The simplest causal inequalities and their violation

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.This approach is more powerful as it can detect all causally nonseparable process matrices, while not all causally nonseparable process matrices can violate a causal inequality [13,14]. Furthermore, physical implementations of certain (multipartite) causally nonseparable process matrices, and of corresponding causal witnesses that detect their causal nonseparability, have been proposed [13][14][15] and even realized experimentally [16], while it is still not known whether there actually exist any physically realizable process that violates a causal inequality. Nevertheless, the device-independent approach is still of interest as it relaxes the requirement to trust the functioning and the operations implemented by one's devices in an experiment. It is furthermore also theoryindependent: causal inequalities can in principle be tested, and correlations with no definite causal order can be identified whatever the description of the physical resource is-whether we use the process matrix framework or any other theory to be discovered in the future. A related open question is whether the ability to violate causal inequalities can-in analogy with Bell nonlocality [17]-be exploited as a resource, just like causally nonseparable process matrices provide advantages for information-theoretical [18], computational [19], and communication complexity [20] tasks.Our paper aims at providing a better understanding of the device-independent characterization of correlations that are compatible with a definite causal order or not. We show that bipartite correlations with a definite causal order form a convex polytope, whose facets correspond to causal inequalities (section 2). We characterize this causal polytope in the simplest scenario where the two parties observe correlations with binary inputs and outputs, which gives us two families of new causal inequalities. We then investigate their possible violation in the framework of process matrices, and find that these can indeed be violated (section 3). This provides an example of 'noncausal' process matrix correlations in a simpler scenario than that considered i...

Causally nonseparable processes admitting a causal model

“…Using the see-saw algorithm, we optimized W ext for a violation of the simplest causal inequalities [4]. We found that W ext is able to violate (by about´-8 10 5 ) the GYNI inequality (9), which proves that W opt is not extensibly causal.…”

“…For the scenario where Alice (Bob) has an input bit x (y) and outputs one bit a (b), one causal inequality is a bound on the probability of success of the 'guess your neighbor's input' (GYNI) [4]:…”

“…Some valid processes and local strategies which result in correlations violating (9) are described in [4]. The relation between causally nonseparable processes and the violation of causal inequalities is not yet fully understood.…”

“…In the bipartite case, all previously known nonseparable processes violate causal inequalities and it is not clear if nonseparable processes with a causal model even exist [4]. In the next section, we explicitly provide a class of nonseparable bipartite processes that allow for a causal model.…”

“…These causal structures can produce correlations incompatible with any definite causal order, violating so-called 4 'causal inequalities' [3,4].…”