Non-Holonomic Tomography: A Method for Assessing Various State-Preparation and Measurement Correlations

Abstract

The following dissertation investigates a problem related to the practice of quantum tomography, where one usually estimates the parameters associated with quantum states or measurements.

In particular, the question answered is whether and how one could detect if states and measurements are correlated.

A similar question answered is how one could detect state-preparation non-localities and measurement non-localities in multiqudit systems.

The solution involves an analysis of certain matrix quantities called \emph{partial determinants}.

Partial determinants are an application of the Born rule that can be interpreted as tomography over a loop in the space of state and measurement settings.

From this perspective, the notion of state and observable become \emph{non-holonomic}

| that is, state and observable parameters can be defined ``locally'' over each setting but not globally over all settings.

As such, state and measurement parameters are not estimated because such estimated values don't exist in correlated systems, but rather the inability to estimate such values is quantified.

Therefore, partial determinants are a measure of the amount of contradiction that would result from any claim of such a estimated values by propagating these estimates through a `tomography loop' of data collected by various experiments.

Such measures of contradiction are generally known as \emph{holonomies}.