What Is an Element of Reality?

[JohnBarchak]

In "Do we really understand quantum mechanics? Strange correlations,
paradoxes and theorems." by F. Laloe, Laboratoire de Physique de l'ENS, LKB, 24 rue Lhomond, F-75005 Paris, France,
Laloe explores the meaning of "element of reality":

"3.2 Of peas, pods and genes
When a physicist attempts to infer the properties of microscopic objects from macroscopic observations, ingenuity (in order to design meaningful experiments) must be combined with a good deal of logic (in order to deduce these microscopic properties from the macroscopic results). Obviously, some abstract reasoning is indispensable, merely because it is impossible to observe with the naked eye, or to take in one's hand, an electron or even a macromolecule for instance. The scientist of past centuries who, like Mendel, was trying to determine the genetic properties of plants, had exactly the same problem: he did not have access to any direct observation of the DNA molecules, so that he had to base his reasoning on adequate experiments and on the observation of their macroscopic outcome. In our parable, the scientist will observe the color of flowers (the "result" of the measurement, +1 for red, -1 for blue) as a function of the condition in which the peas are grown (these conditions are the "experimental settings" a and b, which determine the nature of the measurement). The basic purpose is to infer the intrinsic properties of the peas (the EPR "element of reality") from these observations.

3.2.1 Simple experiments; no conclusion yet.
It is clear that many external parameters such as temperature, humidity, amount of light, etc. may influence the growth of vegetables and, therefore, the color of a flower; it seems very difficult in a practical experiment to be sure that all the relevant parameters have been identified and controlled with a sufficient accuracy. Consequently, if one observes that the flowers which grow in a series of experiments are sometimes blue, sometimes red, it is impossible to identify the reason behind these fluctuation; it may reflect some trivial irreproducibility of the conditions of the experiment, or something more fundamental. In more abstract terms, a completely random character of the result of the experiments may originate either from the fluctuations of uncontrolled external perturbations, or from some intrinsic property that the measured system (the pea) initially possesses, or even from the fact that the growth of a flower (or, more generally, life?) is fundamentally an indeterministic process - needless to say, all three reasons can be combined in any complicated way. Transposing the issue to quantum physics leads to the following formulation of the question: are the results of the experiments random because of the fluctuation of some uncontrolled influence taking place in the macroscopic apparatus, of some microscopic property of the measured particles, or of some more fundamental process?

The scientist may repeat the "experiment" a thousand times and even more: if the results are always totally random, there is no way to decide which interpretation should be selected; it is just a matter of personal taste. Of course, philosophical arguments might be built to favor or reject one of them, but from a pure scientific point of view, at this stage, there is no compelling argument for a choice or another. Such was the situation of quantum physics before the EPR argument.

3.2.2 Correlations; causes unveiled.
The stroke of genius of EPR was to realize that correlations could allow a big step further in the discussion. They exploit the fact that, when the choice of the settings are the same, the observed results turn out to be always identical; in our botanical analogy, we will assume that our botanist observes correlations between colors of flowers. Peas come together in pods, so that it is possible to grow peas taken from the same pod and observe their flowers in remote places. It is then natural to expect that, when no special care is
taken to give equal values to the experimental parameters (temperature, etc.), nothing special is observed in this new experiment. But assume that, every time the parameters are chosen to the same values, the colors are systematically the same; what can we then conclude? Since the peas grow in remote places, there is no way that they can be influenced by the any single uncontrolled fluctuating phenomenon, or that they can somehow influence each other in the determination of the colors. If we believe that causes always act locally, we are led to the following conclusion: the only possible explanation of the common color is the existence of some common property of both peas, which determines the color; the property in question may be very difficult to detect directly, since it is presumably encoded inside some tiny part of a biological molecule, but it is sufficient to determine the results of the experiments.

Since this is the essence of the argument, let us make every step of
the EPR reasoning completely explicit, when transposed to botany. The
key idea is that the nature and the number of "elements of reality"
associated with each pea can not vary under the influence of some
remote experiment, performed on the other pea. For clarity, let us first assume that the two experiments are performed at different times: one week, the experimenter grows a pea, then only next week another pea from the same pod; we assume that perfect correlations of the colors are always observed, without any special influence of the delay between the experiments. Just after completion of the first experiment (observation of the first color), but still before the second experiment, the result of that future experiment has a perfectly determined value; therefore, there must already exist one element of reality attached to the second pea that corresponds to
this fact - clearly, it can not be attached to any other object than the pea, for instance one of the measurement apparatuses, since the observation of perfect correlations only arises when making measurements with peas taken from the same pod. Symmetrically, the first pod also had an element of reality attached to it which ensured that its measurement would always provide a result that coincides with that of the future measurement. The simplest idea that comes to mind is to assume that the elements of reality associated with both peas are coded in some genetic information, and that the values of the codes are exactly the same for all peas coming from the same pod; but other possibilities exist and the precise nature and mechanism involved in the elements of reality does not really matter here. The important point is that, since these elements of reality can not appear by any action at a distance, they necessarily also existed before any measurement was performed - presumably even before the two peas were separated.

Finally, let us consider any pair of peas, when they are already spatially separated, but before the experimentalist decides what type of measurements they will undergo (values of the parameters, delay or
not, etc.). We know that, if the decision turns out to favor time separated measurements with exactly the same parameter, perfect correlations will always be observed. Since elements of reality can not appear, or change their values, depending of experiments that are performed in a remote place, the two peas necessarily carry some elements of reality with them which completely determine the color of the flowers; any theory which ignores these elements of reality is incomplete. This completes the proof.

It seems difficult not to agree that the method which led to these conclusions is indeed the scientific method; no tribunal or detective would believe that, in any circumstance, perfect correlations could be observed in remote places without being the consequence of some common characteristics shared by both objects. Such perfect correlations can then only reveal the initial common value of some variable attached to them, which is in turn a consequence of some fluctuating common cause in the past (a random choice of pods in a bag for instance). To express things in technical terms, let us for instance assume that we use the most elaborate technology available to build elaborate automata, containing powerful modern computers if necessary, for the purpose of reproducing the results of the remote experiments: whatever we do, we must ensure that, somehow, the memory of each computer contains the encoded information concerning all the
results that it might have to provide in the future (for any type of
measurement that might be made).

To summerize this section, we have shown that each result of a measurement may be a function of two kinds of variables:

(i) intrinsic properties of the peas, which they carry along with them.
(ii) the local setting of the experiment (temperature, humidity, etc.);
clearly, a given pair that turned out to provide two blue flowers could have provided red flowers in other experimental conditions. We may also add that:
(iii) the results are well-defined functions, in other words that no
fundamentally indeterministic process takes place in the experiments.
(iv) when taken from its pod, a pea cannot "know in advance" to which sort of experiment it will be submitted, since the decision may not yet have been made by the experimenters; when separated, the two peas therefore have to take with them all the information necessary to determine the color of flowers for any kind of experimental conditions. What we have shown actually is that each pea carries with it as many elements of reality as necessary to provide "the correct answer" to all possible questions it might be submitted to."

The complete paper "Do we really understand quantum mechanics?
Strange correlations, paradoxes and theorems." can be found at:
http://arxiv.org/PS_cache/quant-ph/pdf/0209/0209123.pdf

All the best
John B.

 

[DrChinese]

In "Do we really understand quantum mechanics? Strange correlations,
paradoxes and theorems." by F. Laloe, Laboratoire de Physique de l'ENS, LKB, 24 rue Lhomond, F-75005 Paris, France,
Laloe explores the meaning of "element of reality":

"3.2 Of peas, pods and genes

(comparing particles to peas as a proof...)

Embarrassing. We all understand the "common sense" of the local realistic position. That and a quarter will get you 25 cents.

1) What are you saying, other than quoting other people? Are we to deduce from the quote that it is an exact representation of your position? Or are you being coy, and hoping we will misread your position? If you have something to say, why won't you say it? (That is normally incumbent on those who start threads.)

2) How do peas prove EPR? You are going to have to do better than that. We understand that some people hypothesize the existence of little teeny tiny attributes that we cannot see. Most of us call those "hidden variables" and don't need to call them pea DNA by childish analogy. We also understand that no-one knew about DNA a few hundred years ago. Also a poor analogy.

EPR envisioned that the so-called hidden variables would eventually be uncovered. That hasn't happened in 80 years of looking. Instead, it has become obvious to scientists that there is no combination of hidden variables that can mimic the results of certain experiments (per Bell). Please tell us - SPECIFICALLY and not hand waving - how you conclude otherwise. If there is an "element of reality" we are missing, please, do show us. I, for one, am all ears.

 

[JohnBarchak]

EPR never said ANYTHING about hidden variables - if you can find one instance of EPR talking about hidden variables, I will give you my car.

All the best
John B.

 

[DrChinese]

EPR never said ANYTHING about hidden variables - if you can find one instance of EPR talking about hidden variables, I will give you my car.

All the best
John B.

What kind of car do you have?

"While we have thus shown that the wave function does not provide a complete specification of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible."-EPR

If the more complete description is dependent on finding something which is now hidden, I would call that "hidden variables". The definition of "hidden variables" is usually taken to be those variables which supply the missing description.

It certainly isn't pea DNA, and I notice that you completely sidestep all of my questions as per your usual. Do you have any position? Or is your objective to stir controversy?

 

[JesseM]

EPR never said ANYTHING about hidden variables - if you can find one instance of EPR talking about hidden variables, I will give you my car.

All the best
John B.

They used the phrase "element of reality", and made it clear that they believe there is an element of reality corresponding to the value of both of two physical properties with noncommuting operators, like position and momentum--this exactly what is meant by "hidden variables". From the EPR paper:

Previously we proved that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.

One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the system in any way. No reasonable definition of reality could be expected to permit this.

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

So, they reject the view that position Q and momentum P are not both elements of reality which exist prior to the measurement of the entangled particle--this means they are arguing for hidden variables.

 

[JohnBarchak]

No "hidden variables" were needed. All Bohr had to do in order to complete QM was admit the calculated (unobserved) variable - Einstein claimed that if he could predict it with probabiliy 1, then the calculated (unobserved) variable was an "element of reality", that is, it was as valid as the observed variable. Bohr would not agree to this since it would invalidate Heisenberg uncertainty. So, no hidden variables were ever needed. This is just one more example of QM people putting words in Einsteins mouth.

All the best
John B.

 

[JesseM]

No "hidden variables" were needed. All Bohr had to do in order to complete QM was admit the calculated (unobserved) variable - Einstein claimed that if he could predict it with probabiliy 1, then the calculated (unobserved) variable was an "element of reality", that is, it was as valid as the observed variable. Bohr would not agree to this since it would invalidate Heisenberg uncertainty. So, no hidden variables were ever needed. This is just one more example of QM people putting words in Einsteins mouth.

But that's what the phrase "hidden variables" means--don't get hung up on the word "hidden", it just means "a variable not directly measured, although some may believe its value can be inferred".

Do you agree that if Bell's theorem is violated in an experiment involving spin measurements, then it is impossible to explain the results of the experiment using the idea that each spin-value had a preexisting value without giving up locality?

 

[JohnBarchak]

No, if Einstein's Principle of Local Action is not valid, then neither is experimental science.

 

[JesseM]

No, if Einstein's Principle of Local Action is not valid, then neither is experimental science.

So does that mean you think it is impossible for Bell's theorem to be violated? Or do you disagree that a violation of Bell's theorem discredits any theory that postulates that all these variables have preexisting values and also respects the principle of local action?

Also, what do you think of Bohmian mechanics? This is a deterministic interpretation of QM which says particles have a definite position at all times (even when we measure their momentum), and which includes faster-than-light effects, but nevertheless can be proven to make all the same predictions as ordinary QM.

 

[DrChinese]

No, if Einstein's Principle of Local Action is not valid, then neither is experimental science.

I would say that this is the most moronic thing I have ever seen written, but that wouldn't be a nice thing to say.

Reality is what it is. It certainly does not matter to reality whether your purely semantic argument is correct. Meanwhile, the results of experiments are exactly as Bohr envisioned. So who has the last laugh? How do experiments of entangled particles correlate in violation of Bell?

 

[JohnBarchak]

But that's what the phrase "hidden variables" means--don't get hung up on the word "hidden", it just means "a variable not directly measured, although some may believe its value can be inferred".

But that is the whole point - for Einstein, it was not hidden.

 

[ZapperZ]

But that is the whole point - for Einstein, it was not hidden.

No, for Einstein, it was "hidden" from QM!

There are hidden variables in classical statistical mechanics of coin-tossing. It's perfectly deterministic. But due to our ignorance of the intricate details of its complete dynamics, we lump them all into statistical probabilities. Thus, all those intricate dynamics are hidden from the statistical description of coin-tossing.

Einstein is claiming the same thing. He said that there has to be some underlying mechanism of QM that is not included in its formulation. So these are hidden from the theory. In fact, this idea was later on used by Bohm as the hidden variables.[1]

Irregardless of what you think, there has been no controversies till now that the EPR paper is in fact claiming that QM is incomplete, and that this is due to variables not contained within the formalism. They may not explicitly use the pharse "hidden variables", but the implied presence of them has never been disputed within this paper.

Zz.

[1] D. Bohm, Phys. Rev. v.85, p.166 (1952).

 

[DrChinese]

Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the system in any way. No reasonable definition of reality could be expected to permit this.

"...when you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes (Sir Arthur Conan Doyle)

It is pretty clear from the experimental record: A measurement at one system determines the "reality" of the observable at the other. For spin entanglement, there is complete reality only when the polarizers are at 0 or 90 degrees. That is our "element of reality."

 

[JesseM]

But that is the whole point - for Einstein, it was not hidden.

Aside from what ZapperZ said, "hidden variables" is just a technical term, as long as all physicists understand what this term means in the context of QM, it doesn't matter if the words completely match their ordinary-english meaning. It's like how the "flavor" of a quark doesn't have anything to do with the ordinary meaning of the word "flavor".

And you still haven't answered my question--do you think it is impossible that Bell's theorem can ever be violated, or do you just deny that a violation of Bell's theorem discredits "local hidden variables", meaning the idea that noncommuting variables like position and momentum all have exact values even when we don't measure them, and that no influences can go faster than light?

 

[JohnBarchak]

The fact is that the quantum community has revised the Einstein side of the EPR argument to fit their desires for the past 75 years. I think the TRUTH is a better approach. When someone says that Einstein said something, that something should be what Einstein said.

Bell's 4 dimensional Hilbert space has almost nothing to do with the real world. As far as the Bell Test experiments, until a more mature model of the photon is developed, the Bell Test experiments will prove nothing. Are you familiar with the single photon interference experiments (a photon undergoing interference with itself).

All the best
John B.

 

[JesseM]

The fact is that the quantum community has revised the Einstein side of the EPR argument to fit their desires for the past 75 years. I think the TRUTH is a better approach. When someone says that Einstein said something, that something should be what Einstein said.

They haven't revised anything. Einstein believed that particles have definite values of position and momentum (and other noncommuting variables) at all times, even when we don't measure them, and that's what quantum physicists mean by the technical term "hidden variables". Again, it's unimportant whether the words match their ordinary english meaning, they could have called it "electric ostriches", and as long as all physicists knew what was meant by that term in the context of QM, it wouldn't matter that exact simultaneous values for noncommuting variables have nothing to do with large flightless birds.

Bell's 4 dimensional Hilbert space has almost nothing to do with the real world. As far as the Bell Test experiments, until a more mature model of the photon is developed, the Bell Test experiments will prove nothing. Are you familiar with the single photon interference experiments (a photon undergoing interference with itself).

Bell's inequality has nothing to do with any "4 dimensional Hilbert space", it is just based on basic logic and probability. Here's a quickie explanation of the meaning of Bell's Theorem I wrote up on another forum. First, check out this analogy from the book Time's Arrow and Archimedes' Point:

By modern standards the criminal code of Ypiaria [pronounced, of course, "E-P-aria"] allowed its police force excessive powers of arrest and interrogation. Random detention and questioning were accepted weapons in the fight against serious crime. This is not to say the police had an entirely free hand, however. On the contrary, there were strict constraints on the questions the police could address to anyone detained in this way. One question only could be asked, to be chosen at random from a list of three: (1) Are you a murderer? (2) Are you a thief? (3) Have you committed adultery? Detainees who answered "yes" to the chosen question were punished accordingly, while those who answered "no" were immediately released. (Lying seems to have been frowned on, but no doubt was not unknown.)

To ensure that these guidelines were strictly adhered to, records were required to be kept of every such interrogation. Some of these records have survived, and therein lies our present concern. The records came to be analyzed by the psychologist Alexander Graham Doppelganger, known for his work on long distance communication. Doppelganger realized that among the many millions of cases in the surviving records there were likely to be some in which the Ypiarian police had interrogated both members of a pir of twins. He was interested in whether in such cases any correlation could be observed between the answers given by each twin.

As we now know, Doppelganger's interest was richly rewarded. He uncovered the two striking and seemingly incompatible correlations now known collectively as Doppelganger's Twin Paradox. He found that

(8.1) When each member of a pair of twins was asked the same question, both always gave the same answer;

and that

(8.2) When each member of a pair of twins was asked a different question, they gave the same answer on close to 25 percent of such occasions.

It may not be immediately apparent that these results are in any way incompatible. But Doppelganger reasoned as follows: 8.1 means that whatever it is that disposes Ypiarians to answer Y or N to each of the three possible questions 1, 2, and 3, it is a disposition that twins always have in common. For example, if YYN signifies the property of being disposed to answer Y to questions 1 and 2 and N to question 3, then correlation 8.1 implies that if one twin is YYN then so is his or her sibling. Similarly for the seven other possible such states: in all, for the eight possible permutations of two possible answers to three possible questions. (The possibilities are the two homogeneous states YYY and NNN, and the six inhomogeneous states YYN, YNY, NYY, YNN, NYN, and NNY.)

Turning now to 8.2, Doppelganger saw that there were six ways to pose a different question to each pair of twins: the possibilities we may represent by 1:2, 2:1, 1:3, 3:1, 2:3, and 3:2. (1:3 signifies that the first twin is asked question 1 and the second twin question 3, for example.) How many of these possibilities would produce the same answer from both twins? Clearly it depends on the twins' shared dispositions. If both twins are YYN, for example, then 1:2 and 2:1 will produce the same response (in this case, Y) and the other four possibilities will produce different responses. So if YYN twins were questioned at random, we should expect the same response from each in about 33 percent of all cases. And for homogeneous states, of course, all six posible question pairs produce the same result: YYY twins will always answer Y and NNN twins will always answer N.

[Note--I think Price actually gets the probability wrong here. If both twins are YYN, for example, then if they are questioned at random, the probability both will give the same answer would be P(first twin answers Y)*P(second twin answers Y) + P(first twin answers N)*P(second twin answers N) = (2/3)*(2/3) + (1/3)*(1/3) = 5/9, not 1/3 as Price claims. But this doesn't change the overall argument. edit: as Bartholomew pointed out below, I was misunderstanding Price here, he's actually calculating the probability the twins will give the same answer only in the subset of cases where they were asked different questions, not in all cases as I mistakenly assumed]

Hence, Doppelganger realized, we should expect a certain minimum correlation in these different question cases. We cannot tell how many pairs of Ypiarian twins were in each of the eight possible states, but we can say that whatever their distribution, confessions should correlate with confessions and denials with denials in at least 33 percent of the different question interrogations. For the figure should be 33 percent if all the twins are in inhomogeneous states, and higher if some are in homogeneous states. And yet, as 8.2 describes, the records show a much lower figure.

Doppelganger initially suspected that this difference might be a mere statistical fluctuation. As newly examined cases continued to confirm the same pattern, however, he realized that the chances of such a variation were infinitesimal. His next thought was therefore that the Ypiarian twins must generally have known what question the other was being asked, and determined their answer partly on this basis. He saw that it would be easy to explain 8.2 if the nature of one's twin?'s question could influence one's own answer. Indeed, it would be easy to make a total anticorrelation in the different question cases be compatible with 8.1--with total correlation in the same question cases.

Doppelganger investigated this possibility with some care. He found, however, that twins were always interrogated separately and in isolation. As required, their chosen questions were selected at random, and only after they had been separated from one another. There therefore seemed no way in which twins could conspire to produce the results described in 8.1 and 8.2. Moreover, there seemed a compelling physical reason to discount the view that the question asked of one twin might influence the answers given by another. This was that the separation of such interrogations was usually spacelike in the sense of special relativity; in other words, neither interrogation occurred in either the past or the future light cone of the other. (It is not that the Ypiarian police force was given to space travel, but that light traveled more slowly in those days. The speed of a modern carrier pigeon is the best current estimate.) Hence according to the principle of the relativity of simultaneity, there was no determinate sense in which one interrogation took place before the other.

The situation in one version of the EPR experiment is almost exactly like the situation with these imaginary Ypiarian twins, except that instead of interrogators having a choice of 3 crimes to ask the twins about, experimenters can measure the "spin" of two separated electrons along one of three axes, which we can label a, b, and c (this is not the only type of EPR experiment--the one that is usually tested experimentally is one involving photons called the http://roxanne.roxanne.org/epr/eprS.html explains:

a b c a b c freq
+ + + - - - N1
+ + - - - + N2
+ - + - + - N3
+ - - - + + N4
- + + + - - N5
- + - + - + N6
- - + + + - N7
- - - + + + N8

Each row describes one type of electron pair, with their respective hidden variable values and their probabilites N. Suppose Alice measures the spin in the a direction and Bob measures it in the b direction. Denote the probability that Alice obtains +1/2 and Bob obtains +1/2 by

P(a+,b+) = N3 + N4

Similarly, if Alice measures spin in a direction and Bob measures in c direction, the probability that both obtain +1/2 is

P(a+,c+) = N2 + N4

Finally, if Alice measures spin in c direction and Bob measures in b direction, the probability that both obtain the value +1/2 is

P(c+,b+) = N3 + N7

The probabilities N are always non-negative, and therefore:

N3 + N4 <= N3 + N4 + N2 + N7

This gives

P(a+,b+) <= P(a+,c+) + P(c+,b+)

which is known as a Bell inequality. It must be satisfied by any hidden variable theory obeying our very broad locality assumptions.

But in reality, the Bell inequalities are consistently violated in the EPR experiment--you get results like P(a+, b+) > P(a+,c+) + P(c+,b+). Again, this shows that you can't just assume each pair of electrons had well-defined opposite spins on each axis before you measured them, despite the fact that whenever the two experimenters choose to measure along the same axis, they always find the two electrons have opposite spins on that axis. There are some ways to save the idea that the particle has a well-defined state before measurement, but only at the cost of bringing in ideas like faster-than-light communication between the electrons or the choice of measurements retroactively influencing the states of the two particles when they were created.

 

[Bartholomew]

Note--I think Price actually gets the probability wrong here.

No, he was right. He had the proviso that the questions asked each twin were not the same. This gives the correct probability: 2 cases / 6 cases or 1/3.

 

[JesseM]

No, he was right. He had the proviso that the questions asked each twin were not the same. This gives the correct probability: 2 cases / 6 cases or 1/3.

Ah, I didn't catch that. Still, it seems like the connection with EPR-type experiments would be better if you assume each interrogator picks his question at random right before he asks it to the twin he's interrogating, so that when each twin answers there's no way he could know anything about what question his brother was asked (assuming information can't travel faster than light).

edit: Or perhaps he meant that the experimenters do choose their questions randomly right before they ask them, but that we restrict our attention to the subset of cases where they randomly happened to ask different questions, and throw out the other 1/3 of cases where they happened to ask the same question. Out of this subset, a "hidden-variables" theory where you assume the twins had already decided on answers to all three questions would indeed predict that they'd give the same answer in at least 1/3 of the interrogations.

 

[ttn]

EPR envisioned that the so-called hidden variables would eventually be uncovered. That hasn't happened in 80 years of looking. Instead, it has become obvious to scientists that there is no combination of hidden variables that can mimic the results of certain experiments (per Bell).

You mean, no hidden variable theory can predict the correct answers for the Bell/EPR correlation experiments? That's just plain false. Bohmian mechanics does so.

Or maybe you meant that no hidden variable theory which respects Bell's Locality condition can predict the correct answers for such experiments. That's true; it's Bell's theorem.

But this is no argument against hidden variable theories, since orthodox QM itself violates Bell Locality. Remember, Bell Locality essentially amounts to the idea that joint probabilities for space-like separated events should factorize when you conditionalize on a complete specification (call it "L") of the world in the past light cones of the two events. Mathematically,

P(A,B|a,b,L) = P(A|a,L)*P(B|b,L)

where A and B refer to measurement outcomes, a and b refer to any other relevant parameters local to the two measurements respectively, and L is the complete specification across the past light cones.

Bohr (and all subsequent opponents of hidden variables) invites us to identify L with the quantum mechanical wave function psi. But according to QM,

P(A,B|a,b,psi) = P(A|a,psi)*P(B|b,psi)

is not valid. That is, orthodox QM (considered complete) violates Bell Locality.

And it is therefore a tragic (but admittedly widespread) mistake to argue against hidden variable theories on the grounds that they have to be non-local. Show me a theory that agrees with experiment and *is* local, then that objection might hold some water. But if one's only alternative to the (allegedly) preposterous-because-nonlocal hidden variable theories is orthodox QM itself, well, one would be shooting oneself in one's own foot...

ttn

 

[JesseM]

And it is therefore a tragic (but admittedly widespread) mistake to argue against hidden variable theories on the grounds that they have to be non-local. Show me a theory that agrees with experiment and *is* local, then that objection might hold some water.

An Everett-type interpretation, where there is no nonlocal "collapse of the wavefunction", might be able to do this--this was discussed a bit on the thread Aspect/Innsbruck Interpretation which respects SR locality. I posted some links to papers that argue this:

I came across this paper which seems to argue (although I may be misunderstanding) that you can get such a local description of the universe's state if you use the Heisenberg picture, where it's the operators that change over time rather than the wavefunction:

In the Everett interpretation the nonlocal notion of reduction of the wavefunction is eliminated, suggesting that questions of the locality of quantum mechanics might indeed be more easily addressed. On the other hand, while wavefunctions do not suffer reduction in the Everett interpretation, nonlocality nevertheless remains present in many accounts of this formulation. In DeWitt’s (1970) often-quoted description, for example, “every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on Earth into myriads of copies of itself.” Contrary to this viewpoint, others argue (Page, 1982; Tipler, 1986, 2000; Albert and Loewer, 1988; Albert, 1992; Vaidman, 1994, 1998, 1999; Price, 1995; Lockwood, 1996; Deutsch, 1996; Deutsch and Hayden, 2000) that the Everett interpretation can in fact resolve the apparent contradiction between locality and quantum mechanics. In particular, Deutsch and Hayden (2000) apply the Everett interpretation to quantum mechanics in the Heisenberg picture, and show that in EPRB experiments,1 information regarding the correlations between systems is encoded in the Heisenberg-picture operators corresponding to the observables of the systems, and is carried from system to system and from place to place in a local manner. The picture which emerges is not one of measurement-type interactions “splitting the universe” but, rather, producing copies of the observers and observed physical systems which have interacted during the (local) measurement process (Tipler, 1986).

Likewise, in this paper by the same author, I think he's arguing that the Everett interpretation of quantum field theory can also be understood in terms of information encoded in purely local operators:

In the Everett interpretation, correlations between the two experimenters’ results are not at issue; rather, a different question of causation arises. According to Everett, both possible outcomes, spin-up and spin-down, occur at each analyzer magnet and, at the conclusion of the experiment, there are two copies of each experimenter.2 When they compare their respective results using some causal means of communication, Alice-who-saw-spin-up only talks to Bob-who-saw-spin-down, and Alice-who-saw-spin-down always converses with Bob-who-saw-spin-up. What is the mechanism which brings about this perfect anticorrelation in the possibilities for exchange of information between the Alices and the Bobs?

Deutsch and Hayden(21) have identified this mechanism. In the Heisenberg picture of quantum mechanics, the properties of physical systems are represented by time-dependent operators. When two systems interact, the operators corresponding to the properties of each of the systems may acquire nontrivial tensor-product factors acting in the state space of the other system. These factors are in effect labels, appending to each system a record of the fact that it has interacted with a certain other system in a certain way.(22) So, for example, when the two particles in the EPRB experiment are initially prepared in the singlet state, the interaction involved in the preparation process causes the spin operators of each particle to contain nontrivial factors acting in the space in which the spin operators of the other particle act. When Alice measures the spin of one of the particles, the operator representing her state of awareness ends up with factors which act in the state space of the particle which she has measured, as well as in the state space of the other particle. The operator corresponding to Bob’s state of awareness is similarly modified. When the Alices and Bobs meet to compare notes, it is these factors which lead to the correct pairing-up of the four of them.

The amount of information which even a simple electron carries with it regarding the other particles with which it has interacted is thus enormous. In Ref. 22 I termed this the problem of “label proliferation,” and suggested that the physical question of how all this information is stored3 might receive an answer in the framework of quantum field theory.

More generally, quantum field theory is a description of nature encompassing a wider range of physical phenomena than the quantum mechanics of particles; it is therefore of interest to investigate the degree to which the conceptual picture of the labeling mechanism for bringing about correlations at a distance in a causal manner accords with the field-theoretic formalism.

Indeed, there is a simple line of argument which leads to the conclusion that Everett interpretation Heisenberg-picture quantum field theory must be local. The dynamical variables of the theory are field operators defined at each point in space, whose dynamical evolution is described by local (Lorentz-invariant, in the relativistic case) differential equations. And the Everett interpretation removes nonlocal reduction of the wavefunction from the formalism. So how can nonlocality enter the scene?

This argument as it stands is incorrect, but it can be modified so that its conclusion, the locality of Everett-interpretation Heisenberg-picture quantum field theory, holds. What is wrong is the following: While it is certainly true that operators in Heisenberg-picture quantum field theory evolve according to local differential equations, it is not in general true that all of the information needed to determine the outcomes and probabilities of measurements is contained in these operators. Initial-condition information, needed to determine probabilities, resides in the time-independent Heisenberg-picture state vector. Since not all information is carried in the operators, is incorrect to argue that the local evolution of the operators implies locality of the theory.

However, as discussed in Sec. 4 below, it turns out to be possible to transform from the usual representation of the Heisenberg-picture field theory to other representations in which the operators also carry the initial-condition information. So, in these representations, the simple argument above for the locality of Heisenberg-picture quantum field theory is valid. Bear in mind that in these representations the use of the Everett interpretation still is crucial for the theory to be local. As mentioned above, the Everett interpretation removes a source of explicit nonlocality in the theory (wavefunction collapse); it “defangs” the Bell argument that, notwithstanding the explicitly local transfer of information in the operators, something else of a nonlocal nature must be going on; and it provides labeling as an alternative to the “instruction set” mechanism which in single-outcome interpretations appears as the only explanation for correlations-at-a-distance and which is what ultimately leads to Bell’s theorem. This last issue of instruction sets and labels is no different in field theory than in first-quantized theory, and is discussed in Ref 22. In field theory as in first-quantized theory, interaction-induced transformations of Heisenberg-picture operators (field operators, of course, in the field theory case—see e.g., eq. (154) below) serve to encode the label information.

I also came up with this analogy to think about how an Everett-type interpretation might in principle be able to explain violations of Bell's theorem in a local way:

say Bob and Alice are each receiving one of an entangled pair of photons, and their decisions about which spin axis to measure are totally deterministic, so the only "splitting" necessary is in the different possible results of their measurements. Label the three spin axes a, b, and c. If they always find opposite spins when they both measure their photons along the same axis, a local hidden-variables theory would say that if they choose different axes, the probability they get opposite spins must be at least 1/3 (assuming there's no correlation between their choice of which axes to measure and the states of the photons before they make the measurement). I forgot what the actual probability of opposite spins along different axes ends up being in this type of experiment, but all that's important is that it's less than 1/3, so for the sake of the argument let's say it's 1/4.

So suppose Bob's decision will be to measure along axis a, and Alice's decision will be to measure along axis c. When they do this, suppose each splits into 8 parallel versions, 4 measuring spin + and 4 measuring spin -. Label the 8 Bobs like this:

Bob 1: a+
Bob 2: a+
Bob 3: a+
Bob 4: a+
Bob 5: a-
Bob 6: a-
Bob 7: a-
Bob 8: a-

Similarly, label the 8 Alices like this:

Alice 1: c+
Alice 2: c+
Alice 3: c+
Alice 4: c+
Alice 5: c-
Alice 6: c-
Alice 7: c-
Alice 8: c-

Note that the decision of how they split is based only on the assumption that each has a 50% chance of getting + and a 50% chance of getting - on whatever axis they choose, no knowledge about what the other one was doing was needed. And again, only when a signal traveling at the speed of light or slower passes from one to the other does the universe need to decide which Alice shares the same world with which Bob...when that happens, they can be matched up like this:

Alice 1 (c+) <--> Bob 1 (a+)
Alice 2 (c+) <--> Bob 2 (a+)
Alice 3 (c+) <--> Bob 3 (a+)
Alice 4 (c+) <--> Bob 5 (a-)
Alice 5 (c-) <--> Bob 4 (a+)
Alice 6 (c-) <--> Bob 6 (a-)
Alice 7 (c-) <--> Bob 7 (a-)
Alice 8 (c-) <--> Bob 8 (a-)

This insures that each one has a 3/4 chance of finding out the other got the same spin, and a 1/4 chance that the other got the opposite spin. If Bob and Alice were two A.I.'s running on classical computers in realtime, you could simulate Bob on one computer and Alice on another, make copies of each according to purely local rules whenever each measured a quantum particle, and then use this type of matching rule to decide which of the signals from the various copies of Alice will be passed on to which copy of Bob, and you wouldn't have to make that decision until the information from the computer simulating Alice was actually transmitted to the computer simulating Bob. So using purely local rules you could insure that, after many trials like this, a randomly-selected copy of A.I. Bob or A.I. Alice would record the same type of statistics that's seen in the Aspect experiment, including the violation of Bell's inequality.

Note that you wouldn't have to simulate any hidden variables in this case--you only have to decide what the spin was along the axes each one measured, you never have to decide what the spin along the other 2 unmeasured axes of each photon was.

Now, I realize that the various Everett interpretations are not so straightforward--in my computer simulation above, probability has a clear frequentist meaning, while the problem of getting a notion of "probability" out of any version of the Everett interpretation is notoriously difficult, and perhaps it can't work at all without tacking on extra assumptions. Still, I got the impression that this was the general type of explanation that Mark Rubin was aiming for in his papers, where each observation creates a local splitting of the observer, but the observations of spatially separated observers are only mapped to each other once a signal has had the chance to pass between them.

 

[ttn]

I also came up with this analogy to think about how an Everett-type interpretation might in principle be able to explain violations of Bell's theorem in a local way:

Your discussion is definitely clarifying, but I think the simplest way to understand how MWI gets around Bell's conclusion is simply that Bell's argument is premised on the idea that experiments have definite outcomes. MWI denies this. It says we are radically deluded if we believe that, e.g., in a sequence of polarization measurements, there exists a list (++-+--+-...) of the outcomes of those measurements -- the same kind of list that people analyze statistically in order to confirm agreement with QM, by the way.

In fact, MWI asserts that we are radically deluded about pretty much everything -- not just that physics experiments have outcomes, but that it is daytime now in new england, that certain people have recently died while others have recently been born, that the sun exists, etc. You name it, it ain't true according to MWI. So I have a lot of trouble taking MWI seriously as a scientific theory. It runs dangerously close to the self-refuting character of a claim like "All statements are false, and I can show you a lot of evidence to prove it."

 

[vanesch]

Your discussion is definitely clarifying, but I think the simplest way to understand how MWI gets around Bell's conclusion is simply that Bell's argument is premised on the idea that experiments have definite outcomes. MWI denies this. It says we are radically deluded if we believe that, e.g., in a sequence of polarization measurements, there exists a list (++-+--+-...) of the outcomes of those measurements -- the same kind of list that people analyze statistically in order to confirm agreement with QM, by the way.

This is not entirely true. What you say about MWI is correct in that MWI denies the existence of remote measurements until we got word of it, because that didn't "split your branch". But WHEN we get word of it, we know that we are in one specific "branch" in which it has a meaning to talk about a sequence of polarization measurements.
And a branch (or a world or whatever) is simply one single term in the "wavefunction of the universe" when it is written such that you appear in a sum of product states (Schmidt decomposition) between you and the rest of the universe, and you have to choose one branch.

That's btw where I differ with "true MWI proponents": they try to show that this "choice of your branch" follows from unitary QM, and I think that you cannot do that, and that you have to postulate the Born rule in order to give probabilities to that choice. But WITHIN THAT BRANCH everything has a meaning as if you just used the good old projection postulate.

cheers,
Patrick.

 

[ttn]

This is not entirely true. What you say about MWI is correct in that MWI denies the existence of remote measurements until we got word of it, because that didn't "split your branch". But WHEN we get word of it, we know that we are in one specific "branch" in which it has a meaning to talk about a sequence of polarization measurements.
And a branch (or a world or whatever) is simply one single term in the "wavefunction of the universe" when it is written such that you appear in a sum of product states (Schmidt decomposition) between you and the rest of the universe, and you have to choose one branch.

Perhaps this is one possible version of MWI. (That is another of my frustrations with that theory -- there seem to be so many different versions of it, something its advocates tend to use to parry any criticisms. But I mean the bad advocates, not you... )

Anyway, you'll have to elaborate on your view that "MWI denies the existence of remote measurements until we got word of it". Who exactly is "we"? I suppose you mean, whoever one is talking about. But, since that could be anyone, this seems to mean that each sentient observer possesses his own universe, and that what he takes to be other sentient observers in his universe are really "mindless hulks". (I think that was a term introduced by David Albert in "Interpreting the MWI" or one of his other similar articles... He used the term in a slightly different way, in the context of a slightly different version of MWI, but I think it applies nicely here.)

Come to think of it, I seem to remember you admitting openly on some other thread a while back that this version of MWI you favor basically reduces to solipsism. So I guess my comment above isn't new and won't change your mind. So be it. But if you are actually endorsing some form of solipsism, is this really any more a counterexample to my claim that MWI entails that we are deluded about many things?

That's btw where I differ with "true MWI proponents": they try to show that this "choice of your branch" follows from unitary QM, and I think that you cannot do that, and that you have to postulate the Born rule in order to give probabilities to that choice. But WITHIN THAT BRANCH everything has a meaning as if you just used the good old projection postulate.

Yes, I agree completely on this point. All the alleged derivations of the Born rule from the unitary dynamics end up smuggling in the Born rule, usually in the guise of tracing out certain degrees of freedom from a density matrix.

On a slightly different topic, may I ask your thoughts on the point I originally posted about in this thread? That is, that QM itself violates Bell Locality. This is really a trivial statement that seems to me totally uncontroversial, yet my recent experience is that many people go to extreme lengths of obfuscation to avoid seeing this. Perhaps it is because they recognize at some level that, if QM violates Bell Locality, then Bell's proof that hidden variable theories must violate Bell Locality and are hence to be rejected, leaves them in a pickle. They must either reject regular old QM on those same grounds, or they must confess to having wrongly maligned the hidden variables program for (say) their whole professional life. You seem like a knowledgeable and intelligent person who will give a straight answer on this (admittedly sociological) question.

 

[vanesch]

Come to think of it, I seem to remember you admitting openly on some other thread a while back that this version of MWI you favor basically reduces to solipsism. So I guess my comment above isn't new and won't change your mind. So be it. But if you are actually endorsing some form of solipsism, is this really any more a counterexample to my claim that MWI entails that we are deluded about many things?

You captured very well my viewpoint, I think. And to answer your question:

You have NO IDEA how deluded we all are !

Honestly, isn't this the same kind of criticism that was helt against relativity, in what way we are deluded in our mental picture of what is space, and what is time ?

I do not endorse this view because I like it, I endorse it because I think it is what comes closest to a rigorous application of the current formalism of quantum theory. We seem to have unitary evolution operators ready for about all known physics (except gravity), and nevertheless we somehow need the Born rule to get out numbers for the observed probabilities. I have this strange attitude that I believe more in a formalism than in an interpretation, so the formalism dictates the interpretation, and not vice versa. So I'm not going to postulate new physics to have a nicer, more intuitive interpretation. The only thing I want is a consistent story. And the Copenhagen view is bluntly non-consistent, with its non-defined split in "interaction processes" (unitary) and "observations" (projection).

The day that we have good reasons to change the formalism, I'll change the story without any regrets.

On a slightly different topic, may I ask your thoughts on the point I originally posted about in this thread? That is, that QM itself violates Bell Locality. This is really a trivial statement that seems to me totally uncontroversial, yet my recent experience is that many people go to extreme lengths of obfuscation to avoid seeing this. Perhaps it is because they recognize at some level that, if QM violates Bell Locality, then Bell's proof that hidden variable theories must violate Bell Locality and are hence to be rejected, leaves them in a pickle.

I'm affraid that this leads into a lot of semantics of what exactly is meant by "Bell locality", so depending on how exactly it is defined, the answer will change.

The whole discussion is Einstein's "fault" with his "god doesn't play dice". If you naively think of quantum probabilities as resulting from an underlying classical statistical mechanics, which will predict outcomes with CERTAINTY if we only knew the "hidden variables" then what Bell asserts is that the only way to find the same results as the QM predictions are through genuinly non-local actions upon these hidden variables.

It is the only issue in the whole discussion. From the moment that you accept fundamentally probabilistic systems, which do NOT have a deterministic underlying statistical mechanics explanation, I think the whole point is moot. A "complete description" according to Einstein is a DETERMINISTIC description. So, no, the wave function is not a complete description because it fails to predict deterministically the outcomes of measurements (in MWI: as observed by an observer ; but I won't repeat that for every phrase).

If you go to fundamentally STOCHASTIC systems (such as the observations in QM), Bell's theorem doesn't make much sense because there is no "underlying mechanism" to be analysed, whether it is "local" or not.
The only locality you can require in a stochastic system is whether the LOCAL statistics can be influenced "at a distance" by actions, measurements etc... on another remote system ; and here the answer for QM is no, you can't.
You can, however, obtain correlations which do not obey Bell's inequalities: they are meaningless for a fundamentally stochastic system. It only indicates that a deterministic classical statistical mechanics will not be able to produce the same statistics if that deterministic system is to be local. (and that was what Einstein was looking for)

There are 3 possibilities for fundamentally stochastic theories:
If you can change, say, expectation values of local observables of A by actions, measurements ... at B, you have true action at a distance (and you can build an FTL phone with it). In that case, OF COURSE you will also find correlations not obeying Bell between A and B. This is NOT the case in QM.

But you can also find correlations between A and B which don't necessarily obey Bell, but without influencing remotely local statistics. This respects locality enough to be compatible with relativity, and QM is in this case. It is the only requirement one can put upon a fundamentally stochastic system to be "local".

Finally, you can have a stochastic theory for which Bell is obeyed. This simply means that this stochastic theory can be replaced by an underlying deterministic theory in which we have local interactions and hidden variables.

The big misunderstanding is the failure to see that it is only in classical, deterministic statistical mechanics that correlations imply a causal relationship. Stochastical systems do not have to have this property. So it is not because we find *correlations* between A and B that there needs to be any causal influence. We make this error because we always think in classical, deterministic terms (such as local hidden variable models), and we think of an underlying statistical mechanics which is responsible for the apparently random outcomes. But if no causal relation is implied by correlations, then there is no meaning to be attached to Bell's inequalities. They only have such a meaning in the framework of deterministic systems.

Of course it is tempting to look at the formalism for an explanation of how this comes about, and the "collapse of the wave function" a la Copenhagen gives you the impression that you "collapsed the wavefunction at a distance", so that physically there happened something to the photon.
It is in this circumstance that switching to MWI is enlightening: the "collapse of the wavefunction" just comes down to a local "choice of the branch" ; so nothing happened to your photon. Even if you do not like this view, the very fact that this view exists shows you that no "action at a distance" is necessary.

So to come back to your question: I think that "Bell locality" only has a meaning in the framework of deterministic theories. Indeed, in that framework, respecting Bell's inequalities comes down to opening the possibility of an explanation of the statistical results based upon stochastic distributions of local hidden variables.

In the framework of a fundamental stochastic theory, I don't think that the concept of Bell locality makes sense. Locality here is defined by the impossibility of changing local expectation values remotely. And that's not possible in QM.

cheers,
Patrick.

 

[selfAdjoint]

Patrick, what do you think of the argument presented in http://www.arxiv.org/abs/quant-ph/0502016, that the original Bell theorem is wrong because he argued without using uncertainty, and that if you do include uncertainty then hidden variables theories that are time-varying will show the same correlations that QM does? (Assuming that I haveread it right)? That would be "deterministic" or phenomenological uncertainty, I suppose, in that case.

 

[ttn]

I do not endorse this view because I like it, I endorse it because I think it is what comes closest to a rigorous application of the current formalism of quantum theory. We seem to have unitary evolution operators ready for about all known physics (except gravity), and nevertheless we somehow need the Born rule to get out numbers for the observed probabilities. I have this strange attitude that I believe more in a formalism than in an interpretation, so the formalism dictates the interpretation, and not vice versa. So I'm not going to postulate new physics to have a nicer, more intuitive interpretation. The only thing I want is a consistent story. And the Copenhagen view is bluntly non-consistent, with its non-defined split in "interaction processes" (unitary) and "observations" (projection).

Fair enough, I can certainly respect your overall approach here. I would merely point out that hidden variable theories offer a really elegant and simple and natural solution to the measurement problem. The only half-decent objection to the hv program I know of is the one based on Bell's theorem, but that turns out to be a foolish objection if one's favored alternative is some version of orthodox or Copenhagen QM, which most people's is. So if one is concerned with the measurement problem and with the issue of local causality, one must evidently choose between something like Bohmian mechanics and some version of MWI -- the former being nonlocal but extremely crisp in terms of measurement, the latter being (in some sense?) local but very un-crisp regarding measurement and lots of other stuff.

As much as I prefer the former option to the latter, I must also admit that the latter is still far better than being openly un-concerned with measurement and/or locality, often to the point of outright hostility, the way most advocates of traditional QM seem to be. A less-than-ideal solution is always better than evasion of the problem.

One sarcastic quip I can't resist making: you said you were unwilling to posit new physics merely to generate a clearer interpretation. But you're willing, instead, to postulate a bunch of crazy *philosophy* (solipsism, etc.)?

The day that we have good reasons to change the formalism, I'll change the story without any regrets.

If all you care about is the formalism, why should you prefer any "story" to begin with? This sounds suspiciously like one of the other standard (bad) arguments against hidden variables: "since they share the same formalism and predictions as regular QM, why should they be taken seriously?" Well, OK, but the same thing works in reverse, too.

I'm affraid that this leads into a lot of semantics of what exactly is meant by "Bell locality", so depending on how exactly it is defined, the answer will change.

I think you are confusing the straightforward mathematical condition "Bell Locality" with something bigger having to do with what Bell's Theorem proves, what its assumptions are, etc. Bell Locality simply means that

P(A,B|a,b,L) = P(A|a,L) * P(B|b,L) . . . . (Eq. 1)

where A,B refer to measurement outcomes under detector settings a,b (the measurement events being spacelike separated) and L refers to a complete description of the state of the world across some spacelike surface in the past of the two measurement events.

Let's just ask if orthodox QM (in which we identify "L" with the relevant wave function) satisfies this condition in, say, a typical EPR type case. Well, according to QM, the joint probability for A and B can be written

| <A,B| a b |psi> |^2

where a and b now represent the relevant QM operators corresponding to the properties being measured. This reduces to the form in Eq. (1) only if |psi> is factorizable, i.e., if the two particles aren't entangled. So in the general case, if you assume that quantum mechanics (i.e. the wave function alone) provides a complete description L, Bell Locality is violated.

This has nothing to do with Bell's Theorem. Well, I mean, it has *something* to do with it in that, as soon as you say what I just said, you are led to think: "huh, maybe you can construct a theory that is local by simply dropping the L = |psi> (i.e. completeness) assumption, and positing some sort of hidden variable theory." And as soon as you say that, you remember: Bell's Theorem proves this cannot be done. So one is evidently stuck with the nonlocality that is manifest in orthodox QM. As Bell said, QM doesn't respect the local causality condition, and this cannot be blamed on the 'incompleteness' of the theory.

The whole discussion is Einstein's "fault" with his "god doesn't play dice". If you naively think of quantum probabilities as resulting from an underlying classical statistical mechanics, which will predict outcomes with CERTAINTY if we only knew the "hidden variables" then what Bell asserts is that the only way to find the same results as the QM predictions are through genuinly non-local actions upon these hidden variables.

Agreed. But this in no way undermines my point that QM is already quite manifestly nonlocal (i.e., in violation of the Bell Locality condition) if you treat the wf as a complete description (i.e., if you *don't* supplement it with hidden variables).

It is the only issue in the whole discussion. From the moment that you accept fundamentally probabilistic systems, which do NOT have a deterministic underlying statistical mechanics explanation, I think the whole point is moot. A "complete description" according to Einstein is a DETERMINISTIC description. So, no, the wave function is not a complete description because it fails to predict deterministically the outcomes of measurements (in MWI: as observed by an observer ; but I won't repeat that for every phrase).

No, I'm afraid this is not correct. Orthodox QM itself is a "fundamentally probabilistic system" yet it violates Bell Locality. And since the problem with local deterministic hidden variables is that they are unable to generate sufficiently strong correlations, I think it is rather obvious that a local stochastic hidden variable won't work either. Adding randomness of any kind can only weaken the correlations, so long as things are kept genuinely local. And finally, Einstein did not merely assume that a description had to be deterministic in order to be complete; the EPR argument was, rather, that positing an underlying deterministic or hidden-variables framework was the only way to possibly escape the apparent non-locality of QM. As usual, Bell eloquently railed against this confusion (that the whole problem comes from arbitrarily requiring determinism):

"It is important to note that to the limited degree to which *determinism* plays a role in the EPR argument, it is not assumed but *inferred*. What is held sacred is the principle of 'local causality' -- or 'no action at a distance'... It is remarkably difficult to get this point across, that determinism is not a *presupposition* of the analysis" [Speakable..., 143]

There is a nice discussion of this point in the first couple of sections of this article:

http://plato.stanford.edu/entries/qm-bohm/

and a much more extensive analysis in Tim Maudlin's spectacular book, "Quantum Non-locality and Relativity."

There are 3 possibilities for fundamentally stochastic theories:
If you can change, say, expectation values of local observables of A by actions, measurements ... at B, you have true action at a distance (and you can build an FTL phone with it). In that case, OF COURSE you will also find correlations not obeying Bell between A and B. This is NOT the case in QM.

QM predicts probabilities for individual measurement results; it does not merely make statements about expectation values. And it is plainly obvious that these probabilities are not in accordance with the Bell Locality condition.

Maybe your point is that the Bell Locality condition isn't the right one by which to test for local causality. That may be. And perhaps you are right that the *right* way to test for local causality is to look only at local expectation values, i.e., marginal probabilities in which we average over distant correlated events. You are absolutely correct that by this standard, QM is entirely local. But the problem is: so is, for example, Bohmian mechanics. With *this* definition of "local causality" (which is clearly much weaker than Bell's) there is no Bell Theorem, no proof that hidden variable theories (deterministic or stochastic or whatever) have to be nonlocal.

 

[ttn]

Patrick, what do you think of the argument presented in http://www.arxiv.org/abs/quant-ph/0502016, that the original Bell theorem is wrong because he argued without using uncertainty, and that if you do include uncertainty then hidden variables theories that are time-varying will show the same correlations that QM does? (Assuming that I haveread it right)? That would be "deterministic" or phenomenological uncertainty, I suppose, in that case.

I just read this article last night, so, although I am not Patrick, I will mention a few thoughts.

I believe the author's real point (although he doesn't frame it this way) is that Bell's theorem assumes the existence of "elements of reality" (or "instruction sets" or whatever you want to call them) corresponding to quantum-mechanically non-commuting operators. And this is in conflict with an ontological interpretation of Heisenberg's (thus inaptly named) Uncertainty Principle. So if, with Heisenberg, you want to insist that physical quantities corresponding to non-commuting operators cannot simultaneously exist, then Bell's theorem fails to apply. This is, by the way, not really a new point. People say this kind of thing all the time by way of asserting that Bell's proof doesn't apply to orthodox QM.

But I think all of this misunderstands Bell's argument. Bell never thought his Theorem applied to orthodox QM. It's a theorem about hidden variable theories, about theories in which these pre-measurement "instruction sets" exist. This whole vast literature (Stapp, etc.) arguing about whether Bell's Theorem applies to QM (and includes all sorts of cans of worms regarding "counterfactual definitness" and so forth) is based on this same misunderstanding.

As Bell recognized, one didn't *need* a fancy theorem to demonstrate that orthodox QM is non-local. It just plainly, manifestly violates Bell's Locality condition. As Bell stated, this fact was pointed out long ago by EPR: if you insist that QM is complete, then you cannot escape the conclusion that it is nonlocal. Or, turning that around, if you are bothered by nonlocality, you ought to reject the completeness assumption and consider local hidden variable theories. (Of course, Bell later proved that such theories also don't work, that hv theories must also violate Bell Locality to agree with experiment. But Einstein obviously didn't know about that...)

There is a nice discussion of this whole issue here:

http://www.arxiv.org/abs/quant-ph/0408105

 

[ZapperZ]

Patrick, what do you think of the argument presented in http://www.arxiv.org/abs/quant-ph/0502016, that the original Bell theorem is wrong because he argued without using uncertainty, and that if you do include uncertainty then hidden variables theories that are time-varying will show the same correlations that QM does? (Assuming that I haveread it right)? That would be "deterministic" or phenomenological uncertainty, I suppose, in that case.

Unless I missed it completely, that "paper" is in the same vein as Hess-Philipp's argument of a possible "time-loophole" in Bell theorem.[1] To me, this is the only legitimate challenge to the possibility of a "counterfactual" problem of Bell theorem, and not the detection loophole.

In any case, the possibility of a time-loophole in Bell Theorem isn't a done deal. In fact, I think Gill et al. has sufficiently addressed why there isn't a time loophole in Bell theorem.[2] It would be interesting to wait and see if this paper gets published (he seems to have another one that he cited, but only in the e-print arXiv only, which is usually isn't a good sign) and the responses/rebuttals.

Zz.

[1] K. Hess and W. Philipp, Proc. Natl. Acad. Sci. v.98, p.14224 (2001); K. Hess and W. Philipp, Proc. Natl. Acad. Sci. v.98, p.14228 (2001).

[2] R.D. Gill et al., Proc. Natl. Acad. Sci. v.99, p.14632 (2002).

 

[Cat]

2) How do peas prove EPR? You are going to have to do better than that. We understand that some people hypothesize the existence of little teeny tiny attributes that we cannot see. Most of us call those "hidden variables" and don't need to call them pea DNA by childish analogy. We also understand that no-one knew about DNA a few hundred years ago. Also a poor analogy.

I agree. Peas simply don't have the kind of geometrical properties that are needed to explain real EPR experiments.

EPR envisioned that the so-called hidden variables would eventually be uncovered. That hasn't happened in 80 years of looking. Instead, it has become obvious to scientists that there is no combination of hidden variables that can mimic the results of certain experiments (per Bell). Please tell us - SPECIFICALLY and not hand waving - how you conclude otherwise. If there is an "element of reality" we are missing, please, do show us. I, for one, am all ears.

Aren't you generalising prematurely here? As I understand it the debate on hidden variables is continuing. The experts admit that no "loophole-free" Bell test has yet been done, so it follows that, as far as the experimental evidence is concerned, the possibility remains on the cards.

For an idea as to what hidden variables are needed to explain the majority of EPR experiments, you might like to look at http://en.wikipedia.org/wiki/Local_hidden_variable_theory. For optical Bell tests it is suggested that the polarisation direction itself is the hidden variable. To get results that fit the observations we need to allow, though, for the behaviour of the (imperfect) polarisers and detectors used in actual experiments.

Cat

 

[vanesch]

Fair enough, I can certainly respect your overall approach here. I would merely point out that hidden variable theories offer a really elegant and simple and natural solution to the measurement problem. The only half-decent objection to the hv program I know of is the one based on Bell's theorem, but that turns out to be a foolish objection if one's favored alternative is some version of orthodox or Copenhagen QM, which most people's is. So if one is concerned with the measurement problem and with the issue of local causality, one must evidently choose between something like Bohmian mechanics and some version of MWI -- the former being nonlocal but extremely crisp in terms of measurement, the latter being (in some sense?) local but very un-crisp regarding measurement and lots of other stuff.

I have to say I'm not very knowledgeable of Bohm's theory. I've read a bit about it, but the explicit non-local, hidden mechanism didn't appeal to me. But that's a matter of personal taste. I also heard that Bohm's theory meets quite some difficulties in a relativistic setting such as quantum field theory ; I'm not knowledgeable about it (and, hey, standard QM ALSO has some difficulties in QFT if you scratch the surface). As I said: it is probably a matter of personal opinion.

One sarcastic quip I can't resist making: you said you were unwilling to posit new physics merely to generate a clearer interpretation. But you're willing, instead, to postulate a bunch of crazy *philosophy* (solipsism, etc.)?

Yes. Because it makes you think exactly about what you know, and what you think you know. Even if it will turn out not to be right in the end.

If all you care about is the formalism, why should you prefer any "story" to begin with? This sounds suspiciously like one of the other standard (bad) arguments against hidden variables: "since they share the same formalism and predictions as regular QM, why should they be taken seriously?" Well, OK, but the same thing works in reverse, too.

Why a story ? Because stories are nice :-) Seriously, having a story helps you devellop an intuition for the formalism, and also helps you out when you're confused on how to apply the formalism. Personally, it helps me quite well to see through all these EPR and "quantum eraser" experiments, and when you see how much wrong stuff is said about these things, I think my view, although weird, gives me crystal-clear answers. But that's where it is more important to have a consistent story than an "intuitive" story like the Copenhagen view. MWI viewpoints really give you a very clear view on all these systems, because you're never confronted with "collapse at a distance" and "revival of the collapsed wavefunction" and other idiocies.

I think you are confusing the straightforward mathematical condition "Bell Locality" with something bigger having to do with what Bell's Theorem proves, what its assumptions are, etc. Bell Locality simply means that

P(A,B|a,b,L) = P(A|a,L) * P(B|b,L) . . . . (Eq. 1)

where A,B refer to measurement outcomes under detector settings a,b (the measurement events being spacelike separated) and L refers to a complete description of the state of the world across some spacelike surface in the past of the two measurement events.

If that "complete description" is a stochastic description, then evidently QM, and ALL its equivalent views, are, according to this definition "Bell nonlocal".

My point was that this only has a meaning related to a causality relationship if we intend to work with an underlying deterministic statistical mechanics. If not, the fact that we do not satisfy the Bell locality condition doesn't say anything about a causal non-locality. Correlations then just "are" and do not necessarily imply any causal link. The only way to have a causal link in a purely stochastic model is by a change in local expectation values. This is of course a weaker requirement than Bell locality.

This has nothing to do with Bell's Theorem. Well, I mean, it has *something* to do with it in that, as soon as you say what I just said, you are led to think: "huh, maybe you can construct a theory that is local by simply dropping the L = |psi> (i.e. completeness) assumption, and positing some sort of hidden variable theory." And as soon as you say that, you remember: Bell's Theorem proves this cannot be done.

Ok, but you should agree that the "Bell locality condition" has been designed on purpose for the Bell theorem, no ?

Agreed. But this in no way undermines my point that QM is already quite manifestly nonlocal (i.e., in violation of the Bell Locality condition) if you treat the wf as a complete description (i.e., if you *don't* supplement it with hidden variables).

I told you we would get stuck in semantics

No, I'm afraid this is not correct. Orthodox QM itself is a "fundamentally probabilistic system" yet it violates Bell Locality. And since the problem with local deterministic hidden variables is that they are unable to generate sufficiently strong correlations, I think it is rather obvious that a local stochastic hidden variable won't work either. Adding randomness of any kind can only weaken the correlations, so long as things are kept genuinely local.

No, you're again thinking in deterministic statistical mechanics terms, this time with "added local noise". This will indeed lessen any correlations. But in a truly stochastic system, you cannot require anything about the probabilities. Everything can happen. The only true locality condition

QM predicts probabilities for individual measurement results; it does not merely make statements about expectation values. And it is plainly obvious that these probabilities are not in accordance with the Bell Locality condition.

Probabilities for individual measurement results are of course also the expectation values of the projectors on the corresponding eigenstates. So _all_ probabilities in QM are expectation values.

Maybe your point is that the Bell Locality condition isn't the right one by which to test for local causality. That may be. And perhaps you are right that the *right* way to test for local causality is to look only at local expectation values, i.e., marginal probabilities in which we average over distant correlated events. You are absolutely correct that by this standard, QM is entirely local.

You've got it It is the only requirement by relativity. That's good enough.

But the problem is: so is, for example, Bohmian mechanics. With *this* definition of "local causality" (which is clearly much weaker than Bell's) there is no Bell Theorem, no proof that hidden variable theories (deterministic or stochastic or whatever) have to be nonlocal.

Ah, but that is then if you forget again the hidden variables. Because they DO NOT obey the locality conditions (if I'm not mistaking). If I understood well, the Bell locality condition comes down to the observable effect of the local expectation values condition of an underlying hidden deterministic model, no ? (I'm not 100% sure about that) Ok, they are not observable, you will say. But then there's no point in the first place to introduce them :-)
(unless you absolutely want to get rid of solipsism ...)

cheers,
Patrick.

 

[ttn]

Why a story ? Because stories are nice :-) Seriously, having a story helps you devellop an intuition for the formalism, and also helps you out when you're confused on how to apply the formalism.

I'd go further. I'm a scientific realist. I believe there is an external, physical, objective world that exists independent of human knowledge of it. And I think the purpose of physics is to understand what the world is like. What you are here calling "telling stories" is really the process of building up an evidence-based model of reality -- just like Copernicus was "telling a story" when he said the Earth went around the sun, Maxwell and Boltzmann were "telling stories" when they predicted the distribution of molecular speeds in a gas, and just like, say, contemporary astrophysicists "tell stories" about how shockwaves propagating through infalling matter can result in supernovas.

I don't accept the idea that, in these sorts of cases, the only point of these stories is to help people develop intuition for formalism, etc. If anything, it's just the reverse: the point of the formalism is to help us figure out which story is the correct one, i.e., what the world is like. Isn't that really what science is all about?

If that "complete description" is a stochastic description, then evidently QM, and ALL its equivalent views, are, according to this definition "Bell nonlocal".

I agree with "evidently QM ... [is] Bell nonlocal." But I don't see how this has anything to do with whether a complete description is stochastic. According to QM, the complete description is stochastic; the theory isn't deterministic. So what? It violates the Bell Locality condition regardless, and that's all that matters here.

My point was that this only has a meaning related to a causality relationship if we intend to work with an underlying deterministic statistical mechanics. If not, the fact that we do not satisfy the Bell locality condition doesn't say anything about a causal non-locality. Correlations then just "are" and do not necessarily imply any causal link. The only way to have a causal link in a purely stochastic model is by a change in local expectation values. This is of course a weaker requirement than Bell locality.

So... you're saying any non-deterministic theory is automatically consistent with Bell's local causality requirement, because such theories have no causality in them at all, and hence not even the remotest possibility of verboten non-local causality?

Talk about semantics!

I think it is perfectly reasonable to talk about causality in the context of a stochastic theory. Of course, in such a theory, a complete specification of the causes of some event won't be sufficient to predict with certainty that the event occurs. That's what it means to be stochastic. But you can still talk about the probability distribution of possible events. A complete specification of the causes of a given event would then be sufficient to predict, not the exact outcome, but the exact probability distribution of outcomes. And if you're with me still, it would in addition make perfect sense to ask whether all the elements of this "complete specification of causes" is present in the past light-cone of a given event or whether, instead, some space-like separated event *changes* the probability distribution for the possible outcomes of the event in question. This, I think, is a perfectly reasonable and perfectly appropriate way of deciding whether a non-deterministic theory is locally causal. In fact, this is precisely Bell's Locality criterion.

Ok, but you should agree that the "Bell locality condition" has been designed on purpose for the Bell theorem, no ?

That's a historical question I don't know the answer to. Bell was inspired when he read about Bohm's counterexample to the no-hidden-variables "proofs" but wondered if a local hv theory was possible. Perhaps what we now know as the Bell Locality condition was the first thing he wrote down as an obvious mathematical statement of local causality. Or perhaps he tried some other things first, and only settled on "Bell Locality" when it became clear that the theorem could be based on it. Who knows. And I'm inclined to say: who cares? Bell Locality *is* a natural and reasonable way of deciding between local and nonlocal theories. So even if he did cook it up so as to be able to prove the theorem, I say: he's a great chef!

No, you're again thinking in deterministic statistical mechanics terms, this time with "added local noise". This will indeed lessen any correlations. But in a truly stochastic system, you cannot require anything about the probabilities. Everything can happen.

Stochastic doesn't mean Heraclitean. There are still laws governing what happens, only they are stochastic instead of deterministic. When you roll a fair die, "anything can happen" in the sense that you might get any of the 6 possible outcomes. But it's false that "anything can happen" in the sense that you might see a billion 3's in a row.

So... I think you can put requirement on the probabilities in a stochastic theory. Indeed, writing down specific laws the probabilities obey is precisely what a stochastic theory *does*!

Ah, but that is then if you forget again the hidden variables. Because they DO NOT obey the locality conditions (if I'm not mistaking). If I understood well, the Bell locality condition comes down to the observable effect of the local expectation values condition of an underlying hidden deterministic model, no ?

The Bell Locality condition is really simple. It merely says

P(A|a,b,L) = P(A|a,L)

where "A" is some particular event (say the result of a measurement), "a" is any relevant parameters pertaining to the event (like the orientation of your SG magnets if it's a spin measurement), "L" is a complete specification of the state of the measured object across some spacelike hypersurface in the past of the measurement event, and "b" is any other junk that is spacelike separated from the measurement event. Basically the idea is: once you conditionalize on everything that could possibly affect the outcome in a local manner, specifying in addition information pertaining to space-like separated events will be *redundant* and hence won't change the probabilities.

Ok, they are not observable, you will say. But then there's no point in the first place to introduce them :-)
(unless you absolutely want to get rid of solipsism ...)

Yes, that's exactly what I'll say. =) Since I tend to just disregard MWI as non-serious, I would have said the point of introducing the hidden variables was to solve the measurement problem. The usual argument against this is that, while maybe a hidden variable theory can clear up the measurement problem, the price of doing so is to introduce violations of Bell Locality into theory, and the price is too high. Spoken by advocates of the completeness doctrine (i.e., orthodox QM) that is a preposterous and self-refuting argument since QM itself violates Bell Locality. That is, in terms of locality, QM vs. Bohmian mechanics (say) is a wash. But since the former suffers from a measurement problem and the latter doesn't, Bohmian Mechanics is clearly the superior theory.

Of course, you'll want to bring in MWI as a third candidate. But we've been over that already...

 

[DrChinese]

The Bell Locality condition is really simple. It merely says

P(A|a,b,L) = P(A|a,L)

where "A" is some particular event (say the result of a measurement), "a" is any relevant parameters pertaining to the event (like the orientation of your SG magnets if it's a spin measurement), "L" is a complete specification of the state of the measured object across some spacelike hypersurface in the past of the measurement event, and "b" is any other junk that is spacelike separated from the measurement event. Basically the idea is: once you conditionalize on everything that could possibly affect the outcome in a local manner, specifying in addition information pertaining to space-like separated events will be *redundant* and hence won't change the probabilities.

The way I read this is that the measuring apparatus is outside the scope of the relevant parameters. I do not fully understand how this associates with Bell's formalism though. When you say local, is b required to be outside the light cone? Or simply outside of L?

 

[ttn]

The way I read this is that the measuring apparatus is outside the scope of the relevant parameters. I do not fully understand how this associates with Bell's formalism though. When you say local, is b required to be outside the light cone? Or simply outside of L?

"a" was meant to include any relevant facts about the apparatus used to measure "A". And yes, "b" is to be thought of as outside the past lightcone of A. The idea is just that, as I said in another post, once you've conditionalized a probability on every event on which it might depend in a local way, adding more information isn't going to change the probabilities. Such information would be either irrelevant or redundant.

Here's a really simple example. Say I put a coin in one of my hands and separate them out to my left and right, but without letting you know which hand the coin is in. If that's all you know, then you'll probably attribute a 50% probability to the proposition that the coin is in my left hand. If I then reveal that the coin is in fact in my right hand, the probability you attribute to that earlier proposition (that it was in my left hand) will jump to zero. This may seem like a violation of Bell's local causation condition, since the probability attributed to a certain event changed due to something that happened at a distant location. But it isn't. This merely brings out that we didn't start with a complete description of the state of the coin -- we forgot to conditionalize our probabilities on an appropriate "L" which, here, would obviously consist of some statement about where the coin actually was. Then the conditional probability P(left|L) would be either 1 or 0 from the very beginning, and *this* would not change when you learned whether or not the coin was in my right hand -- i.e., P(left|L) = P(left | L, right?) where by "right?" I mean the outcome of the experiment of looking for the coin in my right hand.

As Patrick and I have been discussing, I think this generalizes in a perfectly straightforward way to non-deterministic theories, i.e., theories in which the various probabilities (conditioned on L) are not restricted to be just zero and one. (Bell thought so too by the way.)

There is a nice article in the current issue of AmJPhys (Feb.) on a thought experiment that is basically the quantum equivalent of the example I just gave with the coin. (The article and the thought experiment are called "Einstein's Boxes".) The idea is, imagine doing this same sort of experiment with a quantum particle (instead of a macroscopic coin), say by splitting a photon's wf in half with a half-silvered mirror, and letting the two halves separate for a while. Then, according to QM, there is a 50% chance of finding it on the left and 50% for the right, just like the coin. But unlike the coin case, and assuming you believe the quantum completeness doctrine, there is no actual position of the photon prior to measurement. It's in a superposition of left and right, neither here nor there. In particular, the wave function represents (by hypothesis) a complete description of the state of the photon. It is, in other words and again assuming the completeness hypothesis for the sake of argument, "L". So we have

P(left|L) = 50%

and

P(right|L) = 50%

But surely it is an experimental fact that the joint probability of finding the photon both on the left and on the right vanishes: P(left&right|L) = 0.

This illustrates that QM, if you believe Bohr that the wave function is a complete description, violates Bell Locality. For according to that condition, the joint probability should factor:

P(left&right|L) = P(left|L) * P(right|L)

which (given the values specified for the three quantities above) it doesn't. So this simple little example is (kind of amazingly, if you think about it) sufficient to show that, if complete, QM violates Bell locality. Of course, this is exactly what motivated people like Einstein to reject the completeness doctrine and try to find a local theory which, of course, means a local hidden variable theory. And since Bell (later) proved that was impossible too, this cute little example with the "quantum coin" actually plays a significant role in establishing that any empirically viable theory (which is to say: Nature!) violates Bell Locality. Not bad for such a trivial little example.

Hopefully that clarifies things a bit. Do check out the article on "Einstein's Boxes." It's quite interesting, if I do say so myself.

 

[JesseM]

this cute little example with the "quantum coin" actually plays a significant role in establishing that any empirically viable theory (which is to say: Nature!) violates Bell Locality.

You still agree that Everett-style interpretations may be an exception to this, even if you personally don't find them plausible, right? I think an advocate of such an interpretation would say your argument is faulty because you assume that after the experimenter makes a measurement there is a single fact about which side the photon was found on, when really there might be different facts observed by different copies of the experimenter.

 

[DrChinese]

...

There is a nice article in the current issue of AmJPhys (Feb.) on a thought experiment that is basically the quantum equivalent of the example I just gave with the coin. (The article and the thought experiment are called "Einstein's Boxes".) The idea is, imagine doing this same sort of experiment with a quantum particle (instead of a macroscopic coin), say by splitting a photon's wf in half with a half-silvered mirror, and letting the two halves separate for a while. Then, according to QM, there is a 50% chance of finding it on the left and 50% for the right, just like the coin. But unlike the coin case, and assuming you believe the quantum completeness doctrine, there is no actual position of the photon prior to measurement. It's in a superposition of left and right, neither here nor there. In particular, the wave function represents (by hypothesis) a complete description of the state of the photon. It is, in other words and again assuming the completeness hypothesis for the sake of argument, "L". So we have

P(left|L) = 50%

and

P(right|L) = 50%

But surely it is an experimental fact that the joint probability of finding the photon both on the left and on the right vanishes: P(left&right|L) = 0.

This illustrates that QM, if you believe Bohr that the wave function is a complete description, violates Bell Locality. For according to that condition, the joint probability should factor:

P(left&right|L) = P(left|L) * P(right|L)

which (given the values specified for the three quantities above) it doesn't. So this simple little example is (kind of amazingly, if you think about it) sufficient to show that, if complete, QM violates Bell locality. Of course, this is exactly what motivated people like Einstein to reject the completeness doctrine and try to find a local theory which, of course, means a local hidden variable theory. And since Bell (later) proved that was impossible too, this cute little example with the "quantum coin" actually plays a significant role in establishing that any empirically viable theory (which is to say: Nature!) violates Bell Locality. Not bad for such a trivial little example.

Hopefully that clarifies things a bit. Do check out the article on "Einstein's Boxes." It's quite interesting, if I do say so myself.

Thanks for taking the time to discuss.

I have always thought that there are many elements of QM that come back to being equivalent to the EPR paradox, it just is easier to see it laid out when looking at entangled particles. Pretty much every variation of the HUP that you can see - the beamsplitting example you give for example - has to be considered as being in opposition to local reality when you really dig down into it. An unentangled photon's polarization is not definite real either unless it is measured.