Entropic limitations on fixed causal order

Abstract: Quantum processes can exhibit scenarios beyond a fixed order of events. We propose information inequalities that, when violated, constitute sufficient conditions to certify quantum processes without a fixed causal order -- causally separable or indefinite causal ordered processes. The inequalities hold valid for a vast class of information measures. Nevertheless, we take under scrutiny the von Neumann, $\alpha-$R\'enyi entropies with parameter $\alpha \in [1/2,1) \cup (1,\infty)$, and max- and min-entropies. We also discuss how the strong subadditivity of quantum (von Neumann) entropy, used along with the information inequality developed here, implies relevant witnesses of causally separable and indefinite causal ordered processes in marginal scenarios. Importantly, we show the violation of these inequalities for the quantum switch, a paradigmatic example of a process with indefinite causal order. Our approach contributes to the important research direction of information-theoretic characterization of quantum processes beyond fixed causal orders.