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In this answer https://physics.stackexchange.com/a/9253/256311, the author said

Usually, we think of quantum mechanics to govern the microscopic world, involving length scales of under a micrometer, i.e. 10^−6 𝑚. Achieving quantum effects on a length scale of 100𝑘𝑚=10^5𝑚 means you span 11 orders of magnitude

My question is: Why do we care about the quantum mechanics scale? Say, two far-separated electrons approximately localized at $x$ and $y$, in system A and B, respectively. We care about how one electron entangle with another electron. Wow does length scale in each system matter? No matter how things going in System A, it has nothing to do with the distance between two systems.

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  • $\begingroup$ QM is NOT about length scales. Quantum effects are spanning the entire length of the universe. as can be seen in spectral lines in the light of objects that are now located beyond the other end of the observable universe. If you want something closer to home, superconductors can be made in any desired size and so can permanent magnets. Neither would be possible without quantum mechanics. The only scales in quantum mechanics that we care about are angular momentum and electric charge scales. Both are 1 in a rational choice of physical units. $\endgroup$
    – FlatterMann
    Commented Jul 20 at 23:16

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Why do we care about the quantum mechanics scale?

Distance is no object for entangled particles. The connection between entangled particles is non-local and instantaneous, so distance does not matter. Where distance does matter is at very short distances. While at a large distance of perhaps 100 kilometers an error of one atom length is almost negligible, but at very short distances approaching the Planck length, the uncertainty in the position of two particles due to the Heisenberg uncertainty principle becomes very significant. This is one reason why classical physics does not describe interactions at very short distance scales.

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  • $\begingroup$ Entanglement is neither non-local nor instantaneous. Both terms would describe physics that breaks relativity. Entanglement does not. The idea that the Heisenberg uncertainty relation somehow dictates length scale is also completely wrong. It has an angular momentum/action scale but NOT a length scale. Heavy objects behave perfectly classical for scales that are much, much smaller than e.g. a proton radius. $\endgroup$
    – FlatterMann
    Commented Jul 20 at 23:11
  • $\begingroup$ @FlatterMann the Planck scale is orders of magnitude smaller than a proton radius. John Bell himself said ""If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local.". Entanglement is non local unless you subscribe to one the weirder interpretations of quantum physics like determinism or the Many Worlds interpretation or Pilot Wave Theory. They are all convoluted attempts to overcome the problem of non locality, when it is not a problem in the first place. $\endgroup$
    – KDP
    Commented Jul 21 at 4:46
  • $\begingroup$ @Flattermenn The non locality in quantum physics most definitely does not violate relativity. You should know better than that. $\endgroup$
    – KDP
    Commented Jul 21 at 4:46
  • $\begingroup$ Nobody has ever seen any physics at the Planck scale. IMHI it's more religion than science at this moment. It also has absolutely nothing to do with quantum mechanics per se. It's gravitational physics whereas even a proton is merely QCD. The length scale at which the motion of a baseball is quantum mechanical has absolutely nothing to do with either. Locality is a statement about the interaction of physical systems that are in different points in space. No such interaction exists. Again, all non-locality violates quantum mechanics. Bell and other are simply using the wrong terminology. $\endgroup$
    – FlatterMann
    Commented Jul 21 at 18:44

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