An investigation into significance and success of information-theoretic principles that aim to single out quantum mechanics from other non-signalling

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My proposed project is an investigation into the significance and success of information-theoretic principles that aim to single out quantum mechanical correlations from other no-signalling theories, in particular, by narrowing down to quantum mechanical correlations from amongst general no-signalling, Bell-inequality violating correlations.The success of quantum mechanics in its prediction of nearly all known phenomena is undeniable, yet we still do not have a deep understanding of many aspects, including its ontology, interpretation, and an insight into phenomena such as entanglement, results of EPR experiments and the Aharonov-Bohm effect. It is sometimes argued that this lack of understanding arises because we lack intuitive, or 'physical' axioms for quantum mechanics. It was proposed by Popescu and Rohrlich that non-locality (where here this is taken to mean Bell-inequality violation) can serve as a physically meaningful axiom for quantum mechanics. Popescu and Rorlich investigated whether quantum mechanics can be arrived at solely through the axioms of the existence of non-locality and no-signalling, i.e., whether quantum mechanics is the unique theory consistent with both non-locality and no-signalling. They found that the answer was negative; there can be theories that are both non-local and allow for stronger violations of Bell inequalities. This fact offers several interesting possibilities: either nature really is more non-local than quantum mechanics suggests, or there is another principle that limits how non-local nature can be. There has since been considerable research effort into possible information-theoretic, device-independent principles that could fulfil this role. Proposals in the literature include non-trivial communication complexity, no advantage for nonlocal computation, information causality, macroscopic locality and local orthogonality. Navascues et al. define a of set of correlations, the 'almost quantum' set, and show that it is strictly larger than the set of quantum correlations but satisfies all the proposed principles, with the exception of the information causality principle.In addition to the challenges in finding a principle that singles out quantum mechanics from other no-signalling theories, there is also the question of how it should be considered. Many authors make remarks about such a principle having fundamental significance. However, as Timpson argues, it is not obvious that knowing the boundaries of a theory should offer any insight into the ontology of quantum mechanics, or what it tells us about the role of measurement. In addition, one might wonder about the notion of fundamental intended. For my DPhil project, I would like to investigate these debates further by considering the following questions: Should we seek a principle theory for quantum mechanics? Does an information-theoretic axiomatic account of quantum mechanics aid its interpretation? If a principle can be found that does single out quantum mechanics from the no-signalling polytope, what status can it be given? In what sense is such a principle fundamental? What light do these considerations shed on information-theoretic approaches to quantum mechanics? Can a physical theory be given for the almost quantum set? In what sense are the mathematical considerations behind the almost quantum set not natural? Can the set or a different set satisfying the proposed information-theoretic principles be derived using considerations that are more natural?