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It has been shown that it is theoretically possible for there to existhigher-order quantum processes in which the operations performed by separateparties cannot be ascribed a definite causal order. Some of these processes arebelieved to have a physical realization in standard quantum mechanics viacoherent control of the times of the operations. A prominent example is thequantum SWITCH, which was recently demonstrated experimentally. However, theinterpretation of such experiments as realizations of a process with indefinitecausal structure as opposed to some form of simulation of such a process hasremained controversial. Where exactly are the local operations of the partiesin such an experiment? On what spaces do they act given that their times areindefinite? Can we probe them directly rather than assume what they ought to bebased on heuristic considerations? How can we reconcile the claim that theseoperations really take place, each once as required, with the fact that thestructure of the presumed process implies that they cannot be part of anyacyclic circuit? Here, I offer a precise answer to these questions: the inputand output systems of the operations in such a process are generally nontrivialsubsystems of Hilbert spaces that are tensor products of Hilbert spacesassociated with systems at different times---a fact that is directlyexperimentally verifiable. With respect to these time-delocalized subsystems,the structure of the process is one of a circuit with a causal cycle. I alsoidentify a whole class of isometric processes, of which the quantum SWITCH is aspecial case, that admit a physical realization on time-delocalized subsystems.These results unveil a novel structure within quantum mechanics, which may haveimportant implications for physics and information processing.

Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics