Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Measurements in two bases are sufficient for certifying high-dimensional entanglement

Nature Physics volume 14, pages 1032–1037 (2018)Cite this article

Subjects

Abstract

High-dimensional encoding of quantum information provides a way of transcending the limitations of current approaches to quantum communication, which are mostly based on the entanglement between qubits—two-dimensional quantum systems. One of the central challenges in the pursuit of high-dimensional alternatives is ascertaining the presence of high-dimensional entanglement within a given high-dimensional quantum state. In particular, it would be desirable to carry out such entanglement certification without resorting to inefficient full state tomography. Here, we show how carefully constructed measurements in two bases (one of which is not orthonormal) can be used to faithfully and efficiently certify bipartite high-dimensional states and their entanglement for any physical platform. To showcase the practicality of this approach under realistic conditions, we put it to the test for photons entangled in their orbital angular momentum. In our experimental set-up, we are able to verify 9-dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Experimental set-up.
Fig. 2: Experimental data certifying 9-dimensional entanglement.

Similar content being viewed by others

References

  1. Lo, H.-K., Curty, M. & Tamaki, K. Secure quantum key distribution. Nat. Photon. 8, 595–604 (2014).

    Article  ADS  Google Scholar 

  2. Bennett, C. H., Shor, P. W., Smolin, J. A. & Thapliyal, A. V. Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48, 2637–2655 (2002).

    Article  MathSciNet  Google Scholar 

  3. Bennett, C. H., Brassard, G. & Mermin, N. D. Quantum cryptography without Bell's theorem. Phys. Rev. Lett. 68, 557–559 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  4. Acín, A. et al. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007).

    Article  ADS  Google Scholar 

  5. Vazirani, U. & Vidick, T. Fully device-independent quantum key distribution. Phys. Rev. Lett. 113, 140501 (2014).

    Article  ADS  Google Scholar 

  6. Cerf, N. J., Bourennane, M., Karlsson, A. & Gisin, N. Security of quantum key distribution using d-level systems. Phys. Rev. Lett. 88, 127902 (2002).

    Article  ADS  Google Scholar 

  7. Barrett, J., Kent, A. & Pironio, S. Maximally nonlocal and monogamous quantum correlations. Phys. Rev. Lett. 97, 170409 (2006).

    Article  ADS  Google Scholar 

  8. Gröblacher, S., Jennewein, T., Vaziri, A., Weihs, G. & Zeilinger, A. Experimental quantum cryptography with qutrits. New J. Phys. 8, 75 (2006).

    Article  ADS  Google Scholar 

  9. Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  10. Eltschka, C. & Siewert, J. Quantifying entanglement resources. J. Phys. A 47, 424005 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  11. Huber, M. & Pawlowski, M. Weak randomness in device independent quantum key distribution and the advantage of using high dimensional entanglement. Phys. Rev. A 88, 032309 (2013).

    Article  ADS  Google Scholar 

  12. Schaeff, C., Polster, R., Huber, M., Ramelow, S. & Zeilinger, A. Experimental access to higher-dimensional entangled quantum systems using integrated optics. Optica 2, 523–529 (2015).

    Article  Google Scholar 

  13. Gutiérrez-Esparza, A. J. et al. Detection of nonlocal superpositions. Phys. Rev. A 90, 032328 (2014).

    Article  ADS  Google Scholar 

  14. Krenn, M., Malik, M., Erhard, M. & Zeilinger, A. Orbital angular momentum of photons and the entanglement of Laguerre–Gaussian modes. Phil. Trans. R. Soc. A 375, 20150442 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  15. Vaziri, A., Weihs, G. & Zeilinger, A. Experimental two-photon, three-dimensional entanglement for quantum communication. Phys. Rev. Lett. 89, 240401 (2002).

    Article  ADS  Google Scholar 

  16. Kulkarni, G., Sahu, R., Magana-Loaiza, O. S., Boyd, R. W. & Jha, A. K. Single-shot measurement of the orbital-angular-momentum spectrum of light. Nat. Commun. 8, 1054 (2017).

    Article  ADS  Google Scholar 

  17. de Riedmatten, H., Marcikic, I., Zbinden, H. & Gisin, N. Creating high dimensional time-bin entanglement using mode-locked lasers. Quant. Inf. Comp. 2, 425–433 (2002).

    MATH  Google Scholar 

  18. Thew, R. T., Acín, A., Zbinden, H. & Gisin, N. Bell-type test of energy-time entangled qutrits. Phys. Rev. Lett. 93, 010503 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  19. Martin, A. et al. Quantifying photonic high-dimensional entanglement. Phys. Rev. Lett. 118, 110501 (2017).

    Article  ADS  Google Scholar 

  20. Jha, A. K., Malik, M. & Boyd, R. W. Exploring energy–time entanglement using geometric phase. Phys. Rev. Lett. 101, 180405 (2008).

    Article  ADS  Google Scholar 

  21. Steinlechner, F. et al. Distribution of high-dimensional entanglement via an intra-city freespace link. Nat. Commun. 8, 15971 (2017).

    Article  ADS  Google Scholar 

  22. Barreiro, J. T., Langford, N. K., Peters, N. A. & Kwiat, P. G. Generation of hyperentangled photon pairs. Phys. Rev. Lett. 95, 260501 (2005).

    Article  ADS  Google Scholar 

  23. Anderson, B. E., Sosa-Martinez, H., Riofrío, C. A., Deutsch, I. H. & Jessen, P. S. Accurate and robust unitary transformations of a high-dimensional quantum system. Phys. Rev. Lett. 114, 240401 (2015).

    Article  ADS  Google Scholar 

  24. Kumar, K. S., Vepsäläinen, A., Danilin, S. & Paraoanu, G. S. Stimulated Raman adiabatic passage in a three-level superconducting circuit. Nat. Commun. 7, 10628 (2016).

    Article  ADS  Google Scholar 

  25. Bertlmann, R. A. & Krammer, P. Bloch vectors for qudits. J. Phys. A 41, 235303 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  26. Krenn, M. et al. Generation and confirmation of a (100 × 100)-dimensional entangled quantum system. Proc. Natl Acad. Sci. USA 111, 6243–6247 (2014).

    Article  ADS  Google Scholar 

  27. Erhard, M., Malik, M., Krenn, M. & Zeilinger, A. Experimental GHZ entanglement beyond qubits. Preprint at http://arxiv.org/abs/1708.03881 (2017).

  28. Dada, A. C., Leach, J., Buller, G. S., Padgett, M. J. & Andersson, E. Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities. Nat. Phys. 7, 677–680 (2011).

    Article  Google Scholar 

  29. Kues, M. et al. On-chip generation of high-dimensional entangled quantum states and their coherent control. Nature 546, 622–626 (2017).

    Article  ADS  Google Scholar 

  30. Bennett, C. H., Di Vincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state entanglement and quantum error correction. Phys. Rev. A. 54, 3824 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  31. Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998).

    Article  ADS  Google Scholar 

  32. Erker, P., Krenn, M. & Huber, M. Quantifying high dimensional entanglement with two mutually unbiased bases. Quantum 1, 22 (2017).

    Article  Google Scholar 

  33. Piani, M. & Mora, C. Class of PPT bound entangled states associated to almost any set of pure entangled states. Phys. Rev. A. 75, 012305 (2007).

    Article  ADS  Google Scholar 

  34. Fickler, R. et al. Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information. Nat. Commun. 5, 4502 (2014).

    Article  Google Scholar 

  35. Arrizón, V., Ruiz, U., Carrada, R. & González, L. A. Pixelated phase computer holograms for the accurate encoding of scalar complex fields. J. Opt. Soc. Am. A. 24, 3500–3507 (2007).

    Article  ADS  Google Scholar 

  36. Wootters, W. K. & Fields, B. D. Optimal state determination by mutually unbiased measurements. Ann. Phys. 191, 363 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  37. Coladangelo, A., Goh, K. T. & Scarani, V. All pure bipartite entangled states can be self-tested. Nat. Commun. 8, 15485 (2017).

    Article  ADS  Google Scholar 

  38. Berkhout, G. C. G., Lavery, M. P. J., Courtial, J., Beijersbergen, M. W. & Padgett, M. J. Efficient sorting of angular momentum states of light. Phys. Rev. Lett. 105, 153601 (2010).

    Article  ADS  Google Scholar 

  39. Mirhosseini, M., Malik, M., Shi, Z. & Boyd, R. W. Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun. 4, 2781 (2013).

    Article  ADS  Google Scholar 

  40. Brougham, T. & Barnett, S. M. Mutually unbiased measurements for high-dimensional time-bin-based photonic states. Europhys. Lett. 104, 30003 (2013).

    Article  ADS  Google Scholar 

  41. Mower, J. et al. High-dimensional quantum key distribution using dispersive optics. Phys. Rev. A. 87, 062322 (2013).

    Article  ADS  Google Scholar 

  42. Spengler, C., Huber, M., Brierley, S., Adaktylos, T. & Hiesmayr, B. C. Entanglement detection via mutually unbiased bases. Phys. Rev. A. 86, 022311 (2012).

    Article  ADS  Google Scholar 

  43. Giovannini, D. et al. Characterization of high-dimensional entangled systems via mutually unbiased measurements. Phys. Rev. Lett. 110, 143601 (2013).

    Article  ADS  Google Scholar 

  44. Tasca, D. S. et al. Testing for entanglement with periodic coarse graining. Phys. Rev. A. 97, 042312 (2018).

    Article  ADS  Google Scholar 

  45. Paul, E. C., Tasca, D. S., Rudnicki, L. & Walborn, S. P. Detecting entanglement of continuous variables with three mutually unbiased bases. Phys. Rev. A. 94, 012303 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  46. Tasca, D. S., Sánchez, P., Walborn, S. P. & Rudnicki, L. Mutual unbiasedness in coarse-grained continuous variables. Phys. Rev. Lett. 120, 040403 (2018).

    Article  ADS  Google Scholar 

  47. Schneeloch, J. & Howland, G. A. Quantifying high-dimensional entanglement with Einstein–Podolsky–Rosen correlations. Phys. Rev. A. 97, 042338 (2018).

    Article  ADS  Google Scholar 

  48. Sauerwein, D., Macchiavello, C., Maccone, L. & Kraus, B. Multipartite correlations in mutually unbiased bases. Phys. Rev. A. 95, 042315 (2017).

    Article  ADS  Google Scholar 

  49. Malik, M. et al. Multi-photon entanglement in high dimensions. Nat. Photon. 10, 248–252 (2016).

    Article  ADS  Google Scholar 

  50. Krenn, M., Malik, M., Fickler, R., Lapkiewicz, R. & Zeilinger, A. Automated search for new quantum experiments. Phys. Rev. Lett. 116, 090405 (2016).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank A. Zeilinger for many fruitful discussions and guidance regarding the experimental set-up. We acknowledge funding from the Austrian Science Fund (FWF) through the START project Y879-N27 and the joint Czech-Austrian project MultiQUEST (I 3053-N27 and GF17-33780L). J.B. and M.H. acknowledge support from the ESQ Discovery Grant of the Austrian Academy of Sciences (ÖAW) project OESQ0002X2. P.E. acknowledges funding by the European Commission (STREP RAQUEL) and the Swiss National Science Foundation (SNF). M.M. acknowledges support from the QuantERA ERA-NET Co-fund (FWF project I3553-N36).

Author information

Authors and Affiliations

  1. Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Vienna, Austria

    Jessica Bavaresco, Natalia Herrera Valencia, Claude Klöckl, Matej Pivoluska, Paul Erker, Nicolai Friis, Mehul Malik & Marcus Huber

  2. Université d’Aix-Marseille, Centre de Saint-Jérôme, Marseille, France

    Natalia Herrera Valencia

  3. Institute of Computer Science, Masaryk University, Brno, Czech Republic

    Claude Klöckl & Matej Pivoluska

  4. Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia

    Matej Pivoluska

  5. Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland

    Paul Erker

  6. Universitat Autonoma de Barcelona, Bellaterra, Barcelona, Spain

    Paul Erker

  7. Institute of Photonics and Quantum Sciences (IPaQS), Heriot-Watt University, Edinburgh, UK

    Mehul Malik

Authors
  1. Jessica Bavaresco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Natalia Herrera Valencia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Claude Klöckl
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Matej Pivoluska
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Paul Erker
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Nicolai Friis
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Mehul Malik
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Marcus Huber
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

M.M. and M.H. conceived and designed the experiments. J.B., N.H. and M.M. performed the experiments. J.B., N.H., M.P., N.F., M.M. and M.H. analysed the data. J.B., C.K., M.P., P.E., N.F., M.M. and M.H. developed theoretical methods. J.B., N.H., C.K., M.P., N.F., M.M. and M.H. wrote the paper.

Corresponding authors

Correspondence to Nicolai Friis, Mehul Malik or Marcus Huber.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Notes 1–10, Supplementary Figures 1–7, Supplementary Table 1, Supplementary References 1–21

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bavaresco, J., Herrera Valencia, N., Klöckl, C. et al. Measurements in two bases are sufficient for certifying high-dimensional entanglement. Nature Phys 14, 1032–1037 (2018). https://doi.org/10.1038/s41567-018-0203-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-018-0203-z

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing