Can the quantum switch be deterministically simulated?

Abstract

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.