Holonomy in Quantum Information Geometry

TL;DR

This thesis develops a unified framework for non-adiabatic geometric phases in mixed quantum states, relating different holonomy theories, exploring topological properties, and applying these concepts to quantum uncertainty and speed limits.

Contribution

It introduces a holonomy theory for Sjöqvist et al.'s geometric phase, relating it to Uhlmann's phase, and extends the theory to higher-order phases and quantum speed limits.

Findings

Unified treatment of two geometric phases in mixed states

Relation established between different holonomy theories

New quantum speed limits derived from geometric phase theory

Abstract

In this thesis we provide a uniform treatment of two non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sj\"{o}qvist et al. We develop a holonomy theory for the latter which we also relate to the already existing theory for the former. This makes it clear what the similarities and differences between the two geometric phases are. We discuss and motivate constraints on the two phases. Furthermore, we discuss some topological properties of the holonomy of `real' quantum systems, and we introduce higher-order geometric phases for not necessarily cyclic dynamical systems of mixed states. In a final chapter, we apply the theory developed for the geometric phase of Sj\"{o}qvist et al. to geometric uncertainty relations, including some new "quantum speed limits".