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I feel like I have done quite a bit of reading about Bell's test experiments and quantum entanglement. I see very well how Bell's test experiments provide different outcomes for a local realism hypothesis vs. what quantum mechanics tells us.

Many articles assert that Bell's test experiments also "prove" or at least validate the Heisenberg uncertainty principle. However, I fail to see any direct connection between Bell's test experiments and the Heisenberg uncertainty principle. All I see is that the reality of quantum entanglement is being validated, as well as the spooky-action-at-a-distance idea.

So, how exactly is it that a Bell's test experiment is any sort of direct validation of the HUP?

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    $\begingroup$ It may not be a direct validation, per se, but the Heisenberg uncertainty principle cannot be true under a local realist framework (as the fundamental uncertainty in the value of characteristics of a particle violates realism). $\endgroup$ – probably_someone Jul 11 '18 at 18:35
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When people say that the Bell's test results validate the HUP, they probably mean that the results invalidate what was until then the most widely accepted argument against the HUP, which was the Einstein-Podolsky-Rosen argument.

In my opinion, the historical EPR paradox is usually badly (or at least anachronistically) presented. Most accounts that I've seen present its main punchline as the nonlocal collapse of the wavefunction, or as the instantaneous creation of nonlocal correlations (albeit ones that cannot be used to transmit information). This presentation is implicitly taking a realist interpretation to quantum mechanics, as has become more popular in recent decades.

But the original EPR argument actually reached a very different conclusion. EPR took it as obvious that no influences, whether capable of transmitting information or not, could propagate faster than light, so they took it as a given that no measure of one particle could possibly affect a spacelike separated particle in any way, including by "collapsing its local wavefunction". In modern language, we would say that they were implicitly taking an epistemological interpretation of QM. They argued that by accurately measuring the position of particle A and the momentum of the spacelike separated particle B, you could gain knowledge of both the position and momentum of particle A (or B), because (they assumed) the measurement process on B could not have possibly affected A or changed its values. This conclusion is incompatible with the HUP.

Roughly speaking, the Bell's test results ruled out "strong locality", and showed that nonlocal correlations can indeed be established, although they can't necessarily transmit information. (That rough summary is actually sweeping a lot of subtleties under the rug. What the Bell test results actually ruled out is counterfactual definiteness theories; they are still compatible with strongly local interpretations such as superdeterminism and many-worlds.) This invalidates EPR's assumption that no measurement on particle B can affect a spacelike separated particle A, and therefore also invalidates their conclusion that the HUP sometimes fails.

It's important not to caricature EPR's assumptions; they were not assuming that nature is completely classical, as is also sometimes implied. They were completely open to the possibility that nature is fundamentally nondeterministic, or that measurements can inevitably change the state of a system in a way that isn't true classically. Their only incorrect assumption was strong locality, but even this modest assumption turned out to be too strong.

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