Gutiérrez-Cuevas, B.J. Little, J.C. Howell, T. Malhotra, A.N. Vamivakas, Quantum and classical optics-emerging links. Physica Scripta 91(6), 063003 (2016). https://doi.org/10.1088/0031-8949/91/6/063003. (Publisher: IOP Publishing. Accessed 2022-01-20)

    Article  ADS  CAS  Google Scholar 

  22. N. Korolkova, G. Leuchs, Quantum correlations in separable multi-mode states and in classically entangled light. Reports Progress Phys. 82(5), 056001 (2019). https://doi.org/10.1088/1361-6633/ab0c6b. (Accessed 2021-11-22)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  23. C. Tzanakis, Discovering by analogy: the case of Schrödinger’s equation. Eur. J. Phys. 19(1), 69–75 (1998). https://doi.org/10.1088/0143-0807/19/1/010. (Accessed 2021-04-15)

    Article  Google Scholar 

  24. S. Liu, S. Liu, C. Yang, Z. Xu, Y. Li, Y. Li, Z. Zhou, G. Guo, B. Shi, Classical simulation of high-dimensional entanglement by non-separable angular and radial modes. Opt. Express 27(13), 18363–18375 (2019). https://doi.org/10.1364/OE.27.018363

    Article  ADS  PubMed  Google Scholar 

  25. C.V.S. Borges, M. Hor-Meyll, J.A.O. Huguenin, A.Z. Khoury, Bell-like inequality for the spin-orbit separability of a laser beam. Phys. Rev. A 82, 033833 (2010). https://doi.org/10.1103/PhysRevA.82.033833

    Article  ADS  CAS  Google Scholar 

  26. X.-F. Qian, J.H. Eberly, Entanglement and classical polarization states. Opt. Lett. 36(20), 4110–4112 (2011). https://doi.org/10.1364/OL.36.004110. (Publisher: Optical Society of America. Accessed 2022-01-20)

    Article  ADS  PubMed  Google Scholar 

  27. R.J.C. Spreeuw, A classical analogy of entanglement. Found. Phys. 28(3), 361–374 (1998). https://doi.org/10.1023/A:1018703709245

    Article  MathSciNet  Google Scholar 

  28. M. McLaren, M. Agnew, J. Leach, F.S. Roux, M.J. Padgett, R.W. Boyd, A. Forbes, Entangled Bessel-Gaussian beams. Opt. Express 20(21), 23589–23597 (2012). https://doi.org/10.1364/OE.20.023589

    Article  ADS  PubMed  Google Scholar 

  29. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, G. Leuchs, Quantum-like nonseparable structures in optical beams. New J. Phys. 17(4), 043024 (2015). https://doi.org/10.1088/1367-2630/17/4/043024

    Article  ADS  Google Scholar 

  30. M. McLaren, T. Konrad, A. Forbes, Measuring the nonseparability of vector vortex beams. Phys. Rev. A 92(2), 023833 (2015). https://doi.org/10.1103/PhysRevA.92.023833

    Article  ADS  CAS  Google Scholar 

  31. C.E.R. Souza, J.A.O. Huguenin, P. Milman, A.Z. Khoury, Topological phase for spin-orbit transformations on a laser beam. Phys. Rev. Lett. 99, 160401 (2007). https://doi.org/10.1103/PhysRevLett.99.160401

    Article  ADS  CAS  PubMed  Google Scholar 

  32. K.F. Lee, J.E. Thomas, Experimental simulation of two-particle quantum entanglement using classical fields. Phys. Rev. Lett. 88, 097902 (2002). https://doi.org/10.1103/PhysRevLett.88.097902

    Article  ADS  CAS  PubMed  Google Scholar 

  33. T. Konrad, A. Forbes, Quantum mechanics and classical light. Contemp. Phys. 60(1), 1–22 (2019). https://doi.org/10.1080/00107514.2019.1580433

    Article  ADS  Google Scholar 

  34. Y. Sun, X. Song, H. Qin, X. Zhang, Z. Yang, X. Zhang, Non-local classical optical correlation and implementing analogy of quantum teleportation. Sci. Reports 5, 9175 (2015). https://doi.org/10.1038/srep09175. (Accessed 2021-07-16)

    Article  ADS  CAS  Google Scholar 

  35. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L.L. Sánchez-Soto, G.S. Agarwal, Experimental violation of a bell-like inequality with optical vortex beams. New J. Phys. 17(11), 113046 (2015). https://doi.org/10.1088/1367-2630/17/11/113046

    Article  ADS  CAS  Google Scholar 

  36. S.K. Goyal, F.S. Roux, A. Forbes, T. Konrad, Implementing quantum walks using orbital angular momentum of classical light. Phys. Rev. Lett. 110, 263602 (2013). https://doi.org/10.1103/PhysRevLett.110.263602

    Article  ADS  CAS  PubMed  Google Scholar 

  37. F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. Lisio, F. Sciarrino, E. Santamato, R.W. Boyd, L. Marrucci, Quantum walks and wavepacket dynamics on a lattice with twisted photons. Sci. Adv. 1(2), 1500087 (2015). https://doi.org/10.1126/sciadv.1500087

    Article  ADS  Google Scholar 

  38. A.N. Oliveira, S.P. Walborn, C.H. Monken, Implementing the Deutsch algorithm with polarization and transverse spatial modes. J. Opt. B: Quantum Semiclass. Opt. 7(9), 288 (2005). https://doi.org/10.1088/1464-4266/7/9/009

    Article  ADS  CAS  Google Scholar 

  39. B. Perez-Garcia, R.I. Hernandez-Aranda, A. Forbes, T. Konrad, The first iteration of Grover’s algorithm using classical light with orbital angular momentum. J. Mod. Opt. 65(16), 1942–1948 (2018). https://doi.org/10.1080/09500340.2018.1459910

    Article  ADS  MathSciNet  Google Scholar 

  40. Z.-L. Zhou, H. Yuan, L.-F. Wei, Entanglement, quantum discord, and non-locality in bell-diagonal states. Int. J. Theor. Phys. 52(2), 420–428 (2013). https://doi.org/10.1007/s10773-012-1348-7. (Accessed 2021-07-19)

    Article  MathSciNet  Google Scholar 

  41. L. Henderson, V. Vedral, Classical, quantum and total correlations. J. Phys. A: Math. Gen. 34(35), 6899–6905 (2001). https://doi.org/10.1088/0305-4470/34/35/315

    Article  ADS  MathSciNet  Google Scholar 

  42. J.H. Eberly, Correlation, coherence and context. Laser Phys. 26(8), 084004 (2016). https://doi.org/10.1088/1054-660X/26/8/084004

    Article  ADS  Google Scholar 

  43. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, New York, 2007)

    Google Scholar 

  44. G. Goubau, F. Schwering, On the guided propagation of electromagnetic wave beams. IRE Trans. Antennas Propag. 9(3), 248–256 (1961). https://doi.org/10.1109/TAP.1961.1144999

    Article  ADS  MathSciNet  Google Scholar 

  45. D. Barberena, G. Gatti, F.D. Zela, Experimental demonstration of a secondary source of partially polarized states. J. Opt. Soc. Am. A 32(4), 697–700 (2015). https://doi.org/10.1364/JOSAA.32.000697

    Article  ADS  CAS  Google Scholar 

  46. F.S. Roux, Y. Zhang, Projective measurements in quantum and classical optical systems. Phys. Rev. A 90, 033835 (2014). https://doi.org/10.1103/PhysRevA.90.033835

    Article  ADS  CAS  Google Scholar 

  47. S. Pirandola, Quantum discord as a resource for quantum cryptography. Sci. Rep. 4(1), 6956 (2014). https://doi.org/10.1038/srep06956

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  48. R. Ghobadi, J.S. Oberoi, E. Zahedinejhad, The power of one qubit in machine learning. ar**v:1905.01390 [quant-ph] (2019). ar**v: 1905.01390. (Accessed 2022-03-21)

  49. B. Dakić, Y.O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, Č Brukner, P. Walther, Quantum discord as resource for remote state preparation. Nat. Phys. 8(9), 666–670 (2012). https://doi.org/10.1038/nphys2377

    Article  CAS  Google Scholar 

  50. K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84(4), 1655–1707 (2012). https://doi.org/10.1103/RevModPhys.84.1655. (Accessed 2021-07-14)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This material is based upon research supported by, or in part by, the U. S. Office of Naval Research under Award Number N000141912374. This work was also supported by the Defense Advanced Research Projects Agency (DARPA) Grant Number D19AP00043 under the mentorship of Dr. Joseph Altepeter. D.I.B. is also supported by the U.S. Army Research Office (ARO) under Grant W911NF-23-1-0288. J. M. L. was supported by the Louisiana Board of Regents’ Graduate Fellowship Program. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of DARPA, ONR, ARO, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Funding

U. S. Office of Naval Research (N000141912374); Defense Advanced Research Projects Agency (D19AP00043); U.S. Army Research Office (W911NF-23-1-0288).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jacob M. Leamer.

Ethics declarations

Conflict of interest

The authors declare no conflicts of interest.

Additional information

The original online version of this article was revised: “On page 2, 3, and 4 of the pdf of this article, the term “—E)(E—” should have read “|E)(E|”. The typesetter apologizes for the mistake.

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

Leamer, J.M., Zhang, W., Savino, N.J. et al. Classical optical analogue of quantum discord. Eur. Phys. J. Spec. Top. 232, 3345–3351 (2023). https://doi.org/10.1140/epjs/s11734-023-00988-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-023-00988-1

Navigation