Questions on Bell's Theorem

[AbstractDreamer]

At first I would like to ask if i understand this topic.

 

ref: http://www.scienceforums.net/topic/87347-why-hidden-variables-dont-work/

 

https://youtu.be/ZuvK-od647c

 

So Bell's Theorem essentially claims to disprove the existence of hidden local variables in entangled photons/electrons; and it concludes that action-at-a-distance is present (or superdeterminism, global variables). Per experiment, by "repeating the procedure over and over" (4:38) and considering the expected unequal distribution of frequencies (6:31) as if there were hidden variables (Bell's inequalities) or "hidden plans" (5:00), and comparing them to the actual recorded distribution of results that are obtained (6:37), these results show that Bell's inequalities are violated. Experimentally, these actual distributions consistently violate Bell's inequalities and also consistently follow Quantum Mechanics "action at a distance".

 

So my problem is with how these hidden variables are modeled (and thus seemingly always in disagreement with results). They seem to be modeled classically, as if there is some explicit agreement between entangled pairs to precisely (classical precision) what values to hold. That is, for instance, electronA "plans" with electronB: "if we are measured like <such and such>..." (in particular, explicit but different directions) "..then I will show definitely UP and you will show definitely UP" or another plan such as "if we are measured in the same direction, then I will show definitely UP and you will show definitely DOWN". Then the experimenter exposes these plans through frequency analysis of all the possible explicit combinations of hidden variables (6:08), and finds that no such plans can exist.

 

But these electrons are modeled classically as if their hidden variables must explicitly and definitively describe one state or other in a certain measurement. If there is anything I have learnt, it is that before anything is measured nothing is certain (superposition, wavefunction etc). That includes any "hidden plans" or "hidden variables" that entangled electrons might be have.

 

So my conclusion then is that why is it not possible for entangled pairs to have uncertain hidden variables, and that these variables are orientated, symmetrical, opposing, and probabilistic in nature? The orientation allows the pair to agree on a frame of reference with respect to direction (actually this variable need not be uncertain or implicit). The symmetry ensures all arbitrary orientations are equivalent. The opposing nature ensures that when a measurement is made in the same direction, it is certain that the result will be opposite. The probabilistic nature of the hidden variables is such that the distribution of frequencies that the experimenters see are a direct measure of this probable nature! E.G.

 

Along any arbitrary orientation/axis/dimension/direction/pole, one entangled electron could have a variable:

[math]

P_{up}=\cos ^2\left(\frac{\alpha }{2}\right) P_{down}=\sin ^2\left(\frac{\alpha }{2}\right)

[/math]

and its entangled partner would have

[math]

P_{up}=\sin ^2\left(\frac{\alpha }{2}\right) P_{down}=\cos ^2\left(\frac{\alpha }{2}\right)

[/math]

Where [math] \alpha [/math] is the angle of measure relative to the axis, and P is the probability of being measured in that state.

 

Given that the pair agree on orientation, they are inherently, mutually, symmetrically, opposingly certain when measured along any same arbitrary axis; and internally, symmetrically, opposingly probabilistic otherwise.

 

That is, arbitrarily orientated, the hidden plan could be:

"if electronA is measured at [math]\alpha=0[/math] and electronB is measured at [math]\alpha=\frac{\pi}{3} [/math] then electronA will show 100% UP 0% DOWN and electronB will show 25% DOWN 75% UP".

They could show both UP if measured at such angles. After many measurements, the individual discrepancies balance out and a probability pattern emerges; and it is this probability that experimenters are comparing to frequency distribution expectations (as long as they are opposing when they must be opposed, such as when measured in the same direction).

 

"if electronA is measured at [math]\alpha=0[/math] and electronB is measured at [math]\alpha=\pi [/math] then electronA will show 100% UP 0% DOWN and electronB will show 0% DOWN 100% UP"

"if electronA is measured at [math]\alpha=\frac{\pi}{4}[/math] and electronB is measured at [math]\alpha=\frac{\pi}{2} [/math] then electronA will show 85.35% UP 14.64% DOWN and electronB will show 50% DOWN 50% UP"

 

That is, the electron's themselves do not know precisely what state/value they will be measured at - there's no explicit plan/variable. But they might have an implicit hidden function/variable that tells them how likely they will be measured in any state relative to a given orientation. And it is this likeliness, over repeated measurements, that consistently violates Bell's inequalities! Over many electron pairs, the recorded distribution of frequencies simply describe the hidden probability function and not any hidden explicit values.

 

So to me, Bell's inequality of expected frequency distribution only apply when the hidden variables are explicit. If the hidden variables are implicitly described through a probability function, then violation of Bell's inequality is only proof against explicit hidden variables and not proof against implicit hidden variables.

 

In other words, when we explicitly list all the possible states and calculate the expected frequencies (Bell's inequalties), we are inadvertently collapsing the probability distribution as described by the hidden state function, which in turn would naturally lead to consistent violations with experimentation, as is apparently the case. This is akin to listing all the possible paths of a photon through a double slit, calculating the expected distribution on the detector (particle-like distribution), and then declaring the interference patterns we consistently see are violating the expected distribution.

 

So an orientated, symmetrical, opposing, and probabilistic hidden state function seems to preserve Quantum Mechanics AND Locality!

 

Einstein would be proud!

 

Ok, I'm ready for my schooling.

 

PS This is not intended as speculation, rather this is likely confusion on my part. Please correct me.

 

Just to clarify:

• I'm not questioning the mathematical derivation of the inequalities.
• I'm not questioning the accuracy of the experimental methods.
• I'm not questioning the results of the experiments.
• I'm not questioning any arguments about loopholes.
• I'm not questioning Quantum Mechanics predictions.

I'm merely querying whether it is appropriate to apply the inequalities to the situation in the experiments, that is, Bell's Theorem.

Edited December 15, 2016 by AbstractDreamer

[imatfaal]

...

I'm merely querying the whether it is appropriate to apply the inequalities to the situation in the experiments, that is, Bell's Theorem.

 

In answering EPR question then yes it was entirely appropriate. The position was that a predetermined but hidden local variable could reproduce the effects of QM without the need for spooky action at a distance. I need to think more about your ideas regarding probabilistic local variables.

 

FYG In your example, from a first (and very hungover) glance, you seem to be conflating electron spin (measured by Stern-Gerlach etc - with entangled being up / down on a chosen axis) and photon polarization (tested by transmission at a polarizer - and entangled normally being orthogonal)

[AbstractDreamer]

Honestly, I'm not aware of the precise details of the different types of experiments. All I have is the pop-science material that is easy to hand such as You-tube, and various linked websites I found. I'm certainly conflating material that I've come across. My ideas only really apply to the experiment presented in You-Tube video at the top of my post, but I hope to have put enough detail in to get my point across. Whether or not they can be generalised across the other experiments, I simply haven't done enough research to comment.

 

Thank you for helping.

 

PS i have just edited photons to electrons in my main post.

Edited December 15, 2016 by AbstractDreamer

[swansont]

If the state is truly undetermined, there is no hidden variable. That's what a hidden variable is — being in a defined state that is some way hidden from us, because we haven't done the measurement.

 

That is, arbitrarily orientated, the hidden plan could be:

"if photonA is measured at [math]\alpha=0[/math] and photonB is measured at [math]\alpha=\frac{\pi}{3} [/math] then photonA will show 100% UP 0% DOWN and photonB will show 25% DOWN 75% UP".

How is that different from normal QM?

[AbstractDreamer]

Im not sure if the following analogy is appropriate:

 

The state of the electron, is analogous to the location of a particle in the double-slit-experiment.

The hidden variable (state-function) is analogous to the wave function.

The state is determined probabilistically by the state-function and orientation.

Analogous to the location determined probabilistically by the wave-function and direction.

 

So call it a hidden variable or call it an discoverable function. The point is QM and Locality are not mutually exclusive.

 

How is that different from normal QM?

 

I don't think it is different, I'm not sure. I used information in the video and interpreted it. The video presents it differently - confusingly. However I cant be certain that my formulas would give the same results as QM. They were just loose examples to demonstrate the idea of a probabilistic function. But then why does QM imply action-at-a-distance?

Edited December 15, 2016 by AbstractDreamer

[Eise]

AbstractDreamer,

 

I think you just repeat the same phenomenon one metaphysical level deeper. If the local function is also indeterminate, then you have gained nothing.

 

I even think that you must think classically here: by doing so, you see that our classical concepts do not work anymore, i.e. our classical concept that causes must be local and determinate.

 

Of course you know that metaphysics is not physics.

 

My 2 cents.

[swansont]

Im not sure if the following analogy is appropriate:

 

The state of the electron, is analogous to the location of a particle in the double-slit-experiment.

The hidden variable (state-function) is analogous to the wave function.

The state is determined probabilistically by the state-function and orientation.

Analogous to the location determined probabilistically by the wave-function and direction.

 

So call it a hidden variable or call it an discoverable function. The point is QM and Locality are not mutually exclusive.

A hidden variable is one that determines the state of the particle before measurement, even though we don't know this. If the particle is in that state, you get different results than if the particle's state was not determined.

 

 

I don't think it is different, I'm not sure. I used information in the video and interpreted it. The video presents it differently - confusingly. However I cant be certain that my formulas would give the same results as QM. They were just loose examples to demonstrate the idea of a probabilistic function.

Compare them to the QM calculation. But generally we don't care much about measurements as in your example. i.e. you want the detectors with the same orientation.

 

But then why does QM imply action-at-a-distance?

Because you can measure the particle states when they are separated by a large distance, and see the correlation. The distance of separation can be larger than ct, there t is the time difference of measurement (e.g. they are measured within 3 nanoseconds of each other, but they are much more than a meter apart)

[AbstractDreamer]

If the local function is also indeterminate, then you have gained nothing.

The local function is not indeterminate. I have given an example of what it could be, though admitted not in any proper form (only as a probability). Only i have no idea if it fits QM results, its just a loose demonstration.

 

Along any arbitrary orientation/axis/dimension/direction/pole, one entangled electron could have a variable:

[math]

P_{up}=\cos ^2\left(\frac{\alpha }{2}\right) P_{down}=\sin ^2\left(\frac{\alpha }{2}\right)

[/math]

and its entangled partner would have

[math]

P_{up}=\sin ^2\left(\frac{\alpha }{2}\right) P_{down}=\cos ^2\left(\frac{\alpha }{2}\right)

[/math]

Where [math] \alpha [/math] is the angle of measure relative to the axis, and P is the probability of being measured in that state.

 

 

 

A hidden variable is one that determines the state of the particle before measurement, even though we don't know this.

Not sure I understand. The state is a probability function before measurement, so i guess in that sense its not a hidden variable. But what does this have to do with QM being incompatible with locality?

 

 

If the particle is in that state, you get different results than if the particle's state was not determined.

Again I don't quite understand. If the particle is measured to be in a certain state [uP], then clearly it could be a different result than if it was not determined (then it could then be [uP] or [DOWN]). Again, how does this make QM irreconcilable with locality

 

 

Compare them to the QM calculation. But generally we don't care much about measurements as in your example. i.e. you want the detectors with the same orientation.

Where can I review the QM calculations? The probabilities I describe above allows detectors in the same orientation. In such a situation [math]\alpha=0 [/math] or [math]\pi [/math] the entangled pairs will always be in opposing states.

"if electronA is measured at and electronB is measured at then electronA will show 100% UP 0% DOWN and electronB will show 100% DOWN 0% UP" , that is, they will certainly be opposed. Similary for [math] \alpha=\pi [/math]

 

For detectors at different angles: for any single pair of entangled waves/particles you can arbitrarily reset the orientation of the axis. If you don't measure either particle, then their states are not defined.

 

Because you can measure the particle states when they are separated by a large distance, and see the correlation. The distance of separation can be larger than ct, there t is the time difference of measurement (e.g. they are measured within 3 nanoseconds of each other, but they are much more than a meter apart)

You can similarly see the correlation with a <hidden> probabilistic function, no? That's what I'm trying to say, why do you need action-at-a-distance, if locality can preserved? Surely, I thought, given the two options, preserving locality is a preferable stance to action-at-a-distance?

Edited December 15, 2016 by AbstractDreamer

[swansont]

Not sure I understand. The state is a probability function before measurement, so i guess in that sense its not a hidden variable. But what does this have to do with QM being incompatible with locality?

If it's a hidden variable, it's not a probability.

 

Again I don't quite understand. If the particle is measured to be in a certain state [uP], then clearly it could be a different result than if it was not determined (then it could then be [uP] or [DOWN]). Again, how does this make QM irreconcilable with locality

If you measure UP, that doesn't mean that the state was UP before the measurement. You have to make multiple measurements to see this.

 

 

The locality argument is whether the correlation you see can occur faster than c. Since it can, the effect can't be local. i.e. there can't be communication between the particles that agrees with relativity.

 

 

Where can I review the QM calculations? The probabilities I describe above allows detectors in the same orientation. In such a situation [math]\alpha=0 [/math] or [math]\pi [/math] the entangled pairs will always be in opposing states.

"if electronA is measured at and electronB is measured at then electronA will show 100% UP 0% DOWN and electronB will show 100% DOWN 0% UP" , that is, they will certainly be opposed. Similary for [math] \alpha=\pi [/math]

 

For detectors at different angles: for any single pair of entangled waves/particles you can arbitrarily reset the orientation of the axis. If you don't measure either particle, then their states are not defined.

 

Again, that sounds exactly like the results QM already predicts. There's no hidden variable.

 

 

You can similarly see the correlation with a <hidden> probabilistic function, no? That's what I'm trying to say, why do you need action-at-a-distance, if locality can preserved? Surely, I thought, given the two options, preserving locality is a preferable stance to action-at-a-distance?

What is the difference between having a hidden probability function and a known one, if they give the same result? How would you test for this?

[AbstractDreamer]

The locality argument is whether the correlation you see can occur faster than c. Since it can, the effect can't be local. i.e. there can't be communication between the particles that agrees with relativity.

The information could have been available to both particles at all times (hidden probability function). Since it could have, why cant it be local? Why must information have to be communicated at the last moment instantaneously? Surely preserving locality agrees with relativity more than action-at-a-distance?

 

What is the difference between having a hidden probability function and a known one, if they give the same result? How would you test for this?

I guess the difference is the known one implies action-at-a-distance and faster-than-light information spanning the entire universe instantaneously; and the hidden probability preserves locality and agrees with relativity.

 

How would I test for this? I don't know I'm not a physicist. I watched a You-tube video and it was apparent to me there was something wrong with the application of Bell's Theory. So I tried to put my thoughts into something coherent, and ask people that know better.

Edited December 15, 2016 by AbstractDreamer

[swansont]

The information could have been available to both particles at all times (hidden probability function). Since it could have, why cant it be local? Why must information have to be communicated at the last moment instantaneously? Surely preserving locality agrees with relativity more than action-at-a-distance?

How do you transmit the information about the state if it's a probability, and hasn't been measured?

 

If it's a probability function and the particles are a km apart, how does the result remain local if the measurement is essentially simultaneous (i.e. in this case, less than a microsecond difference) One particle is measure as UP while the other one is DOWN. That determination only happens at the time of measurement, even in your scenario. No signal traveling at c or slower can be sent from one to the other.

[AbstractDreamer]

How do you transmit the information about the state if it's a probability, and hasn't been measured?

 

If it's a probability function and the particles are a km apart, how does the result remain local if the measurement is essentially simultaneous (i.e. in this case, less than a microsecond difference) One particle is measure as UP while the other one is DOWN. That determination only happens at the time of measurement, even in your scenario. No signal traveling at c or slower can be sent from one to the other.

 

I don't know, I'm trying not to speculate, because really I have not the slightest idea. I was hoping you would give me some ideas how this is possible, rather than the other way around. Surely I'm not the first to think along these lines.

 

But the most obvious way is that there is no transmission of information. Perhaps the information is created at the same time the entangled pair is created? Without any need for communication beyond the moment of separation/creation, each particle has the information it needs - the hidden probability function - ??stored in some inner dimension??

 

The measurement is only simultaneous, if you measure it at the same time. The probability information could be always there, whether a measure is made or not. An orientation variable (which could be local or global) and an internally local probability function is maybe all you need.

 

When a measurement is made in the same direction, which necessitates an opposing measure, then a global orientation variable might need to be referenced i guess. But that would not demand superdeterminism nor action-at-a-distance, because neither the global variable, nor the local probability function is solely responsible for determining the measure. I'm making stuff up now.

 

Alternatively, is it even possible to exactly measure direction to be the same to such precision as to violate conservation of energy, IF measurements made "very close" to the same direction were found to have the same spin?

Edited December 15, 2016 by AbstractDreamer

[swansont]

I don't know, I'm trying not to speculate, because really I have not the slightest idea.

And yet when I point out why things have to be a certain way, you are suggesting alternatives. You can't have it both ways.

 

You can't have a probabilistic system that is local if you get non-local results. The results of entanglement have been experimentally confirmed to be non-local. One has out a limit on the speed of any possible communication at 10,000c, IIRC.

 

That the results are non-local is not in question. Any proposal that requires locality is dead in the water.

 

The probability information could be always there, whether a measure is made or not. An orientation variable (which could be local or global) and an internally local probability function is maybe all you need.

It's not enough to have the probability information. If the probability of UP vs DOWN is 50%, how can you ensure that you don't get UP/UP or DOWN/DOWN, when the conservation law requires UP/DOWN or DOWN/UP? That correlation is why it's called spooky action at a distance. It's non-local.

[AbstractDreamer]

And yet when I point out why things have to be a certain way, you are suggesting alternatives. You can't have it both ways.

 

 

I'm not trying to have it both ways. Can you explain what you mean?

 

Frankly, its difficult to comprehend many of your responses, and how they might correlate to questions i had asked. At the same time, its hard to know whether you have misunderstood what I'm trying to say, or simply way ahead of the conversation. You certainly don't make it easy to follow your train of thought. On the other hand, I have tried to answer each of your responses directly, all the while having no idea what you really asked or what I'm really saying.

 

I'm suggesting alternatives in direct response to your questions.

 

How do you transmit the information about the state if it's a probability, and hasn't been measured?

 

how does the result remain local if the measurement is essentially simultaneous (i.e. in this case, less than a microsecond difference)

 

 

You can't have a probabilistic system that is local if you get non-local results.

Can you explain why non-locality falsifies probability information, or can you explain or show me where I can review the proof for this?

 

The results of entanglement have been experimentally confirmed to be non-local. One has out a limit on the speed of any possible communication at 10,000c, IIRC.

 

That the results are non-local is not in question. Any proposal that requires locality is dead in the water.

Had you said locality is dead in the water at the start, we might have saved a lot of time. However the probability function is not dependent on locality. Though i had hoped that locality was not dead in the water.

 

It's not enough to have the probability information. If the probability of UP vs DOWN is 50%, how can you ensure that you don't get UP/UP or DOWN/DOWN, when the conservation law requires UP/DOWN or DOWN/UP? That correlation is why it's called spooky action at a distance. It's non-local.

 

I explained possible solutions in my previous post and in my OP, in the attempt to reconcile locality. I Indeed i raised the question about the conservation law with respect to how accurately direction can be measured, which you haven't answered. How can you expect me to answer your questions if you do not answer mine? How can i do anything but make further speculations?

 

But if locality is dead in the water, I guess there's no point trying to explain how probability function might work with locality. On the other hand, if action-at-a-distance is the only alternative, then anything goes to be honest.

 

If there are any relevant sources of information that i can access, that would be useful.

I really wanted to stay on track with Bell's Theorem and why it might or might not be appropriate for it to pre-assign expected combinations of values to an entangled pair.

Also I would like to review the QM interpretation as you suggested in #7, and how it uses a known function to explain the violation of Bell's inequalities.

Edited December 15, 2016 by AbstractDreamer

[swansont]

I'm not trying to have it both ways. Can you explain what you mean?

 

Frankly, its difficult to comprehend many of your responses, and how they might correlate to questions i had asked. At the same time, its hard to know whether you have misunderstood what I'm trying to say, or simply way ahead of the conversation.

You are using vocabulary that suggests a certain level of familiarity with the material. If that's not the case, then there are going to be misunderstandings.

 

You use "hidden variable", for example, but then suggest things that indicate that you don't understand what a hidden variable is. Perhaps you should backtrack and ask more fundamental questions.

 

You certainly don't make it easy to follow your train of thought. On the other hand, I have tried to answer each of your responses directly, all the while having no idea what you really asked or what I'm really saying.

 

I'm suggesting alternatives in direct response to your questions.

But these alternatives are wrong, and have been shown to be wrong already. The basic Bell experiments have already been done.

 

 

 

Can you explain why non-locality falsifies probability information, or can you explain or show me where I can review the proof for this?

I've explained this already. Twice. Maybe you could ask specific questions about my answers.

 

I will try one more time. Once the measurement of the first particle has taken place, it dictates what the second particle's state is. You don't have a probability for the second particle anymore. But if the particles can be separated by some distance d > ct (t is the time difference between the measurements) then the result is nonlocal — it would require communication faster than c, if communication of their states is happening.

 

The way to get around that is to have the states determined beforehand, but that's what a hidden variable is, and the Bell tests show that these hidden variables don't exist. The particles really are in an undetermined state before the measurement.

 

 

Had you said locality is dead in the water at the start, we might have saved a lot of time. However the probability function is not dependent on locality. Though i had hoped that locality was not dead in the water.

Again, that's fallout from bringing up the topic as if you were looking for some clarification of a detail, rather than not really understanding the topic much. I had assumed your misconception was about what a hidden variable is, rather than being more widespread.

 

 

I explained possible solutions in my previous post and in my OP, in the attempt to reconcile locality. I Indeed i raised the question about the conservation law with respect to how accurately direction can be measured, which you haven't answered. How can you expect me to answer your questions if you do not answer mine? How can i do anything but make further speculations?

This post is the first time you've mentioned accuracy of direction. If you asked about it before, it was phrased in a way that it wasn't clear what you were asking.

 

The direction can be determined to a much greater accuracy than is needed to do the experiment. That's implied by the uncertainty of the results.

 

But if locality is dead in the water, I guess there's no point trying to explain how probability function might work with locality. On the other hand, if action-at-a-distance is the only alternative, then anything goes to be honest.

 

If there are any relevant sources of information that i can access, that would be useful.

I really wanted to stay on track with Bell's Theorem and why it might or might not be appropriate for it to pre-assign expected combinations of values to an entangled pair.

Also I would like to review the QM interpretation as you suggested in #7, and how it uses a known function to explain the violation of Bell's inequalities.

The video in the link in the OP goes through the basic calculations, IIRC.

[Mordred]

It would be easier to understand entanglement if one understands what occurs when two particles are entangled.

 

To get a better handle study quantum correlation.

 

https://en.m.wikipedia.org/wiki/Quantum_correlation

 

Essentially when two particles become entangled they are quantum correlated. No further is really needed. We don't know the state of either particle until we perform a measurement.

 

Once we measure one particle by its quantum correlation we know the state of the other. There is no hidden action at a distance, Here this should help.

 

http://www.google.ca/url?sa=t&source=web&cd=3&ved=0ahUKEwi48Z6T7fbQAhVX5mMKHRTsD-8QFgglMAI&url=http%3A%2F%2Fwww.theory.caltech.edu%2Fpeople%2Fpreskill%2Fph229%2Fnotes%2Fchap4.pdf&usg=AFQjCNH9C5y14C8VrFVmbmpbIQfb3GIXXg&sig2=4j6T_aO6yNjzzbY2pDAUrQ

 

Everyone has probably heard me stress the importance of terminology. Here is an excellent example.

 

"Action at a distance " now what is the physics definition of action ?

 

If you think about how action is defined there is no action between the entangled pairs when measurement takes place.

 

https://en.m.wikipedia.org/wiki/Action_(physics)

 

None is needed as they are already correlated pairs.

Edited December 15, 2016 by Mordred

[edguy99]

AbstractDreamer - My 2 cents. A real example of a HVT: Entangled photons, nonlocality and Bell inequalities in the undergraduate laboratory.

Modelling Entangled Photons

Dehlinger and Mitchell propose a model (color represents the probability of a photon getting through the filter) where each photon has a polarization angle λ. When a photon meets a polarizer set to an angle γ , it will always register as Vγ if λ is closer to γ than to γ + π/2, i.e.,

• if |γ − λ| ≤ π/4 then vertical
• if |γ − λ| > 3π/4 then vertical
• horizontal otherwise.

Refining the Model – adding Probability

This models the photons as not only having a specific “average” direction, but also as having a “wobble” or “instantaneous” direction (shading represents probability of a photon getting through the filter). Represented as icons, photons present a more “fuzzy” picture of their polarization. The sample 24° photon, with a 30° wobble, will most of the time be picked up as a vertical, but sometimes when the combined angle is over 45°, it will be picked up as a horizontal. To determine polarity, we use these equations.

• Chance of vertical measurement = (cos((γ − λ)*2)+1)/2
• Chance of horizontal measurement = (cos((γ − λ + π/2)*2)+1)/2

I believe this will indeed produce the proper numbers

BUT

carries a new problem. You will get what looks like a lot of noise when you measure at 45 degrees, since many of the split photons will not measure the same polarity because of the randomness. I suspect there is an experiment somewhere that would "close this loophole"?

[Mordred]

good post, covers the topic well. +1