# Bell's Inequality

Is here any progress on explaining Bell's Inequality? I do not mean explaining what it is, I mean how it works.

Symmetry777

jfizzix
Gold Member
Bell's inequality works in the following way (more or less):

If the correlations between measurements on separated systems can be explained locally, then the probability distribution one gets would have to factor in a particular way:

For example, for the correlations of position measurements on particles A and B to be explained locally, the probability density $\rho(x^{A},x^{B})$ would have to be expressible in the following form:
$\rho(x^{A},x^{B})=\int d\lambda \rho(\lambda)\rho(x^{A}|\lambda)\rho(x^{B}|\lambda)$
Here $\lambda$ can represent any variables or information that could be conveyed to both A and B from some point in the past. The form is not limited to this particular interpretation, though.

However, as a consequence of this form, Bell and others proved (for measurements of spin instead of position) that the measurement correlations must obey a certain inequality.

Since we can show experimentally that real measurements can violate Bell inequalities, we can show that sometimes, real measurements cannot be explained by shared information in the past (i.e., cannot be explained locally).

The ultimate conclusions of violating Bell inequalities are tied up with how you choose to interpret quantum mechanics, though.

Jilang and sanpkl
It is that "cannot be explained locally" part that I feel needs explaining. How does that work with or without violating the speed of light?

I'm just in the process of fully developing my understandings of these things myself. In fact, I just read the "Bell's Theorem" wikipedia page about an hour before checking this forum. However, my understanding of the Bell Inequality is as a statement of how classical probabilities must work. For instance, depending on how they're set up, certain classical correlations must be ≤1 or ≤2. However, if we abandon "local realism" (giving into Einstein's disliked "spooky action at a distance"), we find that these inequalities can have upper bounds beyond the ones stated by classical probabilities.

Again, from my understanding, these inequalities aren't exactly a "tool" to be utilized, but rather a way to give support to certain hypotheses within quantum mechanics that they are indeed accurate descriptions of reality. Some have called these quantum correlations "supercorrelations". Again, depending on how they're set up (and comparing them to the numbers stated above), the upper bounds of the correlations can be ≤1√2 or 2√2. As we see, these upper bounds are larger than would be possible if classical mechanics (with local realism) was an accurate description of reality. Experiments have been done which show correlations greater than the classical upper bounds of ≤1 or ≤2.

I look forward to others expanding on my rather broad brushstroke explanation of the Bell Inequalities.

Jilang
Ahhh, I sat on the webpage too long. I'm not sure that the "faster than light" directly has to do with the Bell inequalities. I think that's speaking more to Bell states (or EPR pairs) directly. I believe that Einstein said that "nothing can travel faster than the speed of light". However, it turns out that that's not quite right. What he should have said is that "no meaningful information can travel faster than the speed of light". By "information", one way to think of this is some string of classical bits. We can easily recognize (with ASCII coding, or bitmap pictures) that we can send "information" with a string of bits in a particular pattern of ONEs and ZEROs. However, even with Bell states (i.e., EPR pairs), we can't send this "information" faster than light.

The EPR pairs will be perfectly correlated. However, Bob and Alice will have no way of verifying this without a classical (speed of light) message. Furthermore, Alice can't influence Bob's copy of the EPR pair in any way that would be meaningful for communication. In other words, the individual qubits (forming EPR pairs) will each look like a random string of ONEs and ZEROs with no intrinsic meaning to either Alice or Bob if they "read" them.

Information theorists (who bring in the second law of thermodynamics) ask where the information that's in the EPR pairs went if it can't be used for communication. It can be shown that it's hidden in the fact that the pairs are perfectly correlated. The correlation itself is information, but it's not information that can be used for communication by either Alice or Bob.

(I feel pretty good about all of that, but still look forward to more responses.)

I believe that Einstein said that "nothing can travel faster than the speed of light". However, it turns out that that's not quite right. What he should have said is that "no meaningful information can travel faster than the speed of light".

That is the view Professor Brian Cox advocates.

And, I guess to bring Bell inequalities back in, the fact that they have been violated in many experiments, supports rather strongly that each of the qubits in an EPR pair (i.e., Bell states) is truly in superposition (i.e., not specifically a ONE or ZERO, think Schrodinger's cat) until it is actually "read". What is quite fascinating is that, as soon as Alice (making her the first to "read" her entangled qubit) "reads" hers, then the state of Bob's qubit is determined "instantaneously" (faster than light). The fact that the Bell inequalities are experimentally violated shows that there is some "space fabric" connection between the entangled (EPR pair) qubits.

I can't completely outline the reasoning here, but the fact that the Bell inequalities have been experimentally violated also supports the position that there is no "hidden variable" or "hidden classical communication" between the two entangled qubits.

The fact that the Bell inequalities are experimentally violated shows that there is some "space fabric" connection between the entangled (EPR pair) qubits.
Ah, now we are getting to the part that is fascinating. I agree it seems to say "there is some 'space fabric' connection." If we could derive some measurable consequences from that idea and they panned out we would win the Noble prize. ;)

Jilang
Hmmm, measurable consequences. Well, the quantum mechanics math is pretty much all worked out. However, many people just can't accept the idea that there are "consequences" that travel faster than light (even if they can't be used for communication of information). There are many people trying to work out sub-photon (or sub-electron) models of what's going on. One that I particularly like uses an idea of dividing a photon into something like two balls attached by a string. Sort of like the things used in tailgate golf:

These individual photon "parts" aren't directly observable but are theoretically thought to exist. We might call them "anyons", whereby it takes two anyons to make an observable photon. The anyon pairs are connected with a string that exists in some "strange" dimension (space fabric) that's not like the 3 spacial dimensions we see. Furthermore, these strings can be infinitely stretched without the need for any force, but they stay connected.

Alright, let's think of a qubit as a pair of these anyons. Let's call them QA1 and QA2. Furthermore, let's think of two qubits. Our second qubit has the anyon pair QB1 and QB2. Now, we wish to entangle our qubits into an EPR pair. One way to conceptualize this is to "braid" the strings of the anyons, whereby we form two "new" qubits with the first constructed of anyons QA1 and QB2, with the second constructed from QA2 and QB1. We swapped anyons, but the strings are connected (braided), although possibly stretched out to great lengths as the two qubits possibly separate from each other.

(Again, I welcome any replies/critiques to these thoughts.)

Yes. This is the line of my thinking. I would disagree with the phrase "can be infinitely stretched." I think it is another dimension like string theory uses other dimensions. The two particles do not need to move apart in entangle dimension just because they moved apart in the 3 dimensions of space or to be fancy the 4 dimensions of space-time.

Smolin has a paper where time, energy, momentum, and events are basic and space is emergent. Maybe in that formulation we can have an entanglement dimension of "space" that serves to "communicate" the measurement event of one particle light years away to the other particle with zero or small delay such that the measurement event of the second particle has the probability distribution (many pairs need to be measured to get the distribution not just one) given by standard QM.

Jimster41
Yep, I agree. If we're talking about additional (unobservable) orthogonal dimensions, there doesn't necessarily need to be separation. To take the analogy to simple plane geometry, we can certainly have separation on the X-axis whereby we may (or may not) have separation on the Y-axis.

And yes, I agree about needing to "read" many EPR pairs to accumulate data to test our correlations (and thereby test violations of the Bell inequalities).

small delay

Your (or possibly Smolin's) use of these words is potentially quite fascinating though. With respect to the collapse of superposition (even with entangled states), I personally haven't heard of anyone talk about it in terms of anything but "instantaneous". I suspect that a "small delay" (even if faster than light) would be quite interesting. Maybe you'll get your Nobel prize after all.

(I'm probably going to go quiet for a bit though and get some work done.)

Elroy, the use of small is due to Aspects one million times the speed of light as a lower bound on the speed of communications from his experimental measurements.

Say hello to Astro.

Your (or possibly Smolin's) use of these words is potentially quite fascinating though. With respect to the collapse of superposition (even with entangled states), I personally haven't heard of anyone talk about it in terms of anything but "instantaneous".

There was a paper (which I'll have to go searching for) that indicated that if 'collapse' wasn't instantaneous, you could see deviations (small, though) from QM predictions.

EDIT: it seems posts from August 2013 and earlier have been lost due to the forum upgrade - which I suspect was when the said paper was posted on this site and I replied.

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bhobba
Mentor
Is here any progress on explaining Bell's Inequality? I do not mean explaining what it is, I mean how it works.

Sure.

Its a consequence of the principles of QM. So your question basically is - has any progress been made on the 'why' of those principles.

Indeed there has been progress:
http://arxiv.org/pdf/quant-ph/0101012.pdf
http://arxiv.org/abs/0911.0695

The situation is this. Bells inequality is a consequence of entanglement which is not allowed in standard probability theory. In fact some very reasonable assumptions, as shown above, lead to basically two models that can be used to model physical systems - standard probability theory and Quantum Mechanics. The difference being if you want continuous transformations between pure states or not. But physically if some kind of physical process transforms something in one second it has to go through half a second so continuous transformations seem more or less required. That's the 'why' of QM.

The other interesting thing is its not only continuity that separates the two - its entanglement as well. Only QM allows entanglement - as explained in the second paper. So basically, because we really want continuous processes in modelling physical systems, we must have entanglement and hence Bells inequality.

However if you are asking if any progress has been made on 'how it works' in terms of everyday classical common-sense pictures then I am afraid you are out of luck, because, one thing we know 100% for sure, is the quantum domain does not operate that way.

Thanks
Bill

bhobba
Mentor
It is that "cannot be explained locally" part that I feel needs explaining. How does that work with or without violating the speed of light?

Its tied up with what local means in QM which is really to do with the so called cluster decomposition property:
http://en.wikipedia.org/wiki/Cluster_decomposition_theorem

But that only applies to systems that are not correlated - entangled particles are correlated so evade the cluster decomposition property. Bummer.

Also note we really have to go to QFT for locality to apply because standard QM is based on the Galilean transformations to which locality does not apply. Although it's not usually emphasised classical non relativistic mechanics right at its foundations violates locality - see the first chapter of Landau - Mechanics. Standard QM also has the same issue - see Chapter 3 - Ballentine - Quantum Mechanics - A Modern Development where the dynamics is deduced from the principle of Galilean relativity.

To cut to the chase if QM is non-local or not and influences travel faster than the speed of light depends purely on your conception of local. I believe it does - but that's just me - opinions are like bums - everyone has one - it doesn't make it right.

In the naive reality debate I believe QM violates local realism and counter-factual definiteness - although don't ask me to precisely define those terms off the top of my head - I keep forgetting exactly what they mean and need to look them up.

Thanks
Bill

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Nugatory
Mentor
What is quite fascinating is that, as soon as Alice (making her the first to "read" her entangled qubit) "reads" hers, then the state of Bob's qubit is determined "instantaneously" (faster than light). The fact that the Bell inequalities are experimentally violated shows that there is some "space fabric" connection between the entangled (EPR pair) qubits.

It's almost impossible to resist the temptation to think that Alice's measurement determines the state of Bob's particle through some faster-than-light connection (perhaps messages carried by flying pigs, perhaps as you say "some 'space fabric'"). Nonetheless, you must resist this temptation.

The problem is that if Alice's and Bob's measurements are spacelike-separated, there is no way of saying which one happened first. Some observers moving at some speeds relative to the experimental apparatus will find that Alice measured her particle before Bob measured his; others will find that Bob's measurement came first and determined the state of Alice's particle.

All that entanglement tells us is that if we measure one particle then we know what the result of a measurement of the other particle is, when and if such a measurement is made or has already been made.

Jimster41 and bhobba
bhobba
Mentor
The problem is that if Alice's and Bob's measurements are spacelike-separated, there is no way of saying which one happened first. Some observers moving at some speeds relative to the experimental apparatus will find that Alice measured her particle before Bob measured his; others will find that Bob's measurement came first and determined the state of Alice's particle.

Just to expand on Nugatorys excellent point (I wish I had said it) even if you believe some kind of influences are travelling FTL it cant be used to send information, so clocks cant be synced, and SR still holds.

Thanks
Bill

zonde
Gold Member
It is that "cannot be explained locally" part that I feel needs explaining. How does that work with or without violating the speed of light?
It doesn't work. Quantum mechanics is phenomenological theory that can't have realistic model at it extremes.

Hmmm, okay, I'll jump back in with some degree of confusion. And again, I'll admit that I'm in the process of (carefully) developing my understandings of this stuff. Nugatory didn't like my use of some unobservable "space fabric" that connects things. Let me attempt to make a few points, on which I welcome critique.

First, hopefully, we all agree that we must develop some language to talk about this stuff (or we're forever lost in Alice's wonderland, which we may be anyway). I'll agree that we need to be precise with our language, but we still need it.

Alright, regarding our EPR pair, let's assume that we have solved all problems regarding the maintenance of coherent superposition. (That's, more or less, an entirely separate problem in my mind.)

Ok, we prepare an EPR pair of qubits (possibly sending them both through Hadamard gates, and then a CNOT gate) and then separate them, giving one to Alice and the other to Bob. Let's further assume that both Alice and Bob always "read" their qubits on the |0〉 to |1〉 axis. It's my understanding that things could go as follows:

1. Alice reads her qubit and sees it's a ONE.
2. Alice sends this information to Bob, saying, "ha ha ... ha ha ha, I know that your qubit is a ONE".
3. A couple of hours after receiving Alice's message, Bob reads his qubit and, sure enough, it's a ONE.

Furthermore, they can do this over and over and the results are always the same in that Alice knows what Bob's results will be. The only change will be that Alice "reads" a ONE 50% of the time and a ZERO 50% of the time.

(And, just as an FYI, it's my understanding that the Bell inequalities come into play if Bob rotates his qubit by 45° on either the Bloch sphere's X or Y axis before reading it. This is where classical probability theory and QM predict (maximally) different correlations, with QM winning the day.)

So here are some interesting questions.

1. It's generally agreed that both of the entangled qubits are still in superposition if neither has been read. This is required for the violation of Bell inequalities that has been experimentally observed. So, after Alice reads her qubit, is Bob's still in superposition? To me, we must say "no", or not exactly. It's not in superposition on the |0〉 to |1〉 axis so long as he doesn't manipulate it (i.e., send it through some gate) before reading it. It would seem necessary to assume that it's either a ZERO or ONE (agreeing with Alice's "read" of hers), although Bob must "read" his to verify this.

2. If both Alice's and Bob's qubit are truly in superposition (which we must agree they are) before either is read, what kind of language do we use to talk about how this perfect correlation comes about (assuming no subsequent qubit rotation by Bob)? What is it that instantaneously (possibly across the universe) that causes Bob's qubit to collapse to either a ZERO (|0〉) or ONE (|1〉) the instant that Alice reads hers?

Regards,
Elroy

Why is the correlation factor different to the classical expectation by a factor of √2? Is it to do with geometry?

Here's my understanding of a visual representation of the Bell inequalities. Using classical probability theory, a randomly polarized photon will pass through a polarizing filter 50% of the time. Actually, that's true under QM as well. However, we can think of these things as qubits entangled as an EPR pair.

Also, just an FYI, there are several ways to "set up" the correlations of an EPR pair. Using actual photons, the correlation will be a perfect negative correlation. However, in above posts, I've been talking about qubits in EPR pairs as if they exhibit perfect positive correlations. Please recognize that this is just a matter of "set up" more than anything else. In this explanation, I'm going to go back to EPR pairs exhibiting perfect negative correlations, because that's how the figure below presents it. Just think of this as follows: If Alice reads a ZERO from her qubit, the Bob will read a ONE, and vice-versa.

However, what if Bob rotates his qubit after Alice reads hers, but before he reads his. (In fact, I'm not sure it makes any difference if he rotates his before or after Alice reads hers. I don't think it does.)

If Bob rotates his qubit by 90° (or 270°), this will completely "break" the correlation (making the correlation zero). This will be true for both classical probability theory and QM. In other words, Bob will read completely random values of ZERO and ONE, and they will also have no correlation to what Alice reads.

However, what if Bob rotates his qubit by something other than 90° (or 270°)? This is where things get interesting. Classical probability theory states that the correlations between Alice and Bob will change in a linear fashion. That's the red line in the figure below. However, QM states that the probabilities will change along the cosine function. That's the blue line. If Bob rotates his qubit either 45° (or 135°, 225°, 315°), this is where the two theoretical predictions of the correlations are maximally different. This is the little red and blue X's on the figure.

If you'd rather think in terms of photons rather than qubits, we can imagine Alice using an up/down polarizing filter to read her EPR entangled photon. If it passes through, we can call it a ONE. If it doesn't, we can call it a ZERO. Bob however has decided to rotate his polarizing filter by some number of degrees before "reading" whether or not his photon passes through. This would be precisely equivalent to sending the qubit through some rotation gate before reading it. (Personally, my interests are in quantum computing, so I like talking in terms of qubits.)

The violation of the Bell inequalities are related to the vertical differences (i.e., the correlation axis on the figure below) between the red and blue lines. I'm not of a mind to organize all the math here, but this is where the increase by √2 of the maximal values for the Bell inequalities (thereby supporting the QM interpretation of reality) comes into play. Jilang, I hope this helps. And again, anyone please feel free to critique.

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Hmmm, I just don't seem to be able to help myself. Looking at the above figure, I find it easy to state Bell's inequalities (at least in one form).

Let's assume that Bob is always going to rotate his qubit from EPR pairs by 135° before reading it, but Alice does not. Therefore, over many EPR pairs, according to classical probability theory, their qubits should agree (asymptotically, in the long run) 75% of the time. We must remember that they will agree 50% of the time by pure chance alone. Therefore, a 50% agreement would be equivalent to a zero correlation. In other words, we can work out the math of the vertical axis of the above figure (previous post) as follows:

Correlation = (proportion of agreement - .5) x 2

Working this out, we will see that 50% agreement results in zero correlation, and 100% agreement results in one. And, we can also have negative correlations. This is just when Bob starts getting ZEROs that correspond to Alice's ONEs.

Alright, according to classical probability theory, if Bob rotates his qubits by 135°, there should be a correlation of .5 (according to the above figure). Now this is where we must bring back in decoherence. This is a maximum probability with Bob's 135° rotation. Because of decoherence (which may affect both superposition and entanglement), the correlation may be less than that. Physicists, while conducting their experiments, are constantly fighting decoherence. (This is an entire field in and of itself, quantum error correction.) It's this decoherence that makes Bell's inequality an inequality, rather than an equality.

But let's return to the problem at hand. What does QM say that the maximum possible correlation can be with Bob rotating his qubit by 135°? Taking the appropriate offsets as well as the cosine, we find that QM says the maximum correlation can be 1/√2 (or .707), which is clearly more than .5 (as shown on the figure). Therefore, any experiment that can show correlations above .5 with Bob's reading of the qubit at 135° supports the QM interpretation of reality. Also, Jilang, there's your √2. Depending on how the algorithm is stated, that √2 may appear in different forms.

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Jilang
atyy

Here is one way of seeing Bell's theorem. The arrows indicate possible causal influences in the pictorial language of Bayesian networks, which are a way of indicating the structure of models that use classical probability. A and B are the measurement outcomes, S and T are the measurement settings and λ is the preparation procedure. The arrows in the diagram are consistent with relativistic causal structure because the two possible causes of event A are S and λ, both of which are in the past light cone of A; and the two possible causes of event B are T and λ both of which are in the past light cone of B. The Bell inequality is derived assuming this causal structure. Because quantum mechanics violates the Bell inequalities at spacelike separation, no theory capable of explaining the nonlocal correlations of quantum mechanics (barring some loopholes) can respect relativistic causal structure.

It is important to note that although quantum mechanics does not respect relativistic causal structure if it is used to explain the nonlocal correlations, quantum mechanics does respect the more fundamental relativistic constraint that no classical information is transmitted faster than light.

The diagram is Figure 19 of Wood and Spekkens http://arxiv.org/abs/1208.4119.

Bayesian networks are explained by Wikipedia in http://en.wikipedia.org/wiki/Bayesian_network.

Jilang
atyy