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.

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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


  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


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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.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Nicolai Friis or Mehul Malik or Marcus Huber.

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  1. Supplementary Information

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

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