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Abstract

Entropy may be regarded as the physical quantity with the most facets. It is successfully used to describe aspects of the three intriguing phenomena: equilibrium, uncertainty and entanglement. Despite its central meaning, formulating physical laws in terms of entropy can become unfavorable, for which we give three examples: the lack of well-defined continuum limits often prevents universal descriptions for discrete, continuous and infinite-dimensional degrees of freedom. The entropy of a subregion in a quantum field theory exhibits an ultraviolet divergence, which can not be renormalized. Entropies of marginal distributions do not capture the full information about a global distribution. The first two problems can be traced back to the entropy being an absolute measure of missing information. To overcome these, we propose a more regular use of relative entropy in situations where entropy shows its flaws. Relative entropy allows us to unify entropic descriptions or to extend them into new regimes of validity. In this thesis, we use relative entropy to formulate a new principle of inference, to develop thermodynamics in terms of model states, to derive divergence-free second law-like inequalities for relativistic fluids, to unify entropic uncertainty relations for discrete and continuous variables and to deduce the first entropic uncertainty relation for a quantum field. The third problem becomes relevant in the context of entanglement witnessing for continuous variable systems. In contrast to standard separability criteria, which are based on measuring two observables separately, we start from a phase space representation of the quantum state. We find a perfect witness for pure state entanglement and derive entropic and even more general separability criteria, allowing us to certify entanglement in undetected regions.