I have recently found out1 that the concepts of non-locality and Heisenberg's uncertainty do not live independently of one another in Quantum Mechanics. In other words, roughly the idea goes as: the degree of non-locality in Quantum Mechanics is fundamentally linked to Heisenberg's uncertainty.
This must undoubtedly be of fundamental importance in what forms our understanding of Quantum Mechanics, en par with Bell's theorem and the concept of entanglement. It would be brilliant if someone could explain on a relatively intuitive level, how the concepts of non-locality and uncertainty are connected, in order to hopefully shed some light on some of the following questions:
What is the key idea behind the fact that these two concepts can be quantifiably linked to one another? On the one hand, uncertainty relations are fundamental bounding relations for products of standard deviation of two observables, on the other hand it is far from obvious to see how one quantifies the degree of locality for the state of quantum mechanical system.
Does the locality here refer to that of correlated measurements of many body systems (e.g. two distant photons that are entangled) or that related to topological degrees of freedom of a system that are insensitive to local perturbations (interactions)?
Are these two concepts inter-related only for certain observables? Or is it a generally valid statement?
How do the repercussions (on locality) shift when one considers commuting observables instead of non-commuting ones?
This is merely an attempt to obtain some insight on the discovery of this new property in QM, and please note that the questions here serve more to further clarify my level of confusion and lack of understanding, and that any answer that aims to provide, a basic understanding, be it intuitive or not, without necessarily aiming at these specific questions, will be perfectly acceptable.
tldr: what is linking these two concepts together? Is it another formulation of delocalization of wavefunctions once one considers incompatible observables?
1Oppenheim, Jonathan, and Stephanie Wehner. "The uncertainty principle determines the nonlocality of quantum mechanics." Science 330.6007 (2010): 1072-1074. arXiv:1004.2507