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:

  1. 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.

  2. 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)?

  3. Are these two concepts inter-related only for certain observables? Or is it a generally valid statement?

  4. 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

  • $\begingroup$ I briefly skimmed this paper, and will have study it in detail in order to respond directly to your questions, The paper seems to be proposing a method to compare alternative theories of quantum mechanics (e.g., hidden variables, extensions, etc), and offers a theoretical way to compare them. At first glance there appears to be no changes to standard QM, hence the probable answers to (5) is no. $\endgroup$ – Peter Diehr Feb 5 '16 at 13:33
  • $\begingroup$ For starters, Bell's theorem is of no importance to our understanding of quantum mechanics and you really need to understand why it isn't by looking up the definition of science and comparing that to what Bell's theorem does. That the world is, on a profound level, non-local, follows already from its homogeneity and isotropy in classical mechanics. How else would it "know" to be "the same" "over there" as it is "over here"? Except that it does not actually have to be so. The world may not be non-local at the shortest scale and all of this may just be emergent phenomena at our scale. $\endgroup$ – CuriousOne Feb 5 '16 at 14:39
  • $\begingroup$ What is your quantifiable definition of "non-locality"? Usually I'd just say the HUP shows non-locality by forcing states localized (in the sense of narrow standard deviation) in momentum space to be delocalized (in the sense of having large standard deviation) in position space. E.g. a sharp momentum Gaussian corresponds to a very spread out position Gaussian. But I'm not really sure what you're asking. $\endgroup$ – ACuriousMind Feb 5 '16 at 14:47
  • $\begingroup$ I think it would increase your chances of getting good answers if you would (i) only ask one question (not five), and (ii) remove the "opinion-based" part (e.g. question 5). $\endgroup$ – Norbert Schuch Feb 6 '16 at 9:45
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    $\begingroup$ "serve mainly to clarify on what ideas I m lost." -- Then, you should explain which of the ideas in the paper you understand. Otherwise, you're asking someone to read the paper for you and give you a summary. For my part, I consider that unrealistic. BTW, did you read Oppenheim's blog post on the paper - maybe it helps? $\endgroup$ – Norbert Schuch Feb 6 '16 at 10:10

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