Which causal structures might support a quantum–classical gap?

Abstract

A causal scenario is a graph that describes the cause and effect relationships between all relevant variables in an experiment. A scenario is deemed 'not interesting' if there is no device-independent way to distinguish the predictions of classical physics from any generalised probabilistic theory (including quantum mechanics). Conversely, an interesting scenario is one in which there exists a gap between the predictions of different operational probabilistic theories, as occurs for example in Belltype experiments. Henson, Lal and Pusey (HLP) recently proposed a sufficient condition for a causal scenario to not be interesting. In this paper we supplement their analysis with some new techniques and results. We first show that existing graphical techniques due to Evans can be used to confirm by inspection that many graphs are interesting without having to explicitly search for inequality violations. For three exceptional cases-the graphs numbered #15, 16, 20 in HLP-we show that there exist non-Shannon type entropic inequalities that imply these graphs are interesting. In doing so, we find that existing methods of entropic inequalities can be greatly enhanced by conditioning on the specific values of certain variables. then we say that the observed phenomenon can be explained by the model. Hypothesis testing then becomes the work of designing experiments to rule out different competing explanations.The literature on causal inference provides numerous tools for deciding when one causal model is a better explanation than another. In general, simpler explanations are preferable to more complex ones. Explanations should be faithful, meaning (roughly) that the observed independencies appear in almost all causal models that have the same causal structure as the chosen explanation. If these principles are not enough to differentiate different hypotheses, it is always possible to do so by performing experimental interventions: actively changing the values of some variables and observing the reaction of the remaining variables. A tool of causal inference called the do-calculus tells us which interventions are needed to obtain specific information about the causal structure.The question arises as to whether certain experiments in quantum physics, particularly the so-called 'Belltype' experiments [4-6], can be explained by a causal model. It is widely agreed that any explanation of quantum phenomena in terms of a classical causal model must violate at least one of several assumptions that hold for purely classical phenomena. For example, a classical causal model can explain quantum phenomena if we allow causes to propagate faster than light, as in Bohmian mechanics [7], or if we allow measurement settings to be influenced by a hidden common cause (the super-determinism loophole), and there are many other options. A more recent perspective due to Wood and Spekkens is that classical causal explanations of quantum experiments cannot be faithful [8], i.e. the causal model is not a generic example of the set of models that share it...

The entropic approach to causal correlations

“…In order to see that there are causal bipartite correlations that have entropy vectors not contained in G G (36) and (37) except with bounds on the right-hand side of 0. One can easily verify that equations (36) and (37) can be saturated by causal correlations (for some equal mixtures of correlations  P A B and  P B A ), thus providing such a counterexample.…”

“…In order to study the potential violation of these entropic inequalities, we again need to look at nondeterministic distributions. One can easily see, however, that equations (36) and (37) . This means that the inequalities for counterfactual variables are unable to detect noncausality when both parties are restricted to binary outputs.…”

“…Our goal in this paper is to introduce a new framework for the derivation of causal inequalities and the study of their potential violations: the entropic approach to causal correlations. The idea of using entropies to understand sets of correlations has its origins in the context of Bell inequalities [26][27][28][29] but since then has also found various other applications in quantum contextuality [30][31][32], device-independent applications [33,34], causal inference [9,35,36] and in the characterization of nonsignaling correlations [37]. As for these previous applications, the interest in characterizing the entropies compatible with causal correlations stems not only from practical and technical issues, but also from a more fundamental point.…”

Why initial system-environment correlations do not imply the failure of complete positivity: A causal perspective

“…Pursuing a quantum generalization of the classical framework of causal modelling has already had many interesting applications in quantum foundations, including: revealing a quantum advantage for causal inference [31], uncovering new experimental scenarios wherein there is a gap between quantum and classical correlations [32][33][34][35][36][37][38], uncovering a promising approach to achieving a causal explanation of Bell inequality violations without finetuning [39][40][41][42][43], expanding the set of experimental configurations wherein one can achieve quantum state pooling [44], and exploring the possibility of quantum uncertainty about the causal structure [45][46][47].…”

Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling

“…Entropy is a key concept in our understanding of information theory, thus having a framework that focuses on it naturally leads to new insights and applications [12,29,[39][40][41][42][43][44]. In turn, entropies allow for a much simpler and compact characterization of Bell inequalities -at the cost of not being a complete description-in a variety of scenarios, most notably in the aforementioned quantum networks [28,29,32,[36][37][38]43]. Not surprisingly, however, this approach is also hampered by computational complexity issues.…”