Causal structures from entropic information: geometry and novel scenarios

Abstract

Bellʼs 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.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New J. Phys. 16 (2014) 043001 R Chaves et al 2 1The notion of quantum non-locality is arguably somewhat unfortunate, as it seems to imply that an unspecified super-luminal mechanism is responsible for the strong correlations observed in quantum experiments. Bellʼs theorem only asserts that no local hidden variable (LHV) theory can reproduce these correlations. It does not put forward any alternative mechanism-certainly none involving actions at a distance. It has, however, become customary [1] to refer to statistics that cannot arise in LHV models as 'non-local'. We prefer sticking to commonly accepted language over inventing a possibly better-suited term that will not be widely understood. New J. Phys. 16 (2014) 043001 R Chaves et al New J. Phys. 16 (2014) 043001 R Chaves et al New J. Phys. 16 (2014) 043001 R Chaves et al 6 New J. Phys. 16 (2014) 043001 R Chaves et al 7 New J. Phys. 16 (2014) 043001 R Chaves et al 8 New J. Phys. 16 (2014) 043001 R Chaves et al 9 New J. Phys. 16 (2014) 043001 R Chaves et al 10 T n . We have stated this primarily to clarify the geometric nature of Q n (i.e. as an orthant, up to a linear isomorphism). New J. Phys. 16 (2014) 043001 R Chaves et al New J. Phys. 16 (2014) 043001 R Chaves et al 24

Information-Theoretic Inference of Common Ancestors

“…There, an algorithm is presented that, even though computationally demanding, computes all Shannon-type entropic inequalities for given marginal constraints. Furthermore, it has turned out that entropic inequalities are useful in quantum physics where they restrict possible theories of data generation in more general settings than the ones using Bell inequalities (see, e.g., [27][28][29][30]). Moreover, we would like to mention that meanwhile, information measures for causal inference among strings based on compression length have been proposed [31], thus extending the possible applications of inequalities like the ones presented in this article.…”

On generalized entropies and information-theoretic Bell inequalities under decoherence

“…In generalized Bell scenarios, we deal with the problem of deciding, whether observed data is compatible with a presumed causal relation between the variables [27]. It traditionally focuses on settings, when the region of compatible observations corresponds to some convex polytope.…”

Which causal structures might support a quantum–classical gap?

“…The only cases that do not satisfy Evan's sufficient criteria for interestingness are the Bell and Triangle scenarios, and the graphs # 15, 16, 21. For the latter three, in section 4 we propose a novel extension of the methods of entropic inequalities [18][19][20] that confirms these graphs as interesting (and thereby completes the establishment of HLP's conjecture for graphs with up to six nodes). Section 5 contains our conclusions and outlook.…”

“…In this section we review notation and definitions that will be used in this paper. We refer the reader to [1,2] for background on causal models, and to [18][19][20][21] for background on entropic bounds for causal graphs relevant to this paper.…”