Admissible Causal Structures and Correlations

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. 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 is supported by numerical calculations. We show that these causal models, in a restricted setting, are indeed consistent. We identify two sets of causal structures that, in the classical-deterministic case, forbid and give rise to noncausal correlations, respectively.