Causal structures from entropic information: geometry and novel scenarios
Bell's theorem in physics, as well as causal discovery in machine learning, both face the problem of deciding whether observed data is compatible with a presumed causal relationship between the variables (for example, a local hidden variable model). Traditionally, Bell inequalities have been us...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2014-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/16/4/043001 |
| _version_ | 1797751415779622912 |
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| author | Rafael Chaves Lukas Luft David Gross |
| author_facet | Rafael Chaves Lukas Luft David Gross |
| author_sort | Rafael Chaves |
| collection | DOAJ |
| description | Bell's theorem in physics, as well as causal discovery in machine learning, both face the problem of deciding whether observed data is compatible with a presumed causal relationship between the variables (for example, a local hidden variable model). Traditionally, Bell inequalities have been used to describe the restrictions imposed by causal structures on marginal distributions. However, some structures give rise to non-convex constraints on the accessible data, and it has recently been noted that linear inequalities on the observable entropies capture these situations more naturally. In this paper, we show the versatility of the entropic approach by greatly expanding the set of scenarios for which entropic constraints are known. For the first time, we treat Bell scenarios involving multiple parties and multiple observables per party. Going beyond the usual Bell setup, we exhibit inequalities for scenarios with extra conditional independence assumptions, as well as a limited amount of shared randomness between the parties. Many of our results are based on a geometric observation: Bell polytopes for two-outcome measurements can be naturally imbedded into the convex cone of attainable marginal entropies. Thus, any entropic inequality can be translated into one valid for probabilities. In some situations the converse also holds, which provides us with a rich source of candidate entropic inequalities. |
| first_indexed | 2024-03-12T16:49:12Z |
| format | Article |
| id | doaj.art-07e4fa5ebe05440fa6d6aa4806e43970 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:49:12Z |
| publishDate | 2014-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-07e4fa5ebe05440fa6d6aa4806e439702023-08-08T11:25:32ZengIOP PublishingNew Journal of Physics1367-26302014-01-0116404300110.1088/1367-2630/16/4/043001Causal structures from entropic information: geometry and novel scenariosRafael Chaves0Lukas Luft1David Gross2Institute for Physics, University of Freiburg , Rheinstrasse 10, D-79104 Freiburg, GermanyInstitute for Physics, University of Freiburg , Rheinstrasse 10, D-79104 Freiburg, GermanyInstitute for Physics, University of Freiburg , Rheinstrasse 10, D-79104 Freiburg, GermanyBell's theorem in physics, as well as causal discovery in machine learning, both face the problem of deciding whether observed data is compatible with a presumed causal relationship between the variables (for example, a local hidden variable model). Traditionally, Bell inequalities have been used to describe the restrictions imposed by causal structures on marginal distributions. However, some structures give rise to non-convex constraints on the accessible data, and it has recently been noted that linear inequalities on the observable entropies capture these situations more naturally. In this paper, we show the versatility of the entropic approach by greatly expanding the set of scenarios for which entropic constraints are known. For the first time, we treat Bell scenarios involving multiple parties and multiple observables per party. Going beyond the usual Bell setup, we exhibit inequalities for scenarios with extra conditional independence assumptions, as well as a limited amount of shared randomness between the parties. Many of our results are based on a geometric observation: Bell polytopes for two-outcome measurements can be naturally imbedded into the convex cone of attainable marginal entropies. Thus, any entropic inequality can be translated into one valid for probabilities. In some situations the converse also holds, which provides us with a rich source of candidate entropic inequalities.https://doi.org/10.1088/1367-2630/16/4/043001non-localitymarginal scenarioscausal structuresentropic inequalities |
| spellingShingle | Rafael Chaves Lukas Luft David Gross Causal structures from entropic information: geometry and novel scenarios New Journal of Physics non-locality marginal scenarios causal structures entropic inequalities |
| title | Causal structures from entropic information: geometry and novel scenarios |
| title_full | Causal structures from entropic information: geometry and novel scenarios |
| title_fullStr | Causal structures from entropic information: geometry and novel scenarios |
| title_full_unstemmed | Causal structures from entropic information: geometry and novel scenarios |
| title_short | Causal structures from entropic information: geometry and novel scenarios |
| title_sort | causal structures from entropic information geometry and novel scenarios |
| topic | non-locality marginal scenarios causal structures entropic inequalities |
| url | https://doi.org/10.1088/1367-2630/16/4/043001 |
| work_keys_str_mv | AT rafaelchaves causalstructuresfromentropicinformationgeometryandnovelscenarios AT lukasluft causalstructuresfromentropicinformationgeometryandnovelscenarios AT davidgross causalstructuresfromentropicinformationgeometryandnovelscenarios |