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

我提议的项目是调查信息论原理的意义和成功，旨在从其他无信号理论中分离出量子力学关联，特别是通过从一般的无信号关联、贝尔不等破坏关联中缩小到量子力学关联。不可否认，量子力学在预测几乎所有已知现象方面取得了成功，但我们仍然没有对许多方面有深入的了解，包括它的本体论、解释，以及对纠缠、EPR实验结果和Aharonov-Bohm效应等现象的洞察。有时，人们争辩说，这种理解的缺乏是因为我们缺乏量子力学的直觉公理，也就是“物理”公理。Popescu和Rohrlich提出，非定域性(在这里被认为意味着贝尔不平等破坏)可以作为量子力学的一个物理意义公理。Popescu和Rorlich研究了量子力学是否可以仅仅通过非局域性和无信号存在的公理来得出，即量子力学是否是同时符合非局域性和无信号的唯一理论。他们发现答案是否定的；可能会有既非本地化的理论，又允许更强烈地违反贝尔不平等。这一事实提供了几种有趣的可能性：要么自然界真的比量子力学所暗示的更具非局域性，要么有另一个原理限制了自然界的非局域性。自那以后，人们对可能的信息理论、与设备无关的原理进行了大量的研究工作，以实现这一角色。文献中的建议包括非平凡的通信复杂性、非局部计算没有优势、信息因果关系、宏观局部性和局部正交性。Navascues等人。定义了一组关联，几乎量子的集合，并证明它严格地大于量子关联的集合，但满足所有提出的原理，除了信息因果原理。除了寻找一个将量子力学从其他无信号理论中分离出来的原理的挑战之外，还存在如何考虑它的问题。许多作者对这一具有根本意义的原则发表了评论。然而，正如丁普森所主张的那样，了解一个理论的边界是否应该提供对量子力学本体论的任何洞察，或者它告诉我们关于测量作用的东西，并不是很明显。此外，人们可能会对基本意图的概念感到疑惑。对于我的DPhil项目，我想通过考虑以下问题来进一步研究这些争论：我们应该为量子力学寻求一个原理理论吗？量子力学的信息论公理解释有助于它的解释吗？如果能找到一个从无信号多面体中挑出量子力学的原理，它能被赋予什么地位？在什么意义上，这样的原则是根本的？这些考虑对量子力学的信息论方法有什么启示？能给出几乎量子集的物理理论吗？在什么意义上，几乎量子集背后的数学考虑是不自然的？满足所提出的信息论原理的集合或不同的集合可以使用更自然的考虑来推导吗？

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?