Tsirelson bounds for quantum correlations with indefinite causal order

Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by a set of parties operating in a definite order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson's bound, has remained an open problem. Here we provide a general method for bounding the violation of causal inequalities by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum, and determine a Tsirelson-like bound for the paradigmatic example of the Oreshkov-Brukner-Costa causal inequality. Surprisingly, we find that the algebraic maximum of arbitrary causal inequalities can be achieved by a new type of processes that allow for information to flow in an indefinite direction within the parties' laboratories. In the classification of the possible correlations, these processes play a similar role as the no-signalling processes in Bell scenarios.