Causal hierarchy of multipartite Bell nonlocality

Rafael Chaves1,2, Daniel Cavalcanti3, and Leandro Aolita4

1International Institute of Physics, Federal University of Rio Grande do Norte, 59078-970, P. O. Box 1613, Natal, Brazil
2Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany
3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
4Instituto de F'isica, Universidade Federal do Rio de Janeiro, P. O. Box 68528, Rio de Janeiro, RJ 21941-972, Brazil

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


As with entanglement, different forms of Bell nonlocality arise in the multipartite scenario. These can be defined in terms of relaxations of the causal assumptions in local hidden-variable theories. However, a characterisation of all the forms of multipartite nonlocality has until now been out of reach, mainly due to the complexity of generic multipartite causal models. Here, we employ the formalism of Bayesian networks to reveal connections among different causal structures that make a both practical and physically meaningful classification possible. Our framework holds for arbitrarily many parties. We apply it to study the tripartite scenario in detail, where we fully characterize all the nonlocality classes. Remarkably, we identify new highly nonlocal causal structures that cannot reproduce all quantum correlations. This shows, to our knowledge, the strongest form of quantum multipartite nonlocality known to date. Finally, as a by-product result, we derive a non-trivial Bell-type inequality with no quantum violation. Our findings constitute a significant step forward in the understanding of multipartite Bell nonlocality and open several venues for future research.

► BibTeX data

► References

[1] J. S. Bell, On the Einstein–Podolsky–Rosen paradox, Physics 1, 195 (1964).

[2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Reviews of Modern Physics 86, 419–478 (2014).

[3] C. J. Wood and R. W. Spekkens, The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning, New J. Phys. 17, 033002 (2015).

[4] M. Ringbauer, C. Giarmatzi, R. Chaves, F. Costa, A. G. White, and A. Fedrizzi, Experimental test of nonlocal causality, ArXiv:1602.02767 (2016).

[5] M. J. W. Hall, Local deterministic model of singlet state correlations based on relaxing measurement independence, Phys. Rev. Lett. 105, 250404 (2010).

[6] M. J. W. Hall, Relaxed bell inequalities and kochen-specker theorems, Phys. Rev. A 84, 022102 (2011).

[7] G. Pütz, D. Rosset, T. J. Barnea, Y.-C. Liang, and N. Gisin, Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality, Phys. Rev. Lett. 113, 190402 (2014).

[8] R. Chaves, R. Kueng, J. B. Brask, and D. Gross, Unifying framework for relaxations of the causal assumptions in Bell's theorem, Phys. Rev. Lett. 114, 140403 (2015a).

[9] R. Chaves, Polynomial bell inequalities, Phys. Rev. Lett. 116, 010402 (2016).

[10] C. H. Brans, Bell's theorem does not eliminate fully causal hidden variables, International Journal of Theoretical Physics 27, 219–226 (1988).

[11] G. Brassard, R. Cleve, and A. Tapp, Cost of exactly simulating quantum entanglement with classical communication, Phys. Rev. Lett. 83, 1874–1877 (1999).

[12] N. Gisin and B. Gisin, A local hidden variable model of quantum correlation exploiting the detection loophole, Physics Letters A 260, 323 – 327 (1999).

[13] B. F. Toner and D. Bacon, Communication cost of simulating bell correlations, Phys. Rev. Lett. 91, 187904 (2003).

[14] S. Pironio, Violations of bell inequalities as lower bounds on the communication cost of nonlocal correlations, Phys. Rev. A 68, 062102 (2003).

[15] O. Regev and B. Toner, Simulating quantum correlations with finite communication, SIAM Journal on Computing 39, 1562–1580 (2010).

[16] J. Barrett and N. Gisin, How much measurement independence is needed to demonstrate nonlocality? Phys. Rev. Lett. 106, 100406 (2011).

[17] K. Maxwell and E. Chitambar, Bell inequalities with communication assistance, Phys. Rev. A 89, 042108 (2014).

[18] Jonatan Bohr Brask and Rafael Chaves, Bell scenarios with communication, arXiv preprint arXiv:1607.08182 (2016).

[19] G. Svetlichny, Distinguishing three-body from two-body nonseparability by a bell-type inequality, Phys. Rev. D 35, 3066–3069 (1987).

[20] D. Collins, N. Gisin, S. Popescu, D. Roberts, and V. Scarani, Bell-type inequalities to detect true $\mathit{n}$-body nonseparability, Phys. Rev. Lett. 88, 170405 (2002).

[21] M. Seevinck and G. Svetlichny, Bell-type inequalities for partial separability in $n$-particle systems and quantum mechanical violations, Phys. Rev. Lett. 89, 060401 (2002).

[22] J.-D. Bancal, N. Gisin, Y.-C. Liang, and S. Pironio, Device-Independent Witnesses of Genuine Multipartite Entanglement, Physical Review Letters 106, 250404 (2011), 1102.0197 [quant-ph].

[23] Tobias Moroder, Jean-Daniel Bancal, Yeong-Cherng Liang, Martin Hofmann, and Otfried Gühne, Device-independent entanglement quantification and related applications, Phys. Rev. Lett. 111, 030501 (2013).

[24] Yeong-Cherng Liang, Denis Rosset, Jean-Daniel Bancal, Gilles Pütz, Tomer Jack Barnea, and Nicolas Gisin, Family of bell-like inequalities as device-independent witnesses for entanglement depth, Phys. Rev. Lett. 114, 190401 (2015).

[25] Nick S. Jones, Noah Linden, and Serge Massar, Extent of multiparticle quantum nonlocality, Phys. Rev. A 71, 042329 (2005).

[26] J.-D. Bancal, C. Branciard, N. Gisin, and S. Pironio, Quantifying multipartite nonlocality, Phys. Rev. Lett. 103, 090503 (2009).

[27] M. L. Almeida, D. Cavalcanti, V. Scarani, and A. Acín, Multipartite fully nonlocal quantum states, Physical Review A 81, 052111 (2010), 0911.3559 [quant-ph].

[28] Jean-Daniel Bancal, Nicolas Brunner, Nicolas Gisin, and Yeong-Cherng Liang, Detecting genuine multipartite quantum nonlocality: A simple approach and generalization to arbitrary dimensions, Phys. Rev. Lett. 106, 020405 (2011).

[29] R. Gallego, L. E. Wurflinger, A. Acin, and M. Navascues, Operational framework for nonlocality, Phys. Rev. Lett. 109, 070401 (2012).

[30] Leandro Aolita, Rodrigo Gallego, Adán Cabello, and Antonio Acín, Fully nonlocal, monogamous, and random genuinely multipartite quantum correlations, Phys. Rev. Lett. 108, 100401 (2012a).

[31] J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, Definitions of multipartite nonlocality, Phys. Rev. A 88, 014102 (2013).

[32] D. Saha and M. Pawłowski, Structure of quantum and broadcasting nonlocal correlations, Phys. Rev. A 92, 062129 (2015).

[33] J. Pearl, Causality (Cambridge University Press, 2009).

[34] P. Spirtes, N. Glymour, and R. Scheienes, Causation, Prediction, and Search, 2nd ed. (The MIT Press, 2001).

[35] M. L. Almeida, J.-D. Bancal, N. Brunner, A. Acín, N. Gisin, and S. Pironio, Guess your neighbor's input: A multipartite nonlocal game with no quantum advantage, Phys. Rev. Lett. 104, 230404 (2010).

[36] Andreas Winter, Quantum mechanics: The usefulness of uselessness, Nature 466, 1053–1054 (2010).

[37] Both bi-LHV and input-broadcasting models are particular cases of a generic family called partially paired models in Ref. Jones2005 (see Fig. \ref fig:hierarchy). An input-communication structure is called totally paired if every pair of inputs is sent to some output; otherwise it is partially paired. Partially paired models were shown to satisfy the Svetlichny inequality, which admits quantum violations. In contrast, it was an open question whether there exist totally paired models with quantum violations Jones2005.

[38] The terminology nonsignalling interesting/​boring is inspired by the works HLP14,Pienaar16. There, the terms interesting/​boring are used to denote DAGs that allow/​do not allow one to distinguish – though in a slightly different sense as here – between classical, quantum, and post-quantum models.

[39] In fact, this is a generic property that holds for arbitrary $N$: all causal Bell classes for which all $N$ inputs go to the output of one party while any other locality relaxation involves the latter party are nonsignalling equivalent (see appendix).

[40] 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 (1969).

[41] Nicolas Brunner and Tamás Vértesi, Persistency of entanglement and nonlocality in multipartite quantum systems, Phys. Rev. A 86, 042113 (2012).

[42] Sandu Popescu and Daniel Rohrlich, Quantum nonlocality as an axiom, Foundations of Physics 24, 379–385 (1994).

[43] N. Gisin, A. A. Methot, and V. Scarani, Pseudo-telepathy: input cardinality and Bell-type inequalities, International Journal of Quantum Information 5, 525–534 (2007).

[44] T. Yang, Q. Zhang, J. Zhang, J. Yin, Z. Zhao, M. Żukowski, Z.-B.Chen, and J.-W. Pan, All-Versus-Nothing Violation of Local Realism by Two-Photon, Four-Dimensional Entanglement, Phys. Rev. Lett. 95, 240406 (2005).

[45] L. Aolita, R. Gallego, A. Acín, A. Chiuri G. Vallone, P. Mataloni, and A. Cabello, Fully nonlocal quantum correlations, Phys. Rev. A (2012b).

[46] J. Barrett, A. Kent, and S. Pironio, Phys. Rev. Lett. 97, 170409 (2006).

[47] T. E. Stuart, J. A. Slater, R. Colbeck, R. Renner, and W. Tittel, Experimental Bound on the Maximum Predictive Power of Physical Theories, Phys. Rev. Lett. 109, 020402 (2012).

[48] B. G. Christensen, Y.-C. Liang, N. Brunner, N. Gisin, and P. G. Kwiat, Exploring the Limits of Quantum Nonlocality with Entangled Photons, Phys. Rev. X 5, 041052 (2015).

[49] M. Navascués, S. Pironio, and A. Acín, Bounding the Set of Quantum Correlations, Physical Review Letters 98, 010401 (2007), quant-ph/​0607119.

[50] M. Navascués, S. Pironio, and A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New Journal of Physics 10, 073013 (2008), 0803.4290 [quant-ph].

[51] H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Nonlocality and communication complexity, Rev. Mod. Phys. 82, 665–698 (2010).

[52] M. S. Leifer and Robert W. Spekkens, Towards a formulation of quantum theory as a causally neutral theory of bayesian inference, Phys. Rev. A 88, 052130 (2013).

[53] J. Henson, R. Lal, and M. F. Pusey, Theory-independent limits on correlations from generalized bayesian networks, New J. Phys. 16, 113043 (2014).

[54] T. Fritz, Beyond bell's theorem ii: Scenarios with arbitrary causal structure, Communications in Mathematical Physics 341, 391–434 (2016).

[55] R. Chaves, C. Majenz, and D. Gross, Information–theoretic implications of quantum causal structures, Nat. Commun. 6, 5766 (2015b).

[56] J. Pienaar and C. Brukner, A graph-separation theorem for quantum causal models, New J. Phys. 17, 073020 (2015).

[57] K. Ried, M. Agnew, L. Vermeyden, D. Janzing, R. W. Spekkens, and K. J. Resch, A quantum advantage for inferring causal structure, Nat Phys 11, 414–420 (2015).

[58] Fabio Costa and Sally Shrapnel, Quantum causal modelling, New Journal of Physics 18, 063032 (2016).

[59] T. Christof and A. Löbel, PORTA – POlyhedron Representation Transformation Algorithm, (2009).

[60] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, Nonlocal correlations as an information-theoretic resource, Phys. Rev. A 71, 022101 (2005).

[61] S. Pironio, J.-D. Bancal, and V. Scarani, Extremal correlations of the tripartite no-signaling polytope, J. Phys. A: Math. Theo. 44, 065303 (2011).

Cited by

[1] Debasis Mondal, Jaskaran Singh, and Dagomir Kaszlikowski, "Quantum instrumentality uniquely singles out nonlocal advantage of quantum coherence", Physical Review A 104 4, 042407 (2021).

[2] M. G. M. Moreno, Samuraí Brito, Ranieri V. Nery, and Rafael Chaves, "Device-independent secret sharing and a stronger form of Bell nonlocality", Physical Review A 101 5, 052339 (2020).

[3] Sumit Nandi, Debashis Saha, Dipankar Home, and A. S. Majumdar, "Wigner-approach-enabled detection of multipartite nonlocality using all different bipartitions", Physical Review A 106 6, 062203 (2022).

[4] Joseph Ho, George Moreno, Samuraí Brito, Francesco Graffitti, Christopher L. Morrison, Ranieri Nery, Alexander Pickston, Massimiliano Proietti, Rafael Rabelo, Alessandro Fedrizzi, and Rafael Chaves, "Entanglement-based quantum communication complexity beyond Bell nonlocality", npj Quantum Information 8 1, 13 (2022).

[5] R. Salazar, M. Kamoń, K. Horodecki, D. Goyeneche, D. Saha, R. Ramanathan, and P. Horodecki, "No-go theorem for device-independent security in relativistic causal theories", Physical Review Research 3 3, 033146 (2021).

[6] Ciarán M. Lee and Matty J. Hoban, "Towards Device-Independent Information Processing on General Quantum Networks", Physical Review Letters 120 2, 020504 (2018).

[7] Elie Wolfe, David Schmid, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes", Quantum 4, 280 (2020).

[8] Sayan Gangopadhyay, Tiejun Wang, Atefeh Mashatan, and Shohini Ghose, "Controlled quantum teleportation in the presence of an adversary", Physical Review A 106 5, 052433 (2022).

[9] Joseph Bowles, Flavien Hirsch, and Daniel Cavalcanti, "Single-copy activation of Bell nonlocality via broadcasting of quantum states", Quantum 5, 499 (2021).

[10] Valentin Gebhart, "Creating nonlocality using geometric phases between partially distinguishable photons", Physical Review A 107 6, 062420 (2023).

[11] Paul Skrzypczyk, "Causality: relaxing before exploring", Quantum Views 1, 3 (2017).

[12] Márcio M. Cunha, Alejandro Fonseca, and Edilberto O. Silva, "Tripartite Entanglement: Foundations and Applications", Universe 5 10, 209 (2019).

[13] D. M. Fucci and R. M. Angelo, "Tripartite realism-based quantum nonlocality", Physical Review A 100 6, 062101 (2019).

[14] Ming-Xing Luo, "Nonsignaling causal hierarchy of general multisource networks", Physical Review A 101 6, 062317 (2020).

[15] Amit Kundu, Mostak Kamal Molla, Indrani Chattopadhyay, and Debasis Sarkar, "Maximal qubit violation of n -local inequalities in a quantum network", Physical Review A 102 5, 052222 (2020).

[16] Ciarán M. Lee, "Device-independent certification of non-classical joint measurements via causal models", npj Quantum Information 5 1, 34 (2019).

[17] Alejandro Pozas-Kerstjens, Rafael Rabelo, Łukasz Rudnicki, Rafael Chaves, Daniel Cavalcanti, Miguel Navascués, and Antonio Acín, "Bounding the Sets of Classical and Quantum Correlations in Networks", Physical Review Letters 123 14, 140503 (2019).

[18] Rafael Chaves, George Moreno, Emanuele Polino, Davide Poderini, Iris Agresti, Alessia Suprano, Mariana R. Barros, Gonzalo Carvacho, Elie Wolfe, Askery Canabarro, Robert W. Spekkens, and Fabio Sciarrino, "Causal Networks and Freedom of Choice in Bell’s Theorem", PRX Quantum 2 4, 040323 (2021).

[19] Valentin Gebhart and Augusto Smerzi, "Extending the fair sampling assumption using causal diagrams", Quantum 7, 897 (2023).

[20] Martin Ringbauer and Rafael Chaves, "Probing the non-classicality of temporal correlations", Quantum 1, 35 (2017).

[21] Giancarlo Camilo, Gabriel T. Landi, and Sebas Eliëns, "Strong subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states", Physical Review B 99 4, 045155 (2019).

[22] 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).

[23] Miguel Navascués and Elie Wolfe, "The Inflation Technique Completely Solves the Causal Compatibility Problem", Journal of Causal Inference 8 1, 70 (2020).

[24] R. V. Nery, M. M. Taddei, R. Chaves, and L. Aolita, "Quantum Steering Beyond Instrumental Causal Networks", Physical Review Letters 120 14, 140408 (2018).

[25] Alessia Suprano, Davide Poderini, Emanuele Polino, Iris Agresti, Gonzalo Carvacho, Askery Canabarro, Elie Wolfe, Rafael Chaves, and Fabio Sciarrino, "Experimental Genuine Tripartite Nonlocality in a Quantum Triangle Network", PRX Quantum 3 3, 030342 (2022).

[26] Erika Bene and Tamás Vértesi, "Measurement incompatibility does not give rise to Bell violation in general", New Journal of Physics 20 1, 013021 (2018).

[27] Austin K. Daniel, Yingyue Zhu, C. Huerta Alderete, Vikas Buchemmavari, Alaina M. Green, Nhung H. Nguyen, Tyler G. Thurtell, Andrew Zhao, Norbert M. Linke, and Akimasa Miyake, "Quantum computational advantage attested by nonlocal games with the cyclic cluster state", Physical Review Research 4 3, 033068 (2022).

[28] Francesco Andreoli, Gonzalo Carvacho, Luca Santodonato, Rafael Chaves, and Fabio Sciarrino, "Maximal qubit violation of n-locality inequalities in a star-shaped quantum network", New Journal of Physics 19 11, 113020 (2017).

[29] Paweł Horodecki and Ravishankar Ramanathan, "The relativistic causality versus no-signaling paradigm for multi-party correlations", Nature Communications 10 1, 1701 (2019).

[30] Uta Isabella Meyer, Frédéric Grosshans, and Damian Markham, "Inflated graph states refuting communication-assisted local-hidden-variable models", Physical Review A 108 1, 012402 (2023).

[31] S. G. A. Brito, B. Amaral, and R. Chaves, "Quantifying Bell nonlocality with the trace distance", Physical Review A 97 2, 022111 (2018).

[32] S. G. A. Brito, M. G. M. Moreno, A. Rai, and R. Chaves, "Nonlocality distillation and quantum voids", Physical Review A 100 1, 012102 (2019).

[33] Giancarlo Camilo, Gabriel T. Landi, and Sebas Eliëns, "On the Strong Subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states", arXiv:1810.07070, (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 04:09:39) and SAO/NASA ADS (last updated successfully 2024-04-19 04:09:41). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “Causal hierarchy of multipartite Bell nonlocality

  1. Pingback: Perspective in Quantum Views by Paul Skrzypczyk "Causality: relaxing before exploring"