# Bell's Inequality

#### bhobba

Mentor
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

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#### bhobba

Mentor
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

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#### bhobba

Mentor
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

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#### 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

Mentor
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

Gold Member
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.

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#### 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.

#### 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:
http://physics.stackexchange.com/questions/128848/why-would-classical-correlation-in-bells-experiment-be-a-linear-function-of-ang

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.

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#### bhobba

Mentor
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

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#### bhobba

Mentor
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:

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

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#### DrChinese

Gold Member
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

Gold Member
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.

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