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Bell theorem seems something extremely sophisticated, requiring pages of formulas to prove ... so I wanted to ask for the simplest known proofs and explanations for the violation?

Basing on Bell-type inequality from this note, I have recently prepared mine (paper) with example of violation using just the Born rule - it fits this diagram:

Simple proof of Bell theorem

Top: for any probability distribution among 8 possibilities for 3 binary variables $ABC$, the written inequality is fulfilled. Bottom: example of its violation using Born rule: probabilities being normalized squares of amplitudes.

In other words, we measure 2 out of 3 binary variables for prepared state:

$$\psi=\frac{1}{\sqrt{6}} \sum_{A,B,C=0}^1 \psi_{ABC} |ABC\rangle=\frac{1}{\sqrt{6}}(|001\rangle+|010\rangle+|011\rangle+|100\rangle+|101\rangle+|110\rangle) $$

I have worked on Maximal Entropy Random Walk (MERW, Wikipedia article), which can be seen as quantum correction to diffusion models, e.g repairing the lack of localization for standard diffusion, wrongly concluding that semiconductor is a conductor.

It is basically uniform or Boltzmann distribution among entire paths - like in Feynman's Euclidean path integrals, so it recreates some quantum properties like stationary probability distribution being exactly like for the quantum ground states (with e.g. Anderson localization).

MERW also has the above Born rule, providing a simple explanation for violation of Bell inequalities by our physics. Here is a diagram comparing three philosophies for statistical physics for a single particle:

enter image description here

Only the last philosophy: Boltzmann distribution among full paths agrees with quantum predictions. Assuming ensemble among half-paths toward past or future, the probability distribution turns out to be quantum amplitude. To get randomly same position in a given moment, we need to "draw it" twice: from past and future half-paths, getting the Born rule: probability being normalized square of amplitude.

Is it the correct explanation why our world violates Bell inequalities?

Any other non-magical explanations?

ps. Here I have tried to discuss it in Bell-theorem focused forum.

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closed as off-topic by ACuriousMind Aug 12 '17 at 15:24

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  • $\begingroup$ All your $\psi_{ABC}$ are real numbers, aren't they? Then your Hilbert space is of dimension 1 and there is no way it works. $\endgroup$ – user154997 Aug 12 '17 at 12:38
  • $\begingroup$ This example was originally for MERW, where amplitudes are always real numbers, also quantum ground state amplitude can be always chosen as real (from Frobenius-Perron theorem). Generally, for complex amplitudes we additionally need to take absolute value, but I wanted a simple example of violation of Bell inequalities - where real ones are sufficient. $\endgroup$ – Jarek Duda Aug 12 '17 at 13:16
  • $\begingroup$ I was not opposing real to complex but pointing out that I don't see the Hilbert space vectors, real numbers being scalars. $\endgroup$ – user154997 Aug 12 '17 at 13:34
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    $\begingroup$ I am afraid that without reading your paper it is extremely unclear to me as a reader what your explicit argument for the violation actually is, and you therefore seem to be effectively asking us to review your paper when you ask whether this is correct, but peer review is off-topic here. Additionally, the right answer to "Is this correct?" is potentially "Yes.", which is too short to submit as an answer. $\endgroup$ – ACuriousMind Aug 12 '17 at 15:24
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    $\begingroup$ Soo...what is the question? You've given an example of a violation of the Bell inequality, namely a system of three qubits (which is clearly not the simplest one since the famous Bell states are for two qubits). Then you seem to say that MERWs also violate Bell's inequality, and rather suddenly compare three different approaches to statistical physics. What do you want to know about that, and what has this to do with the initial example of the 3-qubit system? $\endgroup$ – ACuriousMind Aug 12 '17 at 16:16