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The Born rule describes the probability of obtaining an outcome whenmeasuring an observable of a quantum system. As it can only be tested bymeasuring many copies of the system under consideration, it cannot holdstrictly for non-replicable systems. For these systems, we give a procedure topredict the future statistics of measurement outcomes through RepeatedMeasurements (RM). We prove that if the statistics of the results acquired viaRM is sufficiently similar to that obtained by the Born rule, the latter can beused effectively. We apply our framework to a repeatedly measured Unruh-DeWittdetector interacting with a massless scalar quantum field, which is an exampleof a system (detector) interacting with an uncontrollable environment (field)for which using RM is necessary. Analysing what an observer learns from the RMoutcomes, we find a regime where history-dependent RM probabilities are closeto the Born ones. Consequently, the latter can be used for all practicalpurposes. Finally, we study numerically inertial and accelerated detectorsshowing that an observer can see the Unruh effect via RM.

Born rule extension for non-replicable systems and its consequences for Unruh-DeWitt detectors