The fundamental dynamics of quantum particles is neutral with respect to the arrow of time. And yet, our experiments are not: we observe quantum systems evolving from the past to the future, but not the other way round. A fundamental question is whether it is in principle possible to probe a quantum dynamics in the backward direction, or in more general combinations of the forward and the backward direction. To answer this question, we characterise all possible time-reversals that satisfy four natural requirements and we identify the largest set of quantum processes that can in principle be probed in both time directions. Then, we show that quantum theory is compatible with the existence of a new kind of operations where the arrow of time is indefinite. We explicitly construct one such operation, called the quantum time flip, and show that it cannot be realised by any quantum circuit with a definite direction of time. The quantum time flip offers an advantage in a game where a referee challenges a player to identify a hidden relation between two gates, and can be experimentally simulated with photonic systems, shedding light on the information-processing capabilities of exotic scenarios in which the arrow of time is in a quantum superposition.
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