The simplest causal inequalities and their violation

TL;DR

This paper characterizes the causal polytope for bipartite correlations with binary inputs and outputs, identifying inequalities that can be violated within the process matrix framework, thus advancing understanding of causal structures in quantum theory.

Contribution

It fully characterizes the simplest causal polytope and identifies causal inequalities that can be violated in the process matrix framework, extending quantum causal analysis.

Findings

Identified two families of causal inequalities.

Demonstrated violations of these inequalities using process matrices.

Provided a systematic framework for causal inequality analysis.

Abstract

In a scenario where two parties share, act on and exchange some physical resource, the assumption that the parties' actions are ordered according to a definite causal structure yields constraints on the possible correlations that can be established. We show that the set of correlations that are compatible with a definite causal order forms a polytope, whose facets define causal inequalities. We fully characterize this causal polytope in the simplest case of bipartite correlations with binary inputs and outputs. We find two families of nonequivalent causal inequalities; both can be violated in the recently introduced framework of process matrices, which extends the standard quantum formalism by relaxing the implicit assumption of a fixed causal structure. Our work paves the way to a more systematic investigation of causal inequalities in a theory-independent way, and of their violation within the framework of process matrices.