What actually is the definition of locality and non-locality? Does non-locality in Quantum Mechanics mean however far you separate 2 entangled atoms in space, the 2 atoms can still influence each other? Doesn't non-locality mean action at distance? If not, what's the difference between them?
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The short answer is yes, in quantum mechanics quantum non-locality refers to the apparent instantaneous propagation of correlations between entangled systems, irrespective of their spatial separation. In quantum field theory, the notion of locality may have a different meaning, as pointed out already in a comment.
Details: The notions of locality and non-locality in Quantum Mechanics have been originally defined in the context of the EPR controversy between Einstein and Bohr on the phenomenon of quantum entanglement.
Basically, the general "principle of locality" (Wikipedia ref.) requires that "for an action at one point to have an influence at another point, something in the space between the points, such as a field, must mediate the action". In view of the theory of relativity, the speed at which such an action, interaction, or influence can be transmitted between distant points in space cannot exceed the speed of light. This formulation is also known as "Einstein locality" or "local relativistic causality". It is often stated as "nothing can propagate faster than light, be it energy or merely information" or simply "no spooky action-at-a-distance", as Einstein himself put it. For the past 20 years or so it has been referred to also as the "no-signaling" condition.
The phenomenon of entanglement between quantum systems raised the non-locality problem first noted in the EPR paper: A projective measurement on a quantum system at one space location instantly collapses the state of an entangled counterpart at a distant location. Quantum mechanical non-locality refers to this apparent entanglement-mediated violation of Einstein locality.
The remarkable thing about quantum non-locality, however, is that it actually does not imply violation of relativistic causality. Although entanglement correlations are affected instantaneously, they cannot be harnessed for faster-than-light communications. The reason is that the outcome of the local projective measurement is itself statistic and cannot be predicted beforehand. If the same kind of measurement is performed on multiple copies of identically prepared pairs of entangled systems, the overall statistical result is that locally both systems conform to the statistics prescribed by their respective local quantum states. The "spooky action-at-a-distance" of the distant measurement gets wiped out in the total statistics.
There is already a host of related questions on Physics.SE. See for instance Quantum entanglement and spooky action at a distance and similar.
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Well, well, well...some rather predictable phrases, but the one that fascinates me is this one that no one seems to question: "Nothing can propagate faster than light, be it energy or merely information." That is not quantum thinking. That is Newtonian thinking. Quantum physics says, So, what if it did? How many serious traditional physicists are willing to question that? I do. As I have postulated/published elsewhere in the Dallmann-Jones Monograph, we have been too hasty to accept this as fact. It is as simple as being willing to explore in what instances there could be an "instantaneous effect" at a distance, and there is nothing spooky about it. Here is the situation that at least proves my point is as possible as all of the mathematical calculations others have offered up to deny it. (Sometimes, truth is quite simple, and also is despised by the supposed "Deep Thinkers.")
How two things can be entangled and influence instantaneously... SPACE IS A BROOMSTICK
Imagine two people a light year apart: We will call them Mr. Alpha and Mr. Omega. They are facing one another across a lightyear. So, the Newtonian theorists say, "If an electron is spun clockwise where Mr. Alpha is, that it would take at least a light year for the electron in possession of Mr. Omega to spend counter-clockwise." So, it cannot happen faster than traveling 5.88 trillion miles. Let's look at it another quantum way: Between Mr. Alpha and Mr. Omega is a rotating two-inch thick broomstick. Mr. A is looking at it rotate in a clockwise direction. What does Mr. O observe? He sees it spinning counter-clockwise! That explains how the position of the observer changes the effect. Secondly, if Mr. Alpha shoves the broomstick towards Mr. Omega, how long will it take to move at Mr. Omegas's end? Instantly! Ah, but there is no broomstick that long, some would say. True. But what if the space in between Mr. A. and Mr. O acts like a broomstick? We have too long held onto the idea of, "Oh that desk you see is not solid. It is mostly space, i.e., all of humanity, if you removed all the space in their atoms, "would fit inside a sugar cube" That kind of thinking - seeing space as 'emptiness' - is where we got off track. Space is a thing. Space is a broomstick.
