  It has been shown that it is theoretically possible for there to exist
higher-order quantum processes in which the operations performed by separate
parties cannot be ascribed a definite causal order. Some of these processes are
believed to have a physical realization in standard quantum mechanics via
coherent control of the times of the operations. A prominent example is the
quantum SWITCH, which was recently demonstrated experimentally. However, the
interpretation of such experiments as realizations of a process with indefinite
causal structure as opposed to some form of simulation of such a process has
remained controversial. Where exactly are the local operations of the parties
in such an experiment? On what spaces do they act given that their times are
indefinite? Can we probe them directly rather than assume what they ought to be
based on heuristic considerations? How can we reconcile the claim that these
operations really take place, each once as required, with the fact that the
structure of the presumed process implies that they cannot be part of any
acyclic circuit? Here, I offer a precise answer to these questions: the input
and output systems of the operations in such a process are generally nontrivial
subsystems of Hilbert spaces that are tensor products of Hilbert spaces
associated with systems at different times---a fact that is directly
experimentally verifiable. With respect to these time-delocalized subsystems,
the structure of the process is one of a circuit with a causal cycle. I also
identify a whole class of isometric processes, of which the quantum SWITCH is a
special case, that admit a physical realization on time-delocalized subsystems.
These results unveil a novel structure within quantum mechanics, which may have
important implications for physics and information processing.
