Quantum entanglement

We introduce a novel geometric approach to characterize entanglement relations in large quantum systems. Our approach is inspired by Schumacher’s singlet state triangle inequality, which used an entropic-based distance to capture the strange properties of entanglement using geometric-based inequalities. Schumacher uses classical entropy and can only describe the geometry of bipartite states. We extend his approach by using von Neumann entropy to create an entanglement monotone that can be generalized for higher dimensional systems. We achieve this by utilizing recent definitions for entropic areas, volumes, and higher dimensional volumes for multipartite which we introduce in this thesis. This enables us to differentiate systems with high quantum correlation from systems with low quantum correlation and differentiate between different types of multi-partite entanglement. It also enable us to describe some of the strange properties of quantum entanglement using simple geometrical inequalities. Our geometrization of entanglement provides new insight into quantum entanglement. Perhaps by constructing well motivated geometrical structures (e.g. relations among areas, volumes ...), a set of trivial geometrical inequalities can reveal some of the complex properties of higher-dimensional entanglement in multi-partite systems. We provide numerous illustrative applications of this approach.

The angular momentum of light originates from two sources: one is the spin
angular momentum (SAM) of individual photons, which is related to the polarization
of light and the other is the orbital angular momentum (OAM) associated with helical
wavefront of the light if it is helically phased (complex phase front). A beam of light
that is composed of photons possessing both OAM and SAM states can be used in
different areas of study such as rotating microscopic particles, interacting with nonlinear
materials, investigating atom-light interactions, communication and medical
imaging technologies, quantum information, quantum entanglement and etc. In this
dissertation we study coherent beams that convey photons in superposition states
of polarization and complex phase front. Our study includes two fields: (I) classical
wave-like behavior with visible light in the field of singular optics. (II) quantum
particle-like behavior of photons of light in the field of quantum-entangled optics.
The approach is to investigate the state of such photons both mathematically and
experimentally in classical-singular and quantum-entangled fields. We discuss seven projects based on this research. In one project we present
a new method to encode OAM modes into perpendicular polarization components
and making superpositions of polarization and spatial modes mapped by Poincare
sphere. In another project using spatial light modulators (SLM) we realized highorder
disclination patterns in the polarization map of the cross section of the beam.
We also realize new forms of polarization disclination patterns (line patterns where
rotational invariance is violated) known as monstars that were not previously seen.
We proposed a new definition for characterizing these patterns since they can have
zero or negative singularity index. In another project, instead of SLM we used q-plates
to generate new forms of monstars. We proposed a robust and easy method for
determining the topological charge of a complex phase front beam by inspecting the
interference pattern the beam reflected from a wedged optical flat. In another project
we encoded OAM modes onto orthogonal polarization components of a photon from
an entangled pair and investigated the quantum entanglement. We also prepared
a polarization entangled state and calculated some measures of entanglement. We
summarize the projects and discuss the future prospects.