Photon entanglement: why three angles?

[Jabbu]

When the two polarizers are set 60 degrees apart, for example, QM prediction is 25% correlation. It is already different than what is believed to be classical or "expected" result. So what is the point of testing more than one angle in a single experiment? And what difference does it make when those angles are shuffled randomly instead of tested separately?

 

[DrChinese]

When the two polarizers are set 60 degrees apart, for example, QM prediction is 25% correlation. It is already different than what is believed to be classical or "expected" result. So what is the point of testing more than one angle in a single experiment? And what difference does it make when those angles are shuffled randomly instead of tested separately?

The issue is whether local realism is compatible with the predictions of Quantum Mechanics.

Local realists (such as you as best I can tell) believe there are particle attributes even when not observed. If so, there must be counterfactual values. A third reading (even if not taken) would be part and parcel of that belief. Otherwise, there is no point is asserting local realism. So that explains the 3, as 2 are predicted by QM (but a third is not).

Testing with fast switching is not strictly needed, but shows that there is no possibility of communication between the observing devices by an unknown mechanism at light speeds or less. Since that experiment was performed already (Aspect, Weihs, etc) we know that is not an issue. You have previously been given a reference on Weihs et al.

Combine the above with Bell's Theorem, and we know that local realism is not viable.

Now, please do not ask the same question over again. You know already where that leads. If you don't understand my answer, explain what you don't understand. I believe you already have a reference to my web page on Bell, but if not:

http://www.drchinese.com/Bells_Theorem.htm

 

[Nugatory]

When the two polarizers are set 60 degrees apart, for example, QM prediction is 25% correlation. It is already different than what is believed to be classical or "expected" result. So what is the point of testing more than one angle in a single experiment? And what difference does it make when those angles are shuffled randomly instead of tested separately?

Two angles, not perpendicular to one another, are enough to validate the quantum mechanical prediction fot the correlation at two angles. However, Bell's theorem makes a stronger statement: No local hidden variable theory (loosely speaking, one in which the unmeasured properties of the particle have definite values) can reprodiuce the predictions of quantum mechanics. It takes three angles to test this propostion because with two partciles we get two measurements; we need a third angle to have an unmmeasured angle in every set of measurements we make.

Have you read Bell's paper yet? http://www.drchinese.com/David/Bell_Compact.pdf

 

[Jabbu]

Local realists (such as you as best I can tell) believe there are particle attributes even when not observed. If so, there must be counterfactual values. A third reading (even if not taken) would be part and parcel of that belief. Otherwise, there is no point is asserting local realism. So that explains the 3, as 2 are predicted by QM (but a third is not).

Can you describe "must be counterfactual values" in terms of photons, polarizers and data recorded?

One theta already confirms cos^2(theta), it's already non-local. Can you explain in more practical terms what objection or explanation local realists have so more is needed to convince them?

If you don't understand my answer, explain what you don't understand. I believe you already have a reference to my web page on Bell, but if not:

http://www.drchinese.com/Bells_Theorem.htm

Do you have some page about "DrC challenge"? From what I've seen in other posts about it, I don't understand why the "data sample" required is in this format:

a: + + - + - - +
b: - - + - + + -
c: - + - + + - +

...shouldn't it be:

aAlice: + + - + - - +
aBob: - - + - + + -

bAlice: + + - + - - +
bBob: - - + - + + -

cAlice: + + - + - - +
cBob: - - + - + + -


...where a, b, c are three theta relative angles between Alice and Bob polarizers? Can you show how do you calculate results?

 

[Nugatory]

Can you describe "must be counterfactual values" in terms of photons, polarizers and data recorded?

One theta already confirms cos^2(theta), it's already non-local. Can you explain in more practical terms what objection or explanation local realists have so more is needed to convince them?

When I measure the polarization of one member of the pair on one angle and I measure the polarization of the other member of the pair on another angle, then I know the polarization of both members of the pair on those angles. The polarization on any other angle is "counterfactual" - I didn't measure it.

We can construct a local realistic theory for the $cos^2\theta$ result easily enough: just say that the photons are created in a state of definite polarization in every direction, with the values of the polarization in each direction such that the $cos^2\theta$ law holds on average across many pairs . This theory is said to be "counterfactually definite" because it asserts that the polarizations I didn't measure still have definite values that I would have obtained if I had measured them.

I don't understand why the "data sample" required is in this format:

a: + + - + - - +
b: - - + - + + -
c: - + - + + - +

...where a, b, c are three theta relative angles between Alice and Bob polarizers? Can you show how do you calculate results?

The first column of this example above should be read as "for the first pair if we measure the polarization on angle A the left-hand photon will pass and the right-hand one will not; if we measure on angles B or C the left-hand photon will not pass and the right-hand one will". The second column should be read as "for the second pair if we measure the polarization on angle B the left-hand photon will not pass and the right-hand one will; if we measure on angles A or C the left-hand one will pass and the right-hand one will not". This is exactly the local realistic theory I describe above - both photons are created with definite polarization values at all three angles.

You calculate the results by choosing any two of the three possible results because we only get to make two measurements, one on each photon. The challenge is to construct a data set that will lead to a violation of Bell's equality no matter which measurements we choose to make on each pair - and if you try it you'll find that it cannot be done. Therefore, no theory in which the results of measurements on all three angles are predetermined can match the experimental results.

 

[DrChinese]

Do you have some page about "DrC challenge"? From what I've seen in other posts about it, I don't understand why the "data sample" required is in this format:

a: + + - + - - +
b: - - + - + + -
c: - + - + + - +

...shouldn't it be:

aAlice: + + - + - - +
aBob: - - + - + + -

bAlice: + + - + - - +
bBob: - - + - + + -

cAlice: + + - + - - +
cBob: - - + - + + -...where a, b, c are three theta relative angles between Alice and Bob polarizers? Can you show how do you calculate results?

The DrChinese Challenge, yes very good Jabbu! You've been doing some study! Keep reading the references, especially as to how they formulate their setups and theoretical predictions - Weihs, Dehlinger, Zeilinger, etc. Now what I say below is a bit of a different take than Nugatory's but they are fundamentally the same.

Let's talk about a SINGLE photon we will call Alice. Realists believe it has polarization properties at all times. So there must be values for any 3 angles a, b, c (0/120/240 are what I use, which as you have said is the same as 0/60/-60). Further, as pointed out by EPR (1935), for entangled photon pairs, realism implies that they are actually *predetermined*. That is because the outcome of ANY measurement on Alice can be predicted with certainty in advance (by measuring the SAME property a, b or c on entangled partner Bob). This conclusion was very reasonable (until Bell came along). A realist simply believed there was a more complete specification of Alice than QM allows. But otherwise there was no specific contradiction, it's simply a matter of your interpretation. (Back then: Bohr v. Einstein.)

1. But it turned out that the requirement that Alice had simultaneous predetermined polarizations at a, b and c (to be consistent with the above paragraph) led to severe constraints that were missed early on after EPR. Specifically: it should matter not which of a, b and c you measure on Alice, since the outcome is predetermined. In fact, there should be a data set of values that would match up to any measurement you can do on Alice on some set of runs (let's say you are measuring a) where you can also hypothesize b and c. Creating such dataset is easy if we stop here - all you do is provide ANY answer for the unmeasured angles b and c on Alice and no one could disprove it. We'll start with:

a: + + - + - - + +
b: - - + - + + - +
c: - + - + + - + -

2. Of course, Bob's polarizations must be predetermined as well. So by measuring entangled Bob at one of the other angles (let's say b), you actually learn some additional information* about Alice (at least, that's what realists believe). So you could update your dataset above so that it at least kept the attribute that a and b match the cos^2(theta) predictions of QM. That means there would be no disagreement with QM (which presumably also makes experimentally correct predictions). So your dataset has a good relationship between Alice@a and Bob@b (which tells you Alice@b too). So we revise the dataset to fix this:

a: + + - + - - + +
b: - - + - - + - +
c: - + - + + - + -

a-b is 25% (since cos^2(120) is 25% and note that Type I vs Type II is not an issue here)

3. And it shouldn't matter which of the other 2 angles (b or c) you measure for Bob either, for similar reasons (as both Alice AND Bob's polarizations are all predetermined). So now you update your dataset so Alice@a and Bob@b AND Alice@a and Bob@c both match the QM predictions. This is getting progressively harder to do, but you can do it. You will have a 25% match ratio for each in our example. What we are actually doing is describing Alice by measuring one attribute on Alice and inferring another value by measuring Bob. But we are still talking about Alice at the base. Here is our revised sample dataset.

a: + + - + - - + +
b: - - + - - + - +
c: - + + - - + - -

a-b is 25% (since cos^2(120) is 25%)
a-c is 25% (since cos^2(120) is 25%)

4. But now something funny happens. We fixed it so the match ratio for Alice's a and b is 25% (theta=120 degrees), and the match ratio for Alice's a and c is 25%(theta=120 degrees), BUT... the match ratio for Alice's b and c is now 75% (for same theta=120 degrees). What happened? This is an inconsistent result. Because we specified above that all the angle settings were predetermined, so it shouldn't matter that we are examining Alice's a-b, b-c, or a-c. We will try our best to fix this:

a: + + - + - - + +
b: - - + - - + - +
c: - + + + + + - -

a-b is 25% (since cos^2(120) is 25%)
a-c is 25% (since cos^2(120) is 25%)
b-c is 50% (oops - can't get a combo to be 25%)

So for this to work out, there must be something special about the pair we choose to measure - but that violates Observer Independence (i.e. the results were predetermined). The only way to make this work out is to assume there is some force or information moving from Alice's measuring apparatus to Bob's (or vice versa). And if we fixed it so the selection of the angle pair occurred AFTER Alice and Bob separate, we could determine if such signal occurs at light speeds or faster.

And this is the DrChinese challenge, to come up with values for a, b and c for a set of 8 runs that match what you would see if Alice and Bob were entangled and had predetermined outcomes independent of the measurements performed. This time we had a loser, can you do better?

Of course, experiments show that if there is such an effect, it must be at least 10,000 times faster than c. The other thing is that we can drop the assumption (constraints) of realism we started with in 1. above and that resolves things.*In fact, such information exceeds the bounds of the HUP! Because you could simply measure a complementary (non-commuting value on Bob and now you would know both about Alice).

 

[Jabbu]

Further, as pointed out by EPR (1935), for entangled photon pairs, realism implies that they are actually *predetermined*.

Predetermined at what point in time? Flip of my coin may be predetermined at the beginning of time, but that doesn't mean it can be predicted with more than 50% certainty.

That is because the outcome of ANY measurement on Alice can be predicted with certainty in advance (by measuring the SAME property a, b or c on entangled partner Bob).

Non-localists (QM) think measurement on Alice can be predicted with certainty in advance? And local realists (classical physics) think outcome can be predicted by Malus law probability?

 

[RUTA]

Two angles, not perpendicular to one another, are enough to validate the quantum mechanical prediction fot the correlation at two angles. However, Bell's theorem makes a stronger statement: No local hidden variable theory (loosely speaking, one in which the unmeasured properties of the particle have definite values) can reprodiuce the predictions of quantum mechanics. It takes three angles to test this propostion because with two partciles we get two measurements; we need a third angle to have an unmmeasured angle in every set of measurements we make.

Here is an example from Mermin using just two particles and two settings. He acknowledges the idea is originally due to Hardy.

 

[Jabbu]

Here is an example from Mermin using just two particles and two settings. He acknowledges the idea is originally due to Hardy.

Here is example with only one setting: theta= 0 degrees.

QM predicts cos^2(0) = 100% correlation.

Classical physics (Malus law) does not predict anything unless photons polarization relative to their prospective polarizer angle is known at the time of photon-polarizer interaction, or known to be uniformly random.

If it is assumed photon pairs come out unpolarized (uniformly random) then classical physics predicts 50% chance for +/- detection on both sides regardless of any polarizer angle. That is 50% chance AB recorded pairs will match (++ or --), and 50% chance they will be mismatch (+- or -+). Match - mismatch = 0% correlation.

Completely opposite predictions for just one angle setting, how is this not sufficient? The only way Malus law can predict 100% correlation for theta= 0 is if it is known photon pairs come out polarized at 0 degrees as well. Therefore, if you can prove photons come out with uniformly random polarization relative to their polarizer you have proved non-locality.

 

[Nugatory]

Completely opposite predictions for just one angle setting, how is this not sufficient? The only way Malus law can predict 100% correlation for theta= 0 is if it is known photon pairs come out polarized at 0 degrees as well. Therefore, if you can prove photons come out with uniformly random polarization relative to their polarizer you have proved non-locality.

One angle setting is sufficient to verify the quantum mechanical $cos^2\theta$ prediction for that angle and to falsify the classical prediction.

The three-angle experiments you've been reading about are looking (generally successfully) for violations of Bell's inequality for three observables. These lead to an even stronger conclusion: no local hidden-variable theory (not just the classical theory that we've already falsified) can be correct.

 

[RUTA]

Here is example with only one setting: theta= 0 degrees.

QM predicts cos^2(0) = 100% correlation.

Classical physics (Malus law) does not predict anything unless photons polarization relative to their prospective polarizer angle is known at the time of photon-polarizer interaction, or known to be uniformly random.

If it is assumed photon pairs come out unpolarized (uniformly random) then classical physics predicts 50% chance for +/- detection on both sides regardless of any polarizer angle. That is 50% chance AB recorded pairs will match (++ or --), and 50% chance they will be mismatch (+- or -+). Match - mismatch = 0% correlation.

Completely opposite predictions for just one angle setting, how is this not sufficient? The only way Malus law can predict 100% correlation for theta= 0 is if it is known photon pairs come out polarized at 0 degrees as well. Therefore, if you can prove photons come out with uniformly random polarization relative to their polarizer you have proved non-locality.

If both photons' polarizations are totally random with respect to each other, you would find four possible outcomes ++, +-, -+, -- for your single setting distributed evenly which means 50% correlation, i.e., ++ and -- each occurring 25% of the time. Once you find you only get ++ and -- outcomes each 50% of the time, you have entanglement. You can explain this via Mermin's "instruction sets," i.e., local hidden variables. You need another setting to establish counterfactuals and violate Bell's inequality. Read the paper.

 

[stevendaryl]

Here is example with only one setting: theta= 0 degrees.

QM predicts cos^2(0) = 100% correlation.

Classical physics (Malus law) does not predict anything unless photons polarization relative to their prospective polarizer angle is known at the time of photon-polarizer interaction, or known to be uniformly random.

If it is assumed photon pairs come out unpolarized (uniformly random) then classical physics predicts 50% chance for +/- detection on both sides regardless of any polarizer angle. That is 50% chance AB recorded pairs will match (++ or --), and 50% chance they will be mismatch (+- or -+). Match - mismatch = 0% correlation.

Completely opposite predictions for just one angle setting, how is this not sufficient? The only way Malus law can predict 100% correlation for theta= 0 is if it is known photon pairs come out polarized at 0 degrees as well. Therefore, if you can prove photons come out with uniformly random polarization relative to their polarizer you have proved non-locality.

No, that just proves that Malus' law isn't the correct description.

 

[stevendaryl]

Hardy came up with a thought experiment that doesn't involve any inequality.

Prepare two electrons in the composite state \frac{1}{\sqrt{3}}(|U_z\rangle |U_z\rangle + |U_z\rangle |D_z\rangle + |D_z\rangle |U_z\rangle)

where |U_z\rangle means spin-up in the z-direction, and |D_z\rangle means spin-down in the z-direction.

Use this state for an EPR-like experiment involving Alice and Bob. Alice can perform one of two experiments: measure spin in the x-direction, or the z-direction. For each direction, she gets two possible results: spin-up or spin-down. Similarly for Bob.

We can prove, using quantum mechanics:

1. If Alice measures spin-up in the z-direction, then Bob will measure spin-up in the x-direction.
2. If Bob measures spin-up in the z-direction, then Alice will measure spin-up in the x-direction.
3. If Alice measures spin-down in the z-direction, then Bob will measure spin-up in the z-direction.

Let's try to explain this result under the assumption that the outcomes are predetermined. Then there are five possible values for the "hidden variable":

\lambda_1: Both Alice and Bob measure spin-up in either direction.
\lambda_2: Alice measures spin-up in either direction. Bob measures spin-down in z-direction, or spin-up in x-direction.
\lambda_3: Alice measures spin-up in z-direction or spin-down in x-direction. Bob measures spin-down in z-direction, or spin-up in x-direction.
\lambda_4: Alice measures spin-down in z-direction or spin-up in x-direction. Bob measures spin-up in z-direction, or spin-down in x-direction.
\lambda_5: Alice measures spin-down in z-direction or spin-up in x-direction. bob measures spin-up in either direction.

Something that can never happen, according to a hidden-variables theory, is this:
Alice and Bob both measure spin-down in the x-direction. The proof is this:

• Alice either has spin-up in the z-direction, or spin-down in the z-direction.
• If it is spin-up, then Bob must have spin-up in the x-direction. (Fact #1)
• If it is not spin-up in the z-direction, then Bob must have spin-up in the z-direction. (Fact #3)
• But if Bob has spin-up in the z-direction, then Alice will have spin-up in the x-direction. (Fact #2)

So the assumption that the results are predetermined implies that it is impossible for both Bob and Alice to measure spin-down in the x-direction.

Quantum-mechanically, though, it is possible. (I think that happens 1/12 of the time)

 

[Jabbu]

No, that just proves that Malus' law isn't the correct description.

Malus law is the only classical and locally causal solution. Proving that it is not correct (complete) description is the whole point, because every other description is no longer classical or locally causal.

What do you believe is classical prediction for theta= 0 degrees?

 

[Jabbu]

If both photons' polarizations are totally random with respect to each other, you would find four possible outcomes ++, +-, -+, -- for your single setting distributed evenly which means 50% correlation, i.e., ++ and -- each occurring 25% of the time.

If photons polarization is uniformly random with respect to polarizers, then for Malus law it is irrelevant whether relative polarization between photons in each pair is constant (entangled) or random as well, because the chance for each photon to mark either "+" or "-" stays 50% in either case.

I believe in a previous thread we established mismatches are counted as well and that correlation = matches - mismatches, not just number of matches. It's not terribly important since 50% prediction is still far from 100%. Do you have some reference for this "correlation" formula?

Once you find you only get ++ and -- outcomes each 50% of the time, you have entanglement.

Exactly. It's like you and me are tossing two coins and every time it's a match. Can a result be any more "non-local" than that? Your coin flips should not be correlated with my coin flips, and if they are, you can call it "entanglement", but more practically said it's a pretty crazy display of non-local causality. No other relative angle makes this more clear. 100% correlation is the most unambiguously bizarre correlation there can possibly be, testing other angles is almost pointless compared to theta = 0. Is this not true?

 

[stevendaryl]

Malus law is the only classical and locally causal solution. Proving that it is not correct (complete) description is the whole point, because every other description is no longer classical or locally causal.

What do you believe is classical prediction for theta= 0 degrees?

Malus' law is certainly not the only possible local theory. It's not even the only local theory that is consistent with classical E&M in the limit that quantum effects become negligible.

 

[stevendaryl]

If photons polarization is uniformly random with respect to polarizers, then for Malus law it is irrelevant whether relative polarization between photons in each pair is constant (entangled) or random as well, because the chance for each photon to mark either "+" or "-" stays 50% in either case.

Why do you think that the only possibilities are Malus' law, or nonlocal interactions?

 

[stevendaryl]

Exactly. It's like you and me are tossing two coins and every time it's a match. Can a result be any more "non-local" than that?

That isn't necessarily nonlocal. Suppose that the coins worked this way:

If the n^th digit of \pi is an even number, then the n^th coin flip will be heads. Otherwise, the n^th flip will be tails.​

That's a perfectly local rule, in the sense that the two coins don't need instantaneous communication to achieve it. They just need a little microprocessor and some internal mechanism for shifting weight to one side or the other.

 

[Jabbu]

Malus' law is certainly not the only possible local theory. It's not even the only local theory that is consistent with classical E&M in the limit that quantum effects become negligible.

If you mean to suggest there is some other locally causal (classical physics) equation which can calculate probability for photon-polarizer interaction outcome, you only need to name it.

 

[stevendaryl]

If you mean to suggest there is some other locally causal (classical physics) equation which can calculate probability for photon-polarizer interaction outcome, you only need to name it.

If you are claiming that something is impossible, then it's up to you to produce a proof. Otherwise, you're just guessing. You asked "why three angles"? Because with three angles we can prove that the results of QM cannot be reproduced by a local realistic model. With a single angle, there is no such proof. It's as simple as that.

"I can't think of a way to do it" does not logically imply "There is no way to do it".

 

[RUTA]

If photons polarization is uniformly random with respect to polarizers, then for Malus law it is irrelevant whether relative polarization between photons in each pair is constant (entangled) or random as well, because the chance for each photon to mark either "+" or "-" stays 50% in either case.

I believe in a previous thread we established mismatches are counted as well and that correlation = matches - mismatches, not just number of matches. It's not terribly important since 50% prediction is still far from 100%. Do you have some reference for this "correlation" formula?

Exactly. It's like you and me are tossing two coins and every time it's a match. Can a result be any more "non-local" than that? Your coin flips should not be correlated with my coin flips, and if they are, you can call it "entanglement", but more practically said it's a pretty crazy display of non-local causality. No other relative angle makes this more clear. 100% correlation is the most unambiguously bizarre correlation there can possibly be, testing other angles is almost pointless compared to theta = 0. Is this not true?

I'm using the term "correlated" to mean "same results, ++ or --" and "anti-correlated" to mean "opposite results, +- or -+." Then the probability of correlated outcomes in the state |++> + |--> goes as cosine squared and anti-correlated outcomes as sine squared. If you rather start with the singlet state |+-> - |-+>, then sine and cosine are switched.

If you found 100% correlation in a coin flip, then you'd be surprised and wonder what causes the coins to always show heads or tails together. But that's not necessarily true of particles emitted from a common source, since the correlation can result from the event itself, e.g., conservation of angular momentum. The mystery in that case results from having other measurement options available. Here is what happens in the Hardy-Mermin device (two particles, two settings 1 and 2, two outcomes R and G) taken from p 881:


The data exhibit the following important features:

(a) In runs in which the detectors end up with different
settings, they never both flash green: 21 GG and
12 GG never occur.

(b) In runs in which both detectors end up set to 2, one
occasionally finds both flashing green: 22GG sometimes
occurs.

(c) In runs in which both detectors end up set to 1, they
never both flash red: 11RR never occurs.

Since the particles don't know how they'll be measured and can't communicate with each other until after both are measured (measurement events are space-like related), they must coordinate their possible outcomes at the source before heading to the detectors. They occasionally flash GG for setting 22, but can't flash GG if they are measured in 12 or 21. Therefore, they must both leave the source with 1R2G for trials in which 22GG occurred. But, that can't be because if they had been measured in 11 in that trial, they would've produced a RR outcome and that never happens.

"Instruction sets" won't work, so how do the particles make (a) -- (c) happen? There's your mystery.

 

[Jabbu]

That isn't necessarily nonlocal. Suppose that the coins worked this way:

If the n^th digit of \pi is an even number, then the n^{th[/itex] coin flip will be heads. Otherwise, the nth flip will be tails.}​

^{That's a perfectly local rule, in the sense that the two coins don't need instantaneous communication to achieve it. They just need a little microprocessor and some internal mechanism for shifting weight to one side or the other.}

^{That's a magic trick. If the coins were doing it for real the explanation would have to be non-local, that is they would have to be related over distance through some, to us invisible, if not inconceivable, connection defying experience and classical probability theory.}

 

[DrChinese]

1. Predetermined at what point in time? Flip of my coin may be predetermined at the beginning of time, but that doesn't mean it can be predicted with more than 50% certainty.

2. And local realists (classical physics) think outcome can be predicted by Malus law probability?

1. A couple of points to remember before I proceed further: a) This argument was advanced by EPR in 1935. You should already be familiar with this, and I would recommend you re-read it. Look for disussion of "element of reality". b) My post is discussing a SINGLE photon Alice.

I can predict the outcome of ANY measurement you make on Alice IN ADVANCE with 100% certainty. Not 50%. Alice merely needs to be entangled with Bob AND you must tell me what measurement you are planning to make. According to EPR (and any realist), that means that the outcome must have been predetermined UNLESS you require that every possible measurement on Alice be predicted simultaneously in advance. (EPR says the "simultaneous" requirement would be unreasonable.)

So your logic in your statement misses the point of my post. It must be predetermined precisely BECAUSE it can be predicted (according to EPR). 100% Predictable -> EPR Element of Reality -> Predetermined.

2. You REALLY should stop referencing Malus as you do. This is wrong, and it confuses all of us who use the word correctly as it applies to something else entirely. Again, if you don't listen to my advice on this, you will likely close another thread. Please be alerted that you getting dangerously close to having every thread you start get shut down.

 

[Jabbu]

If you are claiming that something is impossible, then it's up to you to produce a proof. Otherwise, you're just guessing.

I'm not guessing, Malus law is the only locally causal or classical physics equation I know of that can calculate probability for photon-polarizer interaction outcome. Do you know of any other?

What do you think is classical physics prediction for theta= 0 degrees?

 

[stevendaryl]

That's a magic trick. If the coins were doing it for real the explanation would have to be non-local, that is they would have to be related over distance through some, to us invisible, if not inconceivable, connection defying experience and classical probability theory.

I don't think you understand the point of a proof. You might believe that there is no way to accomplish such correlations except through nonlocal interaction, but so what? That belief might be true, or that belief might be false. The point of a proof is that you know for certain that if the assumptions are true, then the conclusions are true. In the case of correlated coin-flips, there is no proof that nonlocal interactions are required. (Or at least, I don't know of one.)

 

[stevendaryl]

I'm not guessing, Malus law is the only locally causal or classical physics equation I know of that can calculate probability for photon-polarizer interaction outcome.

By "guessing" I just mean that you're making a claim for which you don't really have any proof that it is true. "I don't know of a way to do something" does not imply "There is no way to do it".

And I gave you a counter-example that proved your claim was wrong.

 

[DrChinese]

1. Here is example with only one setting: theta= 0 degrees.

QM predicts cos^2(0) = 100% correlation.

2. Classical physics (Malus law) does not predict anything unless photons polarization relative to their prospective polarizer angle is known at the time of photon-polarizer interaction, or known to be uniformly random.

3. If it is assumed photon pairs come out unpolarized (uniformly random) then classical physics predicts 50% chance for +/- detection on both sides regardless of any polarizer angle. That is 50% chance AB recorded pairs will match (++ or --), and 50% chance they will be mismatch (+- or -+). Match - mismatch = 0% correlation.

1. This is correct.

2. This is wrong to the extent you are referring to entangled photons, as in 1. Quit mixing references, entangled photons were discovered 150+ years after Malus. Malus describes classical light.

3. This is wrong: polarization entangled pairs are not explained in classical terms. They are not classical! So whatever formula you choose here, it will be your hypothesis as to what it is. Several others have indicated this too. And as I have indicated, no classical formula at all can re-create the predictions of QM. The one you present as 3. is simply one such formula which fails. You are free to try all you like, that is the DrChinese challenge!

 

[Jabbu]

By "guessing" I just mean that you're making a claim for which you don't really have any proof that it is true. "I don't know of a way to do something" does not imply "There is no way to do it".

I'm telling you I simply do not know of any other locally causal or classical physics equation that can calculate probability for photon-polarizer interaction outcome. Do you?

What do you think is classical physics prediction for theta= 0 degrees?

 

[stevendaryl]

I'm telling you I simply do not know of any other locally causal or classical physics equation that can calculate probability for photon-polarizer interaction outcome. Do you?

I gave you one.

Anyway, your original question was: why do we need to consider more than one angle? The answer is: Because otherwise, there is no known proof that the results cannot be obtained by a local realistic theory.

 

[Nugatory]

I'm telling you I simply do not know of any other locally causal or classical physics equation that can calculate probability for photon-polarizer interaction outcome. Do you?

You are missing the point (and I have a nagging suspicion that you still haven't read and understood the essential papers here, so I'll provide the links below).

The EPR argument was that quantum mechanics was incomplete because it only describes the results of measurements in a statistical sense; and that there must be an as-yet-undiscovered locally realistic theory that underlies these predictions in the same way that classical mechanics underlies the statistical predictions of thermodynamics. The paper is here: http://www.drchinese.com/David/EPR.pdf

Bell's theorem (the paper is here: http://www.drchinese.com/David/Bell_Compact.pdf) proves that no such theory can reproduce all the predictions of quantum mechanics in the three-angle case. We did the three-angle experiments to show that the predictions of quantum mechanics are correct, and therefore that there is no unknown local realistic theory underlying the statistical predictions of quantum mechanics.

Thus, your insistence that measurements at one angle are sufficient to refute the classical model is missing the point of the three-angle experiments. The three-angle experiments are done for a different reason to prove a different point.

 

[Jabbu]

If you found 100% correlation in a coin flip, then you'd be surprised and wonder what causes the coins to always show heads or tails together. But that's not necessarily true of particles emitted from a common source, since the correlation can result from the event itself, e.g., conservation of angular momentum.

It looks like only two of us are talking about the same thing. But hold on a second. Even if entangled photons have the same polarization relative to their polarizers, only their probability is the same, but it's the polarizers that will ultimately decide what outcome will actually be, and according to probability theory and local causality that's two separate independent 50% probabilities, so there should not be more than 50% matching pairs in the long run.

Therefore, there is no local or classical explanation for 100% correlation. If there was, QM explanation would be superfluous. Any correlation less than 100% can only be less convincing as it is closer to classical prediction.

 

[Jabbu]

I gave you one.

Anyway, your original question was: why do we need to consider more than one angle? The answer is: Because otherwise, there is no known proof that the results cannot be obtained by a local realistic theory.

If you think your locally causal magic trick explanation is really plausible, then it can just the same disprove any other non-local correlation for any other angle or combination of angles. When experiments are performed with multiple angles the data is recorded separately for each one and the result has to be the same as if each angle was tested independently by itself.

 

[DrChinese]

It looks like only two of us are talking about the same thing. But hold on a second. Even if entangled photons have the same polarization relative to their polarizers, only their probability is the same, but it's the polarizers that will ultimately decide what outcome will actually be, and according to probability theory and local causality that's two separate independent 50% probabilities, so there should not be more than 50% matching pairs in the long run.

The above statements are both ambiguous and relating to different subjects. No one can really be sure what you are discussing at any particular point in time. Are you discussing entangled pairs? Are you talking about what actually happens?

Because when you say "probability theory and local causality that's two separate independent 50% probabilities" you are talking about things that are not applicable. Entangled particles are part of a SINGLE system. They do not follow "product state" (separable) statistics.

 

[stevendaryl]

If you think your locally causal magic trick explanation is really plausible

No, I don't think it's plausible, but it is possible. It is enough to show that there is no proof in the case of a single angle.

Look, this is not complicated: There is a proof in the case of three angles. There is no proof in the case of a single angle. That's all there is to it.

 

[stevendaryl]

If you think your locally causal magic trick explanation is really plausible, then it can just the same disprove any other non-local correlation for any other angle or combination of angles.

That's just not true. It works for one angle, but not for a combination of angles.

Let's take the photon EPR case with three angles:
\theta=0, \theta = 60, \theta = 120

A local deterministic hidden variables model would have a parameter \lambda that would take on 8 different values:

\lambda_{+++}
\lambda_{++-}
\lambda_{+-+}
\lambda_{-++}
\lambda_{+--}
\lambda_{-+-}
\lambda_{--+}
\lambda_{---}

If the hidden variable \lambda had value \lambda_{+++}, then the photon will pass through a filter oriented at \theta = 0, \theta = 60, \theta =. If it had value \lambda_{++-}, then the photon will pas through a filter oriented at \theta = 0, \theta = 60 but would be blocked by a filter at \theta = 120. Etc.

We can prove that there does not exist 8 probabilities
P_{+++}, P_{++-}, ... (where P_{+++} is the probability that \lambda has value \lambda_{+++}, etc.) such that this sort of local realistic theory reproduces the predictions of QM.

Now, if we only have a single angle, \theta=0, then the predictions of QM are that
50% of the time, the photon passes, and 50% of the time, it is blocked. It's EASY to come up with a local realistic model for this case. In this case, there are two possible values for \lambda:

\lambda_{+}
\lambda_{-}

The probabilities for these two values are:

P_{+} = 50%, P_{-} = 50%

That trivial model reproduces the QM predictions for a single angle.

 

[Jabbu]

If the hidden variable \lambda had value \lambda_{+++}, then the photon will pass through a filter oriented at \theta = 0, \theta = 60, \theta =. If it had value \lambda_{++-}, then the photon will pas through a filter oriented at \theta = 0, \theta = 60 but would be blocked by a filter at \theta = 120. Etc.

We can prove that there does not exist 8 probabilities
P_{+++}, P_{++-}, ... (where P_{+++} is the probability that \lambda has value \lambda_{+++}, etc.) such that this sort of local realistic theory reproduces the predictions of QM.

Correlation is not calculated between readings for different angles. Experiments with three angles produce three separate data streams, one for each angle, and correlation is calculated independently for each data stream of AB pairs. For theta= 0, Alice and Bob readings should be 100% correlated just as if theta= 0 was tested independently by itself. Other angles have their completely separate data streams of AB pairs with their own probabilities and their own correlations according to QM's cos^2(theta).

Now, if we only have a single angle, \theta=0, then the predictions of QM are that
50% of the time, the photon passes, and 50% of the time, it is blocked. It's EASY to come up with a local realistic model for this case. In this case, there are two possible values for \lambda:

Proposed local classical equation must be confirmed by classical experiments, for any angle, it has to be able to actually work for photon-polarizer interaction in general, or it simply is not true. It has to give the same result as Malus law, it thus can not be anything else. Either Malus law works or the explanation is non-local, there is no other choice compatible with all the other classical experiments.

 

[RUTA]

Correlation is not calculated between readings for different angles. Experiments with three angles produce three separate data streams, one for each angle, and correlation is calculated independently for each data stream of AB pairs. For theta= 0, Alice and Bob readings should be 100% correlated just as if theta= 0 was tested independently by itself. Other angles have their completely separate data streams of AB pairs with their own probabilities and their own correlations according to QM's cos^2(theta).

"theta" is the difference between the two polarizer/SG magnet settings, so whatever the settings of the two detectors, if they are the same, theta = 0. Read the attached paper and go through all the calculations. There are some mistakes, but they're easy to find. After completing this homework assignment, you'll understand how an experiment showing the violation of Bell's inequality is actually done, to include data analysis. After you've finished, come back with your questions.

 

[stevendaryl]

Proposed local classical equation must be confirmed by classical experiments, for any angle, it has to be able to actually work for photon-polarizer interaction in general, or it simply is not true. It has to give the same result as Malus law, it thus can not be anything else. Either Malus law works or the explanation is non-local, there is no other choice compatible with all the other classical experiments.

I just showed you that a local hidden variables theory can reproduce exactly the predictions of QM for a single filter setting (both Alice and Bob have the filters at the same angle). So what you're saying is false: the predictions for a single angle does not demonstrate that anything nonlocal is going on. In contrast, when you have 3 filter settings to choose from, it is not possible to reproduce the predictions of QM by any local theory.

I don't know why you keep bringing up Malus' law. It is completely useless in predicting correlations for the case of entangled photons, because you can't apply it unless you know the photon polarizations.

 

[Jabbu]

"theta" is the difference between the two polarizer/SG magnet settings, so whatever the settings of the two detectors, if they are the same, theta = 0. Read the attached paper and go through all the calculations. There are some mistakes, but they're easy to find. After completing this homework assignment, you'll understand how an experiment showing the violation of Bell's inequality is actually done, to include data analysis. After you've finished, come back with your questions.

I see. You are not surprised correlation for theta= 0 is 100%, as if that is normal. Instead, what you find more convincing the interaction is non-local is that some combined thetas correlation function "S" is greater than 2. Is that it?

 

[RUTA]

I see. You are not surprised correlation for theta= 0 is 100%, as if that is normal. Instead, what you find more convincing the interaction is non-local is that some combined thetas correlation function "S" is greater than 2. Is that it?

Did you read the paper and reproduce all the calculations? I can help you understand the equations and how they are related to the experiment. What you choose to believe about those equations and the experiment is up to you.

 

[morrobay]

The first column of this example above should be read as "for the first pair if we measure the polarization on angle A the left-hand photon will pass and the right-hand one will not; if we measure on angles B or C the left-hand photon will not pass and the right-hand one will". The second column should be read as "for the second pair if we measure the polarization on angle B the left-hand photon will not pass and the right-hand one will; if we measure on angles A or C the left-hand one will pass and the right-hand one will not". This is exactly the local realistic theory I describe above - both photons are created with definite polarization values at all three angles.

You calculate the results by choosing any two of the three possible results because we only get to make two measurements, one on each photon. The challenge is to construct a data set that will lead to a violation of Bell's equality no matter which measurements we choose to make on each pair - and if you try it you'll find that it cannot be done. Therefore, no theory in which the results of measurements on all three angles are predetermined can match the experimental results.

The data set for above is for Alice:
a: ++-+--+
b: --+-++-
c: -+-++-+

I have found a violation from data set from above post. Converting above data to horizontal format including Bobs values:

..A....B
a b c...a b c
+ - -...- + +
+ -+...- + -
- + -...+ - +
+ - +...- + -
- + +...+ - -
- + -...+ - +
+ - +...- + -

n[a+c+] + n[b+c-] ≥ n[a+b+]

n=1 + n=1 ≥ n= 4 violation

This form of inequality is taken from http://math.ucr.edu/home/baez/physics/Quantum/bells_inequality.html

 

[Jabbu]

Did you read the paper and reproduce all the calculations? I can help you understand the equations and how they are related to the experiment. What you choose to believe about those equations and the experiment is up to you.

I did read it. I didn't reproduce calculations, but I promise I will. You neither confirmed nor denied what I said in my last two post. Please be more specific and at least tell me what are we talking about now.

2 < S = E(a,b) - E(a,b') + E(a',b) + E(a',b')

Is that it, the proof experiment data describe non-local interaction?

 

[RUTA]

I did read it. I didn't reproduce calculations, but I promise I will. You neither confirmed nor denied what I said in my last two post. Please be more specific and at least tell me what are we talking about now.

2 < S = E(a,b) - (a,b') + (a',b) + (a',b')

Is that it, the proof experiment data describe non-local interaction?

If all you're trying to understand are the "non-local" implications of entanglement, then any of Mermin's papers will do. Essentially, it comes down to "no instruction sets." The 1994 paper I posted earlier is the easiest to understand in my opinion because it doesn't involve statistics or inequalities.

If you're trying to understand how data is actually taken concerning "non-locality," then read and work through all the calculations in the last paper I posted. In that case, "non-locality" is represented by violating |S| <= 2. I posted that paper because you seemed confused as to what the cosine squared of theta actually means and it's explained nicely in that article.

In either case, all the material you need to understand what physicists mean by "quantum non-locality" has been posted and explained in this thread. If there is something about the equations you don't understand, people here will help you. If you want to argue about semantics or what physicists should or should not believe concerning ontology, the moderators will likely close the thread.

 

[DrChinese]

Is that it, the proof experiment data describe non-local interaction?

No. It is proof that it is non-local or non-realistic. Or both. No one knows for sure.

 

[RUTA]

No. It is proof that it is non-local or non-realistic. Or both. No one knows for sure.

Careful here, Jabbu. This is a more precise use of the term "non-local" as regards entanglement. The term "quantum non-locality" has come to mean "measurement outcomes with entangled particles that violate classical intuition." One way to characterize exactly what it is in classical intuition that is violated by these QM measurement outcomes is to say they are "non-local" and/or "non-realistic." In this sense, "non-local" means they involve "superluminal influences." In the first 1:30 of this talk by Weinstein , for example, he uses "outcome independence" and "parameter independence" instead of "non-local" and "non-real." He then says "the conjunction of these two is sometimes referred to as strong locality or just locality."

 

[Jabbu]

If you're trying to understand how data is actually taken concerning "non-locality," then read and work through all the calculations in the last paper I posted. In that case, "non-locality" is represented by violating |S| <= 2. I posted that paper because you seemed confused as to what the cosine squared of theta actually means and it's explained nicely in that article.

That. So I'm looking at page 907 and I'm trying to calculate their example:

(Ex) a= -45, a'= 0, b= -25.5, b'= 25.5

...they say result is obtained with equations 10, 20, and 21:

(10) Pvv(a,b) = cos^2(b-a) / 2

(20) E(a,b) = Pvv(a,b) + Phh(a,b) - Pvh(a,b) - Phv(a,b)

(21) S = E(a,b) - E(a,b') + E(a',b) + E(a',b')To evaluate equation 20 there is defined Pvv(a,b) in equation 10, but I'm still missing Phh(a,b), Pvh(a,b), and Phv(a,b). How do I get to these three? Is equation 20 supposed to equal cos^2(b-a)?

 

[Nugatory]

The data set for above is for Alice:
a: ++-+--+
b: --+-++-
c: -+-++-+

I have found a violation from data set from above post. Converting above data to horizontal format including Bobs values:

..A....B
a b c...a b c
+ - -...- + +
+ -+...- + -
- + -...+ - +
+ - +...- + -
- + +...+ - -
- + -...+ - +
+ - +...- + -

n[a+c+] + n[b+c-] ≥ n[a+b+]
n=1 + n=1 ≥ n= 4 violation

Written in this form, the inequality applies to the results on one side (or the other), so we have for Alice $3+2\geq{0}$, for Bob $2+3\geq{0}$, and no violation.

That's just a trick of the notation: n[a+c+] means "Alice would measure + in the a and c directions" which means that either Alice measured a+ and Bob c-, or Alice measured c+ and Bob measured a-.

 

[RUTA]

That. So I'm looking at page 907 and I'm trying to calculate their example:

(Ex) a= -45, a'= 0, b= -25.5, b'= 25.5

...they say result is obtained with equations 10, 20, and 21:

(10) Pvv(a,b) = cos^2(b-a) / 2

(20) E(a,b) = Pvv(a,b) + Phh(a,b) - Pvh(a,b) - Phv(a,b)

(21) S = E(a,b) - E(a,b') + E(a',b) + E(a',b')


To evaluate equation 20 there is defined Pvv(a,b) in equation 10, but I'm still missing Phh(a,b), Pvh(a,b), and Phv(a,b). How do I get to these three? Is equation 20 supposed to equal cos^2(b-a)?

First, you caught the missing minus sign for b, good work.

You can get P_HH, P_VH, and P_VH from Eq (7) using <V| and <H| as needed:

P_HH = cos²(α-β)/2
P_HV = P_VH = sin²(α-β)/2.

Of course it must be the case that P_VV + P_HH + P_HV + P_VH = 1.

Now, from Eq (20) we have E(α-β) = cos²(α-β) - sin²(α-β) = cos2(α-β). This gives

E(a,b) = E(-45, -22.5) = cos(-45) = 1/√2
E(a,b') = E(-45, 22.5) = cos(-135) = -1/√2
E(a',b) = E(0, -22.5) = cos(45) = 1/√2
E(a',b') = E(0, 22.5) = cos(-45) = 1/√2.

Thus, Eq (21) gives S = E(a,b) - E(a,b') + E(a',b) + E(a',b') = 2√2 as shown in Eq (23)

 

[Jabbu]

P_HH = cos²(α-β)/2
P_HV = P_VH = sin²(α-β)/2.

Of course it must be the case that P_VV + P_HH + P_HV + P_VH = 1.

So those are counts of matching and mismatching pairs. Equation 20 is then the same what I called correlation formula: match - mismatch, and QM's cos^2(theta). It means two uniformly random binary sequences should be completely uncorrelated with equal number of matches and mismatches.

0.5:0.5 vs 0.5:0.5
chance for match: (0.5 * 0.5) + (0.5 * 0.5) = 0.5
chance for mismatch: (0.5 * 0.5) + (0.5 * 0.5) = 0.5
correlation: 0.5 - 0.5 = 0%

Now, from Eq (20) we have E(α-β) = cos²(α-β) - sin²(α-β) = cos2(α-β). This gives

Instead of theta = (a-b), shouldn't that be (b-a)? So E(a,b) = cos^2(b-a)?

E(a,b) = E(-45, -22.5) = cos(-45) = 1/√2
E(a,b') = E(-45, 22.5) = cos(-135) = -1/√2
E(a',b) = E(0, -22.5) = cos(45) = 1/√2
E(a',b') = E(0, 22.5) = cos(-45) = 1/√2.

How do you get theta = -45 from (-45, -25.5)? Shouldn't that be theta = (-25.5 - (-45)) = 25.5? Also theta = -135 from (-45, 22.5)?


This is what I get:

E(a,b) = E(-45, -25.5) = cos^2(19.5) = 0.89
E(a,b') = E(-45, 25.5) = cos^2(70.5) = 0.11
E(a',b) = E(0, -25.5) = cos^2(-25.5) = 0.82
E(a',b') = E(0, 25.5) = cos^2(25.5) = 0.82

S = 0.89 - 0.11 + 0.82 + 0.82 = 2.42

 

[Jabbu]

I just showed you that a local hidden variables theory can reproduce exactly the predictions of QM for a single filter setting (both Alice and Bob have the filters at the same angle). So what you're saying is false: the predictions for a single angle does not demonstrate that anything nonlocal is going on. In contrast, when you have 3 filter settings to choose from, it is not possible to reproduce the predictions of QM by any local theory.

What you showed are some statements that do not correspond to anything I've seen anywhere else. You used no any equations, you just asserted "we can prove" without showing any proof.

I don't know why you keep bringing up Malus' law. It is completely useless in predicting correlations for the case of entangled photons, because you can't apply it unless you know the photon polarizations.

Because that's how classical physics prediction is calculated. What do you think is classical physics prediction for theta = 0 degrees?