Progress on Explaining Bell's Inequality

[edpell]

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

 

[jfizzix]

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.

 

[edpell]

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?

 

[Elroy]

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.

 

[StevieTNZ]

A good book you can try is https://www.amazon.com/dp/3319054724/?tag=pfamazon01-20

and this experiment in relation to faster than light communication between entangled systems: http://cms.unige.ch/gap/quantum/wiki/publications:bib:salart2008

 

[Elroy]

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.)

 

[StevieTNZ]

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.

 

[Elroy]

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.

 

[edpell]

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. ;)

 

[Elroy]

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 spatial 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.)

 

[edpell]

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.

 

[Elroy]

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.)

 

[edpell]

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.

 

[StevieTNZ]

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.

 

[bhobba]

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]

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
https://www.physicsforums.com/threads/cluster-decomposition-in-qft.547574/

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

 

[Nugatory]

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.

 

[bhobba]

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 traveling FTL it can't be used to send information, so clocks can't be synced, and SR still holds.

Thanks
Bill

 

[zonde]

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.

 

[Elroy]

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

 

[Jilang]

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

 

[Elroy]

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.

http://upload.wikimedia.org/wikipedia/en/thumb/e/e2/Bell.svg/500px-Bell.svg.png

 

[Elroy]

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.

 

[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.

 

[atyy]

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.

Why not just take quantum mechanics, choose one Lorentz inertial frame to be the preferred frame in which the collapse of the wave function is real? Since the probabilities are Lorentz invariant, observers using other Lorentz inertial frames can calculate in exactly the same way, even though collapse is not real in their frame. In other words, choose the Lorentz Aether interpretation of special relativity. Not advocating this, just considering it as a logical possibility.

 

[bhobba]

It's generally agreed that both of the entangled qubits are still in superposition if neither has been read.

Superposition is not the best way of expressing it - entangled is much better. An observation on either particle entangles it with the observational apparatus and breaks the entanglement with the other particle.

Here is the full technical detail of the inequalities:
http://www.drchinese.com/Bells_Theorem.htm

The whole thing depends crucially on the purely quantum phenomena of entanglement which has no classical analogue.

Thanks
Bill

 

[Ilja]

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.

It proves that a "hidden variable" which transports information only slower than light is not sufficient as an explanation. Thus, there has to be some hidden communication faster than light.

Or, alternatively, we simply have to give up the very idea that some observable correlations require some causal explanation in terms what really, causally happens. But in this case, "Einstein causality" as such becomes almost meaningless. Once there are no causal explanations, what is "Einstein causality" about? Thus, it is not saved, but made meaningless.

Or reduced to what could be named "Einstein correlationality" - that there should be no correlations between a free choice at A and an observation at B if B is outside the future light cone of A. But this "Einstein correlationality" is not questioned anyway, thus, in no need to be saved.

 

[Ilja]

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?

The "it cannot be explained locally" is the generally accepted language, but it is very misleading.

The problem is that they could be explained with some type of information transfer much much faster than the speed of light, but nonetheless with finite maximal speed. In such an explanation everything would be similar to what is named today "local", only the maximal speed has another value, not c but say 10000000000000 c. If the first explanation, with maximal speed c, is named "local", the second one, which is qualitatively of the same type, deserves to be named "local" too, because the word "local" in no way refers to the particular special choice of c as the maximal speed.

So what is meant with "cannot explained locally" is that an explanation requires some information transfer with speed > c.

 

[Ilja]

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.

I would object to naming this "more fundamental". Instead, I would name it "less fundamental", because it has a much more direct connection with observation.

 

[edpell]

If the first explanation, with maximal speed c, is named "local", the second one, which is qualitatively of the same type, deserves to be named "local" too, because the word "local" in no way refers to the particular special choice of c as the maximal speed.

Ilja, good point. I have never heard this before.

 

[atyy]

I would object to naming this "more fundamental". Instead, I would name it "less fundamental", because it has a much more direct connection with observation.

Fundamental from an operational point of view :D which of course is not what is being asked in this thread.

I guess two possibilities are some Bohmian version of QED, or http://arxiv.org/abs/1111.1425?

 

[Elroy]

Superposition is not the best way of expressing it - entangled is much better.

I haven't read the Dr. Chinese page in detail, but I have scanned it. However, I thought I would ask some questions of you regarding this statement.

My agenda is to attempt to formalize a language that can be used for programming and understanding quantum computers. Just to explain a bit further, regarding classical computers, there is a rather sharp distinction between the electronic engineers who design and build them, and the programmers who make them do all the wonderful things we use them for. Sure, it's not a "perfectly" sharp divide, but it's fairly clear. My interests are clearly on the programming side. I'll just assume that the quantum engineers will figure out how to create, transmit (i.e., move), and entangle qubits. I will further just assume that they will figure out how to send them through Pauli gates of any selected angle (which is little more than just rotating the laser that would "read" them).

Given this, I find it useful to think of superposition and entanglement as separate things.

For me, superposition is any qubit that is in a state other than |0〉 or |1〉. It has to do with one, and only one, qubit.

Entanglement, on the other hand, requires a minimum of two qubits (and possibly a large number of qubits). Also, superposition is a yes-or-no thing (although the square of the absolute value probability amplitudes need not be 50/50). However, in contrast, entanglement is a matter of degree (from none to perfect correlation).

(Again, we must remember that when reading pairs of qubits (entangled or not), they will agree with each other 50% of the time by chance alone.)

So here's my definition of entanglement (which I hope agrees with most others). If you have pairs of qubits (with a large number of pairs prepared and measured the same way) and you can find some axis on which to measure them such that (over the long run) they will have greater than a 50% agreement (i.e., absolute value correlation greater than zero), then they are entangled.

It's interesting to think about whether you can have entanglement without superposition, or vice-versa. Also, it's interesting to realize that both entanglement and superposition are lost "instantaneously" faster than C.

Also, if others feel like my definitions of entanglement and superposition are not at least somewhat accepted definitions, I welcome critique.

Regards,
Elroy

 

[atyy]

It's interesting to think about whether you can have entanglement without superposition, or vice-versa.

It is not possible to have entanglement without superposition. A product state is something like |01>. Entanglement means the state cannot be written as a product state in any basis*, so |00> + |11> is an entangled state. So the entangled state is a superposition of the product states |00> and |11>.

*I gave an old-fashioned definition of entanglement there. It means the entanglement entropy is not zero.

 

[Elroy]

Yeah atty, I was pretty much there in my own head, and it makes sense that you can't have entanglement without superposition.

Also, after re-reading it, I can find holes in my definition of entanglement. For instance, if we prepare a large number of qubits, all with a value of |1〉, then separate them into pairs, they will have a perfect correlation, but they won't be entangled. That actually again makes your case that you can't have entanglement without superposition. The value of Alice's qubit must be unknown (i.e., in superposition) before we can even talk about entanglement.

 

[bohm2]

The problem is that they could be explained with some type of information transfer much much faster than the speed of light, but nonetheless with finite maximal speed. In such an explanation everything would be similar to what is named today "local", only the maximal speed has another value, not c but say 10000000000000 c.

According to this paper, the hidden/private quantum signals that exist between entangled particles/systems cannot remain hidden if the speed is anything less than instantaneous:

The new hidden influence inequality shows that the get-out won't work when it comes to quantum predictions. To derive their inequality, which sets up a measurement of entanglement between four particles, the researchers considered what behaviours are possible for four particles that are connected by influences that stay hidden and that travel at some arbitrary finite speed. Mathematically (and mind-bogglingly), these constraints define an 80-dimensional object. The testable hidden influence inequality is the boundary of the shadow this 80-dimensional shape casts in 44 dimensions. The researchers showed that quantum predictions can lie outside this boundary, which means they are going against one of the assumptions. Outside the boundary, either the influences can't stay hidden, or they must have infinite speed.

Quantum nonlocality based on finite-speed causal influences leads to superluminal signaling
http://arxiv.org/pdf/1110.3795v1.pdf
http://www.nature.com/nphys/journal/v8/n12/full/nphys2460.html

The experimental violation of Bell inequalities using spacelike separated measurements precludes the explanation of quantum correlations through causal influences propagating at subluminal speed. Yet, it is always possible, in principle, to explain such experimental violations through models based on hidden influences propagating at a finite speed v>c, provided v is large enough. Here, we show that for any finite speed v>c, such models predict correlations that can be exploited for faster-than-light communication. This superluminal communication does not require access to any hidden physical quantities, but only the manipulation of measurement devices at the level of our present-day description of quantum experiments. Hence, assuming the impossibility of using quantum non-locality for superluminal communication, we exclude any possible explanation of quantum correlations in term of finite-speed influences.

Quantum Nonlocality Based on Finite-speed Causal Influences Leads to Superluminal Signalling
http://pirsa.org/displayFlash.php?id=11110145

 

[bhobba]

Given this, I find it useful to think of superposition and entanglement as separate things.

There are standard definitions of superposition and entanglement in QM. I suggest you stick to those.

They are:

1. Superposition reflects the vector space structure of so called pure states. That is if you have a system that can be in state state |a> and state |b> then it can be in a superposition of those states ie c1*|a> + c2*|b> where c1 and c2 are complex numbers. This is called the principle of superposition and is a fundamental principle of QM. It is not an axiom because it follows from something else - but no need to go into that here.

2. Entanglement applies the principle of superposition to separate systems. Suppose you have a system that can be in state |a> or |b> and another system that also can be in state |a> or |b>. If system 1 is in state |a> and system 2 in state |b> that is written as |a>|b>. Conversely if system 1 is in state |b> and system 2 on state |a> that is written as state |b>|a>. But we can apply the principle of superposition to give a state c1*|a>|b> + c2*|b>|a>. The two systems are then said to be entangled. It is a peculiar non classical situation - system 1 is no longer in state |a> or |b> and the same with system 2 - they are entangled with each other. If you observe system 1 and find it in state |a> by the principles of QM the combined system is in state |a>|b> - so system 2 is in state |b> and conversely. Observing one system immediately has told you about another due to entanglement.

This is the weirdness of entanglement - observing one system immediately tells you about the other system and conversely. The difference classically is that the principle of superposition does not hold and you don't have this peculiar relationship involving complex numbers between states. You can in fact have something similar to entanglement classically (by, for example, putting coloured papers in two envelopes - look at one envelope and you know the colour of the other) but its this complex number thing that distinguishes it. The reason you have complex numbers involved, which distinguishes it from classical probability theory, is the requirement for continuity between pure states:
http://www.scottaaronson.com/democritus/lec9.html

This is the background to my statement right at the beginning of this thread that progress has been made in understanding bells inequalities. We understand this essence of QM is the requirement of continuous transformations between pure states and directly leads to entanglement which simply can't be explained classically - in fact it leads to the overthrow of naive reality.

Thanks
Bill

 

[bhobba]

According to this paper, the hidden/private quantum signals that exist between entangled particles/systems cannot remain hidden if the speed is anything less than instantaneous:

Of course if it was merely very very large rather than instantaneous that would be detectable. What Ilja is saying is that it may be undetectable within current, or even future, experimental technique. Also if true it would likely mean there was a sub-quantum theory to which QM is simply an approximation as classical physics is an approximation to QM. It would fulfil Einstein's belief that QM was incomplete.

Such is not the only proposal along those lines eg primary state diffusion (which I suspect would also require such very very large, but not infinite, superluminal influences):
http://arxiv.org/pdf/quant-ph/9508021.pdf

Thanks
Bill

 

[Elroy]

Oh gosh, bhobba, I wholeheartedly agree. I certainly wasn't attempting to alter the definition of either superposition or entanglement, and thanks outlining standard definitions. I suppose, more than anything, I'm just developing my own way of "saying" those definitions. It's important to me that I'm able to "say" them (possibly with slightly different words but the same meaning). It's also important that the definitions are thoroughly "nailed down."

Regarding your definition of superposition, I have absolutely no argument. It's the standard...
|\psi \rangle = \alpha |0\rangle + \beta |1\rangle
...that is so often stated, where |ψ〉 is the qubit's state, α and β are complex, |α|² + |β|² = 1, and α and β are defined as probability amplitudes.

However, entanglement would seem to be even more complex than what you've stated. (Shucks, I'm being called away but will return tomorrow. I'll say a bit though.) I suppose the primary thing I'd like to say is that it seems that entanglement can be framed in terms of correlations among the qubits as well as superposition across the system of qubits. I did a somewhat poor job with my definition in post #32, but I'll give a more formal definition (in terms of correlations) tomorrow.

Also, I think a definition of entanglement also has to address the situation where qubit #1 is measured on one axis (say Z), but qubit #2 is measured on some other axis (say 45° from Z in either X or Y). This would be the equivalent of rotating qubit #2 by 45° before "reading" it.

Take Care,
Elroy

 

[bhobba]

Also, I think a definition of entanglement also has to address the situation where qubit #1 is measured on one axis (say Z), but qubit #2 is measured on some other axis (say 45° from Z in either X or Y). This would be the equivalent of rotating qubit #2 by 45° before "reading" it.

The standard definitions I gave fully explains Bell. There is no need to go any further.

Dr Chinese's superb link on it gives the detail - and even with easy math:
http://www.drchinese.com/Bells_Theorem.htm

It a result of that damnable principle of superposition with complex numbers.

Thanks
Bill

 

[Jimster41]

I'm reading the book mentioned above. "Quantum Chance" Nicholas Gisin (been mind boggled by EPR and Bell for a long time).

I am a bit confused about the wave collapse between entangled entities. I worry I've been carrying around an incorrect picture of a two slit interference pattern shining on the wall of a dark classroom - The pattern is made up pointillist-like of dots representing samples taken by the screen periodically over some period of time say like over the 6 hours between noon and 6pm. To keep the non-locality notion handy and ready for rumination, I have always held onto the explanation - "The photons that were sampled at 2pm were apparently interfering with the photons sampled at 3 pm"

Now I think I might have that wrong. It really was that every sampling at time t throughout the afternoon, the photon sampled through the left slit was interfering with (uh, itself?) going through the right slit. Still crazy but the entanglement doesn't cross time. I always remembered it as the pattern wasn't visible until the parade of entangled photons were through the slits over time. Maybe I'm just getting confused by the fact that the wall wouldn't look very interesting after only a couple of pairs were sampled - even though the pattern that would eventually be painted was there with each sample. But I think I've asked myself about this and answered myself with "how would you see the pattern with only one pair. It would just be a dot on the wall. Maybe if you had drawn an expected wave interference pattern on the wall you'd notice it was on a trough or a valley but what would that tell you"

I got thrown off into this possibly justified worry again, just now by someone's thought experiment above where Bob reads his detector or let's say flips his "entangled photon pair 1" coin and gets "heads". He then drives over to Alice's house where the the other "entangled photon pair 1' coin is (which hasn't been flipped?) and bets Alice that when she flips it she'll get "heads". Alice still gets to toss her coin right? Or is it laying there dead, stuck to the floor, with only a "heads" side to be had, because Bob already flipped it's entangled twin? And it did this to iteslf, flipped itself, automatically the instant Bob flipped his?

Is there such a thing as an entangled wave collapse that can be smeared across some frame of non-zero time? Or is that kind of the special thing about entangled wave collapse and time, that somehow, though it is not helpful to us in synchronizing our reality across space-time the entangled wave collapse is THE or at least A fundamentally simultaneous thing. I just don't see how if Bob is standing there, obviously having flipped his half of the tangled coin pair and driven all the way over, Alice still has a coin to flip.

I'm totally excited to find this whole PF site. I find it's tough to be alone with this stuff, only books and such. It's just too strange and interesting not to talk about, ask questions about. And my wife's patience with it is... way past flipped.

 

[Elroy]

Hmmm, I really should take this time to mathematically formalize entanglement, so I don't get into (more) trouble with bhobba. But I'll take a shot at some of this.

In my mind, the two-slit experiment is more about superposition than entanglement. It just illustrates the oddities (wave-particle-duality) of a photon quite well. In fact, I've actually done this one here at home. You can take a piece of glass and thoroughly stain it with candle smoke. Then take two very thin razors (like old-fashioned two-sided razors), put them together and then run them down the smoke on the glass (making two very small slits in the smoke). Then, take any laser pointer and shine it at the slits and let it go through onto another surface. You can see the sinusoidal pattern in the light. It's quite cool.

And Jimster, yes, I've always interpreted this as a photon interfering with itself. It has nothing to do with one photon interfering with another photon. In fact, there have been versions of this done where a single photon is emitted every second (temporally very slow in terms of photons). Over time, the interference (sinusoidal) pattern still emerges (so long as there's no way to know which slit the photon went through).

I think I'll hit "post" and make other replies to your comments in another post, so that others don't get woven in.

 

[Jilang]

Elroy, I read your post #22 with interest. Is the difference between the straight lines in the graph and the curved line related from going from 2 dimensions to three? Angles in three dimensions tend to be less than their projections in two dimensions etc.

 

[atyy]

Hmmm, I really should take this time to mathematically formalize entanglement, so I don't get into (more) trouble with bhobba.

There is an old fashioned definition of entropy: not a product state in any basis. This can be formalized using the entanglement entropy.
http://www-thphys.physics.ox.ac.uk/people/JohnCardy/seminars/statphys.pdf

There are other measures, and some proposals are basis dependent.
http://arxiv.org/abs/quant-ph/0507189

 

[Elroy]

Jilang,

Please be sure to read my post #23 as well. You've posted an excellent question and I hope to come up with a well formulated answer that isn't too mathematical, and is also intuitive (at least as intuitive as QM can be).

However, in the interim, here are a couple of good threads:
https://www.physicsforums.com/threa...variables-imply-a-linear-relationship.589923/
http://physics.stackexchange.com/qu...-bells-experiment-be-a-linear-function-of-ang

Also, the Wikipedia page is good: http://en.wikipedia.org/wiki/Bell's_theorem

Regards,
Elroy

 

[Elroy]

I'd like to say a bit more about the two-slit experiment. I think we all agree that a photon is the smallest quanta of light that can be "detected". Anything smaller is rather theoretical, and this is where we must think of waves (rather than particles). The reason the sinusoidal pattern of photons comes about is because each photon (as a wave) goes through both slits simultaneously and then interferes with itself before hitting a back-surface:

What's quite fascinating is that we can put a photon detector at each slit (that minimally "observes" which slit it goes through, still letting it pass), and this destroys the sinusoidal pattern.

One of the most counter-intuitive things about this is that forcing the photon to be a particle at the slits actually increases its options as to where it goes. If the photon is observed going through "one slit or the other" (and not both), then it can hit the darkened stripes on the above figure. Whereas if it has both options, it can not.

Said differently, if we have just one slit, it can get onto the darkened stripes. Two slits (two options) creates areas of decreased probability (with zero probability at the center of the darkened stripes), whereas only one slit allows the photon to go to those areas. Very counter-intuitive compared to classical Newtonian (or even Einsteinian) physics.

 

[bhobba]

I am a bit confused about the wave collapse between entangled entities.

That may be because, popularisations not withstanding, QM does not require collapse. What happens in EPR is you have entangled particles - as soon as one is observed it becomes entangled with the observational apparatus and is no longer entangled with the particle. Collapse is not necessarily involved.

I worry I've been carrying around an incorrect picture of a two slit interference pattern shining on the wall of a dark classroom - The pattern is made up pointillist-like of dots representing samples taken by the screen periodically over some period of time say like over the 6 hours between noon and 6pm. To keep the non-locality notion handy and ready for rumination, I have always held onto the explanation - "The photons that were sampled at 2pm were apparently interfering with the photons sampled at 3 pm"

Interference as in wave-particle duality (which is wrong anyway - it was overthrown when Dirac came up with his transformation theory at the end of 1926 - and likely sooner - but certainly by then) isn't really what's going on in the double slit experiment:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

What happens is each slit acts as a position measurement which means there is an uncertainty in direction after the slit. We have two slits so the state of the particle after the slits is a superposition of the state at each slit as per equation 9 in the above paper due to the symmetry of the situation. The interference pattern is a result of the uncertainty principle and superposition principle.

Is there such a thing as an entangled wave collapse that can be smeared across some frame of non-zero time

I think you are a bit confused about some fundamental ideas and need to read a good book on QM. Unfortunately that will require a bit of math:
https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

There are also some video lectures:
http://theoreticalminimum.com/

If you are dead against math at all check out the following - but your understanding will not be as good:
https://www.amazon.com/dp/0473179768/?tag=pfamazon01-20

Thanks
Bill

 

[bhobba]

I'd like to say a bit more about the two-slit experiment. I think we all agree that a photon is the smallest quanta of light that can be "detected". Anything smaller is rather theoretical, and this is where we must think of waves (rather than particles)

Both theory and observation say the same thing - the smallest quanta of light is the photon. It's the quanta of the underlying EM field. But what field quanta are is not simple - unless its from a textbook on Quantum Field theory its almost certainly wrong - and QFT is a difficult advanced subject - although some good books at the undergraduate level are appearing:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

It is doable after the Susskind books I mentioned in my previous post.

You need to forget this wave particle duality stuff - it was overthrown in 1926 by Dirac - you can do a search on physics forums and find many posts giving the detail. Also see the FAQ:
https://www.physicsforums.com/threads/is-light-a-wave-or-a-particle.511178/

In my previous post I gave the correct explanation of the double slit experiment - it's also very elegantly explained by Feynmans path integral approach - but wave particle duality is a left over from De-Broglies hypothesis that was simply a stepping stone to the correct quantum theory and was quickly done away with.

Thanks
Bill

 

[DrChinese]

One of the most counter-intuitive things about this is that forcing the photon to be a particle at the slits actually increases its options as to where it goes.

The interference pattern shows fringes, while the other pattern does not. So you may have that backwards, not really sure what you mean.

However, you do not need to force a photon to be a "particle" (using that analogy) at the slits to eliminate the interference pattern. Place polarizers at both slits. If they are aligned parallel, there WILL be interference. If they are oriented perpendicular, there will be NO interference pattern.

 

[Elroy]

The interference pattern shows fringes, while the other pattern does not.

I was just trying to convey the idea that, without two slits (with only one slit) (classically, seemingly like fewer options for the photon), it would then be able to get to the center of the troughs in the above sinusoidal (interference pattern) image.

you do not need to force a photon to be a "particle" (using that analogy) at the slits to eliminate the interference pattern

And yes, I didn't mean to imply that detecting the photon going through the slit was the only way to eliminate the interference (sinusoidal) pattern. Simply having only one slit also does it. I'm sure others can come up with a variety of ways to do it.Jilang asked a good question in post #42, which rather directly relates to Bell's inequalities. I see that Dr. Chinese has a page on this (which I admittedly haven't thoroughly read). It seems that there should be a straightforward answer to this question, with an appropriate logical explanation. The two-slit stuff was a bit of a distraction, and I wouldn't mind if we pushed that to another thread.

I'm just working on the easiest way possible to explain the empirically validated violations of Bell's inequalities, showing how the linearity should be replaced with the cosine function when we're dealing with entanglement.

(But got to go for the evening. Y'all take care.)

 

[DrChinese]

I'm just working on the easiest way possible to explain the empirically validated violations of Bell's inequalities, showing how the linearity should be replaced with the cosine function when we're dealing with entanglement...

There really is nothing which is linear at any level that matches the graph. The graph shows a hypothetical local function which most closely approaches the quantum prediction (for entangled cases) and also provides for so-called "perfect" correlations. There aren't any serious models that do this (as they immediately fail to explain Malus). So there is nothing to "replace" per your comment. In other words, forget the linear portion, the graph is just illustrating a concept. You could actually replace it with many different shapes (all of which would be even more ridiculous). For example, a common alternative is: 1/4+(cos^2(theta)/2) which varies between .25 and .75. This matches Malus but is further away on entangled pairs.

If you want a visual, there is one class that most closely matches experiment. Most people reject these because causality is not respected. These are the time symmetric group of interpretations. In these, you have the following 2 key elements:

a) Both the past and the future are elements of the experimental context. So Alice's setting and Bob's setting are both part of the equation when particle pairs are created, even though they are set in the future.

b) Otherwise, locality (c) is always respected. Despite the "non-local" appearance of entangled pairs (which I don't dispute in any way): in these interpretations, everything is cleanly connected by local action.

One of the advantages of this visual is that it naturally explains entanglement which occurs after detection. Sophisticated experiments allows after-the-fact entanglement (hard to believe but true - you can entangle particles that no longer exist). Such is not natural in many other mechanistic explanations.