A Does the Bell theorem assume reality?

[N88]

The assumption is that a "measurement" is something that reveals information about the world. If you flip a coin and look at the coin and see heads, the coin was already "heads" before you looked at it. The assumption is that the same is true of quantum measurements. So A, B, C are properties of the particles. They only become measurement results after you perform the measurement. Therefore, they have statistics even if you haven't measured them.

Of course, there can be things like "measurement results" that don't reveal pre-existing properties. The result could be some kind of cooperative effect of the thing being measured and the thing doing the measurement. Classically, you could describe this more complicated situation this way:

$P(A | \lambda, \sigma)$

Instead of saying that the result $A$ is a deterministic function of some property of the particle's state $\lambda$, it might be randomly produced with a certain probability distribution that depends on both facts about the particle, $\lambda$, and facts about the measuring device, $\sigma$.

However, this more general possibility is not compatible with the perfect anti-correlations observed in the EPR experiment. If Bob already got the result "spin-down in the z-direction", then there is no way for Alice to get anything other than spin-up in the z-direction. So detailed facts about her measuring device, other than the fact that it's measuring the z-component of spin, can't come into play.

The "coin-flip" does not work for me; it's difficult to interpret in the EPRB context. The coin had a Head and a Tail before it was flipped -- via a thumb.

Now I take naive-realism -- from ancient days -- to be that primitive realism which supposes that what is observed is what was real before the observation. Thus the coin had a Head and it showed UP when the coin hit the floor

So, in my terms, only ancient naive realism [though it persists in these modern times] allows that the EPRB particles had A, B, C before they were "flipped" -- via the measurement interaction.

So it seems to me that "modern realism" -- interpreting EPR's realism -- allows that "so-called measurements" do NOT ALWAYS reveal pre-existing properties; instead it allows that "measurement" interactions MAY construct and reveal something new: some correspondence.

Remember that Bell's goal was to provide a "more complete specification" of EPRB. So you seem to be saying that he thought that result could be achieved by naive realism? See next.

Yes, that's part of the challenge. But I have no idea why the word "naive" would be attached to EPR's elements of reality. It was a well made argument, best possible at the time. Bell refuted that (at least showed that it was incompatible with the predictions of QM.

And as I stated earlier in this thread, realism IS an assumption of Bell. And I explained where.

But I am not attaching naive to "EPR's realism". I am attaching it to what you say is "Bell's realism." And from your challenge we know that it does not work; me saying that before Bell it was well-known that it could not work. Thus the classical example that I offered from Malus' time.

To be clearer re how I see it: EPR-realism addresses the modern view: ie, EPR-realism is that realism which allows that there was something corresponding to the observed values.

THUS: A pure measurement [in any field] would reveal that the observed value corresponded 100% to that which pre-existed; like charge. Thus naive-realism holds in such limited cases.

BUT: A perturbative "measurement" [in any field] would reveal that the observed value corresponded < 100% to that which pre-existed. Thus the classical example that I offered from Malus' time. Thus naive-realism does NOT hold in such limited cases. So Bell's realism does not hold here either.

So, to possibly clarify many differing views, and eliminate some: What is the name of the realism that Bell assumes in his famous 1964 paper? And where is it introduced in his mathematics?

 

[bhobba]

So, to possibly clarify many differing views, and eliminate some: What is the name of the realism that Bell assumes in his famous 1964 paper? And where is it introduced in his mathematics?

Its called counterfactual definiteness. A counterfactual theory is one whose experiments uncover properties that are pre-existing:
http://www.johnboccio.com/research/quantum/notes/paper.pdf

Thanks
Bill

 

[stevendaryl]

The "coin-flip" does not work for me; it's difficult to interpret in the EPRB context. The coin had a Head and a Tail before it was flipped -- via a thumb.

Now I take naive-realism -- from ancient days -- to be that primitive realism which supposes that what is observed is what was real before the observation. Thus the coin had a Head and it showed UP when the coin hit the floor

So, in my terms, only ancient naive realism [though it persists in these modern times] allows that the EPRB particles had A, B, C before they were "flipped" -- via the measurement interaction.

Well, that's the realism that was falsified by Bell's theorem.

 

[DrChinese]

1. The "coin-flip" does not work for me;...

2. So, to possibly clarify many differing views, and eliminate some: What is the name of the realism that Bell assumes in his famous 1964 paper? And where is it introduced in his mathematics?

1. Really, your argument bears no relation to either EPR or Bell. So there should be no surprise that its conclusion escapes you.

2. It can be called either Bell realism or EPR realism. EPR uses "elements of reality". If you want to draw a distinction, you can, since Bell does not use the word "realism" in his paper. See my post #3 for details.

 

[lodbrok]

1. Agreed that no one has implied that the results of one toss have a correlation to the results of another...

2. The DrChinese challenge is where you hand pick the results on both sides (Alice and Bob) for the angle settings A=0, B=120, C=240 degrees. Then I pick which pair of angles Alice and Bob actually measure. The challenge is to produce a dataset that will match the statistical predictions of QM.

1. Equation 2 of Steven Daryl's derivation definitely makes that assumption so long as it applies to an EPRB experiment.

2. As decribed, I don't yet see the relevance of your challenge to the discussion. To avoid going off topic please provide a citation, so I can read up the details.

 

[stevendaryl]

... if measured at the same time along the same axis. Aren't the terms in Bells inequality experiment measured in different experiments? Thus, the realism assumption seems to involve the idea that heads from tossing/observing one coin at one moment is anticorrelated with tails from tossing/observing a similar coin at a different time.
Seems obvious from equation 2 of your derivation where you factor AnBn.

The assumption is introduced with the term AnBnBnCn. A term which is impossible in any EPRB experiment for the simple reason that AnBn is one toss, BnCn should be a different toss in EPRB. But you used the same subscript. By factoring out AnBn, you are saying the heads-up correlation persists between tosses which is not true in my humble opinion.

I don't really know what you are talking about. $A_n, B_n, C_n$ are three numbers, each one is either +1 or -1. So it's just a fact of arithmetic that:

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

The assumption is that twin-pair number $n$, has an associated triple of numbers $A_n, B_n, C_n$, where $A_n$ gives the result of a measurement of spin along the first axis, $B_n$ along the second axis, $C_n$ along the third axis. The manipulation above is just arithmetic. There is no additional assumptions involved.

 

[stevendaryl]

BUT: A perturbative "measurement" [in any field] would reveal that the observed value corresponded < 100% to that which pre-existed. Thus the classical example that I offered from Malus' time. Thus naive-realism does NOT hold in such limited cases. So Bell's realism does not hold here either.

Yes, it's certainly possible to have a "perturbative" measurement, where the result is not 100% determined by the pre-existing properties. However, there are no ideas (as far as I know of) how such a perturbative measurement could result in perfect correlation between measurements of distant particles.

That's really Einstein et al's whole point: If measurements DON'T reveal pre-existing properties, then how can two distant measurements produce perfect correlations?

 

[atyy]

So it seems to me that "modern realism" -- interpreting EPR's realism -- allows that "so-called measurements" do NOT ALWAYS reveal pre-existing properties; instead it allows that "measurement" interactions MAY construct and reveal something new: some correspondence.

Yes, it's certainly possible to have a "perturbative" measurement, where the result is not 100% determined by the pre-existing properties. However, there are no ideas (as far as I know of) how such a perturbative measurement could result in perfect correlation between measurements of distant particles.

Isn't the perturbative measurement case dealt with by allowing the measurement apparatus to also have hidden variables? The measurement outcome is the result of interaction between the hidden variables of the apparatus and the hidden variables of the system.

 

[stevendaryl]

Isn't the perturbative measurement case dealt with by allowing the measurement apparatus to also have hidden variables? The measurement outcome is the result of interaction between the hidden variables of the apparatus and the hidden variables of the system.

Yes, I think so. But the fact that in EPR there are perfect correlations between distant measurements implies that in fact, nothing about the measuring apparatus is relevant except the orientation of the spin measurement.

 

[Auto-Didact]

That's really Einstein et al's whole point: If measurements DON'T reveal pre-existing properties, then how can two distant measurements produce perfect correlations?

I quote Bell on this from his well-known paper on Bertlmann's socks:

Could it be that the first observation somehow fixes what was unfixed, or makes real what was unreal, not only for the near particle but also for the remote one ? For EPR that would be an unthinkable 'spooky action at a distance' /8/.

To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism /9/, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story.

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".

Of course, mere correlation between distant events does not by itself imply action at a distance, but only correlation between the signals reaching the two places. These signals, in the idealized example of Bohm, must be sufficient to determine whether the particles go up or down. For any residual undeterminism could only spoil the perfect correlation. It is remarkably difficult to get this point across, that determinism is not a presupposition of the analysis.

There is a widespread and erroneous conviction that for Einstein /10/ determinism was always the sacred principle. The quotability of his famous "God does not play dice" has not helped in this respect.

Among those who had great difficulty in seeing Einstein's position was Born. Pauli tried to help him /11/ in a letter of 1954 :
" ... I was unable to recognize Einstein whenever you talked about him in either your letter or your manuscript. It seemed to me as if you had erected some dummy Einstein for yourself, which you then knocked down with great pomp. In particular Einstein does not consider the concept of "determinism" to be as fundamental as it is frequently held to be (as he told me emphatically many times) ... he disputes that he uses as a criterion for the admissibility of a theory the question : "Is it rigorously deterministic?"..-he was not at all annoyed with you, but only said you were a person who will not listen".

Born had particular difficulty with the Einstein-Podolsky-Rosen argument. Here is his summing up, long afterwards, when he edited the Born-Einstein correspondence /12/ : "The root of the difference between Einstein and me was the axiom that events which happens in different places A and B are independent of one another, in the sense that an observation on the states of affairs at B cannot teach us anything about the state of affairs at A".

Misunderstanding could hardly be more complete. Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other.

These references to Born are not meant to diminish one of the towering figures of modern physics. They are meant to illustrate the difficulty of putting aside preconceptions and listening to what is actually being said. They are meant to encourage you, dear listener, to listen a little harder.

Here, finally, is a summing-up by Einstein himself /13/ :
'If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics, one is first of all struck by the following : the concepts of physics relate to a real outside world.. . It is further characteristic of these physical objects that they are thought of as arranged in a space time continuum. An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects "are situated in different parts of space".

'The following idea characterizes the relative independence of objects far apart in space (A and B) : external influence on A has no direct influence on B...

'There seems to me no doubt that those physicists who regard the descriptive methods of quantum mechanics as definitive in principle would react to this line of thought in the following way : they would drop the requirement ... for the independent existence of the physical reality present in different parts of space ; they would be justified in pointing out that the quantum theory nowhere makes explicit use of this requirement.

'I admit this, but would point out : when I consider the physical phenomena known to me, and especially those which are being so successfully encompassed by quantum mechanics, I still cannot find any fact anywhere which would make it appear likely that (that) requirement will have to be abandoned.

'I am therefore inclined to believe that the description of quantum mechanics ... has to be regarded as an incomplete and indirect description of reality, to be replaced at some later date by a more complete and direct one'.

Bell goes on to say "we will argue that certain particular correlations, realizable according to quantum mechanics, are locally inexplicable. They cannot be explained, that is to say, without action at a distance."

He concludes with a few possible ways to interpret his Theorem:

By way of conclusion I will comment on four possible positions that might be taken on this business - without pretending that they are the only possibilities.

First, and those of us who are inspired by Einstein would like this best, quantum mechanics may be wrong in sufficiently critical situations. Perhaps Nature is not so queer as quantum mechanics. But the experimental situation is not very encouraging from this point of view /19/. It is true that practical experiments fall far short of the ideal, because of counter inefficiencies, or analyzer inefficiencies, or geometrical imperfections, and so on. It is only with added assumptions, or conventional allowance for inefficiencies and extrapolation from the real to the ideal, that one can say the inequality is violated. Although there is an escape route there, it is hard for me to believe that quantum mechanics works so nicely for inefficient practical set-ups and is yet going to fail badly when sufficient refinements are made. Of more importance, in my opinion, is the complete absence of the vital time factor in existing experiments. The analyzers are not rotated during the flight of the particles. Even if one is obliged to admit some long range influence, it need not travel faster than light - and so would be much less indigestible. For me, then, it is of capital importance that Aspect /19, 20/ is engaged in an experiment in which the time factor is introduced.

Secondly, it may be that it is not permissible to regard the experimental settings a and b in the analyzers as independent variables, as we did /21/. We supposed them in particular to be independent of the supplementary variables X, in that a and b could be changed without changing the probability distribution p(X). Now even if we have arranged that a and b are generated by apparently random radioactive devices, housed in separate boxes and thickly shielded, or by Swiss national lottery machines, or by elaborate computer programmes, or by apparently free willed experimental physicists, or by some combination of all of these, we cannot be sure that a and b are not significantly influenced by the same factors X that influence A and B /21/. But this way of arranging quantum mechanical correlations would be even more mind boggling that one in which causal chains go faster than light. Apparently separate parts of the world would be deeply and conspiratorially entangled, and our apparent free will would be entangled with them.

Thirdly, it may be that we have to admit that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An "ether" would be the cheapest solution /22/. But the unobservability of this ether would be disturbing. So would the impossibility of "messages' faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we can actually perform. The exact elucidation of concepts like 'message' and 'we', would be a
formidable challenge.

Fourthly and finally, it may be that Bohr's intuition was right - in that there is no reality below some 'classical' 'macroscopic' level. Then fundamental physical theory would remain fundamentally vague, until concepts like 'macroscopic ' could be made sharper than they are today.

Abner Shimony (of CHSH fame), in Chapter 5 "John S. Bell: Some Reminiscences and Reflections" of Bertlmann, Zeilinger et al. 2002, Quantum (Un)speakables says the following:

5.3 In What Direction Does Bell's Theorem Point?

The conclusion of Bell's famous paper "Bertlmann's Socks and the Nature of Reality" [13], states four possibilities, with no pretense at exhaustiveness, concerning the interpretation of his theorem and of the experiments inspired by it. He expresses reservations about all of them, but seems guardedly to prefer the third: "it may be that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An 'aether' would be the cheapest solution ... But the unobservability of this aether would be disturbing. So would the impossibility of 'messages' faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we can actually perform. The exact elucidation of concepts like 'message' and 'we', would be a formidable challenge."

Similar reservations are expressed in "Beables for Quantum Field Theory" [14], where elements of reality, "beables", are used to construct an alternative to ordinary quantum field theory, which centers on "observables". Near the conclusion he says,

"So I am unable to prove, or even formulate clearly, the proposition that a sharp formulation of quantum field theory, such as that set out here, must disrespect serious Lorentz invariance. But it seems to me that this is probably so."

Bell was evidently willing to entertain radical innovations in physics, and to devote some effort to investigating them mathematically, in order to do justice to the apparent nonlocality revealed by correlation experiments and at the same time to hold on to his conception of a physical reality underlying observations.

We should like Bell, be fearless in the face of such questions. As far as I can tell, there are thus a few ways to logically interpret Bell's Theorem; especially the interpretation that Bell himself preferred seems to be a viable one, albeit dangerously non-standard, flying directly in the face of relativistic QFT. Similar words can be said about Bohmian Mechanics, which I do not necessarily advocate as the solution.

Physicists are actually right to give pause to entertaining such an interpretation, and treading very lightly instead; doing so carelessly - especially if they do not explain why to their students - however leads to them missing something crucial, which I will attempt to explain.

Even if experimentally the answer seems clear, given the well-known mathematical problems of QFT itself, the issue itself however remains mathematically unclear, therefore theoretically also no such clarity can be claimed to be evident. In other words, unless one is willing to step out and claim that the full still unknown physical theory of nature will in principle simultaneously leave QM and relativity fully mathematically intact as we know them, this issue can not be legitimately claimed to be settled, expert consensus or no expert consensus.

I believe that the answers to these foundational questions not only deserve but require clear answers; more strongly, I believe, because of the mathematical problems of QFT that they actually require a mathematical reformulation of both QM and SR/GR, one in which both theories arise as appropriate limits of this new mathematical formulation. I am, of course, not the only one who has taken this stance; Shimony himself offers a similar point of view on the resolution of this matter:

I wish now to say why I think Bell's Theorem points to a yet more radical proposal than that of the foregoing quotations. Bell's Theorem shows that there is some tension between quantum mechanics and the space-time structure of special relativity, even though the impossibility of using quantum mechanical entanglement to send superluminal messages prevents outright inconsistency between the two theories. There are, however, two other areas of tension between quantum mechanics and contemporary space-time theory, concerning general rather than special relativity. First, in general relativity the metric field is a dynamical entity rather than a fixed structure, and therefore it has to be quantized if quantum mechanics applies to physical reality in full generality. But the difficulties encountered in attempting to quantize general relativity have been so great that one suspects them to be not be merely technical and mathematical in character but conceptual. Second, at the Planck level - around $10^{-33} \mathrm{cm}$ - the quantum fluctuations of the space-time metric become as large as or larger than the expectation values of lengths under consideration, so that the metric structure may no longer be well defined. Both of these difficulties suggest that the union of quantum mechanics with general relativity may require the modification of one or both. It is not unreasonable to hope that if these difficulties are resolved, the tension that Bell's Theorem exhibits between quantum mechanics and special relativistic locality may find a deep and natural resolution

I will end this post by citing Clauser (Chapter 6 "Early History of Bell's Theorem", in the same book Quantum (Un)speakables.):

Given historical hindsight, I assert that our basic understanding of quantum theory has been significantly improved via Bell's Theorem and via its associated experimental testing, long after it was confidently asserted by many textbooks to be well understood. It is truly amazing that so many "killer" details slipped through cracks that existed between experimentalists and theorists. It is clearly of continuing importance for experimentalists and theorists to scrutinize each other's work with great care to try to eliminate such cracks.

Given such hindsight, I also assert that it is clearly counterproductive to scientific progress for one camp smugly to hold to a belief that all problems are solved in any given area. It is even more counterproductive for this camp then further to rely on this belief to formulate a religious stigma against others who do not share their cherished belief.

Indeed, history also shows that a prohibition against open discussion and experimental testing of the foundations of quantum theory, in turn, led to a significant delay of
the discovery of important new applications of these foundations. Quantum cryptography, distributed entanglement, etc. undoubtedly would never have been envisioned without the intellectual challenges posed by Bell's Theorem.

My own final conclusion is that the only real loser here has been the "stigma", itself. I hope that John Bell would have agreed.

 

[DrChinese]

Bell goes on to say "we will argue that certain particular correlations, realizable according to quantum mechanics, are locally inexplicable. They cannot be explained, that is to say, without action at a distance."

If you are going to quote Bell so extensively, I am going to assume you agree with the above. And I am going to assume that by action at a distance, we mean either a propagation of a direct cause and its related effect in excess of c - or a direct connection between distant points which is absolutely simultaneous (instantaneous action at a distance, a la Bohmian Mechanics).

The bolded statement is obviously incorrect, unless you suitably redefine terms to make it correct. For example, it is generally agreed that Many Worlds does not involve action at a distance. Retrocausal and certain acausal theories do not feature action at a distance (the context slice is a collection of points which are themselves accessible at c or less. Those couldn't be accepted interpretations if you are correct. Basically, we all have our favorite interpretations (or non-interpretations as the case may be), so we naturally believe our baby is prettier.

Sadly, Bell did not live to see experimental quantum entanglement swapping. If he had, he would realize that entangled particle pairs need not have ever co-existed. Co-existing presumably being a requirement for instantaneous action at a distance - i.e. what a non-local theory purports to explain.

I agree that there is something called "quantum non-locality", which would be *whatever* kind of non-locality that is exhibited in quantum experiments. Such is not constrained by distance in space or time, no does it necessarily involve cause and effect. However, that does not map directly to the kind of action at a distance per the bold above. Lines of "action" in quantum non-locality are constrained to c, and can move either forward or backward in time direction. This is very clear when you look at a diagram of entanglement swapping. Note: The entangled particles themselves can superficially appear to demonstrate effects in excess of a large multiple of c, approaching infinity; while the growth of the cone of action does not grow faster than a traditional light cone.

So my point is that experimental evidence would force the very careful Bell to modify the above statement, were he to have lived longer. I don't think he ever came out as a full blooded Bohmian anyway, although I am not certain about that. I definitely don't get the point of your Clauser quote, which doesn't seem to bear any relation to quantum nonlocality.

 

[Auto-Didact]

As I said before I'm not a Bohmian nor am I advocating BM, certainly not as the correct fundamental theory of physics. I believe we do not yet have such a theory.

I'm merely arguing, like Shimony (who I also quoted extensively) as well as others, that quantum non-locality is something actually occurring in nature and that mathematically explaining it will probably require the modification of both QM and relativity.

 

[DrChinese]

I'm merely arguing, like Shimony (who I also quoted extensively) as well as others, that quantum non-locality is something occurring in nature and that mathematically explaining it will probably require the modification of both QM and relativity.

I quite agree, but I didn't read what you wrote previously in this manner.

 

[lodbrok]

I don't really know what you are talking about. $A_n, B_n, C_n$ are three numbers, each one is either +1 or -1.

Yes. The three numbers are assumed to exist together in some context denoted by the subscript $n$ which could represent a particle pair for example. Not unlike when we have three coins $A, B, C$ which we toss in pairs, so that we have three numbers $A_n, B_n, C_n$ representing the results of the toss each either +1, -1 (Heads or Tails) in the context of a single toss $n$. Note also that in principle you could imagine the third coin would have produced a value had you tossed it at the same time so it makes sense to think of $A_n, B_n, C_n$ counter-factually. In fact, you could toss all three but observe only two values and still satisfy your requirement that the third value exists even if not known. All is well and good up to here.

So it's just a fact of arithmetic that:

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

The assumption is that twin-pair number $n$, has an associated triple of numbers $A_n, B_n, C_n$, where $A_n$ gives the result of a measurement of spin along the first axis, $B_n$ along the second axis, $C_n$ along the third axis. The manipulation above is just arithmetic. There is no additional assumptions involved.

Yes. So long as you are still talking about that specific context, $n$, that is, that specific particle pair, or that specific toss of a pair of coins, your arithmetic is all good, no additional assumptions required. For the coin toss analogy, we can also do simple arithmetic to obtain

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair. It does not matter if both particle pairs are similarly prepared. This means your notation should really be $A_n B_n + A_m C_m$, to acknowledge the different context $m$. Similarly, using the coin toss analogy, we have really have $A_n B_n + A_m C_m$ since we toss only two coins each time, not three and $A_n B_n$ originates from a different toss than $A_m B_m$. And even if we decided to toss three coins each time, we only read two outcomes from each toss and therefore we necessarily use the results from two different context for the terms.

You introduce the analogy by applying your simple arithmetic to the EPRB experiment or to the analogous coin toss experiment. Because according to your arithmetic

$A_n B_n + A_m C_m = A_n B_ n (1 + B_m C_m) = A_n B_n + A_n B_n B_m C_m$

implying $A_n B_n B_m C_m = A_m C_m$
This is only true if $B_n B_m = 1$ and $A_n = A_m$. This means you expect The outcomes from one context to be perfectly correlated with the outcomes from the other. Or in terms of the coin toss experiment. You assume heads from the first toss of coin $B$ is perfectly correlated with heads from the second toss of coin $B$.

I know it is subtle, but the assumption is introduced in the application of the derived expression to the experiment.Nobody would suggest that coins are non-local. But note the problem you face by applying your logic to such a simple coin toss experiment.

 

[stevendaryl]

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair.

I think you are misunderstanding the distinction between what is observed, and a model intended to EXPLAIN those observations. Yes, what is observed is that Alice measures the spin on one particle, and Bob measures the spin on another particle. The class of models that Bell is interested in (the class that Einstein, Podolsky and Rose were interested in) are models of the following form:

1. When a twin-pair is produced, there is an associated state variable $\lambda$ describing the pair.
2. When Alice measures her particle, the result depends on (1) facts about her measuring device, and (2) the value of $\lambda$
3. Similarly, when Bob measures his particle, the result depends on facts about his measuring device, and the value of $\lambda$.

So in general, Alice's result $R_A$ can be written as a function $R_A(\lambda, \alpha, \lambda_A)$, where $\alpha$ is the choice of which measurement to perform, and $\lambda_A$ is other facts about Alice's device. Bob's result $R_B$ is similarly a function $R_B(\lambda, \beta, \lambda_B)$ where $\beta$ is his choice of measurement, and $\lambda_B$ is other facts about Bob's device.

So now, we have to take into account a stark fact about these twin-particles: There is PERFECT anti-correlation (or correlation, depending on the exact type of EPR experiment performed). That means that for each $\lambda$, if it happens to be that $\alpha = \beta$ (that is, if Alice and Bob perform the same measurement), they always get opposite result: No matter what $\lambda_A$ and $\lambda_B$ are, we always have:

$R_A(\lambda, \alpha, \lambda_A) = - R_B(\lambda, \alpha, \lambda_B)$

This implies that $R_A$ and $R_B$ don't actually depend on $\lambda_A$ and $\lambda_B$ at all. If Alice's result is NOT determined by $\lambda$ and $\alpha$, then sometimes she would get a result that would not be anti-correlated with what Bob gets.

So we pick three possible measurements for Alice, $\alpha_1, \alpha_2, \alpha_3$. For each $\lambda$, let $A(\lambda)$ be $R_A(\lambda, \alpha_1)$, let $B(\lambda)$ be $R_A(\lambda, \alpha_2)$ and let $C(\lambda)$ be $R_A(\lambda, \alpha_3)$.

Now, if we create a sequence of twin-pairs, each twin pair is associated with some value of $\lambda$. So we let

$A_n = A(\lambda_n)$
$B_n = B(\lambda_n)$
$C_n = C(\lambda_n)$

where $\lambda_n$ is the value of $\lambda$ for the $n^{th}$ twin pair. These are not measurement results, they are just numbers, unknown functions of $\lambda$ evaluated at $\lambda = \lambda_n$. But we're ASSUMING that the significance of these numbers is that

$A_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_1$.
$B_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_2$.
$C_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_3$.

Under the assumption of perfect anti-correlation, Bob would get $-A_n, -B_n, -C_n$ for the corresponding measurements.

So now, the numbers $A_n, B_n, C_n$ are just three numbers, each are assumed to be $\pm 1$. So we can do manipulations as real numbers to come to the conclusion that:

$|\langle A B \rangle + \langle A C \rangle | \leq 1 + \langle B C \rangle$

where $\langle A B \rangle = \frac{1}{N} \sum_n A_n B_n$ and $\langle A C \rangle = \frac{1}{N} \sum_n A_n C_n$ and $\langle B C \rangle = \frac{1}{N} \sum_n B_n C_n$, and where $N$ is the number of twin pairs produced.

This is simply a mathematical fact about ANY sequence of triples of numbers $A_n, B_n, C_n$ where each number is $\pm 1$.

Now, the question is whether it is possible to measurement the quantities $\langle A B \rangle$, $\langle A C \rangle$, $\langle B C \rangle$. We can't, actually, because the definition of (for example) $\langle A B \rangle$ is that it is the average of $A_n B_n$ over all values of $n$. But we don't measure $A_n$ and $B_n$ over all possible values of $n$. We only measure it on for some of the $n$. So to compare theory with experiment, we have to assume that the average of $A_n B_n$ over some of the $n$ is approximately the same as the average over all $n$.

 

[stevendaryl]

Yes. So long as you are still talking about that specific context, $n$, that is, that specific particle pair, or that specific toss of a pair of coins, your arithmetic is all good, no additional assumptions required. For the coin toss analogy, we can also do simple arithmetic to obtain

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair.

The point of rewriting that expression involving $A_n, B_n, C_n$ is to come up with a relationship about the averages for $A_n B_n$, $A_n C_n$ and $B_n C_n$, averaged over all $n$. Bell's inequality is about the averages, not about particular $n$.

 

[N88]

Yes, it's certainly possible to have a "perturbative" measurement, where the result is not 100% determined by the pre-existing properties. However, there are no ideas (as far as I know of) how such a perturbative measurement could result in perfect correlation between measurements of distant particles.

That's really Einstein et al's whole point: If measurements DON'T reveal pre-existing properties, then how can two distant measurements produce perfect correlations?

"How can two distant measurements produce perfect correlations?"

It is my understanding that the spin-half particles in EPRB are pairwise anti-correlated via the pairwise conservation of total angular momentum.

Support for that such correlation is seen in eqn (14) from Bell (1964):

$A(a,\lambda)_{Alice}=\pm1.\:(1)\:$ $B(b,\lambda)_{Bob}=\pm1=-A(b,\lambda)_{Bob}.\:(2)\:$.

So $A(a,\lambda)_{Alice}=-A(a,\lambda)_{Bob}.\:(3)$ And $A(b,\lambda)_{Alice}=-A(b,\lambda)_{Bob};etc.\:(4).$

And, dropping the subscripts temporarily: such correlation invokes the general product rule:

$P(AB)=P(A)P(B|A).\:(5)$*

* Causation between outcomes would also invoke the same rule, but the outcomes are spacelike separated. So causation between outcomes is not possible.

Thus consistent with Bayesian updating; ie, interpreting (5) epistemically: (5) says that the occurrence of $A(a,\lambda)_{Alice}$ reveals information that is probabilistically relevant to the occurrence of $B(b,\lambda)_{Bob}.$

So we need a deterministic relation that links eqns (3) and (4) and (5), etc., via the angular relation $(a,b)$.

Let $P(A=1,B=1)=P(A=1)P(B=1|A=1)=\frac{1}{2}sin^2\frac{1}{2}(a,b).\:(6)$

Then (6), in agreement with standard QM, supplies the local deterministic particle-detector relations that (in my view) answer your question.

 

[DrChinese]

Yes. The three numbers are assumed to exist together in some context denoted by the subscript $n$ which could represent a particle pair for example. Not unlike when we have three coins $A, B, C$ which we toss in pairs, so that we have three numbers $A_n, B_n, C_n$ representing the results of the toss each either +1, -1 (Heads or Tails) in the context of a single toss $n$. Note also that in principle you could imagine the third coin would have produced a value had you tossed it at the same time so it makes sense to think of $A_n, B_n, C_n$ counter-factually. In fact, you could toss all three but observe only two values and still satisfy your requirement that the third value exists even if not known. All is well and good up to here.

Yes. So long as you are still talking about that specific context, $n$, that is, that specific particle pair, or that specific toss of a pair of coins, your arithmetic is all good, no additional assumptions required. For the coin toss analogy, we can also do simple arithmetic to obtain

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair. It does not matter if both particle pairs are similarly prepared.

Well of course there are *thousands* of particle pairs in an experiment. If you are a (non-contextual) realist, you believe that Alice's outcome does NOT depend on Bob's setting, and vice versa. So it doesn't matter which pair AB, BC or AC you select on a specific iteration. As stevendaryl says, you are looking for an average.

And the purpose of an experiment is simply to confirm the predictions of QM. The outcome would NOT directly affect Bell's Theorem either way, but would only affect it indirectly. Bell answers the question of whether there is a realistic (and local) theory which can match the predictions of QM, and the answer is NO. The experiment shows then that we do not live in a local realistic world.

 

[stevendaryl]

"How can two distant measurements produce perfect correlations?"

It is my understanding that the spin-half particles in EPRB are pairwise anti-correlated via the pairwise conservation of total angular momentum.

Support for that such correlation is seen in eqn (14) from Bell (1964):

$A(a,\lambda)_{Alice}=\pm1.\:(1)\:$ $B(b,\lambda)_{Bob}=\pm1=-A(b,\lambda)_{Bob}.\:(2)\:$.

So $A(a,\lambda)_{Alice}=-A(a,\lambda)_{Bob}.\:(3)$ And $A(b,\lambda)_{Alice}=-A(b,\lambda)_{Bob};etc.\:(4).$

That's right. If Bob's result is a deterministic function of $\lambda$ and his setting, and Alice's result is a deterministic function of $\lambda$ and her setting, then the requirement that $A(a,\lambda)_{Alice} = - A(a, \lambda)_{Bob}$ explains the perfect correlations. But Bell's inequality shows that there are no such functions $A(a, \lambda)_{Alice}$ and $A(b, \lambda)_{Bob}$.

 

[DrChinese]

"How can two distant measurements produce perfect correlations?"
...

EPR answered that question, and yes, Bell walked through the EPR reasoning on the perfect correlations. But that is not the problem! The problem is when you extend that reasoning - as Bell did - to other angle settings. Then the contradiction occurs, because the results are skewed slightly towards Alice's measurement setting affecting the results of Bob.

If you attempt to write down values for results at A=0, B=120 and C=240 degrees for *both* Alice and Bob (6 per trial) - and remember, you think they are predetermined - you will quickly see that the math does not work out. You can hand pick them even, and they will not work out. Try it.

 

[lodbrok]

I think you are misunderstanding the distinction between what is observed, and a model intended to EXPLAIN those observations.

I'm not. Rather my point is precisely that we have to be careful how we apply models to experiment. You can use a model that contradicts the experiment you are modelling.

Yes, what is observed is that Alice measures the spin on one particle, and Bob measures the spin on another particle. The class of models that Bell is interested in (the class that Einstein, Podolsky and Rose were interested in) are models of the following form:

1. When a twin-pair is produced, there is an associated state variable $\lambda$ describing the pair.
2. When Alice measures her particle, the result depends on (1) facts about her measuring device, and (2) the value of $\lambda$
3. Similarly, when Bob measures his particle, the result depends on facts about his measuring device, and the value of $\lambda$.

So in general, Alice's result $R_A$ can be written as a function $R_A(\lambda, \alpha, \lambda_A)$, where $\alpha$ is the choice of which measurement to perform, and $\lambda_A$ is other facts about Alice's device. Bob's result $R_B$ is similarly a function $R_B(\lambda, \beta, \lambda_B)$ where $\beta$ is his choice of measurement, and $\lambda_B$ is other facts about Bob's device.

So now, we have to take into account a stark fact about these twin-particles: There is PERFECT anti-correlation (or correlation, depending on the exact type of EPR experiment performed). That means that for each $\lambda$, if it happens to be that $\alpha = \beta$ (that is, if Alice and Bob perform the same measurement), they always get opposite result: No matter what $\lambda_A$ and $\lambda_B$ are, we always have:

$R_A(\lambda, \alpha, \lambda_A) = - R_B(\lambda, \alpha, \lambda_B)$

This is all true and irrelevant to the point I'm making.

I'm sure you agree that it is wrong to assume perfect anti-correlation between one particle of a pair and another particle of a different pair even if it is prepared similarly. In the same way as it is wrong to assume perfect anti-correlation between heads of one toss and tails of a different toss, even of the exact same coin.

This implies that $R_A$ and $R_B$ don't actually depend on $\lambda_A$ and $\lambda_B$ at all. If Alice's result is NOT determined by $\lambda$ and $\alpha$, then sometimes she would get a result that would not be anti-correlated with what Bob gets.

So we pick three possible measurements for Alice, $\alpha_1, \alpha_2, \alpha_3$. For each $\lambda$, let $A(\lambda)$ be $R_A(\lambda, \alpha_1)$, let $B(\lambda)$ be $R_A(\lambda, \alpha_2)$ and let $C(\lambda)$ be $R_A(\lambda, \alpha_3)$.

Now, if we create a sequence of twin-pairs, each twin pair is associated with some value of $\lambda$. So we let

$A_n = A(\lambda_n)$
$B_n = B(\lambda_n)$
$C_n = C(\lambda_n)$

where $\lambda_n$ is the value of $\lambda$ for the $n^{th}$ twin pair. These are not measurement results, they are just numbers, unknown functions of $\lambda$ evaluated at $\lambda = \lambda_n$. But we're ASSUMING that the significance of these numbers is that

$A_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_1$.
$B_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_2$.
$C_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_3$.

Under the assumption of perfect anti-correlation, Bob would get $-A_n, -B_n, -C_n$ for the corresponding measurements.

So now, the numbers $A_n, B_n, C_n$ are just three numbers, each are assumed to be $\pm 1$. So we can do manipulations as real numbers to come to the conclusion that:

$|\langle A B \rangle + \langle A C \rangle | \leq 1 + \langle B C \rangle$

where $\langle A B \rangle = \frac{1}{N} \sum_n A_n B_n$ and $\langle A C \rangle = \frac{1}{N} \sum_n A_n C_n$ and $\langle B C \rangle = \frac{1}{N} \sum_n B_n C_n$, and where $N$ is the number of twin pairs produced.

This is simply a mathematical fact about ANY sequence of triples of numbers $A_n, B_n, C_n$ where each number is $\pm 1$.

Again this is all trivially true and irrelevant so long as $A_n, B_n, C_n$ arise from the same context $n$. The three averages $\langle A B \rangle, \langle A C \rangle, \langle B C \rangle$ are not simply independent averages without any relationship with each other. Based on the way you derived the expression, the inequality relationship embodies all the assumptions you used in it's derivation, including the fact that they are all based on the same context $n$. Note that by dropping the $n$ subscripts, you are being a little careless and perhaps that is why you are not getting the point. This will be crucial when you apply this relationship to experimental data.

Now, the question is whether it is possible to measurement the quantities $\langle A B \rangle$, $\langle A C \rangle$, $\langle B C \rangle$. We can't, actually, because the definition of (for example) $\langle A B \rangle$ is that it is the average of $A_n B_n$ over all values of $n$. But we don't measure $A_n$ and $B_n$ over all possible values of $n$. We only measure it on for some of the $n$. So to compare theory with experiment, we have to assume that the average of $A_n B_n$ over some of the $n$ is approximately the same as the average over all $n$.

This is the key. You have not clearly stated the assumption. It is not simply that $A_n B_n$ over some of the $n$ is approximately the same as the average over all $n$.

The assumption is in fact that the relationship between three averages $\langle A_n B_n \rangle, \langle A_n C_n \rangle, \langle B_n C_n \rangle$ from the same context $n$ is the same relationship as that between $\langle A_i B_i \rangle$ from one context $i$ and $\langle A_j C_j \rangle$ from a different context $j$ and $\langle B_k C_k \rangle$ from a yet another context $k$ with $i, j, k$ disjoint.

At the very least, you have to agree that this assumption is implied. Do you disagree? Bell's realism assumption definitely includes this "sub-assumption" if you will, as soon as the inequality relationship is applied to any experiment in which simultaneous measurement of $\langle A_n B_n \rangle, \langle A_n C_n \rangle, \langle B_n C_n \rangle$ was not performed (eg EPRB).

If this assumption is true. It should be possible to start from the variables $A_i, B_i, A_j, C_j, B_k, C_k$ and derive the same relationship as what you derived for $A_n, B_n, C_n$ and ask the question, what additional assumptions will be required in that case. It turns out it will be required to assume that $A_i = A_j, B_i = B_k, C_j = C_k$ and $B_i B_k = 1$ which is the same as assuming perfect correlation between heads of one coin toss and tails of another coin toss.

 

[DrChinese]

I'm sure you agree that it is wrong to assume perfect anti-correlation between one particle of a pair and another particle of a different pair even if it is prepared similarly. In the same way as it is wrong to assume perfect anti-correlation between heads of one toss and tails of a different toss, even of the exact same coin.

The only person saying anything about a correlation between one pair toss and another is... you.

You are getting lost in subscripts, and missing the picture Bell presents. Bell shows us that the relationship between 3 pairs of settings (assuming counterfactual definiteness of outcomes for A, B and C) cannot match the quantum expectation. This has nothing whatsoever to do with an experiment. From the Wiki on Bell's Theorem:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

The next question is whether the entangled particle predictions of QM are correct. Any experiment that is run that shows the usual formula is correct will do it. You do not need to test anything at 3 angles (or 4 such as the CHSH) because the only question is whether QM makes the correct prediction (when a local realistic theory predicts something else entirely). So you could do a series of test sat 0 & 120 degrees ONLY, and that would be enough to confirm QM and reject local realism.

So comments about "how we apply models to experiment" are off target. Bell's Theorem sets up a dividing line between QM and Local Realism, and that is independent of an experiment. Any experiment that tests the predictions of QM will be enough to settle things, and that experiment does not need 3 of anything (i.e. AB/BC/AC) to be convincing.

 

[RUTA]

If this assumption is true. It should be possible to start from the variables $A_i, B_i, A_j, C_j, B_k, C_k$ and derive the same relationship as what you derived for $A_n, B_n, C_n$ and ask the question, what additional assumptions will be required in that case. It turns out it will be required to assume that $A_i = A_j, B_i = B_k, C_j = C_k$ and $B_i B_k = 1$ which is the same as assuming perfect correlation between heads of one coin toss and tails of another coin toss.

QM assumes you are using the same rotationally invariant state for every trial and the eigenvalues are +/-1 for all measurements. That does mean there are relationships between the outcomes in different trials.

 

[lodbrok]

The only person saying anything about a correlation between one pair toss and another is... you.

You are getting lost in subscripts, and missing the picture Bell presents. Bell shows us that the relationship between 3 pairs of settings (assuming counterfactual definiteness of outcomes for A, B and C) cannot match the quantum expectation. This has nothing whatsoever to do with an experiment. From the Wiki on Bell's Theorem:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

The next question is whether the entangled particle predictions of QM are correct. Any experiment that is run that shows the usual formula is correct will do it. You do not need to test anything at 3 angles (or 4 such as the CHSH) because the only question is whether QM makes the correct prediction (when a local realistic theory predicts something else entirely). So you could do a series of test sat 0 & 120 degrees ONLY, and that would be enough to confirm QM and reject local realism.

So comments about "how we apply models to experiment" are off target. Bell's Theorem sets up a dividing line between QM and Local Realism, and that is independent of an experiment. Any experiment that tests the predictions of QM will be enough to settle things, and that experiment does not need 3 of anything (i.e. AB/BC/AC) to be convincing.

We are discussing the nature of the realism assumption in Bell's derivation and I'm pointing out the subtle additional assumptions when applying the relationship derived to the experiments performed in the manner of EPRB. The perfect anti-correlation assumption is crucial in the derivation of the relationship. All I'm doing is pointing out the import of that assumption when you now apply the relation to an experiment. I use the coin toss example to illustrate that the problem is not even specific to the EPRB experiment or QM or local realism or non-locality or any other physical concept. It is a problem of incompatible degrees of freedom that is elementary. The question of whether or not a physical hidden variable theory can reproduce the predictions of quantum mechanics is completely irrelevant to the point I'm making.

All I'm saying is that you have to very very careful when you do simple arithmetic with some numbers, and then you try to apply the result to data obtained in an experiment not performed exactly as you assumed when deriving the relationship -- because it always involves introducing additional assumptions which may not always be true. Surely, you aren't arguing that Bell's mathematics are not applied to experiment are you? Otherwise why should the application be off target?

 

[stevendaryl]

I'm not. Rather my point is precisely that we have to be careful how we apply models to experiment. You can use a model that contradicts the experiment you are modelling.

This is all true and irrelevant to the point I'm making.

No, it is very relevant. The point you are making is wrong. We prove a fact about averages:

$|\langle A B \rangle + \langle A C \rangle| \leq 1 + \langle B C \rangle$

This is just a mathematical fact about any sequence of triples of numbers (where each number is $\pm 1$). It doesn't have anything to do with any measurements. It's just a fact.

Then in a real experiment, we measure the averages for measurements: $\langle A B \rangle$, $\langle A B \rangle$ and $\langle A B \rangle$. We find that that inequality is violated. The undeniable conclusion is that the measured quantities did NOT come from a sequence of triples of numbers (one triple for each twin pair). The assumption of local realism is that it did come from such a sequence of triples. (Actually, that there is a sequence of triples associated with the sequence of twin pairs. The triples are assumed to be functions of the values of $\lambda$.) So the violation of the inequality disproves local realism.

The futzing around you're doing with indices is just not relevant. The argument as summarized here does not mention indices at all. It's only talking about averages.

 

[stevendaryl]

We are discussing the nature of the realism assumption in Bell's derivation and I'm pointing out the subtle additional assumptions when applying the relationship derived to the experiments performed in the manner of EPRB.

I think you're confused about what assumptions are needed. The stuff you're saying about indices is not correct. Bell's theorem is about averages, not about specific indices.

Now, there is an additional assumption involved, which is that we're assuming that the average of (for example) $A_n B_n$ over all values of $n$ is approximately the same as the average over those values of $n$ for which we actually measured $A$ and $B$. It would be weird if that were not the case, and such a weirdness would require some explanation. The whole point of a local variables theory is to give an explanation to quantum statistics. If it requires an additional unexplainable effect, then that hardly counts as an explanation.

All I'm saying is that you have to very very careful when you do simple arithmetic with some numbers, and then you try to apply the result to data obtained in an experiment not performed exactly as you assumed when deriving the relationship -- because it always involves introducing additional assumptions which may not always be true. Surely, you aren't arguing that Bell's mathematics are not applied to experiment are you? Otherwise why should the application be off target?

We're saying that Bell's analysis was very very careful, and the results are pretty airtight.

 

[lodbrok]

QM assumes you are using the same rotationally invariant state for every trial and the eigenvalues are +/-1 for all measurements. That does mean there are relationships between the outcomes in different trials.

So you are saying, according to QM, one particle from one pair is perfectly anti-correlated with another particle from a different similarly prepared pair? That is contrary to my understanding but what do I know. My understanding is that there is no correlation between particles from one pair and those of another pair.

 

[lodbrok]

We're saying that Bell's analysis was very very careful, and the results are pretty airtight.

But that is never an argument as opposed to hand-waving. You have to get down to the details. But I've made my point so we can agree to disagree.

 

[stevendaryl]

But that is never an argument as opposed to hand-waving. You have to get down to the details. But I've made my point so we can agree to disagree.

I don't think you've made a point. I think you've just shown that you are confused about what Bell's theorem establishes.

 

[stevendaryl]

So you are saying, according to QM, one particle from one pair is perfectly anti-correlated with another particle from a different similarly prepared pair?

NO. And that is not used in Bell's argument.