The spatiotemporal Born rule is quasiprobabilistic

TL;DR

This paper introduces a quasiprobabilistic framework that unifies the treatment of space and time in quantum theory by extending the Born rule through a pseudo-density operator for sequential measurements.

Contribution

It presents a unique pseudo-density operator that encodes quasiprobabilities for sequential measurements, extending the Born rule into the temporal domain.

Findings

A pseudo-density operator encodes quasiprobabilities for sequential measurements.

The spatiotemporal Born rule provides a unified perspective of space and time in quantum theory.

Application to time-reversal symmetry in open quantum systems.

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.