High dimension and symmetries in quantum information theory

TL;DR

This thesis explores high-dimensional phenomena in quantum information theory, balancing between reducing complexity of quantum processes and understanding universal properties of large quantum systems, with a focus on symmetries and entanglement.

Contribution

It provides a comprehensive analysis of high-dimensional quantum systems, combining complexity reduction techniques with symmetry-based simplifications to better understand quantum channels, measurements, and entanglement.

Findings

Reduction of quantum process complexity while preserving key features

Identification of universal properties in high-dimensional quantum systems

Symmetry-based simplifications for specific quantum problems

Abstract

If a one-phrase summary of the subject of this thesis were required, it would be something like: miscellaneous large (but finite) dimensional phenomena in quantum information theory. That said, it could nonetheless be helpful to briefly elaborate. Starting from the observation that quantum physics unavoidably has to deal with high dimensional objects, basically two routes can be taken: either try and reduce their study to that of lower dimensional ones, or try and understand what kind of universal properties might precisely emerge in this regime. We actually do not choose which of these two attitudes to follow here, and rather oscillate between one and the other. In the first part of this manuscript, our aim is to reduce as much as possible the complexity of certain quantum processes, while of course still preserving their essential characteristics. The two types of processes we are interested in are quantum channels and quantum measurements. Oppositely, the second part of this manuscript is specifically dedicated to the analysis of high dimensional quantum systems and some of their typical features. Stress is put on multipartite systems and on entanglement-related properties of theirs. In the third part of this manuscript we eventually come back to a more dimensionality reduction state of mind. This time though, the strategy is to make use of the symmetries inherent to each particular situation we are looking at in order to derive a problem-dependent simplification.