Quantum diffeomorphisms cannot make indefinite causal order definite (2211.15685v2)

Abstract: The field of indefinite causal order has seen significant advancements in recent years. While classically the causal order of two timelike separated events A and B is fixed - either A before B or B before A - this is no longer true in quantum theory. There, it is possible to encounter superpositions of causal orders. The quantum switch is one of the most prominent processes with indefinite causal order. While the optical quantum switch has been successfully implemented in experiments, some argue that this merely simulates a process with indefinite causal order and that a superposition of spacetime metrics is required for a true realization. Here, we provide a relativistic definition of causal order, show that it encompasses both the optical and gravitational quantum switch, and does not differentiate between them. Moreover, we show that this notion of causal order is invariant under so-called quantum diffeomorphisms and that it is an operationally meaningful observable in both the general relativistic and quantum mechanical sense. Importantly, this observable does not distinguish between the indefinite causal order implemented in the optical and gravitational quantum switch, thus supporting the thesis that the optical quantum switch is just as much a realization of indefinite causal order as its gravitational counterpart.