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Coherent quantum control of causal order is possible too and is a useful resource for several tasks. However, quantum control implies that a controlling system carries the which-order information \u2013 if the control is traced out, the order of events remains in a probabilistic mixture. Can the order of two events be in a pure superposition, uncorrelated with any other system? Here we show that this is not possible for a broad class of processes: a pure superposition of any pair of Markovian, unitary processes with equal local dimensions and different causal orders is not a valid process, namely it results in non-normalised probabilities when probed with certain operations. The result imposes constraints on novel resources for quantum information processing and on possible processes in a theory of quantum gravity.<\/jats:p>","DOI":"10.22331\/q-2022-03-01-663","type":"journal-article","created":{"date-parts":[[2022,3,1]],"date-time":"2022-03-01T10:13:59Z","timestamp":1646129639000},"page":"663","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["A no-go theorem for superpositions of causal orders"],"prefix":"10.22331","volume":"6","author":[{"given":"Fabio","family":"Costa","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia"}]}],"member":"9598","published-online":{"date-parts":[[2022,3,1]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. P. Feynman, ``Space-Time Approach to Non-Relativistic Quantum Mechanics,'' Rev. Mod. 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