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—either a classical mixture or a quantum superposition of causal orders. The inequalities hold valid for a vast class of information measures. Nevertheless, we take under scrutiny the von Neumann, $\alpha$-Rényi 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 nonfixed causal order 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.

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• Quantum correlations in quantum information
• Quantum information theory