Admissible Causal Structures and Correlations

TL;DR

This paper investigates the limitations on quantum causal structures imposed by local quantum theory, introducing a graph-theoretic criterion for admissibility and exploring the nature of causal correlations.

Contribution

It introduces the 'siblings-on-cycles' property as a necessary condition for quantum causal structures and conjectures its sufficiency, supported by explicit models and numerical evidence.

Findings

The 'siblings-on-cycles' property is necessary for admissible quantum causal structures.

Explicit quantum causal models satisfy the proposed property in restricted settings.

Identifies causal structures that produce causal and non-causal correlations in classical-deterministic cases.

Abstract

It is well-known that if one assumes quantum theory to hold locally, then processes with indefinite causal order and cyclic causal structures become feasible. Here, we study qualitative limitations on causal structures and correlations imposed by local quantum theory. For one, we find a necessary graph theoretic criterion--the "siblings-on-cycles" property--for a causal structure to be admissible: Only such causal structures admit a realization consistent with local quantum theory. We conjecture that this property is moreover sufficient. This conjecture is motivated by an explicit construction of quantum causal models, and supported by numerical calculations. We show that these causal models, in a restricted setting, are indeed consistent. For another, we identify two sets of causal structures that, in the classical-deterministic case, give rise to causal and non-causal correlations respectively.