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Higher-order transformations that act on a certain number of input quantum channels with an indefinite causal order, such as the quantum switch, cannot be described by standard quantum circuits that use the same number of calls of the input quantum channels. But could they be simulated, i.e., could their action on their input channels be deterministically reproduced, for all arbitrary inputs, by a quantum circuit that uses on a larger number of calls of the input channels? In this work, we prove that, when only one extra call of each input channel is available, the quantum switch cannot be simulated. We demonstrate the robustness of this result by showing that even when probabilistic and approximate simulations are considered, higher-order transformations that are close to the quantum switch can be at best simulated with a probability strictly less than one. This result stands in stark contrast with the known fact that, when the quantum switch acts exclusively on unitary channels, its action can be simulated. We also show other particular cases where a restricted simulation of the quantum switch is possible. Finally, we discuss the implications of our findings to the analysis of experiments based on the quantum switch.

At the fundamental level, the dynamics of quantum particles and fields is time-symmetric: their dynamical equations are invariant under inversion of the time coordinate, possibly in conjunction with the change of other physical properties, such as charge and parity. At the operational level, the time-symmetry of the fundamental equations implies that certain quantum devices are bidirectional, meaning that the role of their inputs and outputs can be exchanged. Here we characterize the largest set of operations that can in principle be implemented on bidirectional devices, and show that this set includes operations in which the role of the input and output ports of the given devices becomes indefinite. An example of such an operation, called the “quantum time flip,” achieves input-output indefiniteness by adding quantum control to the direction in which a single device is used. We show that quantum operations with indefinite input-output directions can in principle achieve information-theoretic advantages over all possible operations with definite time direction, and can lead to an exetremely strong form of indefinite causal order.

Recent advances in quantum foundations have unveiled the idea that the causal order between quantum events may not always be fixed or even well-defined, allowing for some form of *indefinite quantum causality*. This tutorial will introduce the key concepts and motivations behind this rapidly developing area of research. Focusing on one of the main frameworks developed to explore indefinite quantum causality—the process matrix formalism—I will present key theoretical results, highlight the potential of indefinite causal orders as a resource for quantum information processing, and discuss experimental implementations as well as the physical interpretation of indefinite causal structures.