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.

Random walks have been used in a number of different fields for a long time. With the rise of of quantum random walks a lot of these applications have been made more efficient. Recently there have been a lot of improvements in the realization of these quantum random walks in real world systems. In this research, the effect of uncertainty in the measurement time and measurement frequency are studied on three problems that make use of a quantum random walk. These problems are the network centrality problem, the graph isomorphism problem and the spatial search problem. These effects are studied by first looking at the theory behind these problems after which the theoretical results of these three problems are calculated. These theoretical results will then be compared to simulated results that have a certain level of uncertainty in the measurement time or frequency. For both the network centrality problem and the spatial search problem we concluded that only a small error is made when uncertainties are introduced in the system. This implies that these problems are solvable in realizations of the quantum random walk in real world systems. For the Graph isomorphism problem we saw that the error that is made when uncertainties are introduced was large which implies that this problem would only be solvable for small uncertainties.

We introduce a method of designing NMR pulse sequences, with a specific focus toward complete decoupling of carbon-13 spin qubits coupled to a nitrogen-vacancy (NV) center in diamond. Using Average Hamiltonian Theory, we calculate different intermediate Hamiltonians corresponding to different rotations implemented by NMR pulses. Two novel pulse sequences are obtained containing 24 and 12 evolution periods respectively. Assuming ideal and instantaneous pulses, performance for both sequences is better than the WAHUHA + echo sequence for total evolution times of less than 5 ms and comparable for greater evolution times. Analysis of sensitivity points toward non-robustness for angle errors in the applied pulses when the direction of pulses is not taken into account, however this can be corrected for using chirality sums. Suggestions for further research include the study of non-globally applied pulses and application of the design method to non-zero effective Hamiltonians.

In this thesis, networks of coupled quantum harmonic oscillators are studied. The dynamics of these networks are determined by single-frequency vibrations of the entire network called normal modes. We study the behavior of the nor- mal modes when the network is coupled to a thermodynamical heat bath by looking at the Lindblad Master Equation of the system. From this equation, we determine the rate at which the normal modes decay. Certain normal modes decay very slowly, and some do not decay at all. These normal modes are called quasi-noiseless and noiseless clusters respectively. We determine what happens to the noiseless clusters when the network pa- rameters are very slightly perturbed. We have found that two distinct types of noiseless clusters can be identified. The first type disappears with even the slightest perturbation, making it useless in practice. The second type instead be- comes quasi-noiseless, making it a viable candidate for applications. We show how to determine the degree to which these noiseless clusters become quasi- noiseless by looking at the other normal modes of the network. We also explain how a network of oscillators, including an optional heat bath, can be simulated with an optical setup as described in [3]. We suggest this setup can be used to verify our findings.

The development of quantum computers is a monumental challenge for modern physics. One proposed pathway toward a fully scalable and fault-tolerant quantum computer involves the development of a quantum network. Such a network would have applications ranging from distributed quantum computation to fundamentally secure quantum communication. The building blocks of such a network, the nodes, require optical links with which to generate entanglement with other nodes, as well as memory and data qubits to improve the entanglement between nodes and perform computations. In this thesis, we study the dynamics of spin pairs surrounding nitrogen-vacancy (NV) centres in diamond - a promising proposed node for a quantum network. We build on previous work that has successfully detected and controlled pairs of strongly coupled 13C nuclear spins using the NV centre as a probe, and investigate how these spin pairs can be individually addressed using radio frequency pulses. Next, we consider pairs of P1 centres and demonstrate the detection and control of the electron spins of this pair. We show that dynamical decoupling sequences can be used to initialise and readout the electron pair of the P1 centres with high fidelity (∼94 − 96%), and measure natural dephasing times of ∼ 50 ms with the NV centre in the m s = 0 state. The control we demonstrate over pairs of electron spins in P1 centres is an important proof-of-concept that electronic spin defects in diamond can be coherently controlled, and used as memory or data qubits in a future quantum network node.

Quantum networks play an important role in the fields of quantum information and quantum computation. One of the current problems for these networks concerns nonlocality. Characterizing and detecting nonlocality is relevant to the implementation of quantum networks and quantum repeaters, where Bell inequalities can be used to test if configurations are prepared correctly. This report contains an overview of an iterative method to find Bell inequalities for networks. Starting from a given network, this method constructs a new Bell inequality for a network containing one additional source and one additional party. We use this procedure to find new Bell inequalities for specific network structures and analyse how these can be used to detect nonlocality within a network. In the first part the Bell inequalities are considered from a more theoretical point of view. We focus on star-shaped networks and discuss violations predicted by quantum mechanics. We look for quantitative bounds describing a set of states that lead to violation, giving an indication of the required quality of the sources. Finally this method is applied to a setup similar to the one described by Bernien et al. We consider a network consisting of three parties and two sources. The effect of errors during preparation of an entangled pair of photons on the ability to detect nonlocality is evaluated. The same is done for the effect of measurement errors. Using numerical computations we show that violation of the Bell inequality can be improved by choosing different measurement angles.

The NV-center in diamond is a promising system for quantum information processing. The typical qubits are the electron spin in the NV-center and the surrounding 13C nuclear spins. Fast and precise control of many 13C spins is desirable but challenging due to crosstalk. Crosstalk is the unwanted effect of a control pulse on the other qubits. The coupled NV-13C system is explored, and numerical techniques to optimize the control of multiple and single qubits are used. In this report the effect of the Bloch-Siegert shift in the weak driving regime is investigated first. This is done by studying the effect of a π-pulse when two carbon qubits are driven at the same time on resonance. It turns out that in this weak driving regime the effect of the off resonance is very small, and changes the fidelity not more than 0.3%. Furthermore the effect of a square πpulse on 4 nearby qubits is studied. If the pulse is not optimized (thus square), the pulse cannot be done in less than 100µs while retaining at least 99% fidelity. By optimizing the envelope of the driving field using constrained optimization the pulse can be done in 22µs while retaining at least 99% fidelity.

The Hamiltonian of the NV center contains unknown parameters, such as the zero-field splitting and the magnetic field. Knowing how the frequency measurements depend on the parameters will give greater understanding of the NV center. Including the nitrogen nuclear spin gives us the opportunity to find more parameters such as the quadrupole splitting. Analytically finding expressions for all of these parameters is very time consuming to do, and therefore the method of second order perturbation is used in this report to find approximations. With this method, estimations of the eigenvalue differences and parameters were found, including new estimations for the quadrupole splitting. As we don't exactly know whether the quadrupole splitting is positive or negative, there are two estimates found given by ΔQ+ = 4949.25(2) kHz and ΔQ- = -4949.11(1) kHz.

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.

Solid-state defect centers, such as the nitrogen-vacancy centers in diamond, represent a promising and versatile platform for quantum technologies. This thesis focuses on overcoming the challenge of noise in diamond to facilitate its practical use in various quantum technology applications.@en