This PhD Dissertation collects results of my own work on the topic of continuous variable (CV) quantum teleportation, which is one of the most important applications of quantum entanglement, as well as on the understanding, quantification, detection, and applications of a type of quantum correlations known as Einstein-Podolsky-Rosen (EPR) steering, for both bipartite and multipartite systems and with a main focus on CV systems. For the first results, we examine and compare two fundamentally different teleportation schemes; the well-known continuous variable scheme of Vaidman, Braunstein and Kimble, and a recently proposed hybrid scheme by Andersen and Ralph. We analyse the teleportation of ensembles of arbitrary pure single-mode Gaussian states using these schemes and compare their performance against classical strategies that utilize no entanglement (benchmarks). Our analysis brings into question any advantage due to non-Gaussianity for quantum teleportation of Gaussian states. For the second part of the results, we study bipartite EPR-steering. We propose a novel powerful method to detect steering in quantum systems of any dimension in a systematic and hierarchical way. Our method includes previous results of the literature as special cases on one hand, and goes beyond them on the other. We proceed to the quantification of steering-type correlations, and introduce a measure of steering for arbitrary bipartite Gaussian states, prove many useful properties, and provide with an operational interpretation of the proposed measure in terms of the key rate in one-sided device independent quantum key distribution. Finally, we show how the Gaussian steering measure gives a lower bound to a more general quantifier of which Gaussian states are proven to be extremal. We proceed to the study of multipartite steering, and derive laws for the distribution of Gaussian steering among different parties in multipartite Gaussian states. We define an indicator of collective steering-type correlations, which is interpreted operationally in terms of the guaranteed secret key rate in the multi-party cryptographic task of quantum secret sharing. The final results look at the cryptographical task of quantum secret sharing, whose security has remained unproven almost two decades after its original conception. By utilizing intuition and ideas from steering, we manage to establish for the first time an unconditional security proof for CV entanglement-based quantum secret sharing schemes, and demonstrate their practical feasibility. Our results establish quantum secret sharing as a viable and practically relevant primitive for quantum communication technologies.

We studied the impact of graph structure and temperature has on the sizes and lifetimes of crowds (mesoscopic clusters of consensus) that form when due to Ising Spin dynamics. Spin models have proven successful in modelling the formation of global consensus in opinion for large groups of people. We focus on the case where lots of small groups form consensuses but no global consensus forms. Our results can be interpreted in the light of consumer opinion and give an explanation for fads and brand reputation. Data was obtained by performing Monte Carlo simulations of the Ising Model on various graphs, our key control parameter was the inverse temperature β =1/kBT. In the Ising Model each vertex chooses an opinion −1 or +1 each time step based on the opinions of its neighbours. Low β ≪ 1 exhibit as vertices behaving noisily (flipping opinion often), while β → 1 tend to be quieter (flipping rarely). Interactions between neighbours creates feedback loops which determine crowd properties. The simulations we perform start with β_q = 0 for 0 ≤ t < 500, then the temperature is sharply quenched to a value βq ≥ 0 and kept constant for t ∈ (500, 15000]. We analyse the data in the time period 13000 ≤ t ≤ 15000. The simulations were performed on the 2-d rectangular lattice (RL), K-degree 1-d circular lattice (CL), Erdős-Rényi (ER), and 1-d Watts-Strogatz (WS) graphs. We conclude that crowds with multiple unique feedback loops are more robust against noisy vertices and crowd-crowd interactions. On the CL many feedback loops use the same paths which makes it easy for noisy vertices to disrupt the loops simultaneously. However 1-d structure limits crowd-crowd interactions making crowds strong against absorption by their neighbours. On the RL there are many paths between vertices, thus feedback loops can be easily rerouted around noisy vertices and crowds are robust against noise. On weakly connected ER graphs many feedback loops share the same paths and thus crowds are extremely susceptible to noisy vertices. Disorder in the 1-d lattice structure in WS graphs reduces the number of feedback loops in certain regions, this makes some crowds weak against both noisy vertices and neighbouring crowds. On the RL, CL and WS graphs we observe that size of crowds changes like an anomalous diffusion process. The anomalous diffusion exponent α was determined by plotting the root mean square change in crowd size sqrt(< ΔP² >) as a function of the crowd lifetime L. On the RL changes in crowd size were super-diffusive for 0.35 ≲ β ≲ 0.65; crowds grew very large in short periods of time, however such crowds also had short lifetimes as they were constantly replaced by new crowds. This behaviour is similar to the coming and going of fads. For 0.65 ≲ β ≲ 1 crowd sizes changed sub-diffusively, crowds grew in very small amounts to very large sizes over long time periods, this is analogous to the spread of reputation through word of mouth.