CHAPTER THREE. The Limits of Classical Computing

Gentian Kasa

The purpose of this thesis is to make an analysis of the concept of Hypercomputation and of some hypermachines. This thesis is separated in three main parts. We start in the first chapter with an analysis of the concept of Classical Computability with the Turing Machine and the Church-Turing thesis as a main reference and afterwards, in the second chapter, we continue with an analysis of hypercomputation and some hypermachines. Attention is given to the possible physical realization of these machines and their usefulness. In the third chapter a superficial introduction to Quantum Computing is made and a brief analysis to a quantum hypercomputational model, Tien D. Kieu's Adiabatic Quantum Computing.

Xiping Zheng

The objective of this thesis is to investigate two fundamental problems of quantum optics: macroscopically distinguishable quantum superposition states and quantum chaos. After a brief introduction, Chapter II and Chapter III investigate the problem of superposition states. A method for generating the macroscopically distinguishable superposition states via single two-level atom dispersion was proposed by Savage et a/.[Opt.Lett. 15, 628(1990)]. In chapter II, an extension of this model to a three-level atom is investigated and the result shows that the superposition state can be obtained from the three-level model sytem in the dispersive limit. In chapter III, S ’ the effect of the atomic linewidth is considered. If the atomic linewidth is included, there will be two contradictory conditions for the generation of distinguishable superpositon states. This shows that the atomic linewidth prevents the formation of macroscopically distinguishable superpositon states by the proposed method. From chapter IV to chapter VI, we investigate the transition from a dissipative quantum system to a classically chaotic system. Specifically, wre apply the method of quantum trajectories to the case of optical second harmonic generation. In chapter V, the Q function is obtained numerically by this method and found to agree with the results obtained by solving the master equation. In chapter VI, the field amplitudes and mean photon number associated with a single quantum trajectory are investigated. The results show that the trajectory becomes more classical and the configuration of the classical strange attractor is approached as the field strength is increased. This is because the mean values of operator products factorize into products of mean values in the classical limit.

Gilad Gour

Chapter 14 transitions the focus to multipartite entanglement, a realm that broadens the discussion from bipartite systems to those involving multiple parties. This complex form of entanglement plays a crucial role in quantum computing, cryptography, and communication networks. The chapter introduces the foundational concepts of multipartite entanglement, including its characterization and the challenges associated with its classification. Significant attention is given to the classification of multipartite entangled states through SL-invariant polynomials, which provide tools for understanding the structure and properties of these states. Stochastic Local Operations and Classical Communication (SLOCC) are introduced as a means to classify entanglement. Furthermore, the chapter explores the entanglement of assistance and the monogamy of entanglement, two concepts that illustrate the limitations and potential for distributing entanglement among multiple parties. Through detailed explanations and examples in three and four qubits, this chapter offers insights into the intricate world of multipartite entanglement, revealing both its potential and challenges.