Geometrical aspects of qudits concerning Bell inequalities

TL;DR

This thesis explores the geometric relationship between entanglement and nonlocality in bipartite qudits, using numerical optimization and visualization of state space to analyze Bell inequality violations.

Contribution

It introduces a numerical method to optimize measurement settings for Bell inequalities and visualizes the state space of maximally entangled qudits to compare entanglement and nonlocality.

Findings

Optimized measurement settings reveal nonlocality in specific states.

Geometric visualization clarifies the relation between entanglement and Bell inequality violations.

Numerical algorithms effectively analyze high-dimensional quantum state spaces.

Abstract

The aim of this thesis is to investigate quantum entanglement and quantum nonlocality of bipartite finite-dimensional systems (bipartite qudits). Entanglement is one of the most fascinating non-classical features of quantum theory, and besides its impact on our view of the world, it can be exploited for applications such as quantum cryptography and quantum computing. (...) Although entanglement and nonlocality are ordinarily regarded as one and the same, under close consideration this cannot be taken for granted. The reason for this is that entanglement is defined by the mathematical structure of a quantum state in a composite Hilbert space, whereas nonlocality signifies that the statistical behaviour of a system cannot be described by a local realistic theory. For the latter it is essential that the correlation probabilities of such theories obey so-called Bell inequalities, which are violated for certain quantum states. The main focus of this thesis is on the comparison of both properties with the objective of understanding their relation. (...) Because of the fact that the correlation probabilities in general depend on the measurement settings it is necessary to optimise these in order to reveal nonlocality. This problem is solved for a particular Bell inequality (CGMLP) by means of a self-developed numerical search algorithm. The method is then applied to density matrices of a subspace spanned by the projectors of maximally entangled two-qudit states. This set of states (...) serves to visualise and analyse the state space geometrically.