The Nitrogen-Vacancy (NV) center has demonstrated great potential as a quantum networks platform. While many milestones have been reached, the current hardware implementations have reached their limit in terms of remote entanglement generation rates, which hinders the scalability of the platform. The most dominant limits are that only ∼ 3% of the emission is coherent, and the outcoupling efficiency in state-of-the-art network experiments is ∼ 15%. The coherent emission can be increased via the Purcell effect, which can be achieved by the use of an open Fabry-Pérot microcavity. In such a setup the NV center is embedded in a micrometer-thin diamond membrane, thereby retaining its favourable optical properties. The open microcavity has the ability of in situ spectral and spatial tunability. However, this is accompanied by a high susceptibility to vibrations, which needs to be considered in the experimental design. The goal of this thesis is to simultaneously increase the coherent emission and the outcoupling efficiency by coupling NV centers to an open Fabry-Pérot microcavity. An optimized cavity is found by characterizing different fiber tip mirrors, reaching a bare cavity finesse of 9500. The vibrations in the bare cavity are analyzed, revealing a root mean square cavity length detuning of 22 pm while operating at low temperature. A hybrid cavity is formed with a diamond membrane with NV centers, resulting in cavity with a finesse of 2100, quality factor of 266000, mode volume of 108 λ 3 , and outcoupling efficiency ∼ 30%. Due to a different setup configuration, an increased root mean square cavity length detuning of 190 pm is measured, resulting in a lowered Purcell factor. Off-resonant measurements demonstrate the coupling of NV centers to the cavity. Moreover, excited state lifetimes are measured to obtain a Purcell factor of 2.6 ± 0.5, with a corresponding enhanced coherent emission ratio of 0.07 ± 0.01. This result is in good agreement with simulations that include the effect of vibrations. The ZPL branching ratio and outcoupling efficiency are improved by a factor two compared to state-of-the-art confocal microscope setups. For the first time in open microcavities, the ability to drive the spin of NV centers is shown by an optically detected magnetic resonance measurement with a contrast of (28 ± 2)%. By improving the setup to the already obtained vibration level, and a realistic finesse of 5000, an expected Purcell factor of ∼ 15.7 and outcoupling efficiency of ∼ 80% can be reached, paving the way to the next-generation of quantum network nodes.

"Quantum optimal control is a rapidly growing field with diverse methods and applications. In this work, the possibility of using quantum optimal control techniques to co-optimize the energetic cost and the process fidelity of a quantum unitary gate is investigated. The theoretical definition and quantization of quantum unitary gates, as well as the relationship between the process fidelity and the energetic cost of a quantum unitary gate are explored. Two different quantum optimal control methods to co-optimize both fidelity and energetic cost, i.e., the Gradient Ascent Pulse Engineering method and model-free Deep Reinforcement Learning are investigated. The performance of both quantum optimal control techniques in the presence of noise is probed. We find that the energetic cost of a quantum unitary gate can be quantized by integrating the control pulses and norm of the corresponding Hamiltonian operators over the total time duration of the unitary, and for single qubit gates by calculating the arc length of the quantum unitary gate on the Bloch sphere. A Pareto optimal front between the process fidelity and the energetic cost of a quantum gate is identified, where a lower energetic cost yields an inherently lower process fidelity. A python package called ”EUQOC” (Energy Efficient Universal Quantum Optimal Control) has been created to implement energy optimal quantum gate synthesis, both with the Energy Optimal Gradient Ascent Pulse Engineering (EO-GRAPE) method and by model-free Deep Reinforcement Learning. It is found that the EO-GRAPE method performs better than the reinforcement learning methods, for all noise settings and neural network sizes. For future work, the optimization problem could be translated to the frequency domain to increase the computational efficiency. Furthermore, the relationship between information and energy can be investigated by looking at the complexity of the pulse or the decomposition of the quantum unitary gate."

In this thesis, two topics are studied: mathematical inequalities and non-linear quantum entanglement witnesses. First, various inequalities, like the Cauchy-Schwarz inequality (on finite dimensional vector spaces) and Jensen's inequality, along with their extensions and generalisations, are proved and discussed. The intimate relationship between these inequalities is studied. Because this thesis was restricted to finite dimensional vector spaces, the consequences of generalising the results to infinite dimensional vector spaces are finally determined. Secondly, the topic of entanglement detection is discussed - specifically, non-linear entanglement witnesses are considered. A bipartite and multipartite entanglement criterion based on the previously discussed inequalities are introduced and assessed extensively by considering their optimality, how they relate to other criteria as well as their limitations.

Relaxing the assumptions about the experimental setup in Quantum Key Distribution protocols lays the foundation for Device Independent Quantum Key Distribution (DIQKD). In the finite key regime of DIQKD, the protocols employ the use of only the CHSH inequality, so far. A natural question therefore arises whether are there other Bell inequalities that help achieve better results (in terms of higher rates, greater noise tolerance and lower number of minimum rounds required for positive rates) than those achieved using CHSH inequality? For the inequalities considered, we find that CHSH fares the best on account of noise tolerance. However, considering the other two parameters of interest we present two bipartite Bell-inequalities with three outcomes per party that perform better than CHSH for a certain range of noise involved.