Quantum Reality and the Bell Inequality

This article was obtained from the Internet, and is reproduced here in its entirety. I consider it an excellent overview of the "realist" (materialist, determinist) interpretation and understanding of the standard Quantum Mechanics interpretation of "random reality".    

Author: David Elm

Date: July 24, 1997

Internet Address: http://www.tiac.net/users/davidelm

EPR Paradox - Bell's Inequality

Referring to Bell's paper "On the Einstein-Podolsky-Rosen Paradox" [Physics 1 (1964) p.195-200] and to all EPR experiments in general, it can be shown that there is an error in applying Bell's inequality to the tests which were designed to test it. Therefore all EPR tests which seem to violate the inequality and support non-local effects are faulty and cannot be used to reject all local reality theories of the universe. It can also be shown that the overall logic used in the EPR tests is circular and so the results are non-rigorous.

In the early 1930'S, Einstein and Bohr had been discussing reality at the quantum level. Bohr believed that reality at the quantum level does not exist until it is measured. This view came to be known as 'Copenhagen' quantum mechanics. The usual view of quantum mechanics says that a wave function determines the probabilities of an actual experimental result and that it is the most complete possible specification of the quantum state, and there is no other reason for an event to occur.

Einstein, Podolsky, and Rosen (EPR) set forth a thought experiment which attempted to show that the quantum mechanics could not be not a complete theory if it assumed things happen for no reason. Contrary to what the Copenhagen Interpetation asserted, Einstein, et al. said that properties of quantum particles must be real even before you measure them if you can know exactly what those properties are. Their paper proposes a case where two particles (electrons?) are known to have the same momentum and equivalent positions from the source of their creation. Momentum and position are one of several 'complimentry' properties of matter at the quantum level which Coppenhagen QM Physicists say can only be known to a certain degree. They argued that since there exist ways to know the properties of one particle then you will know the other particle has the exact same property, so by inferrence, those properties of the other particle must be real whether you measure them or not. Einstein et al. believed the predictions of quantum mechanics to be correct, but only as the result of statistical distributions of other unknown but real properties of the particles.

Bohm (1951) presented a paper in which he described a modified form of the Einstein-Podolsky-Rosen thought experiment which he believed to be conceptually equivalent to that suggested by Einstein et al. (1935), but which was easier to treat mathematically. Bohm suggested using two atoms with a known total spin of zero, separated in a way that the spin of each atom points in a direction exactly opposite to that of the other. (If indeed this can be said since QM says the spins don't exist yet!) In this situation, the angular momentum of one particle can be measured indirectly by measuring the corresponding vector of the other particle. (Unfortunately, you cannot actually get a good measurement on a quantum particle, you can only get a probability measurement.) John Bell (1964) subsequently put forth 'Bell's Inequality', which seemed to be a physically reasonable condition of locality. This locality imposed restrictions on the maximum correlations on certain measurements. For example, a pair of spin 1/2 particles formed somehow in the singlet state and moving freely in opposite directions. This inequality appears to be testable in a laboratory experiment because the statistical predictions of quantum mechanics are incompatible with any local hidden-variable theory apparently satisfying only the natural assumptions of 'locality' as shown by the predictions of Bell's Inequality.

Measuring a property of a single quantum particle gives one of two specific readings: 0 or 1, due to probability based the relative angle between the particle and the measuring device. Einstein would say this choice is not just probability, but is due to other real but unknown properties of quantum particles. Bohr says probability is the only thing which exists on this level and that is the complete picture. Einstein did not use the term 'hidden variables' but he believed a deeper reality exists which would someday be knowable and understood. One or the other man was wrong. The debate went on for 30 years before Bell published his inequality.

Bell's paper in 1964 presented a method which seemed to provide a way to test between the two views in the laboratory. By measuring spins, as suggested by Bohm, instead of position and mometum, a real quantitative test could be performed and a situation seemed to exist where QM predicted a correlation will exist in the measurements which should be impossible in a reality as described by Einstein's universe. Many such tests have now been run and the results do seem to violate Bell's upper limit and so most physicists now believe the universe has non-local effects, at least at the quantum level.

Bell's "Theorem' seemed to use only a few common assumptions and simple logic to calculate an upper limit for the maximum possible correlations which are possible in any test of this nature. The simple logic includes the assumption that in any real world situation you cannot have more than the sum of the parts summed. You can think of Bell's inequality as the upper limit of the number of items which can disagree on two lists. For example, two students who take a test will have only a certain number of answers which can disagree. Suppose one student answered 95 questions correctly out of 100 and another student scored 98 out of 100 on the same test, then we can calculate, by a linear addition, that there can be no more than 7 questions that disagree between the two lists. So Bell's Inequality thus plots out to be a straight line with a 'kink' at 0 degrees.?! There are reasons why this "Two list' logic cannot be applied to the 'lists' of data produced in the EPR tests. For one thing, there is no master list in the EPR test so we cannot know, even in principle, which events are actually errors.

The experiment done by Aspect, et al. in 1982 was considered the final nail in the coffin of local causality by many physicisits. In many respects this particular test should NOT be considered a good starting point for the beginner to try to understand, as it was specifically designed to rule out a single possible loophole in all the previous tests, the fact that it was, in principle, possible for the 'effect' to be caused by signals traveling from A to B at speeds below the speed of light. Aspect showed that whatever was happening DOES actually happen faster than the speed of light. The effect which is measured was travelling FASTER THAN the speed of light. Einstein's special relativity asserted that all cause and effect actions for physical objects happen at or below the speed of light, within a sphere which moves outward from the cause at the speed of light. Such effects are called 'local'. Action at a distance beyond the light sphere would be a non-local effect. Aspect appears to have achieved two sets of measurements outside the lightspheres of each other and showed that a correllation still exists. This was accomplished by using acoustial optical couplers to 'randomly' switch both ends of the test into differently aligned detectors, and showing that the excess correlations still show up which seem to violate Bell's upper limit imposed by his inequality.

To really understand the test and the mistake made in each one you should start with some of the earlier tests which were simpler in their setup since what I am refering to is a fundamental misconception in the formulation of the test and not just a loophole. (See "Quantum Reality" by Nick Herbert for a good simplified descriptive explaination of the EPR tests.)

The EPR experiments can take any one of several forms, but the underlying principles are the same. A central source generates photons or particles which have related properties, such as polarization or spin. These particles are separated by some distance and then the properties are measured in analyzers which can detect photons or particles which have passed into the detector.

detector polarizer source polarizer detector

A < -- -///\\\ -- -- -- -- -- -- * -- -- -- -- -- -- -///\\\ -- -> B

| |

| |

` -- -- -- -- -- -- -> coincidence detector < -- -- -- -- -- -- -'

It makes little difference whether the test is done with neutrons or protons or photons, as long as we take into account the known correlations. In some photon tests the analyzers contain a polarizer such as polarizing plates or a calcite crystal to separate photons at 90 degrees from each other. These types of analyzers will detect photons that are polarized in the same orientation as analyzer but photons which are orientated perpendicular to the analyzer will be totally blocked by the polarized plates or deflected into an additional detector with the calcite. Photons which arrive at the analyzer at other angles in between, have the probability of detection which varies with their relative angle to the analyzer, forming a cosine squared type of graph. The 'result' refered to by Bell is a timed count which constitutes a measurement of matches at A and B. It is important to understand that this result is the relative output of the coincidence detector, not the separate results at each analyzer. This small point turns out to cause a major error in the logic of the test because the original reasoning had to do with measurements at A not being affected by actions at B and vice versa. The data which are measured at the coincidence detector (C) is something quite different even though Bell deals with this data as if the same logic can be applied.

The analyzers in some of these tests are designed such that they can be rotated to various angles and in some other tests predetermined settings are fixed. In all cases it can be seen that the angle between the two analyzers is related to the relative output in a shape described as a cos^2 type of graph. The shape of the curve is considered proof by some authors that the inequality is violated since the inequality is a straight line and any curvature on the plus side will 'prove' something spooky going on.

The simplified experiment (See: Herbert) can be paraphrased like this: We can demonstrate a "Spooky' QM effect by aligning both analyzers, lets call them A and B, and counting events which pass through BOTH detectors and using this as your reference count. This count is extrapolated and graphed and it is assumed that it represents 100% correlation between the choices made at A and B. A drop in the measured results refers to mismatches or 'errors'. The test continues by turning one detector through enough of an angle to produce a given percentage of errors, say 5%. Then this analyzer is turned back to normal then the other detector is turned in the opposite direction the same amount, and once again 5% errors occur. Now what percentage of errors should be expected when both detectors are turned and the test is done again? 10 percent?

Bell's theorem is based in the idea that the total measurement for two analyzers cannot be more than the sum of the changes in each, if the local view of Einstein holds since changes at A should have no effect on the results measured at B and vice versa. But when the test is run there are MORE than 10% errors at the given angle. Thus the local views of Einstein had apparently been 'proven' to be wrong.

Bell says:

"Now we make the hypothesis, and it seems at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other." [From Bell's original paper "On the Einstein-Podolsky-Rosen paradox"(1964) in section 2 (Formulation) sentence 5]

Bell puts a footnote in this line refering to a statment made by Einstein: "But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of system S2 is independent of what is done with the system S1, which is spatially separated from the former." ['Albert Einstein, philosopher Scientist', Edited by P. A. Schilp, p.85, Library of Living Philosophers, Evanston, Illinois (1949)].

What Einstein calls S1 and S2 I will refer to as A and B. While Bell's hypothesis certainly seems to follow the same premise of Einstein's comment, there is a mistake in Bell's use of the concept of 'independent' as described by Einstein. Einstein is saying quite clearly that the actual facts and events in system A are not influenced by any change in system B. Bell then applies this same logic to a different situation where 'results' change from actual facts at A and B to the 'RESULTS' measured at the output of the coincidence detector. Einstein's idea of a real factual situation is a detector getting a hit or a miss, it is the information which appears at each end of the experimental setup. But what comes out of the coincidence detector will also depend on whether there is a hit or a miss at the other end of the experiment. Information from each end is compared and then (and only then) can it be said if there is an error. Although the hypothesis of Bell seems only to be a subtle and insignificant change from Einstein's statement, it can be shown to be otherwise. Perhaps it would be easier to see the error if we substitute a more commonly understood situation to see how the logic is being applied:

Suppose two people are playing poker and each cuts and draws five cards from individual decks. Einstein's statements (in poker terms) would be:

"No matter what cards player A draws, it will not change the cards in player B's hand".

Bell's statement (in poker terms) would be:

'therefore, no matter what cards player A draws, it cannot change the 'results' in the comparison of the hands (who wins)".

(What?!!!)

Now it becomes clear that there is a real problem with this kind of logic. Winning the hand at A does depend on what B draws and vice versa since the wins are only measured at C and by definition and physical setup, this will be a 'global' effect. Bell is asking us to believe that on the quantum level two interacting particles behave as though they are still connected. This is like saying I have a magic deck of cards and whatever card player A draws will be the same as player B draws. This is illogical.

Suppose you and I are astronaults and we are now 10 light years distant from each other when we each draw our cards at preset times. Lets say you draw a Jack and one second later I draw a King. Then we travel to a common point to compare our results and compute our winnings. The change in the value of your Jack occured instantly (faster than the speed of light) and at a distance when I drew the King. This can be argued to be a non-local 'effect' which can be measured. It can be said that this is even an effect which has real physical consequances. Suppose we bet a dollar on the outcome. You will (later) feel the 'result' it in your walet. The effect is real and it can be measured and yes it travels faster than the speed of light, but IT'S NOT A PHYSICAL OBJECT OR SIGNAL, it simply a change in 'interpetation' that you are counting. In the case of poker it is easy to see that nothing physical changed and so it would be a mistake to call this a non-local effect. Bell's inequality uses this kind of 'logic' but it is just a bit harder to see.

Consider another card game where two people each have a deck of cards and the rules of this game say when both players cut the same color card then they both get a token from the bank. Now one player cuts a red card and the other cuts a black card. Who made the mistake? The question is meaningless, but the effect of the second card on the value of the first card is instantaneous, faster than the speed of light, yet nothing changes except the 'interpetation' of the VALUE of the first card. Lets call red cards 1 and black cards 0. Now suppose I were trying to cut only red cards and in a run of 6 cards I cut all red: 1,1,1,1,1,1. In the same 6 turns suppose my opponet cut 1,0,1,0,1,0. The resulting tokens from the bank (the results of the coincidence detector) are 1,0,1,0,1,0. Now does it follow that we are really dealing with a clasical situation where 'A does not effect B'? There is a problem with this logic. The same faulty logic is happening in the EPR tests, when the photon at A goes through the 'up' channel and then the photon at B goes through the 'down' channel the effect on A is instantaneous since it suddenly becomes a mismatch instead of a match, but it is important to realize that nothing actually changes at A, only at the coincidence detector.

I believe there is this subtle mistake in all of the tests which causes the inequality to be incorrectly applied to the plotted measurements. Thus the "Simple logic' used by Bell seems to me to contain a vital flaw. I believe the mistake is manifested in the way this reasoning causes one to scale of the data to plot it against the graph of the upper limit as put forth by Bell.

It is argued that changes in the measurements at A should have no effect on the measurements made at B and vice versa. I follow Herbert's lead and call these changes 'errors' since when we make a chart for analyzing the results, we always do something equivalent to plotting 0 degrees at the maximum and using this as our reference point, we plot points of our decreased measurements vs Bell's upper limit. Each mismatch then is considered an error caused by the missalignment of the two polarizers and the poin plots lower on the chart. It is assumed that a turning of analyzer A makes 'errors' at A and a turning of analyzer B makes 'errors' at B. This is wrong. The change which does occur in the output of the coincidence detector is a different kind of effect. It is a 'global' effect. It is something like passing photons through two polarizers in series:

source -- -- -- -> A -- -- -- -- -- -- -- -- -- -- -- -- -> B -- -- -> detector

This is a well known experiment where the measurement at the detector is 1/2 cos^2, where you are concerned with the angle between the orientations of the two polarizers. This result is known as the Law of Malus and plots out a type of a normal sine wave. When A and B are aligned you get the maximum amount of photons. In an ideal experiment 50% of the photons will pass through. When A and B are at 90 degrees from each other you will get no photons. Can the simple logic of Bell's inequality be applied to this? Turn polarizer A through enough of an angle to produce a given percentage of errors, say 5% from the maximum. Then this polarizer is turned back to normal then the other polarizer is turned in the opposite direction the same amount, and once again 5% errors occur. Now what percentage of errors should be expected when both polarizers are turned and the test is done again? 10 percent? Of course not. Anyone would expect a larger change and simply say Bell's limit does not apply. In this case it is obvious that the logic of Bell's inequality does not apply. (Or have I just proven series polarizers are non-local?!) This is know as a global effect, the result depends on the state of both polarizers. A close look at the tests of Bell's Inequality will show you that a similar (but different) global effect is all that is being measured.

source -- -- -- -> X1 -- -- -- -- -- -- -- -- -- -- -- -- -> A -- -- -> detector

source -- -- -- -> X2 -- -- -- -- -- -- -- -- -- -- -- -- -> B -- -- -> detector

Imagine two series experiments set up side by side. Polarizer X1 and X2 are set at the same angle. Polarizer A and B are set at the same angle but different from X1 and X2. Two photons from the two sources which have passed through the first set of polarizers are now known to have the same polarization. When these photons each reach their second polarizer, there will be separate probabilities that each will pass. The local hidden variable view would treat the two branches of the EPR experiment with the same logic. There will be separate probabilities that each photon at A or B will pass or miss. Of course, if you assume there is a QM correlation then the results will plot out differently. Suppose we just consider 1/180 of the pairs, those which happen to arive at the polarizers at an angle of 13 degrees. There is normally one chance in 20 of each photon going into the down channel and 19 chances out of 20 of passing into the up channel. We do not know which one of twenty it will be for either path which will choose the down path, we would not expect it to always be on the 20th photon or at any other specific number. The chance that both choose the up channel is 19/20 x 19/20 or 361 out of 400. The chance that both choose the down channel will be 1/20 x 1/20 or 1 out of 400. So 362 out of 400 pairs will NORMALLY MATCH at A and at B, the other 38 photon pairs will NORMALLY MISMATCH one choosing the up channel and one choosing the down channel. This is normal probability. At the other 179 angles different amounts of matching and mismatching will occur and if you add up all photon pairs at all angles then only 73% normally match with the polarizers aligned. Since QM assumes 100% match at 0 degrees they plot their results diferently that a LHV physicist would and this is what causes the apparent violation of Bell's Inequality.

No special 'entanglement' is required to derive a cos^2 curve with maximua clearly less than 100% and minimums clearly greater than 0%. The computer simulation of the CIRCLES AND SHADOWS game which I ran through 20,000 turns produced a maximum of 73% and a minimum of 37%.

"73% agrees very well with the analytical solution

2-4/pi = 0.72676

that I get for the aligned case (angle = 0)."

-- Bjorn Danielsson

If you erroreously assume that both photons will make the same choice because they came from the same event then you get results which seem to support the QM view. Many people have said that we know both make the same choice due to the results of the EPR tests. It should be clear that this is both an assumption you are making and the result you are proving. If you prove what you assume then you have proven nothing.

The quantum mechanical prediction happens to agree with this scaled up cos^2 curve and so each time the test has been carried out the results seem to follow the curve of Quantum Mechanics and the straight line ascribed to the local reality view of Einstein is apparently exceeded.

Clauser, Horne, Shimony, and Holt

Refering to "experimental test of Local Hidden-Variable Theories" by Freedman and Clauser (1972) Physical Review Letters, 28, 938-41. One derivation of Bell's inequality which has convinced many people that the problems of this test can removed is the idea that all you need is a way to remove the polarizers from each path so that you will have a reference to compare with. This derivation was described by Clauser, Horne, Shimony, and Holt. In 1972 the test performed by Freedman and Clauser used this 'improved' inequality to verify that these tests really do produce the non-local effects. They conclude that their results are strong evidence against local hidden-variable theories. I believe the use of this form of the inequality does not remove the problem inherent in the previous math, it only provides an automatic form of scaling. You have to be especially careful when you try to derive a good equation from a bad one. The problem can be seen in the plot on page 417 of W&Z (Quantum Theory and measurement edited by Wheeler and Zurek) and in equation 3 on page 415. The 1/4 term refers to the fact that the scaled up curve goes from 0 to 100% on a 0 to .5 scale and the range from 22 1/2 to 67 1/2 degrees is 1/2 of the total from 0 to 90 degrees so on a linear Bell type limit the change in the curve should be less than or equal to 1/4. This is the same straight line that Bell's limit plots with the previous versions of the inequality. Of course, the scaled up results form a cosine curve as they should and so the vertical plot covers more than 1/4 in this part of the curve so they conclude that the inequality is violated. Using a reference and ratios does change the scale of the plot but it does not change the bad logic into good logic. Since I have shown that the inequality cannot be plotted so the 100% point coinsides with the maximum measurement in an LHV view you still have to account for that.

Circular Logic

Bell's logic appears to be correct from a QM point of view because of the fact that QM assumes the two photons or particles are somehow entangled with each other and still have a connection with each other after they are separated. They believe that these two photons will both make exactly the same choice if they encounter a polarizer at the same angle.

Nick Herbert gives a clear statment of this assumption on page 215 of "Quantum Reality":

"In the twin state each beam by itself appears completely unpolarized -- an unpredictibly random 50-50 mixture of ups and downs at whatever angle you choose to measure. Though separately unpolarized, each photon's polarization is perfectly correlated with its partner's. If you measure the 'P' of both photons at the same angle (a two photon attribute I call paired polarization), these polarizations always match."

Herbert clearly states (and all other EPR testers in some way assert) that this is assumed at the outset of these important tests. But if you ask how this is known they say "The results of Aspect's test show this". How can the results of the test be used in the assumptions you start with? This circular proof makes all the tests non-rigorous.

Bell's inequality is graphed as a straight line which represents the upper limit in the summation of the 'errors' from both ends of the experiment but Bell mistakenly believes he can plot the maximum measurement of the EPR tests to this same point. The inequality starts at 100% at the upper left corner of these graphs and descends linearly to 0 at the bottom right. So everyone erronously plots the maximum 'extrapolated' measurement at this same point. This is why the normal curve of the results is interpeted as an excess correlation, the upper limit seems to be violated (because a lot of the data is left out when we assume the maximum measurement represents 100% agreement in the choices of ALL pairs when the analyzers are aligned.)

It seems more logical that the detection of each photon arriving at each polarizer is simply a probability based on the relative angles between the polarizer and the spin vector of the photon.

In that case the measurements of these polarizations will NOT always match even when both polarizers are aligned with each other. It means a substantial amount of events will be missing from the data. And Bell's inequality is only applicable if it includes all the data. Detector eficencies are so low that all tests results must be extrapolated to scale. Even Bell explains the need for this. What it means is that they can easily ignore those events which normally miss-match. It is no wonder there are excess correlations if you start by assuming correlations that do not exist.

On the other hand if you do an EPR test an do not assume that both photons will make the same choice then when you extrapolate the results the maximum correlation will be around 73% and the curve will never exceed the inequality which starts at 100%. If you take into account the events which normally will not match, it shifts the graph. The inequality is well above all parts of the measured curve and no violation ever ocurrs.

Since the test only proves QM when you assume QM, it is circular logic and non-rigorous. ALL EPR experiments have used this faulty scaling as the basis of their determination of the validity, or lack of, of the local reality views and it so is clear that the local reality models cannot be rejected using these experiments.

Perhaps Einstein was right all along.

-- David Elm

copyright 1993,1996 by David A. Elm All rights reserved worldwide

Please start your investigation into this study of the EPR tests with this excellent book:

Quantum Reality by Nick Herbert

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