The Born rule as a parallel transport equation: Detecting multiqudit state-preparation and measurement correlations

In the context of quantum tomography, we recently introduced a quantity called a partial determinant \cite{jackson2015detecting}. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurement (SPAM) correlations. Importantly, this is done without any need to estimate state-preparation or measurement parameters. In the present work, we wish to better explain our theoretical perspective behind the PD. Further, we would like to demonstrate that there is an overwhelming variety of applications and generalizations of the PD. In particular we will construct methods for detecting SPAM correlations in multiqudit systems. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a classification of all such methods but focus on explicitly detailing only the most scalable methods, $\mathcal{O}(d^4)$.