# Genuinely multipartite noncausality

1University Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
2Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

### Abstract

The study of correlations with no definite causal order has revealed a rich structure emerging when more than two parties are involved. This motivates the consideration of multipartite "noncausal" correlations that cannot be realised even if noncausal resources are made available to a smaller number of parties. Here we formalise this notion: genuinely N-partite noncausal correlations are those that cannot be produced by grouping N parties into two or more subsets, where a causal order between the subsets exists. We prove that such correlations can be characterised as lying outside a polytope, whose vertices correspond to deterministic strategies and whose facets define what we call "2-causal" inequalities. We show that genuinely multipartite noncausal correlations arise within the process matrix formalism, where quantum mechanics holds locally but no global causal structure is assumed, although for some inequalities no violation was found. We further introduce two refined definitions that allow one to quantify, in different ways, to what extent noncausal correlations correspond to a genuinely multipartite resource.

Causality imposes restrictions on how different parties can communicate: a message sent in the future cannot reach a receiver in the past. It is an intriguing possibility, suggested by both quantum mechanics and general relativity, that nature might not enforce such tight restrictions. With access to the right “noncausal” resources, a group of players could win certain communication games with better odds than with any "causal" resource forcing them to play in a well-defined causal order. Here we consider new types of games, involving more than two players, that cannot be won even with noncausal resources, as long as one group of players is causally ordered with respect to another group. Winning such games would thus certify a "genuinely multipartite" noncausal resource that cannot be reduced to a noncausal resource shared by only some of the players. We show how to mathematically characterise such resources, which will be of help in the quest to understand whether we can one day experimentally observe — and even exploit — noncausal resources.

### ► References

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https:/​/​doi.org/​10.1038/​nphys2930

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https:/​/​doi.org/​10.1038/​ncomms2076

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https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013008

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https:/​/​doi.org/​10.1103/​PhysRevA.90.042106

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https:/​/​doi.org/​10.22331/​q-2017-04-26-10

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https:/​/​doi.org/​10.1088/​1367-2630/​18/​8/​083040

[14] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​10/​102001

[15] L. Hardy, Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure, J. Phys. A: Math. Gen. 40, 3081 (2007).
https:/​/​doi.org/​10.1088/​1751-8113/​40/​12/​S12

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arXiv:1703.00779

[17] M. Araújo, P. A. Guérin, and Ä. Baumeler, Quantum computation with indefinite causal structures, Phys. Rev. A 96, 052315 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.052315

[18] N. Miklin, A. A. Abbott, C. Branciard, R. Chaves, and C. Budroni, The entropic approach to causal correlations, New J. Phys. 19, 113041 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa8f9f

[19] G. Svetlichny, Distinguishing three-body from two-body nonseparability by a Bell-type inequality, Phys. Rev. D 35, 3066 (1987).
https:/​/​doi.org/​10.1103/​PhysRevD.35.3066

[20] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.060401

[21] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.170405

[22] R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Operational framework for nonlocality, Phys. Rev. Lett. 109, 070401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.070401

[23] J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, Definitions of multipartite nonlocality, Phys. Rev. A 88, 014102 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.014102

[24] See Supplementary Material in the arXiv ancillary files' for the full list of 2-causal inequalities in the tripartite lazy scenario and further analysis.
https:/​/​arxiv.org/​src/​1708.07663v2/​anc/​Supplementary_Material.cdf

[1] H. Reichenbach, The direction of time (University of California Press, Berkeley, 1956).

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

[3] Č. Brukner, Quantum causality, Nat. Phys. 10, 259–263 (2014).
https:/​/​doi.org/​10.1038/​nphys2930

[4] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012).
https:/​/​doi.org/​10.1038/​ncomms2076

[5] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, The simplest causal inequalities and their violation, New J. Phys. 18, 013008 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013008

[6] Ä. Baumeler and S. Wolf, Perfect signaling among three parties violating predefined causal order, in 2014 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, NJ, 2014) pp. 526–530.
https:/​/​doi.org/​10.1109/​ISIT.2014.6874888

[7] Ä. Baumeler, A. Feix, and S. Wolf, Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios, Phys. Rev. A 90, 042106 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.042106

[8] O. Oreshkov and C. Giarmatzi, Causal and causally separable processes, New J. Phys. 18, 093020 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​9/​093020

[9] A. A. Abbott, C. Giarmatzi, F. Costa, and C. Branciard, Multipartite causal correlations: Polytopes and inequalities, Phys. Rev. A 94, 032131 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032131

[10] L. Hardy, Probability theories with dynamic causal structure: a new framework for quantum gravity, (2005), arXiv:gr-qc/​0509120.
arXiv:gr-qc/0509120

[11] K. Fukuda, cdd, v0.94g, (2012), https:/​/​www.inf.ethz.ch/​personal/​fukudak/​cdd_home/​.
https:/​/​www.inf.ethz.ch/​personal/​fukudak/​cdd_home/​

[12] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, A purification postulate for quantum mechanics with indefinite causal order, Quantum 1, 10 (2017).
https:/​/​doi.org/​10.22331/​q-2017-04-26-10

[13] A. Feix, M. Araújo, and Č. Brukner, Causally nonseparable processes admitting a causal model, New J. Phys. 18, 083040 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​8/​083040

[14] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​10/​102001

[15] L. Hardy, Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure, J. Phys. A: Math. Gen. 40, 3081 (2007).
https:/​/​doi.org/​10.1088/​1751-8113/​40/​12/​S12

[16] Ä. Baumeler, F. Costa, T. C. Ralph, S. Wolf, and M. Zych, Reversible time travel with freedom of choice, (2017), arXiv:1703.00779 [gr-qc].
arXiv:1703.00779

[17] M. Araújo, P. A. Guérin, and Ä. Baumeler, Quantum computation with indefinite causal structures, Phys. Rev. A 96, 052315 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.052315

[18] N. Miklin, A. A. Abbott, C. Branciard, R. Chaves, and C. Budroni, The entropic approach to causal correlations, New J. Phys. 19, 113041 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa8f9f

[19] G. Svetlichny, Distinguishing three-body from two-body nonseparability by a Bell-type inequality, Phys. Rev. D 35, 3066 (1987).
https:/​/​doi.org/​10.1103/​PhysRevD.35.3066

[20] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.060401

[21] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.170405

[22] R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Operational framework for nonlocality, Phys. Rev. Lett. 109, 070401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.070401

[23] J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, Definitions of multipartite nonlocality, Phys. Rev. A 88, 014102 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.014102

[24] See Supplementary Material in the arXiv ancillary files' for the full list of 2-causal inequalities in the tripartite lazy scenario and further analysis.
https:/​/​arxiv.org/​src/​1708.07663v2/​anc/​Supplementary_Material.cdf

### Cited by

[1] Jessica Bavaresco, Mateus Araújo, Časlav Brukner, and Marco Túlio Quintino, "Semi-device-independent certification of indefinite causal order", arXiv:1903.10526, Quantum 3, 176 (2019).

[2] 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).

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

[4] Kaumudibikash Goswami and Fabio Costa, "Classical communication through quantum causal structures", Physical Review A 103 4, 042606 (2021).

[5] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Cyclic quantum causal models", Nature Communications 12 1, 885 (2021).

[6] Germain Tobar and Fabio Costa, "Reversible dynamics with closed time-like curves and freedom of choice", Classical and Quantum Gravity 37 20, 205011 (2020).

[7] Juan Gu, Longsuo Li, and Zhi Yin, "Two Multi-Setting Causal Inequalities and Their Violations", International Journal of Theoretical Physics 59 1, 97 (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2021-07-31 17:41:01) and SAO/NASA ADS (last updated successfully 2021-07-31 17:41:02). The list may be incomplete as not all publishers provide suitable and complete citation data.