10.000 euro bet on – off – on …

At this moment, the 4^th may at 8:47h the bet seems to be still ‘on’: Joy Christan and Richard Gill are betting whether or not a computer simulation of the exploding balls experiment can be created that can or cannot produce Bell type statistics. This bet is not the 5000 euros one for which the exploding balls experiment should actually be performed (see the earlier post on this subject).

The current bet focusses on the angle settings from Alice and Bob where the difference between classical and QM predictions are most significant: 0, 45, 90 and 135 degrees.

The exact text of the bet as stated by Richard can be found here:

http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=52#p1898

Joy has responded and claimed victory by supplying this model in R: http://rpubs.com/jjc/16415

In his own words:

“Richard Gill has offered 10,000 Euros to anyone who can simulate the N directions of angular momentum vectors appearing in equation (16) of this experimental proposal of mine: http://arxiv.org/abs/0806.3078. Here I am attempting to provide such N directions. They are given by the vectors 'e' in this simulation. He has also offered further 5,000 Euros to me if my proposed experiment is realized successfully. I am hopeful that that will happen someday. The details of these challenges by Richard Gill can be found here: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=46. While this is by no means a perfect simulation of my model, it does meet all the stringent conditions set out by Richard Gill for his challenge.

Since after the explosion the angular momentum vectors 'e' moving along the z direction will be confined to the x-y plane, a 2D simulation is good enough for my proposed experiment. ”

Gill disputes whether Joy's simulation meets the conditions of the challenge.
The agreement now is to have a jury give a verdict (see http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=52&start=10#p1935).

For the non-programmers, and for myself to get acquainted with in the R syntax, I explain this code line  by line below.

Joy’s R code explained

set.seed(9875)

Computers cannot really generate random numbers, so they have a mechanism to pick numbers from a list of previously stored ‘random’ numbers. The seed indicates where to start picking.

angles <- seq(from = 0, to = 360, by = 1) * 2 * pi/360

seq(): generates a sequence of numbers. In seq(from = 0, to = 360, by = 1), an array containing the numbers 0 to 360 is created (the ‘by’ indicates the difference between succeeding numbers). After this all numbers in the array are multiplied by 2 * pi / 360 and stored in angles.  

K <- length(angles)
length(): gives the number of elements in the array, and stores it in a single value K.

corrs <- numeric(K)  ## Container for correlations

declares an array structure named ‘corrs’ which can contain K numbers (?)

M <- 10^5  ## Sample size. Next try 10^6, or even 10^7

The number 10 to the power of 5 = 1000000 is stored in M.

s <- runif(M, 0, pi)

t <- runif(M, 0, pi)

runif(n, min, max) generates an array of random numbers between the min and the max value. Here a list of M numbers between 0 and pi is generated and stored in s, and again in t. 

x <- cos(s)

y <- 1.2 * (-1 + (2/(sqrt(1 + (3 * t/pi)))))

So ‘x’ will contain M random cos() numbers, ‘y’ will contain M numbers with values from the formula above (sqrt() is square root)

 e <- rbind(x, y)  ## 2 x M matrix; M columns of e represent the

## x and y coordinates of points on a circle; y -> -y => e -> -e.

rbind(x,y) simply puts the two lists with numbers in x and y in one 2xM array called ‘e’

for (i in 1:(K - 1)) {

…

}

This is a loop structure: The program iterates over the code between the brackets, having i=1, 2, 3 …, until i =K-1 (K was the number of angles, so 361). Notice that within this loop another loop is used, having ‘j’ going from 1 to K-1.

    alpha <- angles[i]

The i-th angle in the array of angles is stored in ‘alpha’. The angles are expressed in radians, so when i=0 the alpha=0, when i = 1 then alpha=1 *pi /360 etc.

a <- c(cos(alpha), sin(alpha))  ## Measurement vector 'a'

c(): an assignment: So Joy stores the 2 numbers cos(alpha) and sin(alpha) in a 2d array.

for (j in 1:(K - 1)) {

the second loop within the firs loop starts

        beta <- angles[j]

        b <- c(cos(beta), sin(beta))  ## Measurement vector 'b'

‘beta’ will also contain each time a different angle, and b the two numbers cos(beta) and sin(beta)

        ca <- colSums(e * a)  ## Inner products of cols of 'e' with 'a'

        cb <- colSums(e * b)  ## Inner products of cols of 'e' with 'b'

so ‘e’ contains:

x₁  x₂ x₃ …x_M  y₁  y₂ y₃ …y_M

which is multiplies with a (a₁, a₂)

giving

x₁a₁  x₂a₁ x₃a₁ … x_Ma₁  y₁a₂  y₂a₂ y₃a₂ … y_Ma₂

colSums() then sums the numbers per column, giving

x₁a₁₊y₁a₂    x₂a₁₊y₂a₂  x₃a₁₊y₃a₂ …x_Ma_{1 +}y_Ma₂

which is stored in ca. The same calculation is done with ‘b’ giving cb.

        N <- length(ca)

The number of elements in ca is put in N (which will always be M)

        corrs[i] <- sum(sign(-ca) * sign(cb))/N

sign() returns 1 for positive numbers, -1 for negative numbers.

So each number in ca is first multiplied by -1 because of the ‘–ca’, and the sign function results in a list with 1’s and -1’s. These are multiplied by the elements in sign(cb).The resulting list of 1’s and -1’s is summed up and divided by N. The calculated number is stored in the i-th position of corrs.

        Ns[i] <- N

N is stored in the i-th position of Ns

When the code in the loops is completed it continues with the statements below

corrs[K] <- corrs[1]

Ns[K] <- Ns[1]

Here the obtained values for angle=360 is taken the be the same as for angle=0

The rest of the code is for printing the results.

An extended description of R can be found here: http://cran.r-project.org/doc/manuals/R-intro.html

Some concluding remarks

• Joy's model iterates over Alice’s and Bobs angles. This is an efficient mechanism to get results for all the settings. The same results should be obtained using random (integer) angles between 0 and 360 degrees, but one might need a slightly larger 'M' to get nice results for all the angles.
• The conditions for the Bell type simulation seems to be met: All the particles are used in the result set (which excludes the detection loophole) and the measurement for Alice does not use Bob's measurement and vice versa.

Experiment of the century

Joy Christian has proposed a macroscopic experiment to verify his theory in 2012 described in 'Macroscopic Observability of Spinorial Sign Changes under 2pi Rotations'. Now he and Richard Gill seems to have agreed on a bet to have this experiment performed in the presence of mutually trusted physicist. The stakes will be several thousand Euros. 

Joy's experiment is further described on his blog. It consists of balls that can be split in two identical halves. Each half contains a relatively heavy weight randomly attached to the hemisphere. The balls are given a random torsion, and, by heating them a bit, explode in the two hemispheres. The rotation of these halves will be registered by sensors. 

He has been trying to raise funds for this experiment since then. On Feb. the 22th Richard and Joy have settled the first terms for this experiment using the FQXI () forum here.
Andrei Khrennikov and Lucien Hardy, Hans De Raedt, Steve Weinstein (Perimeter), and Christopher Fuchs will be requested to be adjudicators for the experiment. We look forward to this happening!

New models by Richard Gill and Chantal Roth in 'R'

Richard Gill has presented some EPR simulations made in the computer language 'R'.

• Michel Fodje's simulation in Python (https://github.com/minkwe/epr-simple/) of the model from Joy Christian converted to R: http://rpubs.com/gill1109/13325
• Chantal's Java model (see link on the right side of this page) in R: http://rpubs.com/gill1109/13329
  and http://rpubs.com/chenopodium/epr2

The last two implementations give a nice overview of which CHSH can be obtained when allowing a part of the particle pairs not to be detected.

The simulation that implements Joy Christian's model is discussed at http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=11. In short this simulation also leaves out a part of the initial particle pairs, so some say it makes use of the detection loophole. Joy argues that in his model these particle states do not exist, so should not be counted anyway.

There is also a Javascript version available created by Daniel Sabsay: http://libertesphilosophica.info/eprsim/EPR_3-sphere_simulation_test5m.html,
and a mathematic version written by John Reed: http://libertesphilosophica.info/Minkwe_Sim_J_Reed.pdf

A parallelized 3-sphere based simulation

Joy Christian has added a new chapter to his arguments by supplying a Java simulation in his response (1) to James Owen Weatherall (2) that succeeds to yield the results of QM. The program is written by Chantal Roth (who also supplied the earlier EPR simulation framework referenced in the margin of this blog), and can be downloaded at https://github.com/chenopodium/JCS. It can be easily viewed by downloading the NetBeans development environment and  then opening the Java project (after unzipping).

Discussions are going on at

https://groups.google.com/forum/#!topic/sci.physics.foundations/TIic82g2stw

and

http://www.fqxi.org/community/forum/topic/1247

Edit 2013-09-18: At the FQXI al new and interesting posts on this subject are over and over DELETED BY SOMEONE, probably not a fan of the theory of Joy Christian. So there has been quite a discussion, but now only few posts are left. I happened to have a browser open containing the earliest new posts, and saved that as PDF. So if anyone is interested let me know.
 
1) Whither All the Scope and Generality of Bell’s Theorem?, Joy Christian, http://arxiv.org/abs/1301.1653v4
2) The Scope and Generality of Bell’s Theorem, J. O. Weatherall, http://arxiv.org/abs/1212.4854

A Local Realistic simulation of the EPR correlation.

by

Bryan C. Sanctuary

The Simulation Programs

The EPR paradox can be reconciled in a local realistic way by assuming that an isolated spin that makes up a quantum ensemble (the usual quantum state)  has two axes of quantization rather than one.  Using this model the EPR data is exactly simulated one EPR pair at a time. 

In this blog the Java programs that do the simulation are given. The program was written by Chantal Roth and was modified for my model by Mr. Michael Havas.

The research paper is submitted to Physical Review A. 

• A Local Realistic Reconciliation of the EPR Paradox
• A document with most of the derivations; 
• A recorded seminar

These can be found on my blog along with some other information:

http://quantummechanics.mchmultimedia.com/

The following programs contain the source code and compiled versions of the program:

Source code

http://quantummechanics.mchmultimedia.com/EPRfiles/EPR-src.zip

Linux installer (no source)

http://quantummechanics.mchmultimedia.com/EPRfiles/LinInstaller-EPR.sh

Mac installer (no source)

http://quantummechanics.mchmultimedia.com/EPRfiles/MacInstaller-EPR.tgz

Windows installer (no source)

http://quantummechanics.mchmultimedia.com/EPRfiles/WinInstaller-EPR.exe

The program without an installer (no source)

http://quantummechanics.mchmultimedia.com/EPRfiles/EPR-nsrc.zip

The plot below is the simulation that gives exact agreement with the correlation observed experimentally and calculated from quantum mechanics.

Figure 1 A local realistic simulation of the EPR data calculated at every 22.5 degrees and the points connected. The smooth curve is -cosθ_ab plotted as a function of θ_ab.

The model treats a spin in complete isolation before it reaches a filter. The model assumes that an isolated spin has two components of angular momentum, σ_x and σ_z. Since it is impossible to measure them both simultaneously (Heisenberg), two simulations are needed. Each simulation gives half the correlation,

Figure 2  The result of simulating the EPR correlation for one axis of quantization. The quantum coherences are ignored and the correlation from one axis gives half the correlation predicted from quantum mechanics.

The Model

It is assumed that a single spin that makes up the quantum state is described by the state operator,                   

where                    

The body frame,

is related to the laboratory frame by Local Hidden Variables (LHV), θ, φ.  In addition, there are other LHV

and these correspond to the quadrants of the body frame (these quadrants are seen in Figure 3). In complete isolation, the states of this model have eigenstates of ±√2,

Figure 3 Spin depicted in an isotropic environment showing the eight possible pure states with two per quadrant. The 2D spin can occupy only one state at any instant. The states bisecting the quadrants are the eigenstates  |√2,±,q,f>_nx,_nz.

                                       

And these are depicted in Figure 3 showing a “horizontal” and “vertical” component.

Caveats

The details of the model can be found in my research paper.  There are a number of caveats which mainly follow from the Heisenberg Uncertainty Relations: 

• It is impossible to experimentally prove the 2D model is correct because it is impossible to simultaneously measure two spin components, σ_x, σ_z , at the same time. 
• Although the full quantum correlation 2√2 is obtained with my local realistic model, only half the correlation is obtained, √2  per axis, from each simulation.  Therefore the CHSH equation is not violated √2 < 2 whereas Bell’s original inequalities are violated √2<1.
• The reason the CHSH equation is not violated is because in the classical derivation, Hidden Variable space is partitioned into two regions.  As the classical CHSH equation, it is assumed that the two regions commute and can be simultaneously sampled.  My local realistic model does not use any classical notions, and the two regions do not commute and therefore cannot be simultaneously measured.
• Another way to state this is that one spin component is measured; the coincidences from the other components are counterfactual.
• Quantum mechanics is a theory of measurement.  In this treatment, quantum theory is extended to treat one particle in complete isolation.  This leads to some changes in quantum theory that are not addressed in this work.  The model and its simulation are the main focus at this stage.

Since experiment cannot resolve the issue, subjective arguments alone must decide. If quantum mechanics is complete then we have to accept non-local indeterminism.  Non-locality is a concept that no one understands. Indeterminism goes against our basic intuition about Nature.

Successes

In contrast, the local realistic simulation using the 2D spin model:

•    Agrees with both quantum theory and experiment
•   Is consistent with Bell’s correlation and gives more insight into the  quantum correlations
•    Predicts the filter settings that maximize the EPR correlation
•   Agrees with the treatment of Gustafson which identifies a vector of length √2 in the CHSH equation.

The Experimental Data

Here I will only comment on one issue, (please see the paper for more details). Why experiments appear to show violation of the CHSH equation in one experiment?

Heisenberg again reconciles this because half the coincidences that are present cannot be measured when two complementary axes exist. The expression usually used to determine the correlation is,

When there are two axes of quantization, only one can be measured at one time because as one axis lines up with the filter, the other decohers and cannot be detected,

Figure 3. Depiction of how half the correlation is lost in the presence of a probe: (a) The two body fixed axes of spin quantization, z and x (heavy lines).  The two arrows depict two possible orientations of a vector probe field in the laboratory frame, one closer to the z axis and the other closer to the x axis (only one can exist at any instant): (b) The case when the probe is closer to the z axis. Then the 2D spin deterministically nutates until the z axis aligns with the probe while the x component precesses in the XY plane. (c) The same as case (b) but now the probe is closer to the x axis which now aligns in the X direction and the z component precesses in ZY plane. 

                                              

Here a coincidence is defined by a product.  But what happens when there are two axes of quantization?  Then there are two polarizations,

In the case of two axes, the number of coincidences is doubled because there are 2 coincidences for each EPR pair.  However only the Z or the X can be measured, so the correlation associated with the axis that can be measured is 

                                                                       

Where N is the total number of coincidences actually detected. This is half the number that actually exists. Another experiment, with the filter angles rotated to by 90 degrees would reveal the other half of the correlation in principle but due to rotational invariance, the results are indistinguishable.

This means the coincidences from the undetected axis are counterfactual, and it is impossible to experimentally distinguish between the usual description of spin, with a single axis of quantization, and the model here with two orthogonal axes of quantization.

Another way of expressing this is to consider a gedanken experiment where, in spite of non-commutation of the two Pauli spin components, both somehow can be measured simultaneously. In that case a single EPR pair would record a coincidence in the Z channels, and simultaneously another coincidence in the X channels. The total number of coincidences for N EPR pairs is again 2N. With this modification, the simulated correlation per axis is in agreement with the experimental data.

Consider also Figure 3.  As a 2D spin approaches a filter either the Z axis or the X axis aligns with the filter direction, but not both.  Therefore half the coincidences come from the X axis and the other half from the Z axis, in agreement with the above equation.   

Counterfactual events cannot be detected and therefore cannot be experimentally confirmed. This illustrates the limit of quantum measurement, making it impossible to experimentally confirm that two axes exist rather than one.

QM reflection

There are two visual aids I recently came across that i.m.o. can help to visualize what is happening at the sub atomic level. The first is about spin, and how it might be possible that a particle needs a turn of 4 pi to get back in its original position.



It is known as Dirac's belt trick (among many other names).

The second feature that mimics QM at a macroscopic level are the effects seen with walkers. Walkers are tiny droplets bouncing on a vibrating fluid. The droplets in combination with waves on the fluid simulate nicely some QM features, including the interference seen in the double slit experiment. A popularized version of the experiments can be seen here:  



and


It appears that the Austrian Institute for Nonlinear Studies has taken up the challenge imposed by the findings of Yves Clouder and has come up with several computer simulations and calculations based on the walkers or 'bouncers' (3):

"On this basis, it has been explicitly shown how the following quantum mechanical features can be derived from purely classical physics: Planck‘s relation E = hw for the energy of a particle, the Schrödinger equation for conservative and non-conservative systems, the Heisenberg uncertainty relations, the quantum mechanical superposition principle, Born‘s rule, and the quantum mechanical ―decay of a Gaussian wave packet‖. Moreover, also the energy spectrum of a quantum mechanical harmonic oscillator has been derived classically, as well as that of a particle in a box."

 I would like to finish with two statements I consider more or less true. I would like to hear your opinion on these:

"Apart from Theology Quantum Mechanics is the only branch in science in which there are observed phenomena postulated to be inexplicable" (wave-particle duality, Heisenberg’s uncertainty principle, wave function, entanglement, quantum randomness)

"Furthermore it's framework contains several ill- or circular defined objects (measurement, superposition, decoherence) and postulates that cannot even theoretically be disproved" (particle position nonexistent before measurement)

1. Single-Particle Diffraction and Interference at a Macroscopic Scale 2006, Yves Couder and Emmanuel Fort, https://hekla.ipgp.fr/IMG/pdf/Couder-Fort_PRL_2006.pdf
2. A macroscopic-scale wave-particle duality, Toronto 2011, http://www.physics.utoronto.ca/~colloq/Talk2011_Couder/Couder.pdf
3. Sub-Quantum Thermodynamics as a Basis of Emergent Quantum Mechanics, Gerhard Grössing, Entropy 2010, 12, 1975-2044, http://www.mdpi.com/1099-4300/12/9/1975
4. Emergence of Quantum Mechanics from a Sub-Quantum Statistical Mechanics Gerhard Groessing, 2013,  http://arxiv.org/pdf/1304.3719.pdf

 

Weihs disproved?

Donald Graft kindly notified me of his most recent article on EPR (1), in which he describes a computer simulation that models the results of the famous Weihs (2) experiment. The model is local realistic, and neatly mimics the results obtained in the real experiment, while making use of reasonable assumptions on detector calibration etc.

As soon as the source of the simulation has been made publicly available I will add a link in this post.

1. A local realist account of the Weihs et al EPRB experiment, Donald A. Graft, http://arxiv.org/abs/1301.1670
2. Violation of Bell’s inequality under strict Einstein locality conditions
   Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, and Anton Zeilinger, http://arxiv.org/PS_cache/quant-ph/pdf/9810/9810080v1.pdf