Unlimited non-causal correlations and their relation to non-locality

Ämin Baumeler1,2,3, Amin Shiraz Gilani1,4, and Jibran Rashid5

1Institute for Quantum Optics and Quantum Information (IQOQI-Vienna), Austrian Academy of Sciences, 1090 Vienna, Austria
2Faculty of Physics, University of Vienna, 1090 Vienna, Austria
3Facoltà indipendente di Gandria, 6978 Gandria, Switzerland
4Department of Computer Science, University of Maryland, College Park, Maryland 20742, USA
5School of Mathematics and Computer Science, Institute of Business Administration, Karachi, Pakistan

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Abstract

Non-causal correlations certify the lack of a definite causal order among localized space-time regions. In stark contrast to scenarios where a single region influences its own causal past, some processes that distribute non-causal correlations satisfy a series of natural desiderata: logical consistency, linear and reversible dynamics, and computational tameness. Here, we present such processes among arbitrary many regions where each region influences every other but itself, and show that the above desiderata are altogether $\textit{insufficient to limit the amount of "acausality"}$ of non-causal correlations. This leaves open the identification of a principle that forbids non-causal correlations. Our results exhibit $\textit{qualitative and quantitative parallels}$ with the non-local correlations due to Ardehali and Svetlichny.

It is generally assumed that events $A,B,C,\dots$ in space-time cannot influence each other in a cyclic way, i.e., if $A$ influences $B$ ($A$ is in the causal past of $B$), then $B$ cannot influence $A$. This logic, however, breaks down if one assumes quantum theory to hold locally without further assumptions on the causal order among the events: There exist processes where each party (event) influences every other but itself. In previous studies it has been shown that such processes comply with various desiderata: They are logically consistent without restricting free choice, they can be embedded in reversible dynamics, and they are computationally tame. This work reports that causal order is also not enforced by increasing the number of parties. Even more so, this work presents processes among any number of parties where causal order is violated by a larger degree the more parties are considered. Remarkably, these non-causal processes are constructed out of the Ardheali-Svetlichny functions that display the same qualitative and quantitative features for non-local correlations.

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► References

[1] Judea Pearl. ``Causality''. Cambridge University Press. Cambridge (2009).
https:/​/​doi.org/​10.1017/​CBO9780511803161

[2] Lucien Hardy. ``Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity'' (2005). arXiv:gr-qc/​0509120.
arXiv:gr-qc/0509120

[3] Giulio Chiribella, Giacomo Mauro D'Ariano, Paolo Perinotti, and Benoit Valiron. ``Quantum computations without definite causal structure''. Physical Review A 88, 022318 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.022318

[4] Timoteo Colnaghi, Giacomo Mauro D'Ariano, Stefano Facchini, and Paolo Perinotti. ``Quantum computation with programmable connections between gates''. Physics Letters A 376, 2940–2943 (2012).
https:/​/​doi.org/​10.1016/​j.physleta.2012.08.028

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

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

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

[8] Tom Purves and Anthony J Short. ``Quantum Theory Cannot Violate a Causal Inequality''. Physical Review Letters 127, 110402 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.127.110402

[9] Julian Wechs, Hippolyte Dourdent, Alastair A Abbott, and Cyril Branciard. ``Quantum Circuits with Classical Versus Quantum Control of Causal Order''. PRX Quantum 2, 030335 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.030335

[10] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. ``Quantum correlations with no causal order''. Nature Communications 3, 1092 (2012).
https:/​/​doi.org/​10.1038/​ncomms2076

[11] Cyril Branciard, Mateus Araújo, Adrien Feix, Fabio Costa, and Časlav Brukner. ``The simplest causal inequalities and their violation''. New Journal of Physics 18, 013008 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013008

[12] John S Bell. ``On the Einstein Podolsky Rosen paradox''. Physics Physique Fizika 1, 195–200 (1964).
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[13] John Archibald Wheeler. ``World as system self-synthesized by quantum networking''. IBM Journal of Research and Development 32, 4–15 (1988).
https:/​/​doi.org/​10.1147/​rd.321.0004

[14] Stephen W Hawking. ``Chronology protection conjecture''. Physical Review D 46, 603–611 (1992).
https:/​/​doi.org/​10.1103/​PhysRevD.46.603

[15] Albert Einstein. ``Die formale Grundlage der allgemeinen Relativitätstheorie''. In Georg Reimer, editor, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften. Volume Zweiter Halbband, pages 1030–1085. Verlag der Königlichen Akademie der Wissenschaften, Berlin (1914).
https:/​/​doi.org/​10.1002/​3527608958.ch2

[16] Kornel Lanczos. ``Über eine stationäre Kosmologie im Sinne der Einsteinschen Gravitationstheorie''. Zeitschrift für Physik 21, 73–110 (1924).
https:/​/​doi.org/​10.1007/​BF01328251

[17] Kurt Gödel. ``An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation''. Reviews of Modern Physics 21, 447–450 (1949).
https:/​/​doi.org/​10.1103/​RevModPhys.21.447

[18] Ämin Baumeler and Stefan Wolf. ``Perfect signaling among three parties violating predefined causal order''. In 2014 IEEE International Symposium on Information Theory. Pages 526–530. Piscataway (2014). IEEE.
https:/​/​doi.org/​10.1109/​ISIT.2014.6874888

[19] Ämin Baumeler, Adrien Feix, and Stefan Wolf. ``Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios''. Physical Review A 90, 042106 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.042106

[20] Alastair A Abbott, Christina Giarmatzi, Fabio Costa, and Cyril Branciard. ``Multipartite causal correlations: Polytopes and inequalities''. Physical Review A 94, 032131 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032131

[21] Ämin Baumeler and Stefan Wolf. ``The space of logically consistent classical processes without causal order''. New Journal of Physics 18, 013036 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013036

[22] Alastair A Abbott, Julian Wechs, Fabio Costa, and Cyril Branciard. ``Genuinely multipartite noncausality''. Quantum 1, 39 (2017).
https:/​/​doi.org/​10.22331/​q-2017-12-14-39

[23] Christina Giarmatzi. ``Rethinking causality in quantum mechanics''. Springer Theses. Springer. Cham (2019).
https:/​/​doi.org/​10.1007/​978-3-030-31930-4

[24] Juan Gu, Longsuo Li, and Zhi Yin. ``Two Multi-Setting Causal Inequalities and Their Violations''. International Journal of Theoretical Physics 59, 97–107 (2020).
https:/​/​doi.org/​10.1007/​s10773-019-04291-5

[25] Ämin Baumeler and Stefan Wolf. ``Device-independent test of causal order and relations to fixed-points''. New Journal of Physics 18, 035014 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​3/​035014

[26] Mateus Araújo, Adrien Feix, Miguel Navascués, and Časlav 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

[27] Ämin Baumeler. ``Causal Loops: Logically Consistent Correlations, Time Travel, and Computation''. PhD thesis. Università della Svizzera italiana. (2017). url: cqi.inf.usi.ch/​publications/​these_amin.pdf.
https:/​/​cqi.inf.usi.ch/​publications/​these_amin.pdf

[28] Mateus Araújo, Philippe Allard Guérin, and Ämin Baumeler. ``Quantum computation with indefinite causal structures''. Physical Review A 96, 052315 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.052315

[29] Ämin Baumeler and Stefan Wolf. ``Computational tameness of classical non-causal models''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 20170698 (2018).
https:/​/​doi.org/​10.1098/​rspa.2017.0698

[30] Martin J Renner and Časlav Brukner. ``Reassessing the computational advantage of quantum-controlled ordering of gates''. Physical Review Research 3, 043012 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.043012

[31] David Deutsch. ``Quantum mechanics near closed timelike lines''. Physical Review D 44, 3197–3217 (1991).
https:/​/​doi.org/​10.1103/​PhysRevD.44.3197

[32] James B Hartle. ``Unitarity and causality in generalized quantum mechanics for nonchronal spacetimes''. Physical Review D 49, 6543–6555 (1994).
https:/​/​doi.org/​10.1103/​PhysRevD.49.6543

[33] George Svetlichny. ``Effective Quantum Time Travel'' (2009). arXiv:0902.4898.
arXiv:0902.4898

[34] George Svetlichny. ``Time Travel: Deutsch vs. Teleportation''. International Journal of Theoretical Physics 50, 3903–3914 (2011).
https:/​/​doi.org/​10.1007/​s10773-011-0973-x

[35] Seth Lloyd, Lorenzo Maccone, Raul Garcia-Patron, Vittorio Giovannetti, and Yutaka Shikano. ``Quantum mechanics of time travel through post-selected teleportation''. Physical Review D 84, 025007 (2011).
https:/​/​doi.org/​10.1103/​PhysRevD.84.025007

[36] John-Mark A Allen. ``Treating time travel quantum mechanically''. Physical Review A 90, 042107 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.042107

[37] D Ahn, C R Myers, Timothy C Ralph, and R B Mann. ``Quantum-state cloning in the presence of a closed timelike curve''. Physical Review A 88, 022332 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.022332

[38] Todd A Brun, Mark M Wilde, and Andreas Winter. ``Quantum State Cloning Using Deutschian Closed Timelike Curves''. Physical Review Letters 111, 190401 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.190401

[39] Scott Aaronson. ``Quantum computing, postselection, and probabilistic polynomial-time''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, 3473–3482 (2005).
https:/​/​doi.org/​10.1098/​rspa.2005.1546

[40] Scott Aaronson. ``Guest Column: NP-complete problems and physical reality''. SIGACT News 36, 30 (2005).
https:/​/​doi.org/​10.1145/​1052796.1052804

[41] Scott Aaronson and John Watrous. ``Closed timelike curves make quantum and classical computing equivalent''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 631–647 (2009).
https:/​/​doi.org/​10.1098/​rspa.2008.0350

[42] Scott Aaronson, Mohammad Bavarian, and Giulio Gueltrini. ``Computability Theory of Closed Timelike Curves'' (2016). arXiv:1609.05507.
arXiv:1609.05507

[43] George Svetlichny. ``Distinguishing three-body from two-body nonseparability by a Bell-type inequality''. Physical Review D 35, 3066–3069 (1987).
https:/​/​doi.org/​10.1103/​PhysRevD.35.3066

[44] Daniel Collins, Nicolas Gisin, Sandu Popescu, David Roberts, and Valerio Scarani. ``Bell-type inequalities to detect true $\mathit{n}$-body nonseparability''. Physical Review Letters 88, 170405 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.170405

[45] Michael Seevinck and George Svetlichny. ``Bell-type inequalities for partial separability in $\mathit{N}$-particle systems and quantum mechanical violations''. Physical Review Letters 89, 060401 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.060401

[46] Mohammad Ardehali. ``Bell inequalities with a magnitude of violation that grows exponentially with the number of particles''. Physical Review A 46, 5375–5378 (1992).
https:/​/​doi.org/​10.1103/​PhysRevA.46.5375

[47] John F Clauser, Michael A Horne, Abner Shimony, and Richard A Holt. ``Proposed Experiment to Test Local Hidden-Variable Theories''. Physical Review Letters 23, 880–884 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[48] Rodrigo Gallego, Lars Erik Würflinger, Antonio Acín, and Miguel Navascués. ``Operational Framework for Nonlocality''. Physical Review Letters 109, 070401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.070401

[49] Mateus Araújo and Adrien Feix. private communication (2014). The process was communicated to Baumeler before it was found by inspecting the extremal points of the non-causal polytope characterized in Baumeler and Wolf Baumeler2016 (see also Ref. [56] in the latter article).

[50] Ämin Baumeler, Fabio Costa, Timothy C Ralph, Stefan Wolf, and Magdalena Zych. ``Reversible time travel with freedom of choice''. Classical and Quantum Gravity 36, 224002 (2019).
https:/​/​doi.org/​10.1088/​1361-6382/​ab4973

[51] Ämin Baumeler and Eleftherios Tselentis. ``Equivalence of grandfather and information antinomy under intervention''. In Benoı̂t Valiron, Shane Mansfield, Pablo Arrighi, and Prakash Panangaden, editors, Proceedings 17th International Conference on Quantum Physics and Logic. Volume 340, pages 1–12. Electronic Proceedings in Theoretical Computer Science (2021).
https:/​/​doi.org/​10.4204/​EPTCS.340.1

[52] Man-Duen Choi. ``Completely positive linear maps on complex matrices''. Linear Algebra and its Applications 10, 285–290 (1975).
https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0

[53] Andrzej Jamiołkowski. ``Linear transformations which preserve trace and positive semidefiniteness of operators''. Reports on Mathematical Physics 3, 275–278 (1972).
https:/​/​doi.org/​10.1016/​0034-4877(72)90011-0

[54] Howard Barnum, Salman Beigi, Sergio Boixo, Matthew B Elliott, and Stephanie Wehner. ``Local Quantum Measurement and No-Signaling Imply Quantum Correlations''. Physical Review Letters 104, 140401 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.140401

[55] Antonio Acín, Remigiusz Augusiak, Daniel Cavalcanti, Christopher Hadley, Jarosław K Korbicz, Maciej Lewenstein, Lluís Masanes, and Marco Piani. ``Unified Framework for Correlations in Terms of Local Quantum Observables''. Physical Review Letters 104, 140404 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.140404

[56] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov. ``Cyclic quantum causal models''. Nature Communications 12, 885 (2021).
https:/​/​doi.org/​10.1038/​s41467-020-20456-x

[57] Kip S Thorne. ``Do the Laws of Physics Permit Closed Timelike Curves?''. Annals of the New York Academy of Sciences 631, 182–193 (1991).
https:/​/​doi.org/​10.1111/​j.1749-6632.1991.tb52642.x

[58] Kip S Thorne. ``Black Holes & Time Warps: Einstein's Outrageous Legacy''. W.W. Norton & Company. New York (1995). url: wwnorton.com/​books/​9780393312768.
https:/​/​wwnorton.com/​books/​9780393312768

[59] George Svetlichny. ``Nonlinear Quantum Mechanics at the Planck Scale''. International Journal of Theoretical Physics 44, 2051–2058 (2005).
https:/​/​doi.org/​10.1007/​s10773-005-8983-1

[60] John Archibald Wheeler and Kenneth Ford. ``Geons, Black Holes, and Quantum Foam: A Life in Physics''. W.W. Norton & Company. New York (1998).

[61] Germain Tobar and Fabio Costa. ``Reversible dynamics with closed time-like curves and freedom of choice''. Classical and Quantum Gravity 37, 205011 (2020).
https:/​/​doi.org/​10.1088/​1361-6382/​aba4bc

[62] N David Mermin. ``Extreme quantum entanglement in a superposition of macroscopically distinct states''. Physical Review Letters 65, 1838–1840 (1990).
https:/​/​doi.org/​10.1103/​PhysRevLett.65.1838

[63] Andris Ambainis, Dmitry Kravchenko, Nikolajs Nahimovs, and Alexander Rivosh. ``Nonlocal Quantum XOR Games for Large Number of Players''. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Volume 6108 LNCS, pages 72–83. Springer, Berlin, Heidelberg (2010).
https:/​/​doi.org/​10.1007/​978-3-642-13562-0_8

[64] Andris Ambainis, Dmitry Kravchenko, Nikolay Nahimov, Alexander Rivosh, and Madars Virza. ``On symmetric nonlocal games''. Theoretical Computer Science 494, 36–48 (2013).
https:/​/​doi.org/​10.1016/​j.tcs.2013.03.011

[65] Boris S Cirel'son. ``Quantum generalizations of Bell's inequality''. Letters in Mathematical Physics 4, 93–100 (1980).
https:/​/​doi.org/​10.1007/​BF00417500

[66] Jean-Daniel Bancal, Jonathan Barrett, Nicolas Gisin, and Stefano Pironio. ``Definitions of multipartite nonlocality''. Physical Review A 88, 014102 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.014102

[67] Mafalda L Almeida, Jean-Daniel Bancal, Nicolas Brunner, Antonio Acín, Nicolas Gisin, and Stefano Pironio. ``Guess Your Neighbor's Input: A Multipartite Nonlocal Game with No Quantum Advantage''. Physical Review Letters 104, 230404 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.230404

[68] Allen Stairs. ``Quantum Logic, Realism, and Value Definiteness''. Philosophy of Science 50, 578–602 (1983).
https:/​/​doi.org/​10.1086/​289140

[69] Peter Heywood and Michael L G Redhead. ``Nonlocality and the Kochen-Specker paradox''. Foundations of Physics 13, 481–499 (1983).
https:/​/​doi.org/​10.1007/​BF00729511

[70] Renato Renner and Stefan Wolf. ``Quantum pseudo-telepathy and the Kochen-Specker theorem''. In International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings. Pages 322–322. IEEE (2004).
https:/​/​doi.org/​10.1109/​ISIT.2004.1365359

[71] Renato Renner and Stefan Wolf. ``Ernst Specker and the Hidden Variables''. Elemente der Mathematik 67, 122–133 (2012).
https:/​/​doi.org/​10.4171/​EM/​201

[72] Gilles Brassard, Anne Broadbent, and Alain Tapp. ``Quantum Pseudo-Telepathy''. Foundations of Physics 35, 1877–1907 (2005).
https:/​/​doi.org/​10.1007/​s10701-005-7353-4

[73] Adán Cabello. ``Converting Contextuality into Nonlocality''. Physical Review Letters 127, 070401 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.127.070401

[74] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson. ``Quantum Contextuality'' (2021). arXiv:2102.13036.
arXiv:2102.13036

[75] Ognyan Oreshkov. ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics''. Quantum 3, 206 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-02-206

[76] Julian Wechs, Cyril Branciard, and Ognyan Oreshkov. ``Existence of processes violating causal inequalities on time-delocalised subsystems'' (2022). arXiv:2201.11832.
arXiv:2201.11832

[77] Edward Fredkin and Tommaso Toffoli. ``Conservative logic''. International Journal of Theoretical Physics 21, 219–253 (1982).
https:/​/​doi.org/​10.1007/​BF01857727

[78] Michael S Morris, Kip S Thorne, and Ulvi Yurtsever. ``Wormholes, Time Machines, and the Weak Energy Condition''. Physical Review Letters 61, 1446–1449 (1988).
https:/​/​doi.org/​10.1103/​PhysRevLett.61.1446

[79] Igor Dmitriyevich Novikov. ``An analysis of the operation of a time machine''. Journal of Experimental and Theoretical Physics 68, 439 (1989). url: http:/​/​www.jetp.ras.ru/​cgi-bin/​e/​index/​e/​68/​3/​p439?a=list.
http:/​/​www.jetp.ras.ru/​cgi-bin/​e/​index/​e/​68/​3/​p439?a=list

[80] Valery P Frolov and Igor D Novikov. ``Physical effects in wormholes and time machines''. Physical Review D 42, 1057–1065 (1990).
https:/​/​doi.org/​10.1103/​PhysRevD.42.1057

[81] Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner. ``Indefinite causal structures for continuous-variable systems''. New Journal of Physics 18, 113026 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​11/​113026

Cited by

[1] Ravi Kunjwal and Ämin Baumeler, "How quantum nonlocality without entanglement witnesses classical processes without causal order", arXiv:2202.00440.

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