Entanglement and Non-Locality in Continuous-Variable Quantum Systems

The crisis of classical physics, marked by experiments such as Michelson-Morley's and observations on blackbody radiation, led to the development of quantum theory. The concept of energy quantization introduced by Planck, followed by Einstein’s ideas on the quantum nature of light, revolutionized the understanding of microscopic physical phenomena. This thesis explores the connection between entanglement and non-locality, with a particular focus on continuous-variable systems, an area crucial for applications in quantum computing and metrology. The work begins with a discussion on the quantum theory of light, analyzing electromagnetic field quantization and introducing fundamental states such as Fock, coherent, and squeezed states. Squeezed states, in particular, play a key role in reducing quantum noise beyond the standard quantum limit, with implications for precision experiments. Subsequently, the basic concepts of quantum information theory are presented, examining qubits, qudits, and entanglement in both discrete and continuous-variable systems. The analysis focuses on phase-space representation and Gaussian states, exploring how their entanglement structure can be characterized through purity measures and symplectic operators. The thesis then delves into the concept of non-locality, starting from the Einstein-Podolsky-Rosen argument and Bell’s inequalities, demonstrating how quantum mechanics violates the constraints imposed by local hidden variable theories. Quantum correlations, Gisin’s theorem, and the relationship between entanglement and non-locality in pure and mixed states are examined. Multipartite non-locality is also addressed, highlighting the distinction between entanglement and non-locality in complex quantum systems. Finally, the work focuses on non-locality in continuous-variable systems, introducing new Bell-type inequalities that can be tested using parity measurements, particularly in the context of the EPR state. The analysis of phase-space representation and the Wigner function provides additional tools for characterizing non-local correlations. The results presented in this work are fundamental for the development of quantum technologies, from cryptography to quantum computing, and contribute to a deeper understanding of the nature of quantum reality.