Progress on Explaining Bell's Inequality

[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/