One can theoretically conceive of processes in which quantum operations are composed cyclically in a way that is incompatible with a well-defined causal order. Some of these processes can be realized within standard quantum temporal evolutions on systems that are delocalized in time. In this Letter, we introduce the “circuit (super)operator,” a tool that allows one to express how a (possibly cyclic) quantum circuit transforms under arbitrary changes of subsystem decompositions. This framework concisely captures the link between causally indefinite processes and their temporal realizations. We apply it to the quantum switch, a paradigmatic example of a causally indefinite process, and consider two distinct temporal realizations. These correspond to the “causal perspectives” or “causal reference frames” of the two parties involved, describing the process from their respective viewpoints. Surprisingly, we show that these causal perspectives cannot be related by a change of subsystem decomposition, i.e., they cannot be seen as two equivalent descriptions of the same process.

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• Quantum circuits
• Quantum correlations, foundations & formalism
• Quantum foundations
• Quantum information theory