A question about nonlocality

[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.

 

[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.

 

[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.

 

[JesseM]

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?

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.

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.

What assumptions did Hughes refer to as metaphysical? The notion of a local hidden variables theory seems physical enough to me.

 

[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)

Anyway I'd value your comments on points (1)-(3) above. Thanks.

 

[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.

 

[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.

Reference about Bell's inequalities:
http://physics.kenyon.edu/people/schumacher/einstein/docs/EPR.pdf

 

[JesseM]

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."

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.

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?

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.

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.

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

 

[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.

 

[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...) .

 

[alphachapmtl]

Spin 1/2 not easy to imagine

... that spin involves "mathematical properties that are 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

The concept of basic electron spin is that electrons are spin-1/2 particles. ... Because this spin is not something that you can physically view, you must visualize it abstractly. Imagine that you have a deck of cards, and that you take out one of the face cards. How far do you have to turn it before it looks the same again? 180 degrees, right? A face card is a spin-2 object, because it will appear the same twice in a 360 degree rotation. Imagine if you had to turn that face card 720 degrees to get it to look the same! It is not intuitive, nor does it seem natural! But that is what a spin-1/2 particle is like.

So, the spin number can essentially be described as a full rotation, 360 degrees, divided by an intrinsic symmetry angle. This is not a physical angle necessarily, but rather an abstract phase angle. But it is good to try to visualize it. We live in a vector-world, where everything intrinsically appears to have a spin-1 character, so spin-1/2 is especially hard to visualize.

 

[vanesch]

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).

 

[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.

 

[DrChinese]

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? Thanks.

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.

 

[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?

 

[jtbell]

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).

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...

 

[Doc Al]

I've got that same preprint! Classic.

 

[DrChinese]

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?

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).

 

[Rade]

...(sorry I don't know how to post links...)

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

 

[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" , which I just found by googling.

The GHZ scheme uses three particles rather than two, and measurements of spin rather than polarisation. The particles are sent out in different directions, and one of two sorts of spin measurement is made on each – call the first type X and the second type Y. The measurements are made in one of four combinations: Either every particle will be asked X, or two of the particles will be asked Y, and the other one X. Quantum mechanics predicts a 100% probability that if only X measurements are made, an odd number of the particles will be found in the ‘spin-up’ state, whereas if two Y measurements are made, an even number of particles will be measured as ‘spin-up’. The theory says nothing about whether the odd number will be 1 or 3, or the even number 2 or 4 (*).

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!

 

[DrChinese]

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!

Here is one performed by Zeilinger & Pan:

Multi-Photon Entanglement and Quantum Non-Locality

Results: Predictions of QM soundly supported, predictions of Local Realism soundly rejected. Again.

The only thing bad about the GHZ math is that there are a lot of parts, and each one seems so randomly assembled. Of course, it's totally brilliant in the end. I think the Bell Theorem is a lot easier. For the amateur, I don't think the "fair sampling" complaint is that big a deal, but certainly the GHZ Theorem really hits that objection pretty hard.

 

[jtbell]

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...

I just found out that you can download it from the CERN document server. Click on the "Fulltext" link.

I'll keep my yellow photocopy anyway, for sentimental reasons. It brings back memories of the afternoon departmental coffee breaks which took place next to the preprint racks in the back of the colloquium room. Professors and grad students mingled, sipping coffee, munching on cookies and browsing preprints. Try doing that with arxiv.org...

 

[TimH]

Thanks everybody for the extra links and the explanation of GHZ. Can anybody recommend a good and reasonably comprehensive book on entanglement? My math ability is at the basic QM linear algebra level plus vector analysis.

 

[JesseM]

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.

Hi Tim, sorry to take so long in replying, but I've had this post half-finished for a while so I figured I'd finish it up before going off on vacation. Anyway, your statements 1-3 seem to be based on the standard mathematical formulation of QM which represents the state of systems as a vector in Hilbert space, but the question for those interested in the locality or nonlocality of QM should be, is there an alternate mathematical framework that can be used that gives us a purely "local" picture of the dynamics and yet manages to replicate the predictions of the Hilbert space formulation? Here, a "local" theory would be one in which the state of the universe at any given time (or the state of some region containing an isolated system whose behavior we want to predict) can, without any loss of information, be defined entirely in terms of a set of localized facts about the state of every point in space in the universe/region at that time, and with the dynamical laws having the property that the state of a given point in space at a given time is influenced only by information that lies within the past light cone of that position and time.

Our mathematical description of the "state" associated with a given point in spacetime need not be anything simple or intuitive, it can be as weird and abstract as you please, but as long as the theory requires no information about the state of a system beyond localized facts about the state of each point in space, and as long as every localized fact can be affected only by events in its past light cone, then the theory counts as a "local" one. For example, classical electromagnetism is a local theory, because if you know the electromagnetic field vector at every point along with localized facts about particles or electromagnetic fluids at that point, then you won't be missing any information about the state of the region you're looking at, and the field vector and the facts about the charges at a particular point in space and time won't be influenced by anything outside the past light cone of that point. Something similar is true of general relativity, even though the mathematical structure it associates with each point in spacetime is a lot more complicated and difficult to visualize, like the stress-energy tensor at that point and the curvature tensor.

The Hilbert space formulation of QM is not obviously local in this way, since a single state vector is used to describe a system that is not localized to a single point in space. So the question is, can we find a different formulation of QM--perhaps one involving additional state information beyond the state vector, like "hidden variables"--that is local in the above sense, and which yields all the same experimental predictions?

Well, consider the fact that whenever two experimenters measure the spin of entangled particles on the same axis, they always find opposite spins. In a local theory like the one I describe above, when you measure the spin of a particle along a particular axis in a local region, then the answer you get--spin-up or spin-down--must be determined by localized facts about the state of the particle at the moment of measurement. But if neither of the two measurements by the two experimenters lies within past light cone of the other measurement, then neither measurement can have influenced the local state of the other particle when it was measured--the particles can't have communicated at the time of measurement to "coordinate their answers".

So how can you explain the fact that they are always opposite when both experimenters make the same spin measurement? Well, we know "correlation is not causation"--if two events A and B are always correlated, that doesn't necessarily mean that one causes the other, it could be there is some common cause C. So in a local theory, if we can rule out a causal link between the two measurements, the correlation must be explained by something which influences the local state of both particles at the time of measurement in a way that ensures they give opposite results. And because local theories only let local states be influenced by information in their past light cone, the common cause must lie in the region where the past light cone of the two measurement-events overlap. So, as soon as each particle left this overlap region, their states must have already included information which predetermined what results they'd give when measured on that axis, with some common cause in the overlap region making sure that the first particle's predetermined answer will be the opposite of the second particle's predetermined answer (it would make the most sense to have the common cause predetermine each particle's answers at the moment of their creation at a single point in space, since once they depart from each other they can be measured at any time after that).

So what we conclude is that if the two measurement events always give opposite results when measured on a particular axis, then *if* the laws of physics are purely local, it must be that something in the overlap region of the two past light cones of the measurement-events must have influenced the local state of each particle in such a way as to predetermine what answer each particle would give when measured on that axis. But if each experimenter chooses which axis to measure at random, then there wouldn't be enough information in this overlap region to predict in advance what axes the experimenters would choose (since the past light cone of each experimenter's choice can be influenced by events which don't lie in the overlap region). So, we must conclude that the each particles' local state must include a predetermined answer to what spin they will give when measured on *any* possible spin axis, if the laws of physics are local.

So if each experimenter has a choice of 3 spin axes A, B, and C, according to the hypothesis that physics is local, the local state of each particle at the time of measurement must predetermine whether they'll give spin-up or spin-down on each axis. This means that we can characterize any particle's state in terms of these predetermined answers--one particle might have a state of type of {A-up, B-down, C-down} and another might have a state of type {A-down, B-up, C-down}, and so forth. Of course, this type of characterization need not be a *complete* description of the particle's local state, even if we included every possible spin axis--as I said earlier "local states" can be as mathematically complicated as you want, like the tensors characterizing each point in spacetime in general relativity, and it could be there are other aspects of the local state besides its predetermined answers to each possible spin measurement. Nevertheless, every possible local state must have *some* predetermined answers to these spin measurements, if we want to explain the perfect correlations seen between spins of entangled particles when the same axis is chosen, and we want to persist in believing that the laws of physics are totally local. So even though there might be a lot of different possible local states which all fall into a given "type" like {A-down, B-up, C-down}, every possible local state must be of one type or another.

But if each particle has a local state which predetermines its answer to the three spin measurements, then the different "types" are exactly analogous to the different possible scratch lotto cards in my analogy, where each card is assumed to have a predetermined fruit (cherry or lemon) behind every box. And this means that Bell's theorem applies to the particles just like it does to the cards--when we look at the statistics over many measurements, we should be absolutely confident that if physics is purely local, the statistics should satisfy the various Bell inequalities. And yet, when the experiments are actually done, we find that these Bell inequalities are consistently violated! So the whole argument is a sort of "proof by contradiction" to demonstrate the claim that *no* local theory can explain the results seen in QM. There cannot be any way to recast the standard mathematical formalism of QM in a new way that leads to the same predictions as standard QM but which satisfies the requirements for a local theory which I discussed before (which the standard formalism does not satisfy, since it doesn't describe the state of the universe purely in terms of the local state of each point in space, with each point's state being influenced only by what's in its past light cone).

OK, that was long and maybe a little more complicated than it needed to be...but does it help at all with your questions about why Bell's theorem definitively rules out local realism? If not, is there any particular point in my discussion which you have trouble understanding or have other issues with?

 

[DrChinese]

Thanks everybody for the extra links and the explanation of GHZ. Can anybody recommend a good and reasonably comprehensive book on entanglement? My math ability is at the basic QM linear algebra level plus vector analysis.

I would recommend:

"Entanglement" by Amir Aczel

It may not be as technical as you would like, but it covers most of the ground on the subject. On the other hand, the Bell and GHZ original papers should themselves provide sufficient technical detail if the book is not enough.

 

[TimH]

Jesse,

Its been four months, which is too long, but I just came across your last post and want to thank you for taking the time to write it. It definitely helps.