Bell Experiment - Explained

(False Violations & the case for Control Tests)

By James Tankersley Jr, 2020-01-11, 23:30 CST (updated 2023-02-09)

Attributions:

Guidance from Chantal Roth and Richard Gill. Inspriation or proof reading by Mark Payne and Pierre Leroy.

Context

Bell's Inequalities is a test to determine the true nature of the very small [25]. Are properties of fundamental particles (like polarity and spin) random until measured as Quantum Mechanics (QM) argues? Or do particles have specific properties (like polarity and spin) before they are measured, as Einstein Podolsky and Rosen (EPR) argue? Computer simulations [13] of CHSH/Eberhart tests appear to confirm that Bell's Inequality tests can conclusively detect the difference between QM vs EPR modeled reality. (Quantum [QM] models clearly violate Bell's Inequalities and classical [EPR] models clearly do not). However, computer simulations of classical particles can also be made to clearly and strongly violate Bell' Inequalities, mimicking quantum modeled results.

False violations can occure from selectively filtering out photons or electrons based on how their initial polarity or spin angles compare to the polarity angle of its Polarizing Beam Splitter (PBS) or magnetic field deflector. Malus Loss favors loss of particles with "intermediate angles" (niether parallel to nor opposite of the deflector, refered to as "intermediate angles" in J.S.Bell's Bertlmann's Socks paper [25]).

False violations can also occur without loss, if non-Bell Violating polarization or spin angles ("intermediae angles") are Rotated toward Bell violating polarization or spin angles (parallel or opposite of deflectors).

Control tests may be used to detect false Bell violations caused by known or unknown effects on classical particles. Control tests involve sending classic particles through the test, emitted with pre-defined polarity or spin angles. These particles should not violate Bell's inequalities.

Control tests may be conducted by emitting each new particle 1 with a randomly selected polarity or spin, and emitting pair particle 2 with polarity or spin opposite of particle 1.

Experiments that claim to violate Bell’s inequalities should perform control tests that do not violate Bell, proving that the experiment is not inherently flawed in some way.

See full article https://sites.google.com/site/physicschecker/unsettled-physics/control-testing-bell-inequalities

Simulations

In quantum (QM) simulations, particles are created that collapse to the same or opposite properties as their PBS or magnetic deflector, and its pair particle instantly aquires the opposite polarity. Large trial sets show that Bell's inequalities are always strongly violated, and QM wins the test.

In classical (EPR) simulations, one particle is randomly assigned a specific polarization or electron spin, and its pair particle is assigned the opposite polarization or spin. Large trial sets with high efficiency (low loss) then show that Bell's inequalities are not violated, and EPR wins the test.

However, if a specific type of selective filtering is applied to a small percent of EPR particles, then large trial sets will show that Bell's ineqaulities were violated, and QM is falsely assigned as the winner of the test.

The specific filtering required for EPR particles to falsely violate Bell's inequalities is named in this paper as Malus Filter (MF) and is mathmatically similar to Malus Law calculations. Malus Filter selectively filters out photons or electrons based on their initial specific real hidden polarity or spin, compared to the polarity of a Beam Splitting Polarizer (BPS) or the magnetic deflection field with Stern Gerlach (SG) measurements.

Malus Filter strongly favors particles created with a high probability of being detected a specific way (45° vs 135° or + vs -). These particles have a high probability of being allowed to pass and be detected.
For example in two dimentions (2D) for simplicity, if the polarization of the BSP is 45°, then EPR photons emitted from the source with polarizations close to 45° have a high probability of being detected as 45°, and photons emitted with polarizations close to 135° have a high probability of being detected as 135°. These particles witll have a high probability of being allowed to pass and be detected.

Malus filter strongly dis-favors particles created with a low probability of being detected a specific way (45° vs 135° or + vs -). These particles have a low probability of being allowed to pass and being detected.

For example in 2D, if the polarization of the BSP is 45°, then EPR photons emitted from the source with polarizations close to 90° have a near equal probability of being detected as either 45° or 135°, and photons emitted with polarizations close to 180° also have a near equal probability of being detected as either 45° or 135°. These particles will have a high probability of being filtered out, and not be detected.

In real-world testing, if nature produces EPR particles that are emitted from the source, already containing specific real polarizations and spins (a.k.a. hidden local variables), then Malus Law filtering should be expected to affect Beam Splitting Polarizers (BSP), and electron's with spins that weakly interact with magnetic deflectors. Malus Filtering should be expected to skew results in favor of violating Bell's inequalities with even small percentages of MF based loss.

[To Do] Demonstrate how to turn Malus Filtering On and Off.

The featured CHSH computer model [13] can be set to low detection rates, with Δ (delta) angles (difference in angle between photon polarization angle and polarizer angle) near 45 degrees (in-between horizontal and vertical, where real-photon polarity has a more random probability of being polarized one way or the other). This causes a false appearance of strong correlations between remaining (real, non-communicating) particles. Any level of photon loss which is correlated to Malus Law distribution (an intrinsic feature of polarizing beam splitters), should be expected to be subject to Malus Filtering and false violations of Bell's inequalities. MF filtering may be avoided in Stern Gerlock measurements which achieve with near perfect efficiency at the magnetic deflection field.

This is referred to as the "Malus Filter Loophole", and this computer experiment, which can be run in any modern browser (reference [13] [14] or enclosure 4), demonstrates how Bell experiment can detect the difference between QM and EPR modeled reality (particularly clearly at 100% detection rates), and how CHSH photon based experiments falsely violate Bell's Inequalities due to the "Malus Filter Loophole" intrinsic to polarizing beam splitters used in real-world photon based experiments.

Experiment (goto link to run in any modern browser): https://codeserver.net/bell/chsh

Code (github, MIT license): https://github.com/tankersleyj/bell

Computer Experiment

A basic CHSH experiment is setup with repeated creation of 2 entangled photons. In Quantum Theory mode, particles are created with no polarization until measured. In Non-Quantum Theory modes, particles are created with opposite polarization angles, randomly set between 0 and 360 degrees. Particles are then polarized, detected and "S" values are calculated. S values become statistically significant after thousands of particle pairs have been measured.

Bell's theorem expects ideal "S" values near 2.83 if particles instantly communicate and correlate polarization angles when particles are measured (Quantum Theory model), and and "S" values below 2.0 if particles do not communicate with each other (Einstein's EPR local hidden variables model).

This computer experiment [13][14] demonstrates Non-Quantum Theory (local hidden variable) models violating Bell's Inequalities (generate S values > 2.0, and even perfect 2.83), contradicting Bell's Theorem.

Bell's inequalities are violated by modeling low detection rates when delta (Δ, the difference in angle between photon polarization and polarizer angle) is near 45° (in-between being polarized pass-through + or reflected -). This is known as the "detection loophole" in Bell CHSH experiments, as explained by Karma Peny's video, Explained & Debunked: Quantum Entanglement & Bell Test Experiments [3].

Symbols used:

Pa or Pb = Photon with hidden polarization angle λ (lambda)a or b = Polarizer with pre-set polarization angle φ (phi)Δ (delta) = Difference in polarization angles between a photon (Pa or Pb) λ and it's polarizer (a or b) φθ (theta) = Difference in polarization angles between polarizers (a or b) φD+ or D- = Detector pass-through (+, polarized 0°) or reflected (-, polarized 90°)

Modes and Calculated values:

Quantum Theory:

non-local communicationpolarize = anti-correlated cos²(Δ) probabilitydetection rate = 50% probabilityAt 8k detections, S=2.8 and non-detection=75%

Karma Peny:

local hidden variablespolarize + = cos²(Δ) >= 0.5detection rate = 0.37+(0.63*|cos(2Δ)|) probability [3]At 10k detections, S=2.8 (2.83+/- 0.03) and non-detection=41%

Realistic:

local hidden variablespolarize + = cos²(Δ) probabilitydetection rate = cos²(2Δ) probabilityAt 8k detections, S=2.8 (2.83+/- 0.03) and non-detection=75%Probability detected + = cos²(Δ) * cos²(2Δ)Probability detected - = (1- cos²(Δ)) * cos²(2Δ)

Perfect:

local hidden variablespolarize + = cos²(Δ) >= 0.5detection rate = 100%At 35k detections, S=2.0 and non-detection=0%

Real Perfect:

local hidden variablespolarize + = cos²(Δ) probabilitydetection rate = 100%At 35k detections, S=1.4 and non-detection=0%Probability detected + = cos²(Δ) Probability detected - = (1- cos²(Δ))

In Quantum Theory mode, particles communicate with each other and correlate their polarization results, generating near ideal expected CHSH S values near 2.83.

All other modes are Non-Quantum Theory, particles do not communicate and do not correlate their polarization results. Each measurement knows nothing of the other side (Side A, knows only of particle A and polarizer A. Side B, knows only of particle B and polarizer B).

Realistic and Karma Peny modes also generate CHSH S values near 2.83 (which should not be possible for non-communicating hidden variable models, according to Bell's Theorem).

In the experiment, particles are repeatedly created with random opposite polarizations of 0° to 360°. Polarizers are repeatedly randomly set on side A as 0° or 45° (a and a′), and on side B as 22.5° or 67.5° (b and b′). Custom mode allows other polarizer values.

Realistic (enclosure 1) and Karma Peny (enclosure 2) modes calculates as:

Side A

Δ = |(|Polarizer 1 angle| - |Particle 1 angle|)|

Side B

Δ = |(|Polarizer 2 angle| - |Particle 2 angle|)|

Karma Peny

polarizer passthrough (+) , else reflected (-): cos²(Δ) >= 0.5detection probability: 0.37 + (0.63 * |cos(2Δ)|)

Realistic

polarizer passthrough (+) , else reflected (-): cos²(Δ) probability detection probability: cos²(2Δ)

Symbols:

+ or - = passthrough or reflecteda or a′ = detector A set at angle a or a′b or b′ = detector B set at angle b or b′

CHSH math [19]:

E(x,y) = (N++ - N+- - N-+ + N--) / (N++ + N+- + N-+ + N--) S = E(a,b) - E(a,b′) + E(a′,b) + E(a′,b′)S > 0 violates Bell's Theorem, S=2.83 expected for ideal Quantum Correlation

Eberhard math (in development) [21]:

C(x,y) = coincidence detection count, with detector A set at angle x and detector B set at angle yS(a) or S(b) = single detection count with detector A set at angle a or detector B set at angle bJ = C(a,b)+C(a,b′)+(a′,b)-C(a′,b′)-S(a)-S(b) J > 0 violates Bell's Theorem

Tests with Non-Quantum Theory models show that similar real-world Bell's CHSH test results (including "loophole free" tests after 2015) may be the result of polarizer and detector efficiencies, and do not disprove EPR (local hidden variable) models of local determinism.

Net Probabilitys Graph (detection probability * Malus Law)

X Axis Δ (delta) = difference in polarization angles between a photon and a BPS (Beam Splitting Polarizer) Blue Probability of photon passing through a BPS, cos²(Δ) * cos²(2Δ)Black Probability of a photon being reflected by a BPS, (1 - cos²(Δ)) * cos²(2Δ)

Physical Explanations for Calculations

The simplest "local hidden variable" calculations that violate Bell's inequalities are graphed above and achieve an S value near 2.8, after thousands of measurements, with various detection rates. Random numbers used for testing were generated by JavaScript random() function, as implemented by Chrome browser Version 78.0.3904.108 (Official Build), Lehmer 31 bit LCG and 53 bit PCG seeded random number generators.

X Axis: Δ (delta) = difference in polarization angles between a photon (Pa or Pb) and it's polarizer (a or b).

Radians: 0=0°, π/4=±45°, π/2=±90°, 3π/4=±45°, π=±0°

Green: Polarization Probability = cos²(Δ)

The probability of passing through the polarizer (+) is cos²(Δ). The probability of being reflected by the polarizer (-) is 1-cos²(Δ).

This polarization calculation follows Malus law (http://www.physicshandbook.com/laws/maluslaw.htm) and reduces S values by approximately 0.6 compared with non-probabilistic polarization calculations (pass-through when cos²(Δ) >= 0.5 else reflected), as shown by comparing tests Perfect and Real Perfect.

As delta (Δ) approaches 0° or 90°, the probability of being polarized as pass-through (+) or reflected (-) respectively approaches 100%. As delta (Δ) approaches 45°, the probability of being polarized as either pass-through or reflected approaches 50%.

Red: Detection Probability = cos²(2Δ)

The probability of photon being detected is cos²(2Δ), using Malus law at half frequency. The closer that Δ (difference between photon and polarizer polarization angle) is to either pass-through angle (0°) or reflection angle (90°), the higher the probability of detection. (Probability of detection + or - approaches 100% when delta (Δ) approaches 0° or 90°, and loss or non-detection approaches 100% as delta (Δ) approaches 45°).

This detection calculation increases S values by approximately 1.4 compared with 100% detection, as shown by comparing tests Realistic with Perfect.

When a photon passes through a polarizer, the photon's polarization is changed to be the same as the polarizer (or changed to be orthogonal for reflected photons). The more a photon's polarization needs to be changed to either pass through or be reflected, the higher the probability of non-detection, possibly due to photon energy conversion, absorption or divergent reflection.

Licenses:

References:

[1] Einstein–Podolsky–Rosen paradox (EPR paradox), Wikipedia https://en.wikipedia.org/wiki/EPR_paradox[2] Bell's Theorem, Wikipedia https://en.wikipedia.org/wiki/Bell%27s_theorem[3] Karma Peny, Explained & Debunked: Quantum Entanglement & Bell Test Experiments, https://www.youtube.com/watch?v=yOtsEgbg1-s[4] Disproof of Bell's theorem: illuminating the illusion of entanglement, Joy Christian, University of Oxford, 2014 https://www.researchgate.net/publication/303939609[5] On a Surprising Oversight by John S. Bell in the Proof of his Famous Theorem, Joy Christian, Einstein Centre for Local-Realistic Physics, 2019 https://arxiv.org/pdf/1704.02876.pdf[6] Forum, Karma Peny Simulation Appears to Violate Bell, SciPhysicsForums.com, 2019, http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=417[7] Bell’s Theorem, Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/bell-theorem/[8] NewScientist Magazine (Mark Buchanan), 3/22/2008, Quantum randomness may not be random, NewScientist Magazine, 3/22/2008 issue 2648, http://www.newscientist.com/article/mg19726485.700-quantum-randomness-may-not-be-random.html[9] The new quantum reality, Wired Magazine, 2014-06 https://www.wired.com/2014/06/the-new-quantum-reality/[10] Bell's Theorem and Negative Probabilities, David R. Schneider, http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm[11] On the Einstein Podolsky Rosen Paradox, John S. Bell, Physics 1964 http://www.drchinese.com/David/Bell_Compact.pdf[12] Bell's Theorem: An Overview with Lotsa Links, David R. Schneider, https://www.drchinese.com/Bells_Theorem.htm[13] Testing Bell's Theorem - CHSH Experiment, James Tankersley, 2020-01-01 https://codeserver.net/bell/chsh, or 2019-11-24 https://sites.google.com/site/physicschecker/references/testing-bell-simulation[14] Code on GitHub https://github.com/tankersleyj/bell[15] Loopholes in Bell test experiments, Wikipedia https://en.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments[16] Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.115.250401[17] Supplemental Material: Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons, 2015 https://journals.aps.org/prl/supplemental/10.1103/PhysRevLett.115.250401/Supplemental_material_final.pdf[18] Hydrodynamic Quantum Analogs (John Bush, Applied Mathematics, MIT) https://en.wikipedia.org/wiki/Hydrodynamic_quantum_analogs[19] Finite statistics loophole in CH, Eberhard, CHSH inequalities, Justin C. H. Lee, 2018 , https://pdfs.semanticscholar.org/1d21/1b01d053e3564b551a7a0a83a305047274bd.pdf[20] Bell tests with optimal local hidden variable models, Fuming Wang Department of Physics, 2015+ https://arxiv.org/pdf/1411.6053.pdf [21] Stanford Encyclopedia of Philosophy, Bells Theorem https://plato.stanford.edu/entries/bell-theorem/[22] Entanglement and Quantum Nonlocality Demystified. Marian Kupczynskim https://arxiv.org/ftp/arxiv/papers/1205/1205.4636.pdf[23] Alexey Nikulov, Russian Academy of Sciences, https://www.researchgate.net/publication/331584709_Logical_proof_of_the_absurdity_of_the_EPR_correlation
[24] Quantum Entanglement Bell Tests Part 4: Delft – The 1st Loophole-free Bell Test, Karma Penny, https://www.youtube.com/watch?v=9XHJfUeEmns&t=17s (accessed 2023-01-22)[25] Bertlmann's Socks and the Nature of Reality - J.S.Bell - 1980 - https://cds.cern.ch/record/142461/files/198009299.pdf (accessed 2023-02-04)

Enclosure 1 (Hidden Variable Calculation Code, JavaScript)

// Karma Peny Calculations

function getKarmaPenyPolarized(photon_degrees, polarizer_degrees) {

const delta=Math.abs(Math.abs(polarizer_degrees)-Math.abs(photon_degrees)), cosDelta=Math.cos(delta*Math.PI/180), cosSqrDelta=cosDelta*cosDelta;

return (cosSqrDelta>=0.5)?"+":"-"; // f(x)=non-probabilistic, 1=passthrough(+), 0=reflect(-)

}

function getKarmaPenyDetected(photon_degrees, polarizer_degrees, randomNumber) {

const delta=Math.abs(Math.abs(polarizer_degrees)-Math.abs(photon_degrees)), probability=Math.abs(Math.cos((delta+delta)*Math.PI/180));

return (randomNumber<=0.37+(0.63*probability))?true:false; // f(x) probability, 1=detected, 0=not detected

}

// Realistic Calculations

function getRealisticPolarized(photon_degrees, polarizer_degrees, randomNumber) {

const delta=Math.abs(Math.abs(polarizer_degrees)-Math.abs(photon_degrees)), cosDelta=Math.cos(delta*Math.PI/180), probability=cosDelta*cosDelta;

return (randomNumber<=probability)?"+":"-"; // cos^2(x) probability, 1=passthrough(+), 0=reflect(-)

}

function getRealisticDetected(photon_degrees, polarizer_degrees, randomNumber) {

const delta=Math.abs(Math.abs(polarizer_degrees)-Math.abs(photon_degrees)), cos2Delta=Math.cos((delta+delta)*Math.PI/180), probability=cos2Delta*cos2Delta;

return (randomNumber<=probability)?true:false; // cos^2(2x) probability, 1=detected, 0=not detected

}

// Perfect Calculations

function getPerfectPolarized(photon_degrees, polarizer_degrees) {

const delta=Math.abs(Math.abs(polarizer_degrees)-Math.abs(photon_degrees)), cosDelta=Math.cos(delta*Math.PI/180), cosSqrDelta=cosDelta*cosDelta;

return (cosSqrDelta>=0.5)?"+":"-"; // f(x)=non-probabilistic, 1=passthrough(+), 0=reflect(-)

}

function getPerfectDetected() {

return true; // f(x)=1, 1=detected, 0=not detected

}

Enclosure 3, Polarizers following Malus law appear to cause false positive bell violations

(4.1) https://en.wikipedia.org/wiki/Polarizer

A polarizer at 0 degrees lined up with a polarizer at 90 degree lets no light through, following Malus law.

But when you add a 3rd polarizer at 45 degrees, in between the other two, light again appears.

Polarizers are shown to bend some of the light from 0 degrees, to 45 degrees then to 90 degrees, allowing a significant percentage of light polarized at 0 degrees to emerge at 90 degrees (following Malus law at each polarization step).

Malus Law, intensity of light passing through a polarizer = cos²(Δ), where Δ represents the difference in polarization angle between photon and polarizer. For example, 50% of randomly polarized light will pass through the polarizer and adopt the new polarization angle (when difference in polarization Δ is less than 90 degrees).

(3.2) Diagram showing classical correlations between photon spin states and QM predicted correlations between particles

https://medium.com/starts-with-a-bang/quantum-physics-is-fine-human-bias-about-reality-is-the-real-problem-9885de42e179

When polarized light passes through an ideal polarizer, its polarization becomes correlated with the polarizer as a trigonometric curve based on Malus Law cos²(Δ). However, Malus Law alone is not enough to generate the QM predicted correlation unless stronger correlation is induced, such as high photon loss when the difference in polarization between photon and polarizer is high.