Progress on Explaining Bell's Inequality

[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.

 

[Jimster41]

Thanks for that great list of resources! I have ordered a couple of the books, and as of this morning I am planning to take one of those Susskind courses.

I'm not against math. I love math. But, I have no talent for it.

The other image I have carried around is of this almost transparent octopus looking Probability wave-function spread between the laser and the slits and poking through both slits hovering right in front of the screen, but not touching it - just yet. Then when one tentacle touches the screen - all of a sudden you can see the octopus - how's that for an understanding! ;-)

 

[bhobba]

I'm not against math. I love math. But, I have no talent for it.

That's not important. What is more important is perseverance. Take your time and don't give up - its not a race. Post here with any queries - that's what this forum is all about. Soon you will have an understanding way beyond popularisations.

Thanks
Bill

 

[Elroy]

Yes, I'd like to wholeheartedly second bhobba's above comment. For me, it's important that we can develop an (accurate) "conceptual" understanding of these phenomena, possibly with a great deal of use of "everyday" language. Everyday language has the problem in that it's sometimes slippery around the edges, so we do need to be concise with our definitions. However, for me, that's what a great deal of this is about: The development (and acquisition) of the precise language of physics.

Also, I think this approach is important for these concepts to "stick". If they are "purely" abstract, our minds seem to struggle to attach them to the rest of our knowledge. However, this is where a great deal of the struggle comes in. As far as can be understood, all of our "concrete" perceptions are in three spatial dimension and one temporal dimension. As an example, relativity often talks about Minkowski space as a four-dimensional static visual space (including time) to talk about things. It collapses the dynamic dimension of time into a static spatial dimension.

In this depiction, the actual three observable spacial dimensions are only given two dimensions, so the third can be given to time. (Just FYI, the cones are where "causal" events can take place, assuming speed of light limit is obeyed.) To make sense of this depiction, we must "stretch" our minds to recognize that the 2D space is truly attempting to represent the 3D spatial space in which we live. That is why it is labeled a "hypersurface".

Regarding math, we can quite easily represent vectors (or even tensors or spinors) in space (spacial, temporal, or otherwise) of as many dimensions as we like. However, we do often lose the ability to "grasp" the underlying concepts. Furthermore, math can be wrong. In the end, math is yet another (hopefully more accurate and concise) language in which to outline this stuff. As with any other language, it can tell lies. I'm tempted to also include the math-thought-reality tri-image developed by Penrose, but I'll leave it to others to explore that.

p.s. I'm still working on a pictorial way to represent the transition from classical probability theory to the correlations observed with quantum superposition and entanglement (focusing specifically on complete entanglement with EPR pairs, as a first pass). I'm having fun with it.

 

[bhobba]

Relativity is a good example of math giving deep insight.

Check out the following derivation of the Lorentz transformation:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

It's premises are so plausible and fundamental no-one would care to doubt them. In fact we see the speed of light thing in SR is simply fixing a constant that naturally occurs from other more basic symmetry considerations. Yet it has these startling consequences. Its the power and beauty of math.

Math is not about long boring calculations - it about concepts and their consequences.

Thanks
Bill

 

[Jimster41]

Nothing in that Lorentz transform derivation that I'm unanle to follow. I am up on that level of math. And I believe it all. It's just not very portable or compact, at least for me, in terms of trying to get to the next part, or having something I can walk down the sidewalk thinking about. Elroy's 3d space-time diagram (that's what they are called right?) on the other hand, I was already pretty comfortable with...

Reading Penrose' Cycles of Time. I feel like I almost get Confirmal Diagrams, but the strict Conformal Diagrams "don't stick" quite yet.

I did read Dr Chinese' page and I'm enjoying Gisin's "Quantum Chance" Maybe where my question about Alice and Bob went wrong... Gisin's uses The Joystick Metaphor rather than the "coin toss". My confusion is with regard to how a decision made far away/earlier by Bob's Joystick seems to me to be determining what's going to happen to Alice. Bob is standing there betting her what's going to happen (he knows) just to represent the discomfort implied (I'm having) from non-local determinism? Once Bob has moved his joystick, Alice's outcome is set, and her joystick doesn't do what she thinks it does anymore - Or does she always have to move her joystick exactly when Bob does - for the metaphor to be consistent with the math? I get that Bob couldn't control the result of his joystick - it's just that whatever he got the instant He jumped the gun is now some non-local thing playing a deterministic role in Alice's present and future? Is it just the fact that the non-local influence causes an outcome that seems just as random, as far as Aluce can tell, as her Joystick would have given anyway? This is why I wanted to have Bob there gloating, just to drive home the idea that what she thought was random, and looks just like the usual random stuff, from Bob's perspective, which represents another place, in the past, is determined.

Well now that I say it like that, what could be less weird. But then it's an influence out of all causal norm. As far as Alice knows (can tell) nothing causal has happened to her joystick to make it deterministic, rather than random

Crap, what a headache.

In entanglement experiments, I vaguely recall hearing the phrase "delayed choice" is that all this is? Anyway just trying to grok the concept at some level I can enjoy in daily life - which of course may not be possible...

I realize I have a surprising and possibly ridiculous notion that entanglement is somehow a significant, frequent, dispersed, even ubiquitous thing out there. Which is why Bob's gloating is sort of disturbing. I can imagine this is not at all interesting if Bob and Alice's joysticks are so rare, co-located, and short lived, as to be pretty much irrelevant phenomena to existence.

 

[Jimster41]

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.

I totally need a picture. The above (I hadn't understood this post at all the firsttime I read it) is exactly what I mean I think. What I am amazed and confused by. Strange how the sentence "Both the past and the future are elements of the same experiment" Gives me a connotation rich image I can sort of hold onto to represent the meaning the amazing math has uncovered? I just hope it's sort of correct...

So now I would love to have just a couple of more pieces squared away.

Is there any sense in which entanglement is significant feature of space time evolution as we experience it? Or is it an utterly fleeting and rare exception? Or do we not know?

Does the bizarre non-causal a-temporal sounding statement a) above turn out to be utterly innocuous because the "experiment" always provides random results - leading to a conclusion like "well the future is already set, but the result is random". In which case what's the difference between random and uncertain? I'd guess it's about 1 big beer.

But then random seems unsatisfactory, if entanglement is a phenomenon that is involved in our evolution? this feels almost completely fuzzy, but I get this icicle in brain that's asking "where does all the structure come from"

 

[Jimster41]

Someone asked if the photon probability wave front hit the slits at the same time. An innocuous question... That as I try to answer it for myself has me pretty much confused.

Position was uncertain
Can't go faster than c
It was entangled with all the stuff around it forward and backward in time and out to freaking infinity but that matters not because baseballs are made of quantum sh--- and we can manage those pretty good

 

[bhobba]

It was entangled with all the stuff around it forward and backward in time and out to freaking infinity but that matters not because baseballs are made of quantum sh--- and we can manage those pretty good

Where are you getting this from?

Here is a much better analysis:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Thanks
Bill

 

[Jimster41]

Where are you getting this from?

Here is a much better analysis:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Thanks
Bill

Thanks, it looks good. I am pretty weak on the bracket notation, but it's sort of reads intuitively. I understand the setup and the three cases, seems very helpful to break it down that way. I'm going to study it for awhile.

The math defines some pieces of my limit. $e$, $i$ and $\pi$, combinations thereof especially ${ e }^{ -i }$. I have an easier time with Planck's constant of Position and Momentum Relation and Boltzman's constant of Energy Temperature relation. Not that they aren't utterly puzzling. But they don't break my read of equations the way $e$, $i$ and $\pi$ do. I wonder often if you guys really "feel" or "see" those when you read them. I imagine you do but maybe that's not true.

Trying to read this has already been helpful because I remembered this morning how in my DiffEq class and then later in Signals and Systems, we learned how Fourier's theorem shows you can create arbitrary signals by adding together sin waves written in that ${ e }^{ -i }$ notation. Fourier really stuck, even though I was and am still baffled by how you use ${ e }^{ -i }$ to make sin waves. It's the periodicity of -i or something like that and then $\pi$ gets in there with the wack-ness of the very circle itself. I need to refresh my memory on it. So do I understand correctly that those terms represent the points on Schrodinger's Wave Equation, for purposes of integrating over it's point-wise interaction with the... Eigen things...

Eigenstates and Eigenfunctions. I made the grade in Linear Algebra but frankly it was frustrating... because I walked away feeling like it was just dutiful plug and chug, doing the homework, taking good notes. I have almost no 'feel' for what Determinants, Eigenstates and Eigenfunctions mean as operators. They turn matrices into scalars, and ... functions, or vectors of scalar points, or functions of points. I need to drill into that one. It's a real obstacle to reading math.

I can pick it up later with the linearity of the QM operations

So thanks this is helpful. I am still curious at the end of the day about the philosophical implications of this stuff - and I wonder sometimes if maybe you all are saying there really aren't any. Or is it more that it gets weirder the more precise one's understanding?

Sorry for going on and on, and I didn't mean to hijack this thread.

 

[bhobba]

So thanks this is helpful. I am still curious at the end of the day about the philosophical implications of this stuff - and I wonder sometimes if maybe you all are saying there really aren't any. Or is it more that it gets weirder the more precise one's understanding?

There are philosophical implications all right - just not the things you read in some populist accounts eg the overthrow of naive reality by Bells theorem - but not rubbish like conciousness created reality and other mystical twaddle found in trash like What The Bleep Do We Know Anyway.

Thanks
Bill

 

[Elroy]

Alright, I'm still on the idea of straightforward (and visual) explanations of the violations of Bell's inequalities, and how they illustrate the quantum weirdness. I'm going to lay out some pieces here, and open them up to discussion.

Bell’s inequality is often stated as

ρ(a,c) – ρ(b,a) – ρ(b,c) ≤ 1,

where the ρ (rho) values are correlations between events. However, this inequality can be stated in a somewhat simpler form as a straightforward probability problem. Let’s say we have a coin. (We’ll talk a bit later about whether or not it matters that it’s “fair”.) We’ll let “tails” represent a ZERO (0), and “heads” represent a ONE (1). Furthermore, let’s assume that we’re going to flip it three times. The following table provides all the possible (eight) outcomes:

Flip#1(a)  Flip#2(b)  Flip#3(c)
  0          0           0
  0          0           1
  0          1           0
  0          1           1
  1          0           0
  1          0           1
  1          1           0
  1          1           1

We will assume that the event we are interested in is “heads” (a ONE). Let’s identify Flip#1 as event “a”. We will also call Flip#2 event “b”, and Flip #3 event “c”. Furthermore, I’ll use the exclamation point (!) to indicate a “NOT”, such that !b would indicate the NOT b event (or that b didn’t happen, or that we didn’t get heads). I will also use the ∧ symbol to denote a boolean AND. This simply means that both events happened.

Now, I’d like to define three separate possible outcomes for our three flip events:

Outcome#1: a ∧ !b (a happened, b did not happen, and c doesn’t matter)
Outcome#2: b ∧ !c (b happened, c did not happen, and a doesn’t matter)
Outcome#3: a ∧ !c (a happened, c did not happen, and b doesn’t matter)

Let’s go back to the above table of possibilities. We find that the outcomes are identified as:

(a)  (b)  (c)    Outcome#1  Outcome#2  Outcome#3
0    0    0
0    0    1
0    1    0                   *
0    1    1
1    0    0        *                     *
1    0    1        *
1    1    0                   *          *
1    1    1

We can now imagine a situation where we do this three-coin-flip experiment over and over, each time flipping the coin three times. We count (N) how many times each of our outcomes of interest occur, recognizing (according to the above table) that there will be occasions where more than one outcome happens during a single three-flip experiment. With this information, we can state the following inequalities:

N(a ∧ !b) + N(b ∧ !c) ≥ N(a ∧ !c)

(Be sure to think this through. If we trust probability theory, and common sense, this has to be true.)

This is self-evident by the fact that (a ∧ !c) cannot occur unless either (a ∧ !b) or (b ∧ !c) also occurs. It’s straightforward to turn each term in the above inequality into a proportion, by simply dividing by the overall N (number of times we did the experiment). This transforms the inequality into:

p(a ∧ !b) + p(b ∧ !c) ≥ p(a ∧ !c)

This is actually one form of Bell’s inequality, and I hope everyone is somewhat comfortable with the above outlined arguments. Now, I want to extend this argument to something we can rather easily generalized to randomly linearly polarized photons. (For this argument, let’s ignore the cases of circular and elliptical polarization, but just FYI, they do not invalidate any of the arguments.) We might say that our photons are being emitted one-at-a-time from some emitter in direction z, and that they are randomly (linearly) polarized at some angle in the x,y plane, the plane is orthogonal (normal) to the direction of progression (z).

For visualization, let’s say we have a magic (golden) disk that emits brass bars (at the speed of light, if you like) once a second at random angles (orthogonal to progression):
https://lh3.googleusercontent.com/JCEqWeKxz5W4mS64Hubphrn7PATHAYULJS2WPs0pC6vyHFS8yCaJHguxUtNL86xjtZngJpA9aCA=w1709-h719 https://lh6.googleusercontent.com/xXmIk1ydQSSRMJlje6h1S489_pb3yaHjOB3MauhyhXwX3w4QtnNRaS6-QuQmy5rMKZznGdeRlmI=w1709-h719 https://lh4.googleusercontent.com/y-cyBHA72zR2GqNHAfua1GZYIgo_iQ3SXQM1bS5Caw2nUSi1ATcAenTWq6E45MtRUCZeJlAOXkA=w1709-h719

Let’s further assume that we have three “events” (a, b, and c) that we wish to test (very much like flipping our coin. We would like to “test” whether or not our bar passes these events. When placed orthogonal to the angle of progression, we would like to see if our brass bars will pass through disks that look like one or more of the following:

https://lh4.googleusercontent.com/EJVCSCTZqMungEtfWQGIzYuvYXadaQUCeBNlLXvHIOYo5WTwbu9RFTNhPlxxD8Bmoo0i91Wa9Yw=w1709-h719
https://lh4.googleusercontent.com/ibJ3BxDSeWGjDdDBAgNtWxLkblDA0qfprFyYnVbEDxH-KeAShTrf2cVbr6lHr5fbrrgjBt9VSpU=w1709-h719
https://lh5.googleusercontent.com/kGqgB-rOWOUBUieVsAWRVbfpOlP5wtpuSAzX3zlsegGcN7hgwECtqMy9XW3eIY_c8b5byg3hrnI=w1709-h719

The following is an example of the copper rod passing test a (where the z axis is now pointing directly into the picture).

https://lh3.googleusercontent.com/QKwuUMu5RCpqZeUXJXdVWW86K9EK1392URjgZj5rWWMqOveygqTrb4XCGGdGCrR7AqN8H-UO4r0=w1709-h719

Now, to illustrate Bell’s inequality, we can stack our tests, and we’d have something like the following:

https://lh4.googleusercontent.com/YelJ6_aKUfjaMhlr4ejLn6mxvz45PbHGrvDjT_BEhm3k_xbfKdpimBojApk_kwNbcymax0Acr2s=w1709-h7199

We can now imagine an emitted rod that passes a (red) but fails (bumps into) b (green). If the rods are randomly “polarized”, then the proportion of the time that will happen is the area of green we can see compared to the entire circle. This works out to be a linear relationship. It is simply the difference in angle between event a and event b, which is 22.5°, divided by 360°. This works out to be .0625. So, outcome#1 will happen about 6.25% of the time.

We can also work out outcome#2, (b ∧ !c) and not caring about a:
https://lh4.googleusercontent.com/3j848Uwfe9Lv5MdZtdhxx3SONjlDYQvk1BtAsd3iuhEu9pV85QKYxcmwssOg4kRnxrUU9U1IbPI=w1709-h719

This is also a 22.5° difference, which will also happen about 6.25% of the time.

Therefore, returning to Bell’s inequality, outcome#3 cannot happen more than 12.5% of the time.

Bell’s inequality:
p(a ∧ !b) + p(b ∧ !c) ≥ p(a ∧ !c)
.0625 + .0625 ≥ p(a ∧ !c)
.125 ≥ p(a ∧ !c)

Now, let’s look at the (a ∧ !c) situation with our disks (outcome#3):
https://lh5.googleusercontent.com/yxrbGCA23TLMbsmRF9ou3ZsslxtGKP6gexh54NHpBBScj6djp_mMfvg4yrqAnk5O-UC81JG2gYM=w1709-h719
This time, the “exposed” blue (c) area represents a 45° difference in the disks. 45°/360° = .125, just exactly on the border of still being within Bell’s inequality. In fact, it’s the equality version of Bell’s inequality.

Therefore, according to everything I’ve presented, Bell’s inequality holds.

I’ll stop here until another post, but I can absolutely tell you that a randomly linearly polarized photon will get through event a .5 proportion of the time, and a photon will get through event b after getting through event a .1464 proportion of the time (sin²(22.5°) ). Multiplying these together (.5 × .1464) we determine that .0732 is the proportion of observing outcome#1.

The same logic can be worked out for outcome#2 (occurring .0732 proportion of the time) while ignoring event a altogether.

Now this is where things get interesting. Still using photons, it can be shown that outcome#3 (a ∧ !c) will occur with a proportion of .25. We work this out with the knowledge that the photon will get through test a with a .5 proportion. After passing a, it will get through c .5 proportion of the time (sin²(45°)). Multiplying these together, we get .5 × .5 = .25.

Plugging these into Bell’s inequality, we get:

p(a ∧ !b) + p(b ∧ !c) ≥ p(a ∧ !c)
.0732 + .0732 ≥ .25, which is NOT correct. Bell’s inequality is violated.

I’ll explain this more in a subsequent post if there are interesting posts by others. It should be recognized that this does NOT involve EPR pairs. It is a single photon experiment that illustrates a violation in Bell’s inequality in the quantum world.

There is a fairly straightforward “loophole” that people have used to attempt to explain this violation. Can anyone explain it? I’ll give a tip. It has to do with distinctions in what happens with measurements in the classical world versus the quantum world. If interested, I’ll explain it.

And, if interested, I'll carry these explanations forward into the EPR experiments that conclusively show that local causality (locality), even when allowed to travel at the speed of light, can not explain empirically observed quantum phenomenon.

 

[Jilang]

Elroy, I have noticed that these inequalities can be expressed and understood nicely in the form of Venn diagrams.
(A+B+C+D)> (A+D) etc.
I also notice that if you take the square root of each of the terms the inequality still holds.
In your example 0.0732 + 0.0732 > 0.25 is violated, but 0.27 + 0.27 > 0.5 holds.
The difference between the classical and the quantum seems to arise due to the probability being the square of the amplitude. If the amplitudes are considered in the Venn diagrams instead there seems to be no issue. I am wondering if the difference arises due to the equating of the probability with some inherent property rather than equating the amplitude with that property?

 

[Elroy]

Jilang, your idea of taking the square root is interesting, but we can't just do arbitrary mathematical manipulations. In other words, all of our equalities (or inequalities) have to be grounded in empirical experimental evidence. Sure, we can set up mathematical models upon which to form hypotheses, but we must still state how these mathematical models would be empirically (experimentally) tested.

Hopefully, in post #61, I outlined how a bar emitted from a "magic disk" at some angle (orthogonal to direction of progression) would work out to Bell's inequality. In fact, it would be fairly straightforward to do an experiment like this. We could have some motor that spun and then slowed to a stop such that it stopped at completely random orientations (to the axis of the motor). We must also imagine a bar attached orthogonally to the motor's axis. Then, once stopped, the device ejected the bar directly away from the end of the axis.

Next, it came into contact with our red (test a), green (test b), and/or blue (test c) plates. We might want to do this in space, or have the axis of the motor pointing straight down, so that gravity didn't mess us up. However, hopefully, we can see that it could be done.

Then, after "dropping" many bars, we would calculate up our Outcome#1 (a∧!b), Outcome#2 (b∧!b), and Outcome#3 (a∧!c) probabilities, all empirically derived. In this case, over the long run (asymptotically) we would see that Bell's inequalities would hold. Furthermore, we would see that changing our angles would change the probabilities in a linear fashion. In general, it's always just dividing the difference in angles by 360° to find the individual terms of Bell's inequality.

Now, regarding the quantum (photon) situation, we again rely on empirical observations of experiments, and then attempt to derive mathematical "models" of these results. I'll focus on the Outcome#3 (a∧!c) situation because this is the one that's possibly most interesting. Experiments tell us that photons do not act like our brass rods, and this is where things get interesting. Initially, we might imagine our photon as having random linear polarization approaching a vertical polarizer. Experimentally, we know that 50% will get through. This actually is just like the brass rod. However, when a photon is "measured" (tested, test a), it is simultaneously re-oriented to have precisely vertical polarization, even if the polarization was initially off from vertical. This is the whole idea in QM that things can't be measured without also "changing" them. Therefore, once a photon is measured as to whether or not it's vertically polarized, after the measurement it will be precisely vertically polarized (if it passes the test).

Now test c is actually a test of whether or not the photon is polarized at 45°. Classically, we go (45° - 0°)/360° to get a .125 probability of (a∧!c) (Outcome#3). However, this isn't how things work out in the quantum world. Empirically, it has been shown that a photon of previously "known" polarization will pass through a polarization filter of a known angle rotation from the "known" polarization exactly 1-sin²(θ) of the time, where θ is the difference in angles between the "known" polarization of the photon and the angle orientation of the polarization filter.

Therefore, we know our photon will "pass" our test a .5 proportion of the time. And now, using our 1-sin²(θ) rule that has been empirically verified in many experiments, we know that the photons who passed test a (and are then vertically polarized) will "pass" our test b 1-sin²(θ) = 1-sin²(45°) = .5 proportion of the time (and therefore "fail" test b .5 proportion of the time). Therefore, our outcome#3 is .5×.5=.25.

The most interesting thing is that this .25 is different from the classical .125 proportion. In fact, it's doubled. Photons will meet the criteria our proposed outcome#3 at a rate that is twice that of our brass rods. This is precisely due to the fact that "measuring" photons (having them pass through filters) appears to alter them, whereas we assumed that measurements on our brass rods did not alter them (specifically, alter their angular orientation).

Again, this isn't just playing around with math. This is using math to model empirical/experimental findings.

Regards,
Elroy

 

[Jilang]

Elroy, it seems we have all the experimental evidence already and the maths for it. It's the physical mechanism that's missing. Are you suggesting that the act of measuring in some way rotates the plane of polarisation?

 

[Jimster41]

Are you suggesting that the act of measuring in some way rotates the plane of polarisation?

Just trying to understand this myself (to the degree it is possible). I just started Russkind's "Quantum Mechanics: The Theoretical Mininum". In the opening chapters he goes through the difference between classical and QM expectations from measurements of polarization - and I got, for the first time, a mental cartoon of the "Preparation" step, which as I understand it, is irrelevant in the classical expectation, but is inextricably tied to the experimental results in the QM case. So my hip-shot answer to the above was "exactly!". I hope this is right (or at least not wrong). I now have a mental cartoon of the measurement step of a sequence of experiments interacting with the result sequence in a bidirectional way, which I didn't have quite before. Still staring at it.

I'm also reading Gisin's book. On page 88 in a section titled "Alice and Bob each Measure Before Each Other" he goes into the questions when Bob and Alice are in different rest frames. what does simultaneous measurement mean, and if measurement is not simultaneous then who measures first and which result (Bob's or Alice's) determines the other? I haven't read (or tried to read it) but here's a link to the paper they did on the experiment to test the case: http://arxiv.org/abs/quant-ph/0210015. I gather QM won, but I'd be lying if I said I really get what that means - except that GR and QM are in conflict? (Later Edit: No that's wrong I realize, going back some pages in Gisin p50. GR is not in direct conflict with QM because no meaningful information can be transmitted" via the "non-local" thingamajig. He goes on to outline, somewhat euphemistically, their "peaceful coexistence".)

In the next section Gisin goes on to talk about "Superdeterminism and Free Will", so I guess that's a hint.

Which confirms for me somewhat, that the question I was failing to ask coherently, or in proper terms (Bob leering at Alice because he's predetermined the results of her "experiment"), was at least a good one after all (which is a relief). Wish I felt like there was an answer to it though! I should add that I love Gisin's answer, which I gratuitously paraphrase as "If there is no free will, why are there questions?" ;-)

So here's another one that's bothering me today. And I recognize I need to better understand the difference between superposition and entanglement. But what happens to a QM Probability Distribution when the space it occupies is expanding (when there is a + Cosmological constant?). Is the new region "entangled", in "extended superposition", the exact same thing as it was prior to expansion? Is that an incomprehensible (unintelligible) question? Or is it a comprehensible one with a simple answer or, though comprehensible, one that just goes right off into philo-puzzle-land?

Elroy, I got from your description of the sort of simple coin toss model, the same sensation of understanding I got from Gisin's description - which I have, I hope not too incorrectly, taken away as "No two independent random things can be coordinated more than a certain % of the time - just because of what random means - which is, out of two ways, a random thing is one way half the time and the other way half the time". I haven't seen the Venn diagrams of Bell's Inequality. I'm betting I would like them.

 

[Elroy]

Ok, I'll comment on Jimster's post a bit first. Wow, truth be told, I'm still trying to get rock solid on all the implications of the wave function (and the whole wave-particle duality thing). I'll start thinking about Lorentz transformations and different inertial frames of reference once I get these things clear. If I try and sort it out all at once, my head will explode. So, I guess I'm proposing that everything in post #61 is happening in the same inertial frame of reference. In fact, everything I'm talking about has to do with a single photon. So, I guess we could say that we're just talking about Alice, and the heck with Bob.

And now to Jilang.

Are you suggesting that the act of measuring in some way rotates the plane of polarisation?

Yes, that's exactly what I'm proposing. In fact, we know this rather conclusively. As an example, we can take randomly polarized light (such as sunlight) and exactly half of it will get through a linear polarizing filter.

However, now let's imagine that we have two linear polarizing filters, one oriented vertically and the other oriented horizontally. If we put them together, no light will get through. But how is this possible? For any single filter, 50% of the light will get through. So, if the first filter didn't "change" the light (i.e., change the polarization), then 50% should get through the first filter, and then 50% of that 50% should get through the second filter (with 25% coming out after both filters). But again, we know that this doesn't happen. 0% of the light gets through. That pretty much proves that the measurements somehow alter the polarization of the light.

Now, to make things even more weird, let's say we still have our two linear polarizers (one oriented vertical (V) and the other oriented horizontal (H)). Now, let's take a third linear polarizer and orient it at 45° (half way between vertical and horizontal). If we slip it between the V and H polarizers, 12.5% of the light will get through, whereas with only the V and H polarizers, NONE of the light got through. I'll leave the reasoning for that up to you.

Bottom line, in the quantum world, taking measurements virtually ALWAYS alters things.

Regards,
Elroy

 

[Elroy]

Just to say a bit more, that's the "opening of loophole #1" in explaining how QM is different from classical situations. In the classical world, we can measure things without changing them. I can measure the width, depth, and height of my refrigerator without changing it. However, to explain things (just like what I outlined in post #66), we must admit that measurements DO change things in the quantum world.

But all of that opens the door for the EPR pairs paradox. With perfect correlations between Alice and Bob, we can argue that Alice's measurements not only "change things" for her, but that they ALSO "change things" for Bob. How is THAT possible? We can also set things up such that the changes happen faster than the speed of light.

 

[Jimster41]

we must admit that measurements DO change things in the quantum world.

But all of that opens the door for the EPR pairs paradox. With perfect correlations between Alice and Bob, we can argue that Alice's measurements not only "change things" for her, but that they ALSO "change things" for Bob. How is THAT possible? We can also set things up such that the changes happen faster than the speed of light.

Doctor, I concur.

Also, I apologize again for butting in here with my questions. It's just been an interesting discussion to follow. I feel like I'm in your boat.

The book by Gisin, though a "popularization" (by a guy who won the Bell Prize and invented Portable Quantum Cryptography or something) has been a real help for me I might add... though... I can imagine you might also find it disappointing on one level... the conundrum of just how "non-local randomness" (Gisin's words) can be... is not less sharp after reading it... it is more. And I feel betrayed in small part because I thought this guy was going to explain how it could be... he just made it more unavoidably clear... that it is! and the pros all have their brains on bust trying to figure it out... at this very moment... wherever that is.

Carry on...

 

[Jilang]

I haven't seen the Venn diagrams of Bell's Inequality. I'm betting I would like them.

See link 6 in this thread for Susskind's Venn diagram.
https://www.physicsforums.com/threads/bells-theorem-easy-explained.562942/