1. Jan 24, 2008

### TimH

I've been studying EPR and such for about a month and I have a question which may be interesting or may be basic-- I can't tell.

Imagine we have two correlated particles in a spin singlet state. We know that if we measure the spin of one in any particular direction, the spin of the other will be the opposite, in this same direction.

My question is why this situation seems to require faster-than-light communication between the correlated particles. Imagine I take a playing card and rip it in half, mix up the two pieces, and send them to opposite sides of the room. If I look at one piece, at one end of the room, and see that its the King's head, then I know instantly that the other piece, at the other end of the room, is the King's body. Now with quantum spin we can measure in any direction in space, so we have a deck of cards of infinite size. At any direction we get one value, and at the other particle we get the opposite value. So instead of faster-than-light communication between the particles, we should really talk about a property that has an infinite domain (directions in space) mapped into two values (of spin). This property starts at the joined singlet state particles and doesn't change just because they get far apart (which is admittedly weird, but doesn't require any communication between them).

If we accept this kind of property we can drop worry about faster-than-light communication between the particles, yes? Each particle has an infinite amount of information (how its spin is going to measure in any direction) and the data at the other particle is the "reverse."

I have read that some Bell inequality variants rule out this sort of property (this is called a "common cause" type of explanation for the correlation).

So my questions are:
1) if we accept this type of property can we reject any notion of communication between the particles, and
2) is there some variant of Bell Inequality which rules out this sort of property?

Thanks.

2. Jan 24, 2008

### JesseM

Isn't this the same as your previous thread? I gave an analogy on that thread involving scratch lottery cards which you seemed to accept as a way of showing that local hidden variables theories (like your analogy above with the playing card) can be ruled out. Please look over my analogy there again and tell me if there's something about it you're not following.

3. Jan 24, 2008

### TimH

Hi Jesse. Thanks for pointing out to me that I'd posted on this before. I admit I have a hard time grasping the significance of the violation of the Bell inequality. As I asked in my last post in the previous thread, does this necessitate communication between the particles? I know that quantum spin observables are all deeply knit together. So the spin values of correlated particles, even for different spin axes, should be related somehow, albeit very weirdly. What I am curious about is whether the violation can present the appearance of communication without there being an actual physical phenomenon linking the particles. Or, to put it another way, if we assume no coummunication, what does that tell us about quantum spin? I am also somewhat suspicious of Bell inequalities because, as authors like RIG Hughes point out, they are based on premises which are "metaphysical assumptions" about reality. I haven't studies Bell's proof in detail but I wonder about a proof that involves such intangible assumptions.

4. Jan 24, 2008

### JesseM

Bell's theorem just shows that violations of Bell inequalities rule out a certain class of theories in which the outcome of each measurement is determined by local hidden variables (meaning there is some information carried by the particle which determines what spin it gives when measured on a given spin axis, and this information is not influenced by anything outside the particle's past light cone). As to what kind of theory we can use instead, there are a few different interpretations...some kind of FTL communication is one possibility, but many advocates of the many-worlds interpretation say that it gives another possible explanation which doesn't necessitate any FTL signals. Anyway, Bell's theorem doesn't address the question of what we should use as an alternative to local hidden variables theories, it's just a negative result that rules these theories out.
What assumptions did Hughes refer to as metaphysical? The notion of a local hidden variables theory seems physical enough to me.

5. Jan 24, 2008

### TimH

Hughes in his book The Structure and Interpretation of Quantum Mechanics, on page 245, says that Bell-type inequalites all have a common form. They contain facts about the correlation of entagled particles (experimental facts which nobody argues with) joined with facts "of a more metaphysical kind." The Bell-type theorems (he examines several) all then take these two sets of facts and from their union derive Bell-type inequalities. Since QM violates these inequalities he concludes that the "metaphysical" premises (he actually refers to them as Pmet, so he really views them this way) are false. As he says on page 246, in these Bell-type theorems "there seems to be little doubt that we are genuinely, and remarkably, putting metaphysical theses to experimental test."

The reason I have trouble with Bell is because 1) translating Pmet into something as conclusive as a proof seems hard to do, 2) I associate theorems with math, where a theorem is ideally based on axioms, and Pmet seem very general to be thought of as like axioms, 3) Hughes says that incompatible observables (i.e. observables that don't commute), are "deeply related" to each other (I can find the quote if you want) (i.e. they are represented by subspaces that are oblique to each other, etc.), and it seems simpler to say that spin is a property that has mathematical properties that are not easy to imagine, leading to violation of the equalities, than to resort to faster-than-light or to many worlds. (3) is really my main point-- by Occams razor it seems more reasonable to say that quantum spin is mind-bendingly weird and creates weird statistical results, than to resort to complex explanations. Does this seem like a reasonable position?

Part of my problem is that I can visualize the correlations easily, but not the violatations, since they are statistical. I have to imagine a run of measures and think about what I expect versus the actual result. I wish there was a simpler way to grasp the weirdness--that would help me. Mermin's paper describes a situation where there is a Bell violation with a statistical sample of one particle, if I recall correctly (I'll check)

6. Jan 24, 2008

### TimH

Sorry I didn't mean to duck your question about the content of Pmet. Basically he covers lots of different versions of Bell inequalities with conditions like causality, a common cause, completeness, etc. Each of these has long definitions so I couldn't describe them here without writing a long essay. Part of what I wonder about is the process of taking features of macroscopic reality (Pmet), seeing that this is violated by microscopic reality (QM), and then concluding something about the kind of explanations you can have in QM. Since macroscopic reality is composed of QM reality I feel there may be something wrong with this type of argument.

7. Jan 26, 2008

### alphachapmtl

Remarks 1,2

Remark 1:
Just a remark about infinity. (it may not be relevant here)
A deck of cards of infinite size is a countable infinity (1,2,3,4,...).
Direction in space is a continuum, so any direction in space is an uncountable infinity.
They can't be put in a 1 to 1 correspondence.
"One of Cantor's most important results was that the cardinality of the continuum () is greater than that of the natural numbers (); that is, there are more real numbers R than whole numbers N." http://en.wikipedia.org/wiki/Infinity

Remark 2:
I don't think the deck of cards analogy is correct.
Seeing a card for A tells you the corresponding card for B, but tells nothing about B's other cards.
In the case of particles, measuring spin z-up for A tells we would measure z-down for B, but it also alter the probabilities for measuring all other orientations for B, according to the projection law sin(t)2+cos(t)2=1.
If A is z-up, then measuring B at 10 degree from z we get up with probability cos(10 degree)^2 and down with probability sin(10 degree)^2.

http://physics.kenyon.edu/people/schumacher/einstein/docs/EPR.pdf [Broken]

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8. Jan 27, 2008

### JesseM

It sounds to me as if he was just using "metaphysical" to talk about the idea that the outcome of each measurement is determined by preexisting hidden variables--if the variables are assumed to be "hidden" then this isn't really an ordinary physical hypothesis, so one can call it metaphysical.
What do you mean by "mathematical properties that are not easy to imagine"? The issue is simply that the mathematical properties are not compatible with local hidden variables theories--basically, any theory where the world can be described entirely in terms of localized facts (like the momentum of a particular particle at a given point in spacetime), and where any given fact (including the result of a measurement) can only be influenced by other facts which lie in its past light cone. You can call this type of theory "metaphysical" to the extent that some of these facts may be impossible to determine experimentally, but I think this is a description that is sufficiently well-defined that you could take any mathematical theory of physics and decide if it fits the description or not; your theory can involve any weird mathematical machinery you like, but as long as it is "local" in this sense, Bell's theorem shows that it can't give the kind of correlations seen with entangled particles.
Maybe it would help to discuss a different Bell inequality? As in my example on the other thread, imagine that we have pairs of scratch lotto cards given to Alice and Bob, each with three boxes (call them box A, box B and box C) that, when scratched, reveal either a cherry or a lemon. Imagine that this time, we find that every single time that Alice and Bob choose to scratch the same box on their respective card, they find the same fruit (in my previous example they found opposite fruits, but this example will be a little easier to follow when it's the same, and the argument is basically identical if you assume they always find opposite ones). The "hidden variables" explanation for this would be that on each trial they received identical cards with identical "hidden fruits" under each one, so if Alice got a card with hidden fruit [box A: cherry, box B: lemon, box C: lemon], then Bob also got a card with the same hidden fruit.

Now, suppose we know the hidden fruits for a large number of trials, and we consider the total number of trials where each of the following was true:

1: Total number of trials where Bob's card had "box A: cherry" and "box B: lemon"
2. Total number of trials where Bob's card had "box B: cherry" and "box C: lemon"
3: Total number of trials where Bob's card had "box A: cherry" and "box C: lemon"

Note that these cases are not all mutually exclusive--a trial that fell into category 3 could also fall into category 1 (if Bob's card was [box A: cherry, box B:lemon, box C: lemon]), and likewise a trial that fell into category 3 could also fall into category 2, although no trial can fall into both 1 and 2 together since they say opposite things about what's behind box B.

So, we're interested in the total number of trials that fall into 1, the total number that fall into 2, and the total number that fall into 3. And the inequality we get here is that the sum of (number that fall into #1) + (number that fall into #2) must always be greater than or equal to (number that fall into #3). Why? Well, simply because any trial that falls into #3 must either fall into #1 or #2...the only possibilities for #3 are [box A:cherry, box B: lemon, box C: lemon] which also falls into #1, or [box A:cherry, box B: cherry, box C: lemon] which also falls into #3. So, every time you have a new trial that adds to the running total of #3, it also adds to the running total of #1 + #2, meaning that no matter how many trials you do and what the statistics of different cards are, the total of #1 + #2 will always be greater than or equal to the total of #3.

So, we have:

(Number of trials where Bob's card has box A: cherry and box B: lemon) + (Number of trials where Bob's card has box B: cherry and box C: lemon) >= (Number of trials where Bob's card has box A: cherry and box C: lemon)

Now, remember that according to this hidden-variables theory, in order to account for the fact that Alice and Bob always get the same fruit when they choose the same box to scratch, we are assuming they both have the same hidden fruit under each box on a given trial. So, it should also be true that:

(Number of trials where Bob's card has box A: cherry and Alice's card has box B: lemon) + (Number of trials where Bob's card has box B: cherry and Alice's card has box C: lemon) >= (Number of trials where Bob's card has box A: cherry and Alice's card has box C: lemon)

Here we are still talking about the truth about what hidden fruits are behind each of the boxes on their cards, not which cards they actually choose to scratch. However, if they each choose which box to scratch randomly, scratching each with equal frequency, and there is no correlation between what box they choose to scratch and what combination of hidden fruits are on their card on that trial (this is one of the conditions of Bell's theorem, that the state of the particles when they're created and sent on their merry way can't anticipate or control what choice the experimenter makes), then the above should lead us to conclude:

Probability(Bob scratches box A and gets cherry, Alice scratches box B and gets lemon) + Probability(Bob scratches box B and gets cherry, Alice scratches box C and gets lemon) >= Probability(Bob scratches box A and gets cherry, Alice scratches box C and gets lemon)

So if we then do a large number of trials and find that this inequality is consistently violated, we know the explanation where each experimenter has a "hidden fruit" behind each box (i.e. any explanation where the card has local properties that predetermine what result it will give for each measurement) cannot be correct. And there is a basically identical inequality for the statistics of spins of entangled particles when the experimenters can measure on 3 possible axes, and the inequality can similarly be violated in quantum mechanics with certain choices of angles for the three axes. This type of inequality is discussed in more detail here:

http://www.upscale.utoronto.ca/PVB/Harrison/BellsTheorem/BellsTheorem.html

9. Jan 29, 2008

### TimH

Jesse, thanks so much for taking the time to explain all this. I've been away but I'm going to take another stab at understanding the lotto-ticket version this week.

You asked me what I meant by saying that spin involves "mathematical properties that are not easy to imagine." What I was thinking of was: 1) the representation of spin as vectors in a complex two-dimensional Hilbert Space, which cannot be visualized, 2) the fact that spin observables of a system are "knit together"(Hughes term) making them incompatible with each other (which Hughes says is what makes QM fundamentally different and strange compared with classical mechanics), and 3) the general fact that spin as a quantum phenomenon seems to be elemental and ultimately mysterious-- with only a loose connection to its classical counterpart. Please let me know if I'm off on any of (1)-(3). Thanks.

10. Jan 29, 2008

### TimH

My background is in philosophy not physics, so I may be thinking about spin in a way that isn't familiar. The question that motivates me in studying nonlocality is this: under what circumstances would the appearance of nonlocality and actual instantaneous action-at-a-distance be indistinguishable? In other words, is nonlocality simply an appearance, based on the way we see and think about quantum events, or is it something real? In the quantum cakes article, which I like because it involves a very simple (hypothetical) set-up, to see a nonlocal influence one has to make assumptions about the states of quantum particles that have not been measured. I wonder if this is the key to really understanding this phenomenon. Peter Morgan (a Yale physicist who posts to this site) referenced me in making this observation, which makes me think I'm on to something (this is in the thread "Classical interacting random field models" in Beyond the Standard Model) (sorry I don't know how to post links...) .

Last edited: Jan 29, 2008
11. Jan 29, 2008

### alphachapmtl

Spin 1/2 not easy to imagine

Like the fact that the electron (a spin 1/2 particle) must turn twice to be back in its original position (so that its complex wave function is identically restore).
This is certainly not easy to imagine if we base our imagination on the behavior of usual macroscopic objects.

Next is a quote from: https://www.physicsforums.com/archive/index.php/t-10596.html

12. Jan 29, 2008

### vanesch

Staff Emeritus
To the OP: what you described in your original post is often the "discovery" people who learn first of things like the EPR paradox, make: if you have a correlated system, "learning" about it on one side immediately learns us something about the other side, without having to communicate.

Now, obviously, physicists are not such a naive lot as having overlooked that. It turns out (and JesseM pointed you to some tedious explanations) that for the EPR situations, no such a priori correlation can exist. That is exactly what Bell found out (to his great surprise: he was trying to show the opposite!). That is, the set of ALL correlations of 3 possible measurements on each side (3 angular directions), have to satisfy certain inequalities (Bell's inequalities) for such an explanation to hold, and the quantum-mechanical predictions do not satisfy these inequalities.

How does Bell proceed ? Exactly as you supposed: for each angle, you "hide" a card in each particle, and the opposite card in the other particle. So IF people decide to measure along the same angles, THEN they will find opposite results. THAT, by itself, can be explained by the hidden cards. But now Bell went on calculating the statistics of what happens when people measure along DIFFERENT directions. And he found that, if we are to have the perfect anti-correlation we just found for equal directions, then the correlations for non-equal directions cannot behave totally freely: they have to satisfy certain inequalities. And it are these inequalities that are violated by the predictions of quantum mechanics.
So, in short, we cannot "hide" a set of cards for each angle in each particle, and hope to recover the statistics as predicted by quantum theory. That's Bell's theorem.

Now, if you really want to read about this, the best thing to do is to read Bell's own little book: "speakable and unspeakable in quantum mechanics", where he explains exactly all this in great pains (especially with Professor Bertlemann's socks).

13. Jan 29, 2008

### TimH

Many thanks to everyone who's posted on this. I've spent a morning reading Jesse's excellent description of the lotto tickets and its beginning to make sense. Jesse thanks so much for taking the time to give such a detailed explanation. I'll also check out Bell's book.

I drew a Venn diagram showing the Bell inequality (A not B) + (B not C) >= (A not C). Seeing it spatially really helped. Thanks also for the discussion of spin 1/2 as a 720 degree rotation-- it sort of suggests that rotation is somehow essential to the nature of the electron, doesn't it?

A few observations: I have no reason to want to believe that hidden variables exist, so I'm not hostile to Bell's reasoning. I just want to understand the phenomenon of nonlocality. I noticed that in Jesse's explanation (and others) the variables aren't rigorously non-commuting (as I understand non-commuting). Basically what I mean is, if Bell's inequality is true for a population where the properties are, say, gender (M/F), size (big/small) and age (young/old), why is it surprising that Bell is violated when dealing with spin measurements? As I understand it (and I'm probably missing something...) spin probabilities at different angles to each other are related by a cos^2(separation angle/2) formula. So unlike the scratch ticket example where there are three distinct boxes (A,B,C), with spin there is a continuum of possible measures (directions in space). This seems to be a big qualitative difference, with the violation of the inequality coming out of this difference. Is this accurate?

To put it another way: is the violation of Bell by QM just a "weird, naked, experimental fact" or can you explain/justify the violation by looking at the mathematical representation of multiple spin measures and how they relate to each other? Is nonlocality a distinct property of non-commuting (incompatible) quantum observables, that is expressed clearly in their mathematical representation?

Thanks.

Last edited: Jan 29, 2008
14. Jan 29, 2008

### DrChinese

The cos^2 theta formula is accurate, but that shouldn't be the formula if there are hidden variables which are observer independent. Instead it would be perhaps .25+((cos^2 theta)/2), which is what you get when you assume that Alice and Bob resolve independently. That formula is far different than what is actually observed. In other words, the perfect correlations make sense classically as long as you don't consider any other combinations of angles. When you do try to make sense of those, you see what a mess you have on your hands. Because predictions drawn from a classical analogy won't work, as they are internally inconsistent. Quantum predictions are not inconsistent.

15. Jan 29, 2008

### TimH

Thank you for the post, though I feel from your explanation that I'm missing something. Since my background is in Philosophy I may just "think different" about this. You can only have a Bell violation in QM if you are dealing with non-commuting variables, yes? I guess I'm asking a big "essay type question" which is: how does the non-commuting character generate the violation of the inequality, if it can be thought of that way? Doesn't the fact that your three observables A,B,C can be arbitrarily close to each other (by measuring directions that are arbitrarily close) somehow "make" the violation happen?

16. Jan 29, 2008

### Staff: Mentor

From my days in graduate school, I still have a yellowed 27-year-old photocopy of a preprint of one of Bell's lectures about M. Bertlmann...

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17. Jan 29, 2008

### Staff: Mentor

I've got that same preprint! Classic.

18. Jan 29, 2008

### DrChinese

Non-commuting observables are the ones that exhibit the "strange" behavior.

I prefer to think of it in terms of the Heisenberg Uncertainty Principle (HUP). If the HUP is "real", then the violation of Bell Inequality makes plenty of sense. If the HUP is just an artifact of the limits of measurement apparati, then the violation makes no sense.

Although you may not be as interested in following the math, that is what really does the convincing. Belief in pre-existing values for the observables leads to inconsistencies, such as "negative probabilities" (likelihood less than 0) for certain outcomes.

If you want to see easy math for Mermin's version, see this link: Bell's Theorem with Easy Math

If you want to see the negative probabilities, see this link: Bell's Theorem and Negative Probabilities

The math in either of these is not too hard too follow, requires only some very simple algebra and probability theory. It only sounds complicated. It is pretty convincing, I think Einstein would have had to acknowledge it if he had lived to see it (since he was a skeptic).

19. Jan 29, 2008

The way I do it is to highlight the http://[name] [Broken] in the tool bar, then used "edit" copy, then "paste" into your reply. You can also add files--see "additional options" below.

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20. Jan 29, 2008

### gel

An alternative to Bell's theorem is the the Greenberger-Horne-Zeilinger scheme, which is predicted by quantum mechanics, and clearly rules out the playing card analogy. I think this is simpler in some respects as it doesn't require any inequalities or probabilities (other than probs of 0 or 100%) to see the non-local effect. The following quote is from http://oolong.co.uk/causality.htm" [Broken], which I just found by googling.

It should be clear by thinking about it for a bit, that there is no possible underlying joint state for the X and Y-spins which would give the results mentioned here.
I'm not sure if this experiment has actually been accurately performed, but it's a simple prediction of QM.

(*) I assume this should say 0 or 2!

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