The spatiotemporal Born rule is quasiprobabilistic

Abstract

Contrary to general relativity, quantum theory treats space and time in fundamentally different ways. In particular, while joint probabilities associated with spacelike separated measurements are defined in terms of the Born rule, joint probabilities associated with measurements performed in sequence are defined in terms of the state-update rule. In this work, we show that one obtains a more unified perspective of space and time in quantum theory by embracing a quasiprobabilistic description of sequential measurements. More precisely, we show that there exists a unique \emph{pseudo}-density operator encoding canonical quasiprobabilities associated with sequential measurements in precisely the same manner that a density operator encodes joint probabilities associated with spacelike separated measurements, thus providing a natural extension of the Born rule into the temporal domain. As an application, we show how such a spatiotemporal Born rule combined in conjunction with a quantum Bayes' rule yields an operational notion of time-reversal symmetry for sequential measurements on an \emph{open} quantum system.