Abstract
The existence of a global causal order between events places constraints on the correlations that parties may share. Such ‘causal correlations’ have been the focus of recent attention, driven by the realization that some extensions of quantum mechanics may violate so-called causal inequalities. In this paper we study causal correlations from an entropic perspective, and we show how to use this framework to derive entropic causal inequalities. We consider two different ways to derive such inequalities. Firstly, we consider a method based on the causal Bayesian networks describing the causal relations between the parties. In contrast to the Bell-nonlocality scenario, where this method has previously been shown to be ineffective, we show that it leads to several interesting entropic causal inequalities. Secondly, we consider an alternative method based on counterfactual variables that has previously been used to derive entropic Bell inequalities. We compare the inequalities obtained via these two methods and discuss their violation by noncausal correlations. As an application of our approach, we derive bounds on the quantity of information—which is more naturally expressed in the entropic framework—that parties can communicate when operating in a definite causal order.
Highlights
When describing most physical phenomena it seems natural to assume that physical events take place in a well-defined causal structure
As shown by Bell’s Theorem [5], quantum correlations obtained by measurements on distant entangled parties are incompatible with Reichenbach’s principle [6, 7] or, more generally, with classical theories of causality, forcing us to generalize the notion of causal models [8,9,10,11,12,13]
We review the entropic characterization of marginal scenarios [28] using two complementary methods, the first considering the entropies of the variables composing a given causal model, and the second based on the counterfactual approach to correlations
Summary
When describing most physical phenomena it seems natural to assume that physical events take place in a well-defined causal structure. Earlier events can influence later ones but not the opposite, or, if two events are distant enough (typically, space-like separated) from each other, any correlation between them can only be due to some common cause in their past This intuition is formalized in Reichenbach’s principle [1] and generalized by the mathematical theory of causal models [2] that form the basis for our current understanding of how to infer causation from empirically observed correlations. 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. V we use this approach to derive bounds on mutual informations in causal games
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