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I'm watching some archived video lectures on QM in Coursera given by Umesh Vazirani from UC Berkeley and I have a question regarding a Bell's experiment (I guess something close to this) described in the lecture. I'm going to link to some of the videos, hoping that I'm not infringing any copyrights. If you think I am, kindly let me know and I'll remove the links immediately.

I don't know if it's an oft-used one, but the experiment is described as such: There are two boxes far apart with inputs 0 and 1 for which both of these boxes should output either 0 or 1 so, that if the inputs are both 1 then the output bits should be different, otherwise the output bits should be the same (as in same bits for both boxes, it doesn't matter if the bits are both 0s or 1s).

Now it's said that if it were for some hidden variable, the highest rate of success for this experiment could be 75% (both boxes always outputting 0 or something like that). This is the claim I can't agree with, but I'll get there in a bit. The second thing that's said, is that if we use QM then the success rate can be higher (85% is mentioned which is roughly equal to $cos^{2}(\frac{\pi}{4})$). The video overview of this experiment can be found here.

There's also another video which explains how the 85% success rate is found. Although, my own calculations show that the maximal possible achievable success rate is approximately 89.5%. This can be achieved by slightly adjusting the $\theta$ used in defining the measuring basis. But this is besides the point.

Now my question is: Does such an experiment really disprove all the local hidden variable theories, or is it simply Wittgenstein's ladder in action? I think the local hidden variable theories weren't given a fair chance. Would the success rate be lower than the 89.5% I mentioned, when we "entangle" the particles as we did in the QM model and run the same measurements, but expect no actual entanglement between these particles. Which means that instead of entanglement we describe these two particles as synced up so that their local hidden variables will cause the measurement readings to yield certain values. This way we can use the knowledge that they're synced up to build two boxes which might have the success rate as high as in the QM model.

Now reading up on Bell's theorem from Wikipedia I'm given the understanding that in an actual experiment the success rates might differ, but ever so slightly. That difference, I understand, comes from some kind of a Bell's inequality defined by the actual measurements done on the system. Is this more or less correct?

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  • $\begingroup$ Could you explain precisely what you mean by 'synced up'? At first glance, it seems you might be wanting to allow what happens to one particle to affect the other... this violates the assumption of local hidden variables. A local hidden variable is some quantity that determines the outcomes of measurements on a particle, depending only on the physical properties of the particle and measuring device at the time and location of the measurement. $\endgroup$ – Mark Mitchison Dec 30 '12 at 17:39
  • $\begingroup$ What I mean by syncing up is that the local hidden variables get the same (or opposite, or maybe even some other relationship) values if these particles are brought together. That way, if you now separate these particles, you still have the information that they were together at some point and based on this knowledge you can expect these particles to yield similar (or, once again, opposite or whatever) results under measurement. Using this expectation, you, by my understanding, could design two boxes that get better success rates than 75%, but that don't rely on QM. $\endgroup$ – Deiwin Dec 30 '12 at 18:10
  • $\begingroup$ Maybe a comment to accompany the -1? A suggestion on how to improve the question? Or at least a note on which parts of it are weak. I'm quite new to QM, but my lack of knowledge in the field alone shouldn't merit a downvote. I'm here to learn and you (whoever you are) are definitely not helping. (Maybe you're helping the community by filtering out bad questions, but I'm obviously too close to it to view the action as such) $\endgroup$ – Deiwin Dec 31 '12 at 1:03
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The statement "The observed violations of the Bell inequalities disprove local hidden variable theories" is a profound one, and certainly does not qualify as a didactically over-simplification.

In essence, what these violations tell us is that you can't build quantum physics from some hidden variable theory, unless you are happy to include into this theory some magic that 1) makes the hidden variable theory unphysical (non-locality) or 2) makes it at least as strange as quantum mechanics (negative probabilities).

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    $\begingroup$ I wouldn't dismiss non-local theories out of hand: string theory comes with its own flavour of non-locality via the holographic principle and atemporal/time-symmetric theories (transactional interpretation) are imo quite elegant from a philosophical point of view (even though personally I don't really buy the identification of the wave function and its conjugate as offer and confirmation wave) $\endgroup$ – Christoph Dec 30 '12 at 18:08
  • $\begingroup$ There is nothing which makes a theory which is not Einstein-causal (which is misleadingly named "non-local", as if a theory with, say, a maximal speed of information transfer of, say, 1000 c would not be a local theory) unphysical. If a theory is physical or not depends on it making testable predictions. The ("nonlocal") de Broglie-Bohm interpretation makes them, the same as QT itself, thus, is a physical theory. $\endgroup$ – Schmelzer Jun 10 '16 at 11:41
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In QM the measurement process is done instantly. If the two particles of the system are separate in space the two particles change their state instantly (from a linear combination of spin to a specific state). QM said that the state before the measurement is not a determined state but a linear combination of the two states of spin.

$$| \phi \rangle = {1\over \sqrt 2 }|\uparrow \rangle+ {1\over \sqrt 2 }|\downarrow \rangle $$

This is the real state, that is, the reality of these particles when they travel through space.

When the observation process occurs the state is $|\uparrow \rangle$ or $|\downarrow \rangle$. QM does not say that the spin state is determined when the particles travel through space.

Bell's Inequalities is a theorem that allows one to statistically calculate the difference between the QM evolution of state and the classical evolution in which the state of evolution is determinated $ |\uparrow \rangle$ or $|\downarrow \rangle$ even if we don't know the real state of the particle.

Alain Aspect in 1982 definitely resolved the EPR paradox experimentally by saying that QM is correct and classical hidden variables don't exist.

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  • $\begingroup$ From a wiki article: However, his results were not completely conclusive, since there were so-called loopholes that allowed for alternative explanations that comply with local realism. $\endgroup$ – Deiwin Oct 27 '13 at 16:27

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