Experimental entanglement of temporal order

Giulia Rubino1,2, Lee A. Rozema1, Francesco Massa1, Mateus Araújo1,3, Magdalena Zych4, Časlav Brukner1,3, and Philip Walther1

1Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, Vienna A-1090, Austria
2Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering, University of Bristol, Bristol BS8 1FD, United Kingdom
3Institute for Quantum Optics & Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, Vienna A-1090, Austria
4Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The study of causal relations has recently been applied to the quantum realm, leading to the discovery that not all physical processes have a definite causal structure. While indefinite causal processes have previously been experimentally shown, these proofs relied on the quantum description of the experiments. Yet, the same experimental data could also be compatible with definite causal structures within different descriptions. Here, we present the first demonstration of indefinite temporal order outside of quantum formalism. We show that our experimental outcomes are incompatible with a class of generalised probabilistic theories satisfying the assumptions of locality and definite temporal order. To this end, we derive physical constraints (in the form of a Bell-like inequality) on experimental outcomes within such a class of theories. We then experimentally invalidate these theories by violating the inequality using entangled temporal order. This provides experimental evidence that there exist correlations in nature which are incompatible with the assumptions of locality and definite temporal order.

In all standard physical theories, causal relations among events are inferred from a background temporal order, which is assumed as fixed. This assumption requires rigorous experimental tests which should be independent of the particular theory under study. Conversely, it was recently discovered that quantum mechanics permits the existence of processes without a predefined temporal order. Thus far, experimental evidence for this effect relied on the quantum description of the experiments. In this work we go beyond previous results by experimentally violating a Bell inequality for temporal order, which is fulfilled by a large class of theories in which a condition on locality and a predefined order of events are assumed. We thus provide the first proof of the indefiniteness of the temporal order without presupposing the quantum description of the experiment. Furthermore, within quantum theory, our results demonstrate for the first time quantum entanglement between two causal processes.

► BibTeX data

► References

[1] We test hypothesis IIb by placing the two operations individually in a single quantum switch, and we show that each operation is local. Under the assumption of a definite causal order, this test implies that the sequential action of the two operations is also local. We could not have directly tested the combined action of the two operations because such a test could only show that control and target are coupled, but would not provide any information about whether this is due to the violation of IIb, or due to the indefinite causal order, or both.

[2] Alastair A. Abbott, Julian Wechs, Fabio Costa, and Cyril Branciard. Genuinely multipartite noncausality. Quantum, 1: 39, December 2017. ISSN 2521-327X. 10.22331/​q-2017-12-14-39. URL https:/​/​doi.org/​10.22331/​q-2017-12-14-39.
https:/​/​doi.org/​10.22331/​q-2017-12-14-39

[3] Philippe Allard Guérin and Časlav Brukner. Observer-dependent locality of quantum events. New Journal of Physics, 20 (10): 103031, oct 2018. 10.1088/​1367-2630/​aae742.
https:/​/​doi.org/​10.1088/​1367-2630/​aae742

[4] Mateus Araújo, Fabio Costa, and Časlav Brukner. Computational advantage from quantum-controlled ordering of gates. Phys. Rev. Lett., 113: 250402, Dec 2014. 10.1103/​PhysRevLett.113.250402.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.250402

[5] Mateus Araújo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, and Časlav Brukner. Witnessing causal nonseparability. New Journal of Physics, 17 (10): 102001, 2015. 10.1088/​1367-2630/​17/​10/​102001.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​10/​102001

[6] Alain Aspect, Jean Dalibard, and Gérard Roger. Experimental test of bell's inequalities using time-varying analyzers. Phys. Rev. Lett., 49: 1804–1807, Dec 1982. 10.1103/​PhysRevLett.49.1804. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.49.1804.
https:/​/​doi.org/​10.1103/​PhysRevLett.49.1804

[7] Jonathan Barrett. Information processing in generalized probabilistic theories. Phys. Rev. A, 75: 032304, Mar 2007. 10.1103/​PhysRevA.75.032304. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.75.032304.
https:/​/​doi.org/​10.1103/​PhysRevA.75.032304

[8] J. S. Bell. On the einstein-podolsky-rosen paradox. Physics, 1 (3): 195–200, 1964.

[9] Cyril Branciard. Witnesses of causal nonseparability: an introduction and a few case studies. Scientific Reports, 6: 26018, May 2016. 10.1038/​srep26018.
https:/​/​doi.org/​10.1038/​srep26018

[10] Cyril Branciard, Mateus Araújo, Adrien Feix, Fabio Costa, and Časlav Brukner. The simplest causal inequalities and their violation. New Journal of Physics, 18 (1): 013008, 2016. 10.1088/​1367-2630/​18/​1/​013008. URL http:/​/​stacks.iop.org/​1367-2630/​18/​i=1/​a=013008.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013008
http:/​/​stacks.iop.org/​1367-2630/​18/​i=1/​a=013008

[11] Časlav Brukner. Quantum causality. Nat. Phys., 10: 259–263, Apr 2014. 10.1038/​nphys2930.
https:/​/​doi.org/​10.1038/​nphys2930

[12] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/​RevModPhys.86.419. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.86.419.
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[13] Giulio Chiribella. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A, 86: 040301, Oct 2012. 10.1103/​PhysRevA.86.040301.
https:/​/​doi.org/​10.1103/​PhysRevA.86.040301

[14] Giulio Chiribella, Giacomo Mauro D'Ariano, Paolo Perinotti, and Benoit Valiron. Quantum computations without definite causal structure. Phys. Rev. A, 88: 022318, Aug 2013. 10.1103/​PhysRevA.88.022318.
https:/​/​doi.org/​10.1103/​PhysRevA.88.022318

[15] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880–884, Oct 1969. 10.1103/​PhysRevLett.23.880. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.23.880.
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[16] Borivoje Dakic and Časlav Brukner. Quantum Theory and Beyond: Is Entanglement Special? Contribution to ``Deep beauty'', Editor Hans Halvorson (Cambridge University Press, 2010. 10.1017/​CBO9780511976971.011.
https:/​/​doi.org/​10.1017/​CBO9780511976971.011

[17] Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller. Inseparability criterion for continuous variable systems. Phys. Rev. Lett., 84: 2722–2725, Mar 2000. 10.1103/​PhysRevLett.84.2722. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.84.2722.
https:/​/​doi.org/​10.1103/​PhysRevLett.84.2722

[18] Adrien Feix, Mateus Araújo, and Časlav Brukner. Quantum superposition of the order of parties as a communication resource. Phys. Rev. A, 92: 052326, Nov 2015. 10.1103/​PhysRevA.92.052326.
https:/​/​doi.org/​10.1103/​PhysRevA.92.052326

[19] Stuart J. Freedman and John F. Clauser. Experimental test of local hidden-variable theories. Phys. Rev. Lett., 28: 938–941, Apr 1972. 10.1103/​PhysRevLett.28.938. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.28.938.
https:/​/​doi.org/​10.1103/​PhysRevLett.28.938

[20] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Morgan W. Mitchell, Jörn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger. Significant-loophole-free test of bell's theorem with entangled photons. Phys. Rev. Lett., 115: 250401, Dec 2015. 10.1103/​PhysRevLett.115.250401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.250401.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250401

[21] 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, Aug 2018. 10.1103/​PhysRevLett.121.090503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.121.090503.
https:/​/​doi.org/​10.1103/​PhysRevLett.121.090503

[22] Philippe Allard Guérin, Adrien Feix, Mateus Araújo, and Časlav Brukner. Exponential communication complexity advantage from quantum superposition of the direction of communication. Phys. Rev. Lett., 117: 100502, Sep 2016. 10.1103/​PhysRevLett.117.100502.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.100502

[23] Lucien Hardy. Quantum theory from five reasonable axioms. https:/​/​arxiv.org/​abs/​quant-ph/​0101012, 2001.
arXiv:quant-ph/0101012

[24] Lucien Hardy. Foliable Operational Structures for General Probabilistic Theories, page 409–442. Cambridge University Press, 2011. 10.1017/​CBO9780511976971.013.
https:/​/​doi.org/​10.1017/​CBO9780511976971.013

[25] B. Hensen, H. Bernien, A. E. Dreau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellan, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526: 682–686, Oct 2015. 10.1038/​nature15759. URL http:/​/​dx.doi.org/​10.1038/​nature15759.
https:/​/​doi.org/​10.1038/​nature15759

[26] John C. Howell, Ryan S. Bennink, Sean J. Bentley, and R. W. Boyd. Realization of the einstein-podolsky-rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion. Phys. Rev. Lett., 92: 210403, May 2004. 10.1103/​PhysRevLett.92.210403. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.92.210403.
https:/​/​doi.org/​10.1103/​PhysRevLett.92.210403

[27] P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao. High-visibility interference in a bell-inequality experiment for energy and time. Phys. Rev. A, 47: R2472–R2475, Apr 1993. 10.1103/​PhysRevA.47.R2472. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.47.R2472.
https:/​/​doi.org/​10.1103/​PhysRevA.47.R2472

[28] M. Lamehi-Rachti and W. Mittig. Quantum mechanics and hidden variables: A test of bell's inequality by the measurement of the spin correlation in low-energy proton-proton scattering. Phys. Rev. D, 14: 2543–2555, Nov 1976. 10.1103/​PhysRevD.14.2543. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevD.14.2543.
https:/​/​doi.org/​10.1103/​PhysRevD.14.2543

[29] Jean-Philippe W. MacLean, Katja Ried, Robert W. Spekkens, and Kevin J. Resch. Quantum-coherent mixtures of causal relations. Nature Communications, 8: 15149, May 2017. 10.1038/​ncomms15149.
https:/​/​doi.org/​10.1038/​ncomms15149

[30] Ll. Masanes, M. P. Müller, D. Pérez-García, and R. Augusiak. Entanglement and the three-dimensionality of the bloch ball. Journal of Mathematical Physics, 55 (12): 122203, 2014. 10.1063/​1.4903510. URL https:/​/​doi.org/​10.1063/​1.4903510.
https:/​/​doi.org/​10.1063/​1.4903510

[31] Lluís Masanes and Markus P. Müller. A derivation of quantum theory from physical requirements. New Journal of Physics, 13 (6): 063001, 2011. 10.1088/​1367-2630/​13/​6/​063001. URL http:/​/​stacks.iop.org/​1367-2630/​13/​i=6/​a=063001.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​6/​063001
http:/​/​stacks.iop.org/​1367-2630/​13/​i=6/​a=063001

[32] Nikolai Miklin, Alastair A Abbott, Cyril Branciard, Rafael Chaves, and Costantino Budroni. The entropic approach to causal correlations. New Journal of Physics, 19 (11): 113041, 2017. 10.1088/​1367-2630/​aa8f9f. URL http:/​/​stacks.iop.org/​1367-2630/​19/​i=11/​a=113041.
https:/​/​doi.org/​10.1088/​1367-2630/​aa8f9f
http:/​/​stacks.iop.org/​1367-2630/​19/​i=11/​a=113041

[33] Ognyan Oreshkov. Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics. Quantum, 3: 206, December 2019. ISSN 2521-327X. 10.22331/​q-2019-12-02-206. URL https:/​/​doi.org/​10.22331/​q-2019-12-02-206.
https:/​/​doi.org/​10.22331/​q-2019-12-02-206

[34] Ognyan Oreshkov and Christina Giarmatzi. Causal and causally separable processes. New Journal of Physics, 18 (9): 093020, 2016. 10.1088/​1367-2630/​18/​9/​093020.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​9/​093020

[35] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. Quantum correlations with no causal order. Nat. Commun., 3: 1092, Oct 2012. 10.1038/​ncomms2076.
https:/​/​doi.org/​10.1038/​ncomms2076

[36] Lorenzo M. Procopio, Amir Moqanaki, Mateus Araújo, Fabio Costa, Irati Alonso Calafell, Emma G. Dowd, Deny R. Hamel, Lee A. Rozema, Časlav Brukner, and Philip Walther. Experimental superposition of orders of quantum gates. Nat. Commun., 6 (7913), Aug 2015. 10.1038/​ncomms8913.
https:/​/​doi.org/​10.1038/​ncomms8913

[37] J. G. Rarity and P. R. Tapster. Experimental violation of bell's inequality based on phase and momentum. Phys. Rev. Lett., 64: 2495–2498, May 1990. 10.1103/​PhysRevLett.64.2495. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.64.2495.
https:/​/​doi.org/​10.1103/​PhysRevLett.64.2495

[38] Martin J. Renner and Časlav Brukner. Reassessing the computational advantage of quantum-controlled ordering of gates. Phys. Rev. Research, 3: 043012, Oct 2021. 10.1103/​PhysRevResearch.3.043012. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.3.043012.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.043012

[39] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland. Experimental violation of a bell's inequality with efficient detection. Nature, 409: 791, Feb 2001. 10.1038/​35057215. URL http:/​/​dx.doi.org/​10.1038/​35057215.
https:/​/​doi.org/​10.1038/​35057215

[40] Giulia Rubino, Lee A. Rozema, Adrien Feix, Mateus Araújo, Jonas M. Zeuner, Lorenzo M. Procopio, Časlav Brukner, and Philip Walther. Experimental verification of an indefinite causal order. Science Advances, 3 (3), 2017. 10.1126/​sciadv.1602589. URL http:/​/​advances.sciencemag.org/​content/​3/​3/​e1602589.
https:/​/​doi.org/​10.1126/​sciadv.1602589
http:/​/​advances.sciencemag.org/​content/​3/​3/​e1602589

[41] Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst, Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel, Michael S. Allman, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E. Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yanbao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw, Jeffrey A. Stern, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Thomas Jennewein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin, Emanuel Knill, and Sae Woo Nam. Strong loophole-free test of local realism. Phys. Rev. Lett., 115: 250402, Dec 2015. 10.1103/​PhysRevLett.115.250402. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.250402.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250402

[42] R. Simon. Peres-horodecki separability criterion for continuous variable systems. Phys. Rev. Lett., 84: 2726–2729, Mar 2000. 10.1103/​PhysRevLett.84.2726. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.84.2726.
https:/​/​doi.org/​10.1103/​PhysRevLett.84.2726

[43] Magdalena Zych, Fabio Costa, Igor Pikovski, and Časlav Brukner. Bell's theorem for temporal order. Nature Communications, 10: 3772, 08 2019. 10.1038/​s41467-019-11579-x. URL https:/​/​doi.org/​10.1038/​s41467-019-11579-x.
https:/​/​doi.org/​10.1038/​s41467-019-11579-x

Cited by

[1] Veronika Baumann, Marius Krumm, Philippe Allard Guérin, and Časlav Brukner, "Noncausal Page-Wootters circuits", Physical Review Research 4 1, 013180 (2022).

[2] Gaoyan Zhu, Yuanbo Chen, Yoshihiko Hasegawa, and Peng Xue, "Charging Quantum Batteries via Indefinite Causal Order: Theory and Experiment", Physical Review Letters 131 24, 240401 (2023).

[3] Aurélien Drezet, "Indefinite causal order with fixed temporal order for electrons and positrons", Quantum Studies: Mathematics and Foundations 10 1, 101 (2023).

[4] Ricardo Faleiro, Nikola Paunkovic, and Marko Vojinovic, "Operational interpretation of the vacuum and process matrices for identical particles", Quantum 7, 986 (2023).

[5] Julian Wechs, Cyril Branciard, and Ognyan Oreshkov, "Existence of processes violating causal inequalities on time-delocalised subsystems", Nature Communications 14 1, 1471 (2023).

[6] Huan Cao, Jessica Bavaresco, Ning-Ning Wang, Lee A. Rozema, Chao Zhang, Yun-Feng Huang, Bi-Heng Liu, Chuan-Feng Li, Guang-Can Guo, and Philip Walther, "Semi-device-independent certification of indefinite causal order in a photonic quantum switch", Optica 10 5, 561 (2023).

[7] Zhu Cao, "Sequential device-independent certification of indefinite causal order", Physical Review A 108 1, 012428 (2023).

[8] Martin J. Renner and Časlav Brukner, "Computational Advantage from a Quantum Superposition of Qubit Gate Orders", Physical Review Letters 128 23, 230503 (2022).

[9] Seid Koudia, Angela Sara Cacciapuoti, Kyrylo Simonov, and Marcello Caleffi, "How Deep the Theory of Quantum Communications Goes: Superadditivity, Superactivation and Causal Activation", IEEE Communications Surveys & Tutorials 24 4, 1926 (2022).

[10] Nick Ormrod, Augustin Vanrietvelde, and Jonathan Barrett, "Causal structure in the presence of sectorial constraints, with application to the quantum switch", Quantum 7, 1028 (2023).

[11] Ngo Phuc Duc Loc, "Time-system entanglement and special relativity", Modern Physics Letters A 39 01, 2350183 (2024).

[12] M. H. Raddadi, Abdallah A. Nahla, and D. A. M. Abo-Kahla, "Quantized study for asymmetric two two-level atoms interacting with intensity-dependent coupling regime", Indian Journal of Physics 97 5, 1345 (2023).

[13] Marco Fellous-Asiani, Raphaël Mothe, Léa Bresque, Hippolyte Dourdent, Patrice A. Camati, Alastair A. Abbott, Alexia Auffèves, and Cyril Branciard, "Comparing the quantum switch and its simulations with energetically constrained operations", Physical Review Research 5 2, 023111 (2023).

[14] Marcello Caleffi, Kyrylo Simonov, and Angela Sara Cacciapuoti, "Beyond Shannon Limits: Quantum Communications Through Quantum Paths", IEEE Journal on Selected Areas in Communications 41 8, 2707 (2023).

[15] Kacper Dębski, Magdalena Zych, Fabio Costa, and Andrzej Dragan, "Indefinite temporal order without gravity", Physical Review A 108 6, 062204 (2023).

[16] Tein van der Lugt, Jonathan Barrett, and Giulio Chiribella, "Device-independent certification of indefinite causal order in the quantum switch", Nature Communications 14 1, 5811 (2023).

[17] Aaron Z. Goldberg, Khabat Heshami, and L. L. Sánchez-Soto, "Evading noise in multiparameter quantum metrology with indefinite causal order", Physical Review Research 5 3, 033198 (2023).

[18] Seid Koudia, Angela Sara Cacciapuoti, and Marcello Caleffi, "Deterministic Generation of Multipartite Entanglement via Causal Activation in the Quantum Internet", IEEE Access 11, 73863 (2023).

[19] Fabio Costa, "A no-go theorem for superpositions of causal orders", Quantum 6, 663 (2022).

[20] Teodor Strömberg, Peter Schiansky, Robert W. Peterson, Marco Túlio Quintino, and Philip Walther, "Demonstration of a Quantum Switch in a Sagnac Configuration", Physical Review Letters 131 6, 060803 (2023).

[21] Shuqing Lin, Yanfeng Zhang, Zhaoyang Wu, Shihao Zeng, Qing Gao, Jiaqi Li, Xiaoqun Yu, and Siyuan Yu, "Power-efficient programmable integrated multiport photonic interferometer in CMOS-compatible silicon nitride", Photonics Research 12 3, A11 (2024).

[22] P Chamorro-Posada and J C Garcia-Escartin, "The SWITCH test for discriminating quantum evolutions", Journal of Physics A: Mathematical and Theoretical 56 35, 355301 (2023).

[23] Lachlan Parker and Fabio Costa, "Background Independence and Quantum Causal Structure", Quantum 6, 865 (2022).

[24] Michael Antesberger, Marco Túlio Quintino, Philip Walther, and Lee A. Rozema, "Higher-Order Process Matrix Tomography of a Passively-Stable Quantum Switch", PRX Quantum 5 1, 010325 (2024).

[25] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, "Indefinite Causal Order in a Quantum Switch", Physical Review Letters 121 9, 090503 (2018).

[26] Magdalena Zych, Fabio Costa, Igor Pikovski, and Časlav Brukner, "Bell's theorem for temporal order", Nature Communications 10, 3772 (2019).

[27] Kejin Wei, Nora Tischler, Si-Ran Zhao, Yu-Huai Li, Juan Miguel Arrazola, Yang Liu, Weijun Zhang, Hao Li, Lixing You, Zhen Wang, Yu-Ao Chen, Barry C. Sanders, Qiang Zhang, Geoff J. Pryde, Feihu Xu, and Jian-Wei Pan, "Experimental Quantum Switching for Exponentially Superior Quantum Communication Complexity", Physical Review Letters 122 12, 120504 (2019).

[28] Giulia Rubino, Lee A. Rozema, Daniel Ebler, Hlér Kristjánsson, Sina Salek, Philippe Allard Guérin, Alastair A. Abbott, Cyril Branciard, Časlav Brukner, Giulio Chiribella, and Philip Walther, "Experimental quantum communication enhancement by superposing trajectories", Physical Review Research 3 1, 013093 (2021).

[29] Philippe Allard Guérin, Giulia Rubino, and Časlav Brukner, "Communication through quantum-controlled noise", Physical Review A 99 6, 062317 (2019).

[30] Ognyan Oreshkov, "Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics", Quantum 3, 206 (2019).

[31] Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard, "Quantum Circuits with Classical Versus Quantum Control of Causal Order", PRX Quantum 2 3, 030335 (2021).

[32] K. Goswami and J. Romero, "Experiments on quantum causality", AVS Quantum Science 2 3, 037101 (2020).

[33] Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard, "Communication through coherent control of quantum channels", Quantum 4, 333 (2020).

[34] Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard, "Communication through coherent control of quantum channels", arXiv:1810.09826, (2018).

[35] Julian Wechs, Alastair A. Abbott, and Cyril Branciard, "On the definition and characterisation of multipartite causal (non)separability", New Journal of Physics 21 1, 013027 (2019).

[36] Nikola Paunković and Marko Vojinović, "Causal orders, quantum circuits and spacetime: distinguishing between definite and superposed causal orders", Quantum 4, 275 (2020).

[37] Teodor Strömberg, Peter Schiansky, Marco Túlio Quintino, Michael Antesberger, Lee Rozema, Iris Agresti, Časlav Brukner, and Philip Walther, "Experimental superposition of time directions", arXiv:2211.01283, (2022).

[38] Aaron Z. Goldberg and Khabat Heshami, "Breaking the limits of purification: postselection enhances heat-bath algorithmic cooling", Journal of Physics Communications 7 1, 015003 (2023).

[39] Jessica Bavaresco, Mateus Araújo, Časlav Brukner, and Marco Túlio Quintino, "Semi-device-independent certification of indefinite causal order", Quantum 3, 176 (2019).

[40] Martin J. Renner and Časlav Brukner, "Reassessing the computational advantage of quantum-controlled ordering of gates", Physical Review Research 3 4, 043012 (2021).

[41] Natália S. Móller, Bruna Sahdo, and Nelson Yokomizo, "Quantum switch in the gravity of Earth", Physical Review A 104 4, 042414 (2021).

[42] C. T. Marco Ho, Fabio Costa, Christina Giarmatzi, and Timothy C. Ralph, "Violation of a causal inequality in a spacetime with definite causal order", arXiv:1804.05498, (2018).

[43] Philippe Allard Guérin, Giulia Rubino, and Časlav Brukner, "Communication through quantum-controlled noise", arXiv:1812.06848, (2018).

[44] Timothée Hoffreumon and Ognyan Oreshkov, "The Multi-round Process Matrix", Quantum 5, 384 (2021).

[45] Aurélien Drezet, "Indefinite causal order with fixed temporal order for electrons and positrons", arXiv:2202.12886, (2022).

[46] Nikola Paunkovic and Marko Vojinovic, "Challenges for extensions of the process matrix formalism to quantum field theory", arXiv:2310.04597, (2023).

[47] Fabio Costa, Jonathan Barrett, and Sally Shrapnel, "A de Finetti theorem for quantum causal structures", arXiv:2403.10316, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 03:42:38) and SAO/NASA ADS (last updated successfully 2024-03-28 03:42:39). The list may be incomplete as not all publishers provide suitable and complete citation data.