# Non-holonomic tomography I: The Born rule as a connection between experiments

## TL;DR

This paper introduces a geometric interpretation of partial determinants in quantum tomography, framing them as holonomies related to non-holonomic constraints, with analogies to gauge theories and thermodynamics.

## Contribution

It presents a novel theoretical perspective that views partial determinants as holonomies, connecting quantum tomography with geometric and gauge-theoretic concepts.

## Key findings

- Partial determinants are shown to be holonomies in a geometric framework.
- Analogies are drawn between quantum tomography and gauge field theory.
- The approach generalizes the notions of state and measurement to non-holonomic constraints.

## Abstract

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-measurment (SPAM) correlated errors. As such, PDs bypass the need to estimate state-preparation or measurement parameters individually. In the present work, we suggest a theoretical perspective for the PD. We show that the PD is a holonomy and that the notions of state, measurement, and tomography can be generalized to non-holonomic constraints. To illustrate and clarify these abstract concepts, direct analogies are made to parallel transport, thermodynamics, and gauge field theory. This paper is the first of a two part series where the second paper [2] is about scalable applications of the PD to multiqudit systems.