Photon entanglement: why three angles?

[johana]

Switching angles at the last possible minute does relate to local or classical explanations in the sense that it proves that there are no such explanations. If you DON'T switch, then that leads to the possibility that the settings affect the hidden variable in a local way.

Varying is only necessary if you are asserting that the measurement devices are (or might be) communicating with each other so as to affect the outcome of the correlation tests.

Do you disagree with this? Are you suggesting supposed communication between distant measurement devices is a local theory?

 

[DrChinese]

Do you disagree with this? Are you suggesting supposed communication between distant measurement devices is a local theory?

When you say "distant measurement devices" in this context, we all know that means devices that are sufficiently far apart so that no signal could move from one to the other at a speed less than c so that Alice's setting could be transmitted to Bob (and vice versa). A typical batch of observations takes a few minutes. So that only helps if Alice and Bob are FAR removed from each other because after 1/20 of a second, there is no place on Earth that far removed. So you must randomly select and change settings very fast: fast switching.

To put things in context: a nanosecond is about a foot. Bell tests are usually done perhaps 5 to 500 feet apart (although much larger distances have been done too). And PDC pair production is on the magnitude of 1,000-10,000 per second.

The idea of fast switching was to PROVE there could be no communication between distant measuring devices via some sub-c mechanism which did not fit into any theory at all. If you can imagine, the local realist was trying to say: "There COULD be something local occurring that QM does not contemplate." Fast switching proves that wrong.

 

[johana]

We all know that the Bohmian group (de Broglie-Bohm, Bohmian Mechanics) and several others are explicitly non-local. So I call anything that is not EXPLICITLY non-local to be "non-realistic" by definition (to comply with Bell). That would then include: Many Worlds, Relational Blockworld (ask RUTA about that), Cramer's Absorber, Aharanov's Time Symmetric QM, and a few others.

I'm not sure "non-realistic" is adequate substitute for "non-local". How about Newton's gravity, would you say it's non-local just because interactions are instantaneous?

 

[DrChinese]

I'm not sure "non-realistic" is adequate substitute for "non-local". How about Newton's gravity, would you say it's non-local just because interactions are instantaneous?

"Non-realistic" is not a substitute for "non-local" in any sense I am aware.

Traditional Newtonian gravity is non-local, yes definitely.

 

[johana]

The idea of fast switching was to PROVE there could be no communication between distant measuring devices via some sub-c mechanism which did not fit into any theory at all. If you can imagine, the local realist was trying to say: "There COULD be something local occurring that QM does not contemplate." Fast switching proves that wrong.

The problem is that it's usually described as an essential part of the experiment, which makes it far from obvious it's just there to close some loophole no one even cares about.

Traditional Newtonian gravity is non-local, yes definitely.

Classical electromagnetism too. So the whole of classical physics was already non-local even before there was any QM. Why are we surprised then? What is different between QM non-locality and classical physics non-locality?

 

[stevendaryl]

Classical electromagnetism too. So the whole of classical physics was already non-local even before there was any QM. Why are we surprised then? What is different between QM non-locality and classical physics non-locality?

I wouldn't say that classical electromagnetism was nonlocal. Classical electromagnetism is the paradigm example of a local theory---no effect can propagate faster than the speed of light.

 

[johana]

I wouldn't say that classical electromagnetism was nonlocal. Classical electromagnetism is the paradigm example of a local theory---no effect can propagate faster than the speed of light.

I wouldn't call it non-local myself, but Coulomb and Lorentz force equations assume instantaneous interaction over distance just like Newton's law of gravity. Whether instantaneous action over distance indeed implies "non-local", I'm not sure, that's the question.

 

[billschnieder]

Fast random switching has nothing to do with non-locality. It is done *only* to eliminate the possibility that Alice and Bob have conspired in advance to manipulate the results. As to the topic question, "Why 3 angles". Actually almost no experiments measure 3 angles. They measure 4. And that is because nobody uses the original Bell's inequalities, they all use the CHSH version which is based on 4 angles. So the reason why they use 3 angles rather than just 2 is because they are not primarily trying to test the QM prediction, but also to test the inequality which was derived using 3 angles (Bell) or 4 angles (CHSH).

BTW: I do not share the believe that those inequalities actually apply to the experiments they are being used for, as has been discussed many times here already, and in the literature.

 

[Avodyne]

Coulomb and Lorentz force equations assume instantaneous interaction over distance just like Newton's law of gravity. Whether instantaneous action over distance indeed implies "non-local", I'm not sure, that's the question.

The complete EM interaction is not instantaneous. If two stationary charges are one light-year apart, and then you move one of them, the total force felt by the other charge does not change until one year later. The same is true of the gravitational force in general relativity.

What is different between QM non-locality and classical physics non-locality?

See this article by David Mermin from 1985: https://cp3.irmp.ucl.ac.be/~maltoni/PHY1222/mermin_moon.pdf
Begin with "A gedanken demonstration" on page 4.

 

[DrChinese]

What is different between QM non-locality and classical physics non-locality?

As a matter of convention, "classical physics" usually considers General Relativity rather than Newtonian gravity, making c fundamental. Further, causes precede effects and the observer does NOT have a fundamental role in defining reality ("the moon is there when no one looks").

We now realize that this neat and pretty picture of our universe is not accurate. Of course, beauty is in the eye of the beholder.

 

[johana]

Fast random switching has nothing to do with non-locality. It is done *only* to eliminate the possibility that Alice and Bob have conspired in advance to manipulate the results.

Is it about determinism, choice and free will? Is it actually a part of the inequality derivation?

BTW: I do not share the believe that those inequalities actually apply to the experiments they are being used for, as has been discussed many times here already, and in the literature.

There you said this:

In other words, as Alice or Bob rotates their polarizers, the coincidence counts change.

Is that coincidence count the same thing as the number of matching pairs? Isn't it supposed to change as Alice or Bob rotate their polarizers, what's your objection about?

 

[DrChinese]

Is that coincidence count the same thing as the number of matching pairs? Isn't it supposed to change as Alice or Bob rotate their polarizers, what's your objection about?

You are asking billschnieder's viewpoint. As he and I have had many discussions about this point, I will pass this on - and which you should take as a fair summary:

1. billschnieder is a local realist. Post-Bell, local realism is generally not considered viable.

2. Discussing the pro's and con's of local realism is outside of the scope of this thread. If you want to discuss his viewpoint and reasoning further, that should be in a new thread.

3. Even in a new thread, you should be aware that this is a moderated forum in which generally accepted science is discussed. No one has the right to put forth their own personal opinions when such viewpoint does not have suitable references to support same. You can check the forum guidelines for details, but that policy is enforced.

 

[billschnieder]

Is it about determinism, choice and free will? Is it actually a part of the inequality derivation?

Neither. It is only for avoiding a conspiracy in which Alice and Bob freely chose in advance to manipulate the results and fool everyone else.

Is that coincidence count the same thing as the number of matching pairs? Isn't it supposed to change as Alice or Bob rotate their polarizers, what's your objection about?

Yes, coincidence count means the same as number of matching pairs. Trick question: is coincidence a local result, Why can't you use coincidence to send information?

As to the other question, you can find my answer here.

 

[johana]

Neither. It is only for avoiding a conspiracy in which Alice and Bob freely chose in advance to manipulate the results and fool everyone else.

Why are there three angles in the derivation then?

Yes, coincidence count means the same as number of matching pairs. Trick question: is coincidence a local result, Why can't you use coincidence to send information?

Because each side receives only half of the whole information?

 

[stevendaryl]

Why are there three angles in the derivation then?

Because three works and two doesn't.

 

[billschnieder]

Why are there three angles in the derivation then?

Because the particles come in pairs and the "magic trick" requires talking about outcomes we did not measure but could have, so we need at least 3 angles.

Because each side receives only half of the whole information?

Yes, because you need information from both sides to determine coincidences (ie, coincidence is "nonlocal" information). Isn't that the same reason why you can't use entanglement or "nonlocality" to transmit information? I'll leave it up to the reader to figure out the implications.

 

[DrChinese]

Why are there three angles in the derivation then?

Please check out my post #6 in this thread. I explain in detail why 3 angles are needed (as stevendaryl states in the previous post). You can also do with 4 or more. Don't be confused about 3 angles vs 2 measurements. That is not an issue in the science of this.

Keep in mind the local realist position: particle attributes exist and are well defined at all times, immune from the changes of other particles at some distance. That means one photon has many predetermined "elements of reality" (to use EPR wording). They must be predetermined because they can be predicted with certainty, the logic goes, even though they cannot ALL be predicted with certainty simultaneously. Read EPR and you will see this explicitly stated.

When you compare the possible values of those elements of reality for 3 angles, you realize that no ensemble of them can reproduce the Malus relationship (we are still talking about a single stream of photons, not pairs). Since the reality of particle attributes must be subjective in some respect (dependent on the nature of measurements made and NOT fully predetermined), our premises fail.

 

[billschnieder]

Keep in mind the local realist position: particle attributes exist and are well defined at all times, immune from the changes of other particles at some distance.

Yes. Local realists make a clear distinction between particle attributes and observables in an experiment involving particles.

That means one photon has many predetermined "elements of reality" (to use EPR wording). They must be predetermined because they can be predicted with certainty

Yes, one photon has many real attributes, but the outcome if a measurement which also includes a measuring device can not be said to "belong" to the photon. It belongs to the whole experimental setup. It can not be said to exist before the experiment has been done, even if the particle attributes do exist before the experiment. This is the local realist view. We've discussed this previously here

When you compare the possible values of those elements of reality for 3 angles, you realize that no ensemble of them can reproduce the Malus relationship (we are still talking about a single stream of photons, not pairs).

You probably mean values of observables and not values of particle attributes. But what malus relationship for a single stream of unpaired photons at 3 angles ??

 

[DrChinese]

Yes, one photon has many real attributes, but the outcome if a measurement which also includes a measuring device can not be said to "belong" to the photon. It belongs to the whole experimental setup. It can not be said to exist before the experiment has been done, even if the particle attributes do exist before the experiment.

This is a cockamamie description of the EPR viewpoint. (And we do not need to hear your view, since it is not a generally accepted viewpoint.) The combo of the photon attributes AND the measuring device is an ELEMENT OF REALITY in the EPR local realist view. That is because the outcome of ANY measurement can be predicted with certainty PRIOR to actually performing that measurement. Further, EPR explicitly says that it is not reasonable to require all possible outcomes to be simultaneously predictable. Of course, that is their definition of realism. 1 angle, 2 angles, 3 angles, 360 angles, they all are pre-existing to EPR. The measurement device itself plays a role, sure, but that role must be very limited to get the same answer every time. If it added some element of randomness, we wouldn't be able to predict the outcome in advance with certainty.

Bill, stick with the straight historical interpretation of EPR/Bell/Aspect. Don't derail the thread with your pet ideas, or the outcome will be the same as the other times.

 

[Dale]

I wouldn't call it non-local myself, but Coulomb and Lorentz force equations assume instantaneous interaction over distance just like Newton's law of gravity.

Coulombs law is not a law of classical EM. The actual laws of classical EM differ from Coulombs law to prevent instantaneous action at a distance.

 

[DrChinese]

But what malus relationship for a single stream of unpaired photons at 3 angles ??

A single stream of photons, not pairs Bill. A stream from Alice alone. Those photons, according to the view espoused by EPR, have particle attributes independent of the act of observation. At all angles, say 0, 120 and 240 degrees. The polarization attributes of ANY photon stream polarized some way at 0 degrees has a % polarization relationship discovered by Malus at any other angle such as 120 or 240 degrees.

Pairs have nothing to do with this. The pairs vs triples thing is simply a ruse you execute to confuse others. Stop and instead, please assist others with the standard program. You have enough knowledge to help. Everyone has their own pet ideas, but they are not welcome here as you well know from past experience.

 

[billschnieder]

A single stream of photons, not pairs Bill. A stream from Alice alone. Those photons, according to the view espoused by EPR, have particle attributes independent of the act of observation. At all angles, say 0, 120 and 240 degrees. The polarization attributes of ANY photon stream polarized some way at 0 degrees has a % polarization relationship discovered by Malus at any other angle such as 120 or 240 degrees.

Pairs have nothing to do with this. The pairs vs triples thing is simply a ruse you execute to confuse others. Stop and instead, please assist others with the standard program. You have enough knowledge to help. Everyone has their own pet ideas, but they are not welcome here as you well know from past experience.

Please calm down and read my question again, I'm simply asking you to elaborate what you mean. You said

When you compare the possible values of those elements of reality for 3 angles, you realize that no ensemble of them can reproduce the Malus relationship (we are still talking about a single stream of photons, not pairs).

So I asked you what malus relationship are you talking about which involves a single stream of unpaired photons at 3 angles? Are you saying no ensemble of photons can reproduce the classical malus law? But malus law involves 2 angles not three so you will have to explain what you mean because it is not clear from your statement. No one other than you has mentioned pairs vs triples etc, and I'm not sure what alleged pet idea has you riled up. I'm talking pretty standard stuff here. Everyone knowledgeable in this field knows the difference between $\lambda$ and $A,B$, The former are the hidden variables which are claimed to exist prior to measurement, while the latter are the observables which only exist after measurement. Don't confuse the two as you appear to be doing.

 

[DrChinese]

So I asked you what malus relationship are you talking about which involves a single stream of unpaired photons at 3 angles? Are you saying no ensemble of photons can reproduce the classical malus law? But malus law involves 2 angles not three so you will have to explain what you mean because it is not clear from your statement.

Simple, and I am referring to EPR as a starting point. The following is not the view of QM.

Any set of Alice's photons (a stream) has polarization at all angles independent of the act of observation. That is because the polarization can be predicted in advance by looking at matching Bob (in the ideal case of course). Those angles would include the 3: 0/120/240 degrees.

1. According to Malus, the statistical match rate M() between any two of those angles (of Alice) is 25% (cos^2(theta or 120 degrees difference in this case). And further: M(0,120) = M(120,240) = M(0,240).

2. Since Alice and Bob are polarization clones (demonstrated by the perfect correlations), we can measure any element of Alice by measuring Bob. This allows us to accurately determine 2 simultaneous elements of Alice - one by measuring Bob, the other by measuring Alice. This would even allow us to know more than the HUP allows (this was the EPR reasoning).

3. So we now know Alice's match rate for any of the 3 pairs of angles of Alice. Since the nature of our observation, BY DEFINITION, cannot change the underlying reality, it does not matter which of the three match rates we choose to observe, M(0,120), M(120,240) or M(0,240).

But there is no underlying data set of values which will satisfy Malus at all three sets of angles for the Alice stream, as required by 3. Ergo, one of our assumptions must be wrong. The only one added for local realism is the requirement that Alice have simultaneous polarization values independent of the act of observations (realism). So that must be false. Or, as EPR points out, there is spooky action at a distance.

 

[billschnieder]

Simple, and I am referring to EPR as a starting point. The following is not the view of QM.

Any set of Alice's photons (a stream) has polarization at all angles independent of the act of observation. That is because the polarization can be predicted in advance by looking at matching Bob (in the ideal case of course). Those angles would include the 3: 0/120/240 degrees.

1. According to Malus, the statistical match rate M() between any two of those angles (of Alice) is 25% (cos^2(theta or 120 degrees difference in this case). And further: M(0,120) = M(120,240) = M(0,240).

2. Since Alice and Bob are polarization clones (demonstrated by the perfect correlations), we can measure any element of Alice by measuring Bob. This allows us to accurately determine 2 simultaneous elements of Alice - one by measuring Bob, the other by measuring Alice. This would even allow us to know more than the HUP allows (this was the EPR reasoning).

3. So we now know Alice's match rate for any of the 3 pairs of angles of Alice. Since the nature of our observation, BY DEFINITION, cannot change the underlying reality, it does not matter which of the three match rates we choose to observe, M(0,120), M(120,240) or M(0,240).

But there is no underlying data set of values which will satisfy Malus at all three sets of angles for the Alice stream, as required by 3. Ergo, one of our assumptions must be wrong. The only one added for local realism is the requirement that Alice have simultaneous polarization values independent of the act of observations (realism). So that must be false. Or, as EPR points out, there is spooky action at a distance.

You have a single stream of Alice's photons, there is no such thing as match rate for a single photon. What is matching what?

 

[johana]

Because three works and two doesn't.

Can you be more specific, work towards what goal? Are you talking about some equation, some law of physics, mathematics or logic?

 

[johana]

Because the particles come in pairs and the "magic trick" requires talking about outcomes we did not measure but could have, so we need at least 3 angles.

What is "magic trick", some equation? It requires 3 angles to achieve what goal?

 

[Nugatory]

Can you be more specific, work towards what goal? Are you talking about some equation, some law of physics, mathematics or logic?

From much earlier in this thread: https://www.physicsforums.com/showpost.php?p=4836252&postcount=3

We're trying to set up a situation in which the quantum mechanical prediction differs from any local hidden-variable theory that might have satisfied the EPR trio. That's the goal.

Bell's theorem shows that certain three-angle setups will work for that purpose.

 

[billschnieder]

What is "magic trick", some equation? It requires 3 angles to achieve what goal?

For 3 angles $a,b,c$ with outcomes $A,B,C$ each of which can be +1 or -1, you can do the following algebra
$AB - AC = A(B - C)$
Remembering that $BB = 1$
$A(B - C) = A(B - BBC) = AB(1 - BC)$
therefore
$AB - AC = AB(1 - BC)$
Taking absolute values
$|AB - AC| \leq |AB||(1 - BC)|$
since $|AB| = 1$ and $(1 - BC)$ is always positive anyway
$|AB - AC| \leq (1 - BC)$
and therefore
$|AB - AC| + BC \leq 1$
This is a Bell inequality.

Notice the absence of "locality" or "realism" in the above derivation. The "magic trick" is how we started with just AB and AC, and all of a sudden you have BC in the final expression. You can't do this trick without a third angle.

 

[DrChinese]

You have a single stream of Alice's photons, there is no such thing as match rate for a single photon. What is matching what?

Apparently you do not understand a basic application of Malus, circa 1809.

A stream of Alice photons polarized at 0 degrees as + will have a 25% chance of being polarized + at 120 degrees. A stream of Alice photons polarized at 120 degrees as + will have a 25% chance of being polarized + at 240 degrees. A stream of Alice photons polarized at 0 degrees as + will have a 25% chance of being polarized + at 240 degrees. So if it passes the polarizer, it is matched.

To the EPR local realist, a single photon has polarization properties at all angles which are definite at all times independent of the act of observation. The classical relationship between these values was determined long ago to be statistical in nature (a la Malus). That there is no possible dataset that could account for this was never considered because it was not clear that the polarization would be pre-determined at all possible angles. The advent of entanglement, as pointed out in EPR, to add this critical point.

Of course, EPR intended to provide a counter-example to the HUP to disprove the completeness of QM. They didn't realize that QM's observer dependent predictions would upset their apple cart, so to speak.

 

[DrChinese]

What is "magic trick", some equation? It requires 3 angles to achieve what goal?

You really need to read or understand Bell's Theorem, which reveals the "magic trick". You can find it here, although it is in a form which is a lot more difficult to follow than most lay derivations:

On the Einstein Podolsky Rosen paradox
http://www.drchinese.com/David/EPR_Bell_Aspect.htm

It explains everything, see his [14] where the third angle is introduced. Or see another of my Bell derivations that shows the impossibility of certain local realistic predictions (specifically a negative probability) using a modified form of the Bell reasoning:

Bell's Theorem and Negative Probabilities
http://www.drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

You have been given the explanation in lay terms here. But there is no shortcut to the understanding of the 3 angles beyond what has been presented already. You must work it through at some point yourself.

 

[stevendaryl]

Can you be more specific, work towards what goal? Are you talking about some equation, some law of physics, mathematics or logic?

Okay, one version of the EPR experiment uses spin-1/2 particles: Through some process, an electron-positron pair is created and it is found that for any direction $\vec{a}$, if the electron is measured to be spin-up in direction $\vec{a}$, then the corresponding positron will measured to be spin-down in that direction. So the hypothesis is that for each electron produced, and for each possible direction $\vec{a}$, it is somehow pre-determined whether the electron is spin-up or spin-down in that direction.

What this hypothesis means is that associated with the $n^{th}$ electron/positron pair, there is a function $F_n(\vec{a})$ that returns $+1$ if the electron has spin up in direction $\vec{a}$ and returns $-1$ if the electron has spin-down in that direction. The corresponding function for the positron is just the negative of $F_n$.

What Bell's theorem shows is that there is no such function. Or rather, that no such function can possibly reproduce the predictions of quantum mechanics.

We can make the problem discrete by considering, not the full range of vectors $\vec{a}$, but some finite set of $M$ possibilities: $\vec{a}_1, \vec{a}_2, ..., \vec{a}_M$. Let $R_{i,j}$ be $F_i(\vec{a}_j)$. So $i$ refers to which electron/positron pair, and $j$ refers to which direction its spin is measured with respect to.

Then the question of hidden variables becomes the question of whether it is possible to fill in the values $R_{i,j}$ of a $N\times M$ matrix such that:

1. For each $i$ and $j$, $R_{i,j}$ is either +1 or -1.
2. For a fixed $j$ (that is, a fixed choice of direction $\vec{a}_j$), the average value of $R_{i,j}$ over all possible $i$ is 0. (Just as many spin-up as spin-down.)
3. For any pair of directions $\vec{a}_j$ and $\vec{a}_{j'}$, the average over all $i$ of $R_{i,j} R_{i, j'}$ is the quantum prediction of $\frac{1}{2}(cos^2(\frac{\theta_{j, j'}}{2}) - sin^2(\frac{\theta_{j,j'}}{2}))$, where $\theta_{j,j'}$ is the angle between $\vec{a}_j$ and $\vec{a}_{j'}$.

So the "one angle" versus "two angle" versus "three angle" is just this:

• It's always possible to fill in a one-column matrix (and satisfy the above rules)
• It's always possible to fill in a two-column matrix (and satisfy the above rules).
• For certain choices of directions $\vec{a}_j$, it is impossible to fill in a matrix with 3 or more columns (and satisfy the above rules).

 

[johana]

For 3 angles $a,b,c$ with outcomes $A,B,C$ each of which can be +1 or -1, you can do the following algebra
$AB - AC = A(B - C)$
Remembering that $BB = 1$
$A(B - C) = A(B - BBC) = AB(1 - BC)$
therefore
$AB - AC = AB(1 - BC)$
Taking absolute values
$|AB - AC| \leq |AB||(1 - BC)|$
since $|AB| = 1$ and $(1 - BC)$ is always positive anyway
$|AB - AC| \leq (1 - BC)$
and therefore
$|AB - AC| + BC \leq 1$
This is a Bell inequality.

Notice the absence of "locality" or "realism" in the above derivation. The "magic trick" is how we started with just AB and AC, and all of a sudden you have BC in the final expression. You can't do this trick without a third angle.

I don't see absence of locality, but I see it's general, so if it is indeed true it should not be violated regardless of whether data came from QM experiment, classical experiment, or from my dream.

Can you show an example QM dataset that can violate that inequality?

 

[stevendaryl]

I don't see absence of locality, but I see it's general, so if it is indeed true it should not be violated regardless of whether data came from QM experiment, classical experiment, or from my dream.

Can you show an example QM dataset that can violate that inequality?

In an EPR experiment, there are two particles produced, and for each particle, you get one opportunity to measure the spin relative to some angle. So in each "run" of the experiment, you only get the results of 2 angles.

So in terms of the matrix that I mentioned, that means that if you have 3 possible angles, then you have to fill in a 3-column matrix. But experimentally, you only test 2 values. So for each row, you only can fill in 2 of the three columns by experimental values. The third matrix element must be left blank.

Bell's inequality shows that there is no way to fill in the "blanks" by values in a way that satisfies the predictions of QM.

 

[stevendaryl]

In an EPR experiment, there are two particles produced, and for each particle, you get one opportunity to measure the spin relative to some angle. So in each "run" of the experiment, you only get the results of 2 angles.

So in terms of the matrix that I mentioned, that means that if you have 3 possible angles, then you have to fill in a 3-column matrix. But experimentally, you only test 2 values. So for each row, you only can fill in 2 of the three columns by experimental values. The third matrix element must be left blank.

Bell's inequality shows that there is no way to fill in the "blanks" by values in a way that satisfies the predictions of QM.

So in terms of Bill's notation, for every round of the EPR experiment, you can only learn the values of two of the three quantities $A, B, C$. So it's not really a dataset violating Bell's inequality. It's a partial dataset which cannot possibly be made complete.

 

[johana]

So in terms of Bill's notation, for every round of the EPR experiment, you can only learn the values of two of the three quantities $A, B, C$. So it's not really a dataset violating Bell's inequality. It's a partial dataset which cannot possibly be made complete.

I need to confirm what exactly is meant by "angle", "dataset", and "partial dataset". Say Alice and Bob can turn their polarizers to 0, 20, and 30 degrees, and we are testing for these three combinations:

a= (0,20) = 20°
b= (30,0) = 30°
c= (30,20) = 10°

With relative angle a = 20° we get for example this dataset A = --, +-, ++, -+, ++
With relative angle b = 30° we get for example this dataset B = +-, ++, -+, -+, +-
With relative angle c = 10° we get for example this dataset C = ++, -+, +-, -+, --

Correct? What partial dataset are you talking about?