The spatiotemporal Born rule is quasiprobabilistic

Abstract: Contrary to the joint probabilities associated with measurements performed on spacelike separated quantum systems -- which are defined in terms of the Born rule -- the joint probabilities associated with the outcomes of \emph{sequential} measurements on a quantum system may not in general be consistently computed in terms of the trace of a fixed operator multiplied by a tensor product of projectors. However, in this work we show that if one modifies the probabilities associated with two-point sequential measurements by a correction term which accounts for state disturbance due to the initial measurement, then one obtains a quasiprobability distribution which is uniquely encoded by a bipartite operator which may be viewed as a spatiotemporal quantum state, thus yielding a spatiotemporal extension of the Born rule. Moreover, we prove that such a correction term may be viewed as an obstruction to the existence of a Born rule for the joint probabilities associated with sequential measurements, as we show that such a Born rule exists only when the probabilities and the aforementioned quasiprobabilities in fact coincide. We also show that when the notion of Bayesian inversion for quantum channels is applied in conjunction with the spatiotemporal Born rule, one naturally arrives at a quasiprobabilistic Bayes' rule for sequential measurements.