I have posted a paper on the vixra website at http://vixra.org/abs/1610.0327
with the title:
Correlation of – Cos θ Between Measurements in a Bell’s Inequality Experiment Simulation Calculated Using Local Hidden Variables
and abstract:
This paper shows that the theoretical correlation between elementary particles’ hidden variable unit vector spin axes p, projected onto Alice’s and Bob’s respective detector angle unit vectors a and b, in a Bell’s Inequality experiment, is – cos θ. This equates with the quantum correlation value and exceeds the “Bell’s Inequality” attenuated correlation in absolute magnitude. Further, aggregates of elementary particles’ hidden variable unit vector spin axes p, when projected onto appropriate detector angle vectors, give values which break the Bell’s Inequalities in exact accordance with values given by Quantum Mechanics calculations. On the other hand, Bell’s attenuated correlations correspond to correlations calculated without fractionalising the raw integer measurements A and B made by Alice and Bob. Also, when aggregating the raw integer measurements, the Bell’s Inequalities are not broken.
To be clear, neither in the paper nor in this blog am I claiming to have broken Bell’s Inequality with a model based on hidden variables. To make such a claim would at the least require the use of integer measurements, that is involve fuzzy vectors rather than exact vectors in my terminology below.
To summarise my findings. There is a 2 x 2 way of looking at my two computer simulations in this paper.
| Exact vectors _ | Fuzzy vectors on a hemisphere | |
| My Simulation #1: correlations | (a) The Quantum Mechanical correlation of 0.707 for theta = 45o cannot be directly calculated via QM but
(b) has been directly calculated in my Simulation#1, so there is nothing spooky going on here despite 0.707 breaking Bell’s Inequality. The magnitude of this correlation would be equivalent to a CHSH statistic S=2.828 except that the CHSH statistic only applies to fuzzy vectors, not to exact vectors. |
The Bell correlation of 0.5 for theta = 45o is the mundane attenuated correlation associated with failure to break Bell’s Inequality. Equivalent to a CHSH statistic of S=2.
The most recent CHSH experiment finds S=2.4 based on 245 pairs of particles. See https://arxiv.org/abs/1508.05949
|
| My Simulation #2: proportions | (a) There exist truly amazing QM calculations of proportions which are projections on exact vectors by Susskind which are accurately matched by
(b) my Simulation#2 calculations, so there is nothing spooky going on here despite them breaking Bell’s Inequality See https://www.youtube.com/watch?v=XlLsTaJn9AQ&p=A27CEA1B8B27EB67 AND http://www.lecture-notes.co.uk/susskind/quantum-entanglements/lecture-5/violation-of-bells-theorem/ |
Bell proportions, mundane values which do not break Bell’s Inequality |
The fuzzy vector column results are straightforward, and my simulations show these as failures to break the Bell’s Inequality. They both use fuzzy vectors on a hemisphere. Fuzzy vectors on a hemisphere correspond to raw measurements by Alice and Bob at their detectors. They have unit magnitude but the vector is only known to be pointing at either one hemisphere or the other. Hence a fuzzy vector. Proportions and correlations both use simple counts and sums of 1s and -1s to obtain the mundane results. The exact particle vector direction is irrelevant to these ‘fuzzy’ calculations.
Next on to Column 1: exact vectors. Amazingly, QM allows calculations of proportions which break the Bell Inequality. I used Susskind’s online example of a Bell Inequality in my Simulation 2 for this.
Even more amazingly, I obtained the QM values for these proportions very accurately using my hidden variable Simulation 2!!!
What my simulation 2 does is start with the hidden particle unit vectors and use the standard dot product calculation of projections or loadings onto the detectors’ exact vectors. These projections are not integer values in general and so fractional values are being added along exact vectors. As this agrees accurately with QM calculations and I conclude that QM calculations are measuring the same thing. No real surprise as these QM calculations use Projection Operators! Bell’s Inequalities however are designed to be applied only to integer measurements.
So far so good.
It is the correlations using exact vectors which causes all the problems of misunderstanding in this field.
I have found in my Simulation 1 correlation = 0.707 for theta = 45 degrees.
My simulation 1 starts with the exact particle unit vectors and again uses the standard dot product calculation of non-integer projections onto the detectors’ exact vectors. So again fractional projection values are being added along exact vectors.
AFAIK, no QM calculation can go directly to correlation= 0.707 because it requires the knowledge of the hidden variables i.e. the particles’ exact directions, for individual particles, and these are never, ever known in a real experiment. So calculating a correlation for an exact vector needs a lot more information than does calculating an overall proportion along an exact vector. Too much information for a real experiment.
However, my Simulation 1 gives 0.707 because I can generate artificial particles with known (to my computer) hidden variables. So I and everyone else knows that the quantum correlation really exists for theta = 45 degrees and is 0.707. After all, I have obtained it in Simulation 1 based on exact vectors.
So on to the misunderstanding. QM has no access to the hidden variables for calculating the quantum correlation, so it cannot do it. But it is known with certainty that the quantum correlation actually exists. So somewhere along the line over the years someone must have decided that the quantum correlation must be able to arise from the fuzzy vector data. But this is a complete misunderstanding of the situation. The situation is that the quantum correlation corresponds to a correlation on exact vectors whereas the CHSH statistic using real experiments is derived using correlation between fuzzy vectors.
This mis-match of what the correlations are measuring underlies all the supposed mystery of the quantum correlation. There is no spookiness attached to these quantum correlations as I have simulated them non-spookily for exact vectors. The only spookiness is why anyone should chase the supposedly spooky correlation in the wrong cell above, searching in the top right hand cell of correlations between fuzzy vectors, for which my Simulation 1 gives the mundane and attenuated value of correlation = 0.5. It is no surprise that the correlation is attenuated as the vectors are fuzzy, and fuzziness indicates lack of reliability of measurement which is well-known in reliability theory to attenuate a correlation.
Of course there could be something going on in nature which is very very spooky and much more spooky than anything above, and that I have not covered in my local, real simulations. My own preon model (http://vixra.org/abs/1511.0115) subdivides the electron into many components which does allow a glimmer of scope for non-local effects. Also my preon model has 24 dimensions as the preons contain strings, and it is also possible that what looks like a local effect in multi-dimensions could look like a non-local effect when viewed in three dimensions.
The really strange outcome for me is that using exact vectors, for theta = 45 degrees, and non-integer measurements by Alice and Bob in a particle-at-a-time simulation (which is impossible in practice) one can obtain a correlation of 0.707 between Alice and Bob’s measurements. This same correlation is obtained via Quantum Mechanics in a summary way using integer measurements from Alice and Bob as starting points though the subsequent calculations involve Projection Operators and abstract mathematical space.
So there is a kind of parallel here between these two approaches.
But in a particle-at-a-time simulation using fuzzy vector measurements, that is integer measurements, by Alice and Bob, the correlation is reduced to 0.5. However in real-life experiments, of necessity using integer measurements by Alice and Bob, the higher correlation of 0.707 is obtained. (To be more exact, the correlation of 0.5 is exceeded at a statistically significant level.)
I can write off any mystery in the 0.707 correlation in QM calculations as being a side effect of using higher mathematics in the calculations, and moreover QM does not produce the raw data of the individual integer measurements by Alice and Bob. But in real experiments we do have both the raw data of integer measurements by Alice and Bob and we also have the correlation greater in absolute magnitude than 0.5. It is in the real experiments where the mystery lies as one has the cake (integer measurements) and eats it (disattenuated correlation greater than 0.5 using an ordinary statistical calculation)!
(Revised 14 March 2019)
Ben6993's blog 