Non-classical correlations in quantum mechanics and beyond

Is entanglement an exclusive feature of quantum systems, or is it common to all non-classical theories? And if this is the case, how strong is quantum mechanical entanglement as compared to that exhibited by other theories? The first part of this thesis deals with these questions by considering quantum theory as part of a wider landscape of physical theories, collectively called general probabilistic theories (GPTs). Among the other things, this manuscript contains a detailed introduction to the abstract state space formalism for GPTs. We start with a comprehensive review of the proof of a famous theorem by Ludwig that constitutes one of its cornerstones (Ch. 1). After explaining the basic rules of the game, we translate our questions into precise conjectures and present our progress toward a full solution (Ch. 2). In Ch. 3 we consider entanglement at the level of measurements instead of states, focusing on one of its main implications, i.e. data hiding. We determine the maximal data hiding strength that a quantum mechanical system can exhibit, and also the maximum value among all GPTs, finding that the former scales as the square root of the latter. In the second part of this manuscript we look into quantum entanglement. In Ch. 4 we discuss the entanglement transformation properties of a class of maps that model white noise acting either locally or globally on a bipartite system. In Ch. 5 we employ matrix analysis tools to develop a unified approach to Gaussian entanglement. The third part of this thesis concerns more general forms of non-classical correlations in bipartite continuous variable systems. In Ch. 6 we devise a general scheme that allows to consistently classify correlations of bipartite Gaussian states into classical and quantum ones. Finally, Ch. 7 explores some problems connected with a certain strong subadditivity matrix inequality.