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  • Open Access

Indefinite Causal Order and Quantum Coordinates

Anne-Catherine de la Hamette*, Viktoria Kabel, Marios Christodoulou, and Časlav Brukner

  • University of Vienna, Faculty of Physics, Vienna Doctoral School in Physics, and Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, A-1090 Vienna, Austria and Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

  • *Contact author: acdelahamette@gmail.com
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Phys. Rev. Lett. 135, 141402 – Published 1 October, 2025

DOI: https://doi.org/10.1103/bnkn-4p3f

Abstract

Classically, the causal order of two timelike separated events A and B is fixed—either A before B or B before A. This is no longer true in quantum theory, where it is possible to encounter superpositions of causal orders. The quantum switch is one of the most prominent processes with indefinite causal order. Optical realizations of the quantum switch have been successfully implemented in experiments, but some argue that this merely simulates a process with indefinite causal order and that a superposition of spacetime metrics is required for a true realization. Here, we provide a relativistic definition of causal order between operationally defined events that defines a meaningful observable in both the general relativistic and quantum mechanical sense. Importantly, this observable does not distinguish between the indefinite causal order implemented on an optical bench and the gravitational quantum switch, a gedanken experiment where the indefinite causal order is achieved by a quantum superposition of gravitational fields. Therefore, our results support the thesis that the optical quantum switch is just as much a realization of indefinite causal order as its gravitational counterpart, which makes use of the quantum mechanical behavior of spacetime.

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References (62)

  1. L. Hardy, Probability theories with dynamic causal structure: A new framework for quantum gravity, arXiv:gr-qc/0509120.
  2. G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013).
  3. O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012).
  4. M. Zych, F. Costa, I. Pikovski, and Č. Brukner, Bell’s theorem for temporal order, Nat. Commun. 10, 3772 (2019).
  5. L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. Alonso Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, Experimental superposition of orders of quantum gates, Nat. Commun. 6, 7913 (2015).
  6. G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, Experimental verification of an indefinite causal order, Sci. Adv. 3, e1602589 (2017).
  7. G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, Experimental entanglement of temporal order, Quantum 6, 621 (2022).
  8. K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, Indefinite causal order in a quantum switch, Phys. Rev. Lett. 121, 090503 (2018).
  9. K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H. Li, L. You, Z. Wang, Y.-A. Chen, B. C. Sanders, Q. Zhang, G. J. Pryde, F. Xu, and J.-W. Pan, Experimental quantum switching for exponentially superior quantum communication complexity, Phys. Rev. Lett. 122, 120504 (2019).
  10. M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. Lima, Computational advantage from the quantum superposition of multiple temporal orders of photonic gates, PRX Quantum 2, 010320 (2021).
  11. P. A. Guérin and Č. Brukner, Observer-dependent locality of quantum events, New J. Phys. 20, 103031 (2018).
  12. H. Kristjánsson, T. Odake, S. Yoshida, P. Taranto, J. Bavaresco, M. T. Quintino, and M. Murao, Exponential separation in quantum query complexity of the quantum switch with respect to simulations with standard quantum circuits, arXiv:2409.18420.
  13. O. Oreshkov and N. J. Cerf, Operational quantum theory without predefined time, New J. Phys. 18, 073037 (2016).
  14. N. Paunković and M. Vojinović, Causal orders, quantum circuits and spacetime: Distinguishing between definite and superposed causal orders, Quantum 4, 275 (2020).
  15. C. Marletto and V. Vedral, Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity, Phys. Rev. Lett. 119, 240402 (2017).
  16. S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn, Spin entanglement witness for quantum gravity, Phys. Rev. Lett. 119, 240401 (2017).
  17. A. Belenchia, R. M. Wald, F. Giacomini, E. Castro-Ruiz, Č. Brukner, and M. Aspelmeyer, Quantum superposition of massive objects and the quantization of gravity, Phys. Rev. D 98, 126009 (2018).
  18. M. Christodoulou and C. Rovelli, On the possibility of laboratory evidence for quantum superposition of geometries, Phys. Lett. B 792, 64 (2019).
  19. C. Anastopoulos and B. L. Hu, Quantum superposition of two gravitational cat states, Classical Quantum Gravity 37, 235012 (2020).
  20. A.-C. de la Hamette, V. Kabel, E. Castro-Ruiz, and Č. Brukner, Quantum reference frames for an indefinite metric, Commun. Phys. 6, 231 (2023).
  21. M. Aspelmeyer, When Zeh meets Feynman: How to avoid the appearance of a classical world in gravity experiments, in From Quantum to Classical: Essays in Honour of H.-Dieter Zeh, edited by C. Kiefer (Springer International Publishing, Cham, 2022), pp. 85–95.
  22. O. Oreshkov, Time-delocalized quantum subsystems and operations: On the existence of processes with indefinite causal structure in quantum mechanics, Quantum 3, 206 (2019).
  23. D. Felce, N. T. Vidal, V. Vedral, and E. O. Dias, Indefinite causal orders from superpositions in time, Phys. Rev. A 105, 062216 (2022).
  24. V. Vilasini and R. Renner, Embedding cyclic information-theoretic structures in acyclic space-times: No-go results for indefinite causality, Phys. Rev. A 110, 022227 (2024).
  25. N. Ormrod, A. Vanrietvelde, and J. Barrett, Causal structure in the presence of sectorial constraints, with application to the quantum switch, Quantum 7, 1028 (2023).
  26. L. Hardy, Implementation of the quantum equivalence principle, in Progress and Visions in Quantum Theory in View of Gravity, edited by F. Finster, D. Giulini, J. Kleiner, and J. Tolksdorf (Springer International Publishing, Cham, 2020), pp. 189–220.
  27. F. Giacomini, E. Castro-Ruiz, and Č. Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames, Nat. Commun. 10, 494 (2019).
  28. L. Loveridge, T. Miyadera, and P. Busch, Symmetry, reference frames, and relational quantities in quantum mechanics, Found. Phys. 48, 135 (2018).
  29. A. Vanrietvelde, P. A. Höhn, F. Giacomini, and E. Castro-Ruiz, A change of perspective: Switching quantum reference frames via a perspective-neutral framework, Quantum 4, 225 (2020).
  30. E. Castro-Ruiz, F. Giacomini, A. Belenchia, and Č. Brukner, Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems, Nat. Commun. 11, 2672 (2020).
  31. P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Trinity of relational quantum dynamics, Phys. Rev. D 104, 066001 (2021).
  32. A.-C. de la Hamette and T. D. Galley, Quantum reference frames for general symmetry groups, Quantum 4, 367 (2020).
  33. M. Krumm, P. A. Höhn, and M. P. Müller, Quantum reference frame transformations as symmetries and the paradox of the third particle, Quantum 5, 530 (2021).
  34. E. Castro-Ruiz and O. Oreshkov, Relative subsystems and quantum reference frame transformations, Commun. Phys. 8, 187 (2025).
  35. A.-C. de la Hamette, T. D. Galley, P. A. Höhn, L. Loveridge, and M. P. Müller, Perspective-neutral approach to quantum frame covariance for general symmetry groups, arXiv:2110.13824.
  36. A.-C. de la Hamette, S. L. Ludescher, and M. P. Müller, Entanglement-asymmetry correspondence for internal quantum reference frames, Phys. Rev. Lett. 129, 260404 (2022).
  37. C. Cepollaro and F. Giacomini, Quantum generalisation of Einstein’s equivalence principle can be verified with entangled clocks as quantum reference frames, Classical Quantum Gravity 41, 185009 (2024).
  38. V. Kabel, A.-C. de la Hamette, E. Castro-Ruiz, and Č. Brukner, Quantum conformal symmetries for spacetimes in superposition, Quantum 8, 1547 (2024).
  39. F. Giacomini and Č. Brukner, Einstein’s equivalence principle for superpositions of gravitational fields, arXiv:2012.13754.

  40. Note that Eq. (3) should be understood as suggestive since the Hilbert space assigned to the metric has not properly been introduced. However, if classically distinguishable spacetimes are represented by orthogonal states, one can form linear superpositions thereof and define a suitable inner product in the Hilbert space. They would correspond to the superposition of semiclassical states in a complete theory.

  41. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, (W.H. Freeman and Co., San Francisco, 1973).
  42. O. Bengyat, A. Di Biagio, M. Aspelmeyer, and M. Christodoulou, Gravity-mediated entanglement between oscillators as quantum superposition of geometries, Phys. Rev. D 110, 056046 (2024).

  43. Note that this notion of causal order can be generalized straightforwardly to encompass the causal ordering of any finite number N of timelike events in an arbitrary number of branches M. Denoting the sign of the proper time difference between two events El and Ek by slk=sign(τk−τl), we say that a process has definite causal order if all pairwise products slk(i)slk(j),1≤l<k≤N,1≤i<j≤M are positive. Otherwise, the process displays some form of ICO.

  44. When considering ICO on a fixed spacetime background due to a superposition of crossing points, we consider two copies of the same manifold and metric, (M1,g(1))=(M2,g(2)), in Definition t2.

  45. F. Giacomini and Č. Brukner, Quantum superposition of spacetimes obeys Einstein’s equivalence principle, AVS Quantum Sci. 4, 015601 (2022).

  46. Note that the action of the diffeomorphisms on the fields is the inverse of their actions on the manifold points.

  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/bnkn-4p3f for a mapping between different descriptions of the quantum switch as well as an operational encoding of the causal order, which includes Refs. [13,39,41,45,48–53].
  48. V. Kabel, A.-C. de la Hamette, L. Apadula, C. Cepollaro, H. Gomes, J. Butterfield, and Č. Brukner, Quantum coordinates, localisation of events, and the quantum hole argument, Commun. Phys. 8, 185 (2025).
  49. M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015).
  50. C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, The simplest causal inequalities and their violation, New J. Phys. 18, 013008 (2015).
  51. C. Branciard, Witnesses of causal nonseparability: An introduction and a few case studies, Sci. Rep. 6, 26018 (2016).
  52. J. Bavaresco, M. Araújo, Č. Brukner, and M. T. Quintino, Semi-device-independent certification of indefinite causal order, Quantum 3, 176 (2019).
  53. T. van der Lugt, J. Barrett, and G. Chiribella, Device-independent certification of indefinite causal order in the quantum switch, Nat. Commun. 14, 5811 (2023).
  54. M. Zych, F. Costa, I. Pikovski, and Č. Brukner, Quantum interferometric visibility as a witness of general relativistic proper time, Nat. Commun. 2, 505 (2011).
  55. A. R. H. Smith and M. Ahmadi, Quantum clocks observe classical and quantum time dilation, Nat. Commun. 11, 5360 (2020).
  56. J. Paczos, K. Dębski, P. T. Grochowski, A. R. H. Smith, and A. Dragan, Quantum time dilation in a gravitational field, Quantum 8, 1338 (2024).
  57. K. Dȩbski, P. T. Grochowski, R. Demkowicz-Dobrzański, and A. Dragan, Universality of quantum time dilation, Classical Quantum Gravity 41, 135014 (2024).
  58. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, England, 1995).
  59. R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (Princeton University Press, Princeton, NJ, 2001).
  60. E. Castro-Ruiz, F. Giacomini, and Č. Brukner, Dynamics of quantum causal structures, Phys. Rev. X 8, 011047 (2018).
  61. J. H. Selby, A. B. Sainz, and P. Horodecki, Revisiting dynamics of quantum causal structures—when can causal order evolve?, Entropy 26, 643 (2024).
  62. E. Castro-Ruiz, F. Giacomini, and Časlav Brukner, Comment on “Revisiting dynamics of quantum causal structures–when can causal order evolve?”, arXiv:2009.10027.

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Phys. Rev. Lett. 135, 141402– Published 1 October, 2025
  • Received 14 December 2022
  • Revised 7 August 2025
  • Accepted 4 September 2025
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