This thesis is concerned with a seemingly naive question: what happens when you separate two physical systems that were previously together? One of the greatest discovery of the last century is that systems that obey quantum mechanical instead of classical laws remain inextricably linked even after they are physically separated, a phenomenon known as entanglement. This leads immediately to another, deep question: 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 the thesis deals with these questions by considering quantum theory as part of a wider landscape of physical theories, collectively called general probabilistic theories (GPTs). Chapter 1 reviews the compelling motivations behind the GPT formalism, preparing the ground for Chapter 2, where we translate the above questions into precise conjectures, and present our progress toward a full solution. In Chapter 3 we consider entanglement at the level of measurements instead of states, which leads us to the investigation of one of its main implications, data hiding. In this context, 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 explore some problems connected with quantum entanglement. In Chapter 4 we discuss its resistance to white noise, as modelled by channels acting either locally or globally. Due to the limited number of parameters on which these channels depend, we are able to answer all the basic questions concerning various entanglement transformation properties. The following Chapter 5 presents our view on the topic of Gaussian entanglement, with particular emphasis on the role of the celebrated ‘positive partial transposition criterion' in this context. Extensively employing matrix analysis tools such as Schur complements and matrix means, we present unified proofs of classic results, further extending them and closing some open problems in the field along the way.
The third part of this thesis concerns more general forms of non-classical correlations in bipartite continuous variable systems. In Chapter 6 we look into Gaussian steering and problems related to its quantification, moreover devising a general scheme that allows to consistently classify correlations of bipartite Gaussian states into ‘classical’ and ‘quantum’ ones. Finally, Chapter 7 explores some problems connected with a ’strong subadditivity’ matrix inequality that plays a crucial role in our analysis of correlations in bipartite Gaussian states. Among the other things, the theory we develop allows us to conclude that a Rényi-2 Gaussian version of the elusive squashed entanglement coincides with the corresponding entanglement of formation when evaluated on Gaussian states.