Analysing causal structures in generalised probabilistic theories
1Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, AT
2Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK
Published: | 2020-02-27, volume 4, page 236 |
Eprint: | arXiv:1812.04327v4 |
Doi: | https://doi.org/10.22331/q-2020-02-27-236 |
Citation: | Quantum 4, 236 (2020). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Causal structures give us a way to understand the origin of observed correlations. These were developed for classical scenarios, but quantum mechanical experiments necessitate their generalisation. Here we study causal structures in a broad range of theories, which include both quantum and classical theory as special cases. We propose a method for analysing differences between such theories based on the so-called measurement entropy. We apply this method to several causal structures, deriving new relations that separate classical, quantum and more general theories within these causal structures. The constraints we derive for the most general theories are in a sense minimal requirements of any causal explanation in these scenarios. In addition, we make several technical contributions that give insight for the entropic analysis of quantum causal structures. In particular, we prove that for any causal structure and for any generalised probabilistic theory, the set of achievable entropy vectors form a convex cone.

Popular summary
In this work we consider modified notions of causation within a broad class of theories (generalised probabilistic theories) that include classical and quantum theory as special cases but also theories whose correlations are more 'spooky' than those in quantum mechanics. We show how to derive causal constraints within such theories, and apply our technique to a variety of causal structures. By studying causation in the most general post-quantum theory, we derive constraints that must hold for any causal explanations of correlations, and so can be understood as minimal requirements of causation itself. Furthermore, by comparing general constraints to those in quantum theory provides a way to investigate what is special about quantum mechanics.
► BibTeX data
► References
[1] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1 (3): 195–200, 1964. ISSN 01923188. 10.1002/prop.19800281202.
https://doi.org/10.1002/prop.19800281202
[2] 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 Journal of Physics, 17 (3): 033002, 2015. ISSN 1367-2630. 10.1088/1367-2630/17/3/033002.
https://doi.org/10.1088/1367-2630/17/3/033002
[3] M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Zukowski. Information causality as a physical principle. Nature, 461 (7267): 1101–1104, 2009. ISSN 0028-0836. 10.1038/nature08400.
https://doi.org/10.1038/nature08400
[4] S. W. Al-Safi and A. J. Short. Information causality from an entropic and a probabilistic perspective. Physical Review A, 84 (4): 042323, 2011. ISSN 1050-2947. 10.1103/PhysRevA.84.042323.
https://doi.org/10.1103/PhysRevA.84.042323
[5] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23 (15): 880–884, 1969. ISSN 0031-9007. 10.1103/PhysRevLett.23.880.
https://doi.org/10.1103/PhysRevLett.23.880
[6] D. M. Greenberger, M. A. Horne, and A. Zeilinger. Going beyond Bell's theorem. In M. Kafatos, editor, Bell's Theorem, Quantum Mechanics and Conceptions of the Universe, pages 69–72. Kluwer Academic, Dordrecht, The Netherlands, 1989. 10.1007/978-94-017-0849-4.
https://doi.org/10.1007/978-94-017-0849-4
[7] T. Fritz. Beyond Bell's theorem: correlation scenarios. New Journal of Physics, 14 (10): 103001, 2012. ISSN 1367-2630. 10.1088/1367-2630/14/10/103001.
https://doi.org/10.1088/1367-2630/14/10/103001
[8] T. Fritz and R. Chaves. Entropic inequalities and marginal problems. IEEE Transactions on Information Theory, 59 (2): 803–817, 2013. ISSN 0018-9448. 10.1109/TIT.2012.2222863.
https://doi.org/10.1109/TIT.2012.2222863
[9] R. Chaves, C. Majenz, and D. Gross. Information-theoretic implications of quantum causal structures. Nature communications, 6: 5766, 2015. ISSN 2041-1723. 10.1038/ncomms6766.
https://doi.org/10.1038/ncomms6766
[10] M. Weilenmann and R. Colbeck. Non-Shannon inequalities in the entropy vector approach to causal structures. Quantum, 2, 2018. 10.22331/q-2018-03-14-57.
https://doi.org/10.22331/q-2018-03-14-57
[11] E. Wolfe, R. W. Spekkens, and T. Fritz. The Inflation Technique for Causal Inference with Latent Variables. Journal of Causal Inference, 2016. 10.1515/jci-2017-0020.
https://doi.org/10.1515/jci-2017-0020
[12] B. S. Tsirelson. Quantum generalizations of Bell's inequality. Letters in Mathematical Physics, 4 (2): 93–100, 1980. ISSN 1573-0530. 10.1007/BF00417500.
https://doi.org/10.1007/BF00417500
[13] T. Van Himbeeck, J. Bohr Brask, S. Pironio, R. Ramanathan, A. B. Sainz, and E. Wolfe. Quantum violations in the Instrumental scenario and their relations to the Bell scenario. Quantum, 3: 186, 2019. ISSN 2521-327X. 10.22331/q-2019-09-16-186.
https://doi.org/10.22331/q-2019-09-16-186
[14] J. Henson, R. Lal, and M. F. Pusey. Theory-independent limits on correlations from generalized Bayesian networks. New Journal of Physics, 16 (11): 113043, 2014. ISSN 1367-2630. 10.1088/1367-2630/16/11/113043.
https://doi.org/10.1088/1367-2630/16/11/113043
[15] R. Chaves and C. Budroni. Entropic nonsignaling correlations. Physical Review Letters, 116 (24): 240501, 2016. ISSN 0031-9007. 10.1103/PhysRevLett.116.240501.
https://doi.org/10.1103/PhysRevLett.116.240501
[16] A. J. Short and J. Barrett. Strong nonlocality: a trade-off between states and measurements. New Journal of Physics, 12 (3): 033034, 2010. ISSN 1367-2630. 10.1088/1367-2630/12/3/033034.
https://doi.org/10.1088/1367-2630/12/3/033034
[17] H. Barnum, J. Barrett, L. O. Clark, M. Leifer, R. Spekkens, N. Stepanik, A. Wilce, and R. Wilke. Entropy and information causality in general probabilistic theories. New Journal of Physics, 12 (3): 033024, 2010. ISSN 1367-2630. 10.1088/1367-2630/12/3/033024.
https://doi.org/10.1088/1367-2630/12/3/033024
[18] S. L. Braunstein and C. M. Caves. Information-theoretic Bell inequalities. Physical Review Letters, 61 (6): 662–665, 1988. ISSN 0031-9007. 10.1103/PhysRevLett.61.662.
https://doi.org/10.1103/PhysRevLett.61.662
[19] B. Steudel and N. Ay. Information-theoretic inference of common ancestors. Entropy, 17 (4): 2304–2327, 2015. ISSN 1099-4300. 10.3390/e17042304.
https://doi.org/10.3390/e17042304
[20] R. Chaves and T. Fritz. Entropic approach to local realism and noncontextuality. Physical Review A, 85 (3): 032113, 2012. ISSN 1050-2947. 10.1103/PhysRevA.85.032113.
https://doi.org/10.1103/PhysRevA.85.032113
[21] J. Pienaar. Which causal structures might support a quantum–classical gap? New Journal of Physics, 19 (4): 043021, 2017. 10.1088/1367-2630/aa673e.
https://doi.org/10.1088/1367-2630/aa673e
[22] M. Weilenmann and R. Colbeck. Analysing causal structures with entropy. Proceedings of the Royal Society A, 473 (2207), 2017. 10.1098/rspa.2017.0483.
https://doi.org/10.1098/rspa.2017.0483
[23] A. J. Short and S. Wehner. Entropy in general physical theories. New Journal of Physics, 12 (3): 033023, 2010. ISSN 1367-2630. 10.1088/1367-2630/12/3/033023.
https://doi.org/10.1088/1367-2630/12/3/033023
[24] Z. Zhang and R. W. Yeung. A non-Shannon-type conditional inequality of information quantities. IEEE Transactions on Information Theory, 43 (6): 1982–1986, 1997. ISSN 00189448. 10.1109/18.641561.
https://doi.org/10.1109/18.641561
[25] J. Barrett. Information processing in generalized probabilistic theories. Physical Review A, 75 (3): 032304, 2007. ISSN 1050-2947. 10.1103/PhysRevA.75.032304.
https://doi.org/10.1103/PhysRevA.75.032304
[26] H. P. Williams. Fourier's method of linear programming and its dual. The American Mathematical Monthly, 93 (9): 681–695, 1986. 10.2307/2322281.
https://doi.org/10.2307/2322281
[27] D. Monniaux. Quantifier elimination by lazy model enumeration. In International Conference on Computer Aided Verification, pages 585–599. Springer, 2010. 10.1007/978-3-642-14295-6_51.
https://doi.org/10.1007/978-3-642-14295-6_51
[28] J. Cadney and N. Linden. Measurement entropy in generalized nonsignalling theory cannot detect bipartite nonlocality. Physical Review A, 86 (5): 052103, 2012. ISSN 1050-2947. 10.1103/PhysRevA.86.052103.
https://doi.org/10.1103/PhysRevA.86.052103
[29] M. Weilenmann and R. Colbeck. Inability of the entropy vector method to certify nonclassicality in linelike causal structures. Physical Review A, 94: 042112, 2016. 10.1103/PhysRevA.94.042112.
https://doi.org/10.1103/PhysRevA.94.042112
[30] N. Pippenger. The inequalities of quantum information theory. IEEE Transactions on Information Theory, 49 (4): 773–789, 2003. 10.1109/TIT.2003.809569.
https://doi.org/10.1109/TIT.2003.809569
[31] J. Pearl. On the testability of causal models with latent and instrumental variables. In Proceedings of the Eleventh conference on Uncertainty in artificial intelligence, pages 435–443. Morgan Kaufmann Publishers Inc., 1995. 10.5555/2074158.2074208.
https://doi.org/10.5555/2074158.2074208
[32] R. Chaves, G. Carvacho, I. Agresti, V. Di Giulio, L. Aolita, S. Giacomini, and F. Sciarrino. Quantum violation of an instrumental test. Nature Physics, 14 (3): 291, 2018. 10.1038/s41567-017-0008-5.
https://doi.org/10.1038/s41567-017-0008-5
[33] R. Chaves, L. Luft, T. Maciel, D. Gross, D. Janzing, and B. Schölkopf. Inferring latent structures via information inequalities. In Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, pages 112–121, Corvallis, Oregon, 2014. AUAI Press. 10.5555/3020751.3020764.
https://doi.org/10.5555/3020751.3020764
[34] M. Weilenmann. Quantum causal structure and quantum thermodynamics. PhD thesis, University of York, 2017. Also available as arXiv:1807.06345.
arXiv:1807.06345
[35] M. Navascues and E. Wolfe. The inflation technique solves completely the classical inference problem. e-print arXiv:1707.06476 , 2017.
https://doi.org/10.1515/jci-2018-0008
arXiv:1707.06476
[36] C. Branciard, N. Gisin, and S. Pironio. Characterizing the nonlocal correlations created via entanglement swapping. Physical Review Letters, 104 (17): 170401, 2010. ISSN 1079-7114. 10.1103/PhysRevLett.104.170401.
https://doi.org/10.1103/PhysRevLett.104.170401
[37] C. Branciard, D. Rosset, N. Gisin, and S. Pironio. Bilocal versus nonbilocal correlations in entanglement-swapping experiments. Physical Review A, 85 (3): 032119, 2012. ISSN 1050-2947. 10.1103/PhysRevA.85.032119.
https://doi.org/10.1103/PhysRevA.85.032119
[38] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70 (13): 1895–1899, 1993. ISSN 0031-9007. 10.1103/PhysRevLett.70.1895.
https://doi.org/10.1103/PhysRevLett.70.1895
[39] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller. Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Physical Review Letters, 81 (26): 5932–5935, 1998. ISSN 0031-9007. 10.1103/PhysRevLett.81.5932.
https://doi.org/10.1103/PhysRevLett.81.5932
[40] M. Zukowski, A. Zeilinger, M. A. Horne, and A. Ekert. "Event-ready-detectors" Bell experiment via entanglement swapping. Physical Review Letters, 71 (26): 4287–4290, 1993. ISSN 1079-7114. 10.1103/PhysRevLett.71.4287.
https://doi.org/10.1103/PhysRevLett.71.4287
[41] N. Gisin. The elegant joint quantum measurement and some conjectures about N-locality in the triangle and other configurations. e-print arXiv:1708.05556, 2017.
arXiv:1708.05556
[42] R. Chaves. Entropic inequalities as a necessary and sufficient condition to noncontextuality and locality. Physical Review A, 87 (2): 022102, 2013. ISSN 1050-2947. 10.1103/PhysRevA.87.022102.
https://doi.org/10.1103/PhysRevA.87.022102
[43] V. Vilasini and R. Colbeck. On the sufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure. e-print arXiv:1912.01031 , 2019.
https://doi.org/10.1103/PhysRevResearch.2.033096
arXiv:1912.01031
[44] O. Klein. Zur quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre. Zeitschrift für Physik, 72 (11): 767–775, 1931. ISSN 0044-3328. 10.1007/BF01341997.
https://doi.org/10.1007/BF01341997
Cited by
[1] Nicolas Gisin, Jean-Daniel Bancal, Yu Cai, Patrick Remy, Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Sandu Popescu, and Nicolas Brunner, "Constraints on nonlocality in networks from no-signaling and independence", Nature Communications 11 1, 2378 (2020).
[2] Michael J. Grabowecky, Christopher A. J. Pollack, Andrew R. Cameron, Robert W. Spekkens, and Kevin J. Resch, "Experimentally bounding deviations from quantum theory for a photonic three-level system using theory-agnostic tomography", Physical Review A 105 3, 032204 (2022).
[3] Paulo J Cavalcanti, John H Selby, Jamie Sikora, and Ana Belén Sainz, "Decomposing all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories", Journal of Physics A: Mathematical and Theoretical 55 40, 404001 (2022).
[4] Mohammad Ali Javidian, Vaneet Aggarwal, and Zubin Jacob, "Quantum causal inference in the presence of hidden common causes: An entropic approach", Physical Review A 106 6, 062425 (2022).
[5] Lorenzo Catani and Matthew Leifer, "A mathematical framework for operational fine tunings", Quantum 7, 948 (2023).
[6] Thomas D. Galley, Flaminia Giacomini, and John H. Selby, "A no-go theorem on the nature of the gravitational field beyond quantum theory", Quantum 6, 779 (2022).
[7] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality", Physical Review A 107 6, 062203 (2023).
[8] V. Vilasini and Roger Colbeck, "Impossibility of Superluminal Signaling in Minkowski Spacetime Does Not Rule Out Causal Loops", Physical Review Letters 129 11, 110401 (2022).
[9] Martin Plávala, "General probabilistic theories: An introduction", arXiv:2103.07469, (2021).
[10] V. Vilasini and Roger Colbeck, "Analyzing causal structures using Tsallis entropies", Physical Review A 100 6, 062108 (2019).
The above citations are from Crossref's cited-by service (last updated successfully 2023-09-27 22:45:57) and SAO/NASA ADS (last updated successfully 2023-09-27 22:45:58). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.