Quantum teleportation experiments are certified by an averaged teleportation fidelity exceeding the classical bound of 2/32/3, evaluated on events selected by multi-photon coincidence detections. Analyzing proposed teleportation and entanglement-swapping experiments between the International Space Station and the Lunar Gateway, Lin and Hu (Class. Quantum Grav. 38, 165002 (2021)) showed that in the high-loss, long-baseline regime this certification is vulnerable to the same loopholes that affected Bell tests before 2015: with a sufficiently narrow coincidence window, an entirely classical event-based simulation with no entanglement can push the averaged fidelity above 2/32/3. They posed the open problem of finding a certification criterion for quantum teleportation that plays the role the Clauser-Horne (CH) inequality plays in Bell tests: a bound on the raw recorded data, requiring no fair-coincidence or fair-sampling assumption, that all classical models obey even under extreme loss and arbitrary selection bias, while genuinely quantum teleportation violates it. Adjacent lines of work - device-independent certification of qubit teleportation (Ho, Bancal and Scarani), teleportation witnesses beyond average fidelity (Cavalcanti et al.; Carvacho et al.), refined fidelity benchmarks and criteria (Bussandri et al.; Gong and Wang; Opatrný et al.), and loss-robust event-ready teleportation experiments (Peng et al.) - address neighboring questions.

Background

In quantum teleportation (Bennett et al., Phys. Rev. Lett. 70, 1895 (1993)), an unknown quantum state is destroyed at a sender (Alice) and reconstructed at a distant receiver (Bob) using a shared entangled pair, a Bell-state measurement (BSM - a joint measurement projecting two qubits onto a maximally entangled basis), and a classical message. In entanglement swapping, the same BSM applied to one photon from each of two independent entangled pairs projects the two remaining, never-interacted photons into an entangled state. Photonic experiments certify success by the averaged fidelity FF between input and output states over an ensemble of test states: any classical strategy limited to measuring the input and classically communicating a re-preparation ('measure-and-prepare') achieves at most F=2/3F=2/3 for uniformly distributed qubits, the benchmark rooted in Massar and Popescu's analysis of optimal information extraction from finite ensembles (Phys. Rev. Lett. 74, 1259 (1995)). Because photonic BSMs are probabilistic and detectors are lossy, the fidelity is in practice evaluated only on runs in which all photons are registered within a coincidence window (four-fold coincidences in swapping-type setups); observed four-fold rates grew from 0.00065 Hz (Jennewein et al., Phys. Rev. Lett. 88, 017903 (2002)) through 0.016 Hz (Ma et al., Nat. Phys. 8, 480 (2012)) to 100 Hz over 143 km - though under conditions where a Bell test was not feasible (Herbst et al., PNAS 112, 14202 (2015)); the state of the art is reviewed in Pirandola et al., Nat. Photonics 9, 641 (2015).

In Bell tests, conditioning the statistical analysis on detected pairs is known to open two loopholes: the detector-efficiency loophole (if detectors register only a small and possibly biased fraction of the emitted pairs, the detected subset can fake a violation of the CHSH inequality, the standard two-party Bell inequality of Clauser, Horne, Shimony and Holt) and the coincidence-time loophole (the freedom to choose the coincidence window can bias the selected ensemble; Larsson and Gill, Europhys. Lett. 67, 707 (2004)). The Clauser-Horne (CH) inequality (Phys. Rev. D 10, 526 (1974)) is the standard remedy: a bound on the raw detected counts that remains valid at any detector efficiency, with no fair-sampling or fair-coincidence assumption, so that postselection cannot manufacture a violation; this is how the coincidence-time loophole was closed in Bell experiments (Larsson et al., Phys. Rev. A 90, 032107 (2014); reviewed in Larsson, J. Phys. A 47, 424003 (2014)). For entanglement swapping, Gisin and Gisin (Phys. Lett. A 297, 279 (2002); arXiv:quant-ph/0201077) showed that a local-variable model exploiting the detection loophole reproduces the quantum predictions exactly on postselected data - so postselected swapping statistics alone cannot certify quantumness.

Lin and Hu (Class. Quantum Grav. 38, 165002 (2021); arXiv:2012.07590) analyzed teleportation and entanglement-swapping experiments on a very long baseline between the International Space Station (ISS) and the Lunar Gateway (LG), with spacelike-separated measurement events and a light travel time of about 1.3 seconds. In this regime the link loss is so severe that the detector-efficiency and coincidence-time loopholes become significant, and they demonstrated the danger concretely: an event-based computer simulation, with no entanglement and no quantum nonlocality, in which each local photon detection carries a random trigger time distributed according to the angular difference between incoming and outgoing polarizations, can push the averaged teleportation fidelity above the classical bound 2/32/3 once a sufficiently narrow coincidence window is used to select the data. Their concluding remarks pose the open problem studied here: go beyond the averaged fidelity and find a criterion for quantum teleportation that is the counterpart of the CH inequality in Bell tests and is coincidence-time loophole-free. This question is the teleportation analogue of the loophole-free program that transformed Bell testing after 2015, and it is the certification bottleneck for the proposed class of space experiments (the Deep Space Quantum Link, a proposed program of long-baseline quantum-optics experiments linking ground stations, satellites, and lunar-distance spacecraft, whose roadmap includes long-range teleportation tests).

Problem Statement

Construct - or prove the impossibility of - a certification criterion for quantum teleportation (and its entanglement-swapping variant) that is the counterpart, for teleportation, of the Clauser-Horne inequality for Bell tests, in the following sense:

  1. The criterion is an inequality or witness evaluated on the raw record of a teleportation experiment - all emitted trials, detected and undetected events, with detection time tags and the recorded Bell-state-measurement and analyzer outcomes - with no fair-coincidence, fair-sampling, or no-enhancement assumption (a no-enhancement assumption would assert that undetected events behave statistically like detected ones) beyond those explicitly stated in the model.

  2. Every classical model in a precisely stated adversarial class satisfies the bound even with arbitrary detector inefficiency, unannounced photon loss, and coincidence-window or other selection bias; the class must include local and event-based simulations, measure-and-prepare channels, and strategies that use only separable (classically correlated, unentangled) resources, and the bound must hold for each of these three families. In particular, postselection on detection coincidences must not be able to manufacture a violation, closing the coincidence-time and detector-efficiency loopholes for teleportation data as the CH inequality closes them for Bell data; the criterion must reject the type of classical event-based simulation with which Lin and Hu pushed the averaged fidelity above 2/32/3.

  3. A quantum teleportation implementation with realistic resources (non-maximally entangled pairs, partial Bell-state measurement, and the link losses characteristic of Earth-Moon-class baselines such as the ISS-Lunar Gateway proposal) violates the bound, so that a successful teleportation or entanglement-swapping experiment can certify the genuinely quantum, nonlocal character of the process beyond the averaged fidelity. If no criterion with properties 1 and 2 can be violated in this regime, the answer may instead be an impossibility theorem.

The common objective of all requested results is a loophole-free certification theory for teleportation data in the high-loss, long-baseline regime. Lin and Hu posed the problem in this regime because there the selection bias is unavoidable and the averaged-fidelity benchmark demonstrably fails.

References

  1. Shih-Yuin Lin and Bei-Lok Hu, 'Quantum teleportation and entanglement swapping with long baseline in outer space', Classical and Quantum Gravity 38, 165002 (2021). DOI: 10.1088/1361-6382/ac1080; preprint arXiv:2012.07590. https://doi.org/10.1088/1361-6382/ac1080
  2. Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, 'Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels', Physical Review Letters 70, 1895 (1993). DOI: 10.1103/PhysRevLett.70.1895. https://doi.org/10.1103/PhysRevLett.70.1895
  3. S. Massar and S. Popescu, 'Optimal Extraction of Information from Finite Quantum Ensembles', Physical Review Letters 74, 1259 (1995). DOI: 10.1103/PhysRevLett.74.1259. https://doi.org/10.1103/PhysRevLett.74.1259
  4. Thomas Jennewein, Gregor Weihs, Jian-Wei Pan, and Anton Zeilinger, 'Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping', Physical Review Letters 88, 017903 (2002). DOI: 10.1103/PhysRevLett.88.017903. https://doi.org/10.1103/PhysRevLett.88.017903
  5. N. Gisin and B. Gisin, 'A local variable model for entanglement swapping exploiting the detection loophole', Physics Letters A 297, 279 (2002). DOI: 10.1016/S0375-9601(02)00428-0; preprint arXiv:quant-ph/0201077. https://arxiv.org/abs/quant-ph/0201077
  6. Xiao-song Ma, Stefan Zotter, Johannes Kofler, Rupert Ursin, Thomas Jennewein, Časlav Brukner, and Anton Zeilinger, 'Experimental delayed-choice entanglement swapping', Nature Physics 8, 480 (2012). DOI: 10.1038/nphys2294. https://doi.org/10.1038/nphys2294
  7. Thomas Herbst, Thomas Scheidl, Matthias Fink, Johannes Handsteiner, Bernhard Wittmann, Rupert Ursin, and Anton Zeilinger, 'Teleportation of entanglement over 143 km', PNAS 112, 14202 (2015). DOI: 10.1073/pnas.1517007112. https://doi.org/10.1073/pnas.1517007112
  8. S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, 'Advances in quantum teleportation', Nature Photonics 9, 641 (2015). DOI: 10.1038/nphoton.2015.154. https://doi.org/10.1038/nphoton.2015.154
  9. John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, 'Proposed experiment to test local hidden-variable theories', Physical Review Letters 23, 880 (1969). DOI: 10.1103/PhysRevLett.23.880. https://doi.org/10.1103/PhysRevLett.23.880
  10. John F. Clauser and Michael A. Horne, 'Experimental consequences of objective local theories', Physical Review D 10, 526 (1974). DOI: 10.1103/PhysRevD.10.526. https://doi.org/10.1103/PhysRevD.10.526
  11. Jan-Åke Larsson and Richard D. Gill, 'Bell's inequality and the coincidence-time loophole', Europhysics Letters 67, 707 (2004). DOI: 10.1209/epl/i2004-10124-7. https://doi.org/10.1209/epl/i2004-10124-7
  12. Jan-Åke Larsson, 'Loopholes in Bell inequality tests of local realism', Journal of Physics A: Mathematical and Theoretical 47, 424003 (2014). DOI: 10.1088/1751-8113/47/42/424003. https://doi.org/10.1088/1751-8113/47/42/424003
  13. Jan-Åke Larsson, Marissa Giustina, Johannes Kofler, Bernhard Wittmann, Rupert Ursin, and Sven Ramelow, 'Bell-inequality violation with entangled photons, free of the coincidence-time loophole', Physical Review A 90, 032107 (2014). DOI: 10.1103/PhysRevA.90.032107. https://doi.org/10.1103/PhysRevA.90.032107
  14. Melvyn Ho, Jean-Daniel Bancal, and Valerio Scarani, 'Device-independent certification of the teleportation of a qubit', Physical Review A 88, 052318 (2013). DOI: 10.1103/PhysRevA.88.052318; preprint arXiv:1308.0084. https://doi.org/10.1103/PhysRevA.88.052318
  15. Daniel Cavalcanti, Paul Skrzypczyk, and Ivan Šupić, 'All Entangled States can Demonstrate Nonclassical Teleportation', Physical Review Letters 119, 110501 (2017). DOI: 10.1103/PhysRevLett.119.110501. https://doi.org/10.1103/PhysRevLett.119.110501
  16. Gonzalo Carvacho, Francesco Andreoli, Luca Santodonato, Marco Bentivegna, Vincenzo D'Ambrosio, Paul Skrzypczyk, Ivan Šupić, Daniel Cavalcanti, and Fabio Sciarrino, 'Experimental Study of Nonclassical Teleportation Beyond Average Fidelity', Physical Review Letters 121, 140501 (2018). DOI: 10.1103/PhysRevLett.121.140501; preprint arXiv:1802.10056. https://doi.org/10.1103/PhysRevLett.121.140501
  17. Luca Mazzarella, Makan Mohageg, Dmitry V. Strekalov, Aileen J. Zhai, Ulf E. Israelsson, Andrey Matsko, Nan Yu, Charis Anastopoulos, Bradley Carpenter, Jason R. Gallicchio, Bei-Lok Hu, Thomas Jennewein, Paul Kwiat, Shih-Yuin Lin, Alexander Ling, Christoph Marquardt, Matthias Meister, Brian S. Moffat, Raymond T. Newell, Albert Roura, Christian Schubert, Giuseppe Vallone, Paolo Villoresi, and Lisa Wörner, 'Goals and feasibility of the deep space quantum link', Proc. SPIE 11839, Quantum Communications and Quantum Imaging XIX (2021). DOI: 10.1117/12.2593986. https://doi.org/10.1117/12.2593986
  18. Makan Mohageg, Luca Mazzarella, Charis Anastopoulos, Jason Gallicchio, Bei Lok Hu, Thomas Jennewein, Spencer Johnson, Shih-Yuin Lin, Alexander Ling, Christoph Marquardt, Matthias Meister, Raymond Newell, Albert Roura, Wolfgang P. Schleich, Christian Schubert, Dmitry V. Strekalov, Giuseppe Vallone, Paolo Villoresi, Lisa Wörner, Nan Yu, Aileen Zhai, and Paul Kwiat, 'The deep space quantum link: prospective fundamental physics experiments using long-baseline quantum optics', EPJ Quantum Technology 9, 25 (2022). DOI: 10.1140/epjqt/s40507-022-00143-0; preprint arXiv:2111.15591. https://doi.org/10.1140/epjqt/s40507-022-00143-0
  19. Jan-Michael Mol, Luisa Esguerra, Matthias Meister, David Edward Bruschi, Andreas Wolfgang Schell, Janik Wolters, and Lisa Wörner, 'Quantum memories for fundamental science in space', Quantum Science and Technology 8, 024006 (2023). DOI: 10.1088/2058-9565/acb2f1; preprint arXiv:2303.11810. https://doi.org/10.1088/2058-9565/acb2f1
  20. Makan Mohageg, Charis Anastopoulos, Olivia Brasher, Jason Gallicchio, Bei Lok Hu, Thomas Jennewein, Spencer Johnson, Shih-Yuin Lin, Alexander Ling, Alexander Lohrmann, Christoph Marquardt, Luca Mazzarella, Matthias Meister, Raymond Newell, Albert Roura, Giuseppe Vallone, Paolo Villoresi, Lisa Wörner, and Paul Kwiat, 'Towards satellite tests combining general relativity and quantum mechanics through quantum optical interferometry: progress on the deep space quantum link', EPJ Quantum Technologies 12, 78 (2025). DOI: 10.1140/epjqt/s40507-025-00370-1. https://doi.org/10.1140/epjqt/s40507-025-00370-1
  21. D. G. Bussandri, G. M. Bosyk, and F. Toscano, 'Challenges in certifying quantum teleportation: Moving beyond the conventional fidelity benchmark', Physical Review A 109, 032618 (2024). DOI: 10.1103/PhysRevA.109.032618; preprint arXiv:2403.07994. https://doi.org/10.1103/PhysRevA.109.032618
  22. Diego G. Bussandri, Gustavo M. Bosyk, Pablo Crespo Del Amo, and Karol Życzkowski, 'Quality assessment of quantum teleportation through the distribution of fidelity', Physical Review Applied 25, 064047 (2026). DOI: 10.1103/3gvb-qqx1; preprint arXiv:2510.13600. https://doi.org/10.1103/3gvb-qqx1
  23. Neng-Fei Gong and Tie-Jun Wang, 'General Criteria for Certifying Genuine High-Dimensional Quantum Teleportation' (2026). arXiv:2605.11593. https://arxiv.org/abs/2605.11593
  24. Tomáš Opatrný, Allison Brattley, and Kunal K. Das, 'Benchmark for Quantum Teleportation with Non-Uniform Prior Distributions' (2026). arXiv:2605.24861. https://arxiv.org/abs/2605.24861
  25. Li-Chao Peng, Dian Wu, Xue-Mei Gu, Jian Qin, Ke-Mi Xu, Han-Sen Zhong, Hui Wang, Yu-Ming He, Ming-Chen Chen, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan, 'Unconditional quantum teleportational advantage of single photons' (2025). arXiv:2511.08951. https://arxiv.org/abs/2511.08951
  26. Xue-Mei Gu, Liang Huang, Alejandro Pozas-Kerstjens, Yang-Fan Jiang, Dian Wu, Bing Bai, Qi-Chao Sun, Ming-Cheng Chen, Jun Zhang, Sixia Yu, Qiang Zhang, Chao-Yang Lu, and Jian-Wei Pan, 'Experimental full network nonlocality with independent sources and strict locality constraints', Physical Review Letters 130, 190201 (2023). DOI: 10.1103/PhysRevLett.130.190201; preprint arXiv:2302.02472. https://doi.org/10.1103/PhysRevLett.130.190201