Why the De Raedt Local Realistic Computer Simulations are wrong

[DrChinese]

In another thread, we were discussing Zonde's and the De Raedt's model for simulating Bell tests using a purported local realistic computer simulation. I have broken out into this new thread some results which will be interesting to those of you interested in this subject. I will start out by including a few relevant posts I made in that thread, and then you can skip to post #4 below to see my critique of the De Raedt models.

To give some quick background, the De Raedt simulation is intended to take the format of a typical Bell test using Alice and Bob and polarizing beam splitters. They then use a formula that acts independently on Alice and Bob to reproduce the quantum mechanical results. The idea is that independence of the formulas proves that there could be physical independence as well. If so, a local hidden variable program is possible - at least that is the claim.

 

[DrChinese]

Their (Fortran) code is at the end of http://rugth30.phys.rug.nl/pdf/COMPHY3339.pdf"

Can you help me decipher this statement:

k2=ceiling(abs(1-c2*c2)**(d/2)*r0/tau) ! delay time

this looks to me like:

k2=ceiling(abs(1-(c2*c2))**((d/2)*(r0/tau))) ! delay time

and since d=2 and static reduces to:

k2=ceiling( abs(1-(c2*c2))**(r0/tau) ) ! delay time

------------------------------------------------------------------------

After examining this statement, I believe I can find an explanation of how the computer algorithm manages to produce its results. It helps to know exactly how the bias must work. The De Raedt et al model uses the time window as a method of varying which events are detected (because that is how their fair sampling algorithm works). That means, the time delay function must be - on the average - such that events at some angle settings are more likely to be included, and events at other angle setting are on average less likely to be included. It actually does not matter what physical model they propose, because eventually they must all accomplish the same thing. And that is: the bias function must account for the difference between the graphs of the QM and LR correlation functions.

Which is simply that we want the difference between the LR correlation function and the QM correlation function to be zero at 0, 45, 90, 135 degrees. That is because there is no difference in the graphs at those angles. But there is a difference at other angles. That same difference must be positive and maximum at angles like 22.5, 157.5 etc, and be negative and minimum at angles like 67.5 and 112.5 etc. (Or maybe vice versa )

So we need an embedded bias function that has those parameters, and if their computer program is to work, we will be able to find it. Once we find it, we can then assess whether it truly models the actual experimental data. If we see it does, they win. Otherwise, they lose. Of course, my job is to challenge their model. First, I must find out how they do it.

So we know that their function must: i) alternate between positive and negative bias, ii) it must have zero crossings every 45 degrees (pi/4), and iii) it must have a period of 90 degrees (pi/2). It does not need to be perfect, because the underlying data isn't going to be perfect anyway. Any of this starting to look familiar? Why yes, that is just the kind of thing we saw in zonde's model.

 

[DrChinese]

So now, per my prior post on the De Raedt model:

Let's assume I can demonstrate how the bias function uses the delay to do its work (by affecting which events are within the time window and therefore counted). The next question is: does it model all of the data of relevant Bell tests? Well, yes and no. Obviously they claim to produce QM-like data as far as was reported - YES in this regard. But most likely we will see that the traditional Bell test experimenters did not consider this clever twist - some perhaps NO in some way. It should be possible to extend the actual experiments to show whether the De Raedt model is accurate or not. In fact, I believe I can show this without performing an experiment once I run their algorithm myself.

I think I can safely give the De Raedts an A for coming up with a simulation that works as it does. As I have said previously, a simulation which produces a QM-like result is NOT the same as a local realistic theory. So such a simulation - ALONE and BY ITSELF - is NOT a disproof of the Bell Theorem. Because there are additional consequences of any local realistic theory, and if those are not considered then it cannot be a candidate. Again, this is why Santos has failed with stochastic models.

 

[DrChinese]

HERE IS WHY THE DE RAEDT MODEL IS WRONG:

I looked at the computer simulation in some detail, and you can follow the link above to the code itself. It does in fact simulate the QM predictions, exploiting the time window/detection/unfair sampling methodology. It is common in actual Bell tests to match pairs of events using a relatively small time window. The choice of the window size determines which pairs of events are considered. They use a formula to simulate which pairs are considered, and that has the effect of creating an unfair (biased) sample. That sample then matches QM expectations, even though the full universe would not. If that actually occurred in the physical experiment itself, then it would hypothetically explain the QM results with a Local Realistic model.

Now, I have a lot of criticisms of their model. I will detail those as needed in our discussion as relevant. But what I am reporting now is that the model flat out is wrong. Here is why:

1. In the model, they do succeed in getting the Type II PDC simulation to yield results compatible with QM. They get an A for that.

2. However, those results require a polarization entangled input source. It is also possible to use their formula on a source which is NOT polarization entangled. A source which is NOT polarization entangled will not yield the QM expectation values, it will yield the Local Realistic expectation values. However, their simulation yields instead the same results as an entangled source.

3. There are several ways to get such a source as I describe. You can take a Type II PDC source and put an H filter over the Alice stream, and a V filter over the Bob stream. Or you can simply use a single Type I PDC crystal, instead of the usual 2. Either way, you have a source of paired photons which are not polarization entangled.

Therefore, their results yield the SAME expectation values regardless of whether or not the source is polarization entangled. When the actual experiement is performed in these two cases, the results are actually different. Therefore, the De Raedt model does not accurately model what is seen, while QM does.

Your comments are welcome.

 

[ajw1]

HERE IS WHY THE DE RAEDT MODEL IS WRONG:
...
2. However, those results require a polarization entangled input source. It is also possible to use their formula on a source which is NOT polarization entangled. A source which is NOT polarization entangled will not yield the QM expectation values, it will yield the Local Realistic expectation values. However, their simulation yields instead the same results as an entangled source.
...

I'm not sure what you mean. When I disable the entanglement relation (and thereby making the polarization for both photons random, I get this attached graph.

                //polarization relation
                //Particle2.Polarization = Particle1.Polarization + h.PiOver2; // polarization of particle 2

(for those who haven't read the other thread, I have changed the code from de Readt to more object orientated code, without changing the logic)

 

[DrChinese]

I'm not sure what you mean. When I disable the entanglement relation (and thereby making the polarization for both photons random, I get this attached graph.

                //polarization relation
                //Particle2.Polarization = Particle1.Polarization + h.PiOver2; // polarization of particle 2

Ah, sorry, that does NOT solve the problem at all. In fact, the De Raedts put that result in the following paper as well:

http://arxiv.org/abs/0712.2565

But that is not the problem I am referring to. In my point, the relation:

Particle2.Polarization = Particle1.Polarization + h.PiOver2; // polarization of particle 2

...holds. In other words, you cannot comment out that line! See my next post which will explain the PDC types in a little more detail. When you see this, you will realize that there are 2 separate ways to have the relation above: one which IS polarization entangled, one which is NOT polarization entangled. Only the polarization entangled version should reproduce the quantum mechanical results. The other should yield the classical curve that results from a product (separable) state.

 

[DrChinese]

For those not familar with the PDC crystal types:

Parametric down conversion (PDC) is accomplished by particular crystals with non-linear optical properties. The process is not completely understood. You see PDC crystal used in most photon entanglement experiments because all you need is a laser and a PDC crystal or 2, the crystal is cut to handle a specific input wavelength.

There are 2 types:

Type I: Produces an HH output from a V input. If you take a second Type I crystal and rotate it 90 degrees, you get VV output from an H input. Neither of these are polarziation entangled. If you put these 2 together and match the phases properly, you get polarization entanglement.

Type II: Produces an HV output from a H input, or a VH output from a V input. Neither of these are polarziation entangled. If you give the input a 45 degree tilt (half V, half H), you get polarization entanglement.

With either of the above, it is possible to have a known output with a fixed relationship between the outputs. The photons come out in pairs that are NOT entangled in polarization, but are entangled in other degrees of freedom. These do NOT produce the same statistics as polarization entangled photons although they otherwise have similar characteristics.

These pairs produce identical predictions to the polarization entangled pairs in the De Raedt model. That contradicts experiment.

 

[ajw1]

For those not familar with the PDC crystal types:

Parametric down conversion (PDC) is accomplished by particular crystals with non-linear optical properties. The process is not completely understood. You see PDC crystal used in most photon entanglement experiments because all you need is a laser and a PDC crystal or 2, the crystal is cut to handle a specific input wavelength.

There are 2 types:

Type I: Produces an HH output from a V input. If you take a second Type I crystal and rotate it 90 degrees, you get VV output from an H input. Neither of these are polarziation entangled. If you put these 2 together and match the phases properly, you get polarization entanglement.

Type II: Produces an HV output from a H input, or a VH output from a V input. Neither of these are polarziation entangled. If you give the input a 45 degree tilt (half V, half H), you get polarization entanglement.

With either of the above, it is possible to have a known output with a fixed relationship between the outputs. The photons come out in pairs that are NOT entangled in polarization, but are entangled in other degrees of freedom. These do NOT produce the same statistics as polarization entangled photons although they otherwise have similar characteristics.

These pairs produce identical predictions to the polarization entangled pairs in the De Raedt model. That contradicts experiment.

I don't think this can be a valid argument against the de Raedt model other then saying that the model doesn't describe the complete reality (as a model never does). There are a lot circumstances where it will yield results that are different from experimental observation.
As they say in several publications on this model: they don't postulate an interpretation with this model.

The model could therefore easily be adjusted to include the results for the PDC experiments as you mention, for instance by assigning a boolean property 'PDC' to each of the photons on creation (again without assigning any ontology to this property) and again treat each of them local realistic in the filters.

 

[DrChinese]

I don't think this can be a valid argument against the de Raedt model other then saying that the model doesn't describe the complete reality (as a model never does). There are a lot circumstances where it will yield results that are different from experimental observation.
As they say in several publications on this model: they don't propose an interpretation with this model.

The model could therefore easily be adjusted to include the results for the PDC experiments as you mention, for instance by assigning a boolean property 'PDC' to each of the photons on creation (again without assigning any ontology to this property) and again treat each of them local realistic in the filters.

No, that won't work either because this type of setup matches their initial assumption exactly. Both the polarization entangled and the non-polarization-entangled photons emerge from the PDC with this attribute - that the polarizations are orthogonal. They are coming out of the same crystal either way!

And sorry, but theories/hypotheses that do not match experiment usually get dropped in favor of candidates that do.

 

[ajw1]

No, that won't work either because this type of setup matches their initial assumption exactly. Both the polarization entangled and the non-polarization-entangled photons emerge from the PDC with this attribute - that the polarizations are orthogonal. They are coming out of the same crystal either way!

Maybe I wasn't clear enough in what I was trying to argue. The PDC property is not related to the Parametric down conversion process itself. It just refers to the type of particle that is produced (I might have called it 'IsEntangled' but that would have been confusing in a different way)
The point is that, apart from the polarization, I think I am allowed to use as many hidden variables as convenient, and use them in the selection process, as long as I respect locality.

And it doesn't seem difficult to extend the model for the proposed change in experimental setup to have it produce all the expected results.

 

[DrChinese]

Maybe I wasn't clear enough in what I was trying to argue. The PDC property is not related to the Parametric down conversion process itself. It just refers to the type of particle that is produced (I might have called it 'IsEntangled' but that would have been confusing in a different way)
The point is that, apart from the polarization, I think I am allowed to use as many hidden variables as convenient, and use them in the selection process, as long as I respect locality.

And it doesn't seem difficult to extend the model for the proposed change in experimental setup to have it produce the expected results.

You cannot be serious.

You may as well write a program that outputs the original Weihs et al data, and label it as a "LR" simulation program. If you have a switch in the program that changes it from "compliant dataset 1" to "compliant dataset 2" you haven't accomplished anything.

The fact is, the all of the photons are entangled and all of them are perpendicular. Some of them are also polarization entangled. If I understand you correctly, you want light to be delayed going through a filter if it is polarization entangled, but not otherwise. Right. Please do not insult the intelligence of the readers on this board. As you yourself say, we are now simply adjusting the model until the results match our conclusion, disregarding the facts.

 

[zonde]

So such a simulation - ALONE and BY ITSELF - is NOT a disproof of the Bell Theorem.

Yes, but ...
This simulation alone and by itself is clear proof about limits of applicability for Bell Theorem. And that is because Bell Theorem is mathematical no-go theorem so one mathematical counter example for the case with unfair sampling is sufficient.

2. However, those results require a polarization entangled input source. It is also possible to use their formula on a source which is NOT polarization entangled. A source which is NOT polarization entangled will not yield the QM expectation values, it will yield the Local Realistic expectation values. However, their simulation yields instead the same results as an entangled source.

The model does not claim that it explains non-entangled photons with correlated polarizations. So your argument is not valid. If you place such requirement then the model should be modified (if possible) to cover this new situation.
To me it seems that the easiest way to do this is to use two separate variables for polarization and detection delay with the same offset between them in entangled state but uncorrelated offset for detection delay variable in non-entangled state.

I think that more up to the point argument is that with this model there should be white noise outside coincidence window. But there does not seem to be such (unfortunately do not know about published results of such analysis).

 

[DrChinese]

1. Yes, but ...
This simulation alone and by itself is clear proof about limits of applicability for Bell Theorem. And that is because Bell Theorem is mathematical no-go theorem so one mathematical counter example for the case with unfair sampling is sufficient.2. The model does not claim that it explains non-entangled photons with correlated polarizations. So your argument is not valid. If you place such requirement then the model should be modified (if possible) to cover this new situation.
To me it seems that the easiest way to do this is to use two separate variables for polarization and detection delay with the same offset between them in entangled state but uncorrelated offset for detection delay variable in non-entangled state.

1. The "loophole" has a loophole.

2. Actually, it should if you follow their reasoning. However, after reading more of the De Raedt's works, it looks like they later discovered this exact issue and in fact did make changes to their models. But that is not completely clear to me at this point, as they have several papers with very similar models and nearly identical arguments.

The simulation model with code (thanks ajw1!) was entitled "A computer program to simulate Einstein–Podolsky–Rosen–Bohm experiments with photons" and was accepted for publication on 10 Jan 2007. This has the flaw present which I identified above.

Another paper, entitled "Event-based computer simulation model of Aspect-type experiments strictly satisfying Einstein’s locality conditions" was accepted for publication on 6 Aug 2007. This specifically refers to two experimental setups, called Experiment I/Type I and Experiment II/Type II, which must both be correctly described by their model. Experiment I is polarization entangled pairs which violates the Bell Inequality, and Experiment II is polarized but not entangled pairs which are separable (and therefore do not violate the Inequality). In their words (from the newer ref, page 3):

"The sources used in EPRB experiments with photons emit photons with opposite but otherwise unpredictable polarization. We refer to this experimental set-up as Experiment I. Inserting polarizers between the source and the observation stations changes the pair generation procedure such that the two photons have a fixed polarization. We refer to this set-up as Experiment II. As a result of the fixed polarization of the photons the photon intensity measured in the detectors behind the polarizers in each observation station obeys Malus’ law. Our simulation model reproduces the correct quantum mechanical behavior for the single-particle and two-particle correlation function for both types of experiments."

Further, they recognize explicitly the following two points:

a) "The difference between this model and the model described in Ref.,9 is the algorithm to simulate the polarizer. In Ref.9 we used a model for the polarizers that is too simple to correctly describe experiments of type II." I will need to do additional research to determine if the revised algorithm in this newer paper solves the issue I am identifying, or if in fact they made some other completely unrelated change. So I have some more reading to do... :)

b) They explcitly recognize that a single algorithm MUST always apply to the PDC setup, stating "...the event-by-event simulation reproduces the single- and two particle results of quantum theory for both Experiment I and II, without any change to the algorithm that simulates the polarizers." In other words, you cannot hand add the "ENTANGLED" property to the algorithm as ajw1 suggested (thankfully they acknowledge this). However, I now don't think I have the current code for this simulation model. awj1? Know where I should be looking? By the way, I did download a program from their website which does the simulation. Unfortunately, it is an EXE file and there is no source.

Regardless, I should have something more on this later today.

 

[DrChinese]

Ok, I have completed my review of the paper linked above as "Event-based computer simulation model of Aspect-type experiments strictly satisfying Einstein’s locality conditions" by De Raedt et al. It does use the same algorithm as the earlier paper.

I. In the newer paper, we have the PE (polarization-entangled) pairs as Experiment I and the results violate the Bell Inequality - and this is exactly (sometimes verbatim) the same as the earlier paper (this is the desired result).

II. In the newer paper, we have the NPE (non-polarization-entangled) pairs as Experiment II and the results do not violate the Bell Inequality (this is the desired result).

What we need to consider are the following 2 special cases of Experiment II, which I will call Experiment III and Experiment IV:

III. A subset of the Experiment II in which the angle settings of the polarizers used to get NPE pairs are set at 0 and 90 degrees only (or alternately 90 degrees and 0 degrees only). In their Experiment II, they do this but also consider other settings which are not perpendicular. So we are simply accepting those Experiment II results as valid, and being a subset of a more general rule. As before, of course, the results do not violate the Bell Inequality (this is the desired result).

IV. A setup is considered that exactly matches Experiment III above, with the exception that the polarizers used to get the NPE pairs are removed. The NPE pairs are still produced, because we also tilt the source laser by 45 degrees. This has the effect of producing the same NPE pairs but without the extra polarizers to get in the way - and which do affect the simulation. The results should still be in accordance with Experiment III, but they no longer are. Instead, they now match Experiment I. Such results are in contradiction to what is actually observed, so this is NOT the desired result.

The only problem with the above is that the authors of the paper do not include the computational algorithm for Experiments II and III, so we must take their word on the results. However, they do provide the logic for Experiments I and IV, so that we can see for ourselves. It is IV that is problematic, and matches neither the predictions of QM or of actual physical experiment.

 

[PTM19]

Can you help me decipher this statement:

k2=ceiling(abs(1-c2*c2)**(d/2)*r0/tau) ! delay time

this looks to me like:

k2=ceiling(abs(1-(c2*c2))**((d/2)*(r0/tau))) ! delay time

and since d=2 and static reduces to:

k2=ceiling( abs(1-(c2*c2))**(r0/tau) ) ! delay time

I've never used fortran but from what I was able to google the operator precedence seems to be standard with exponentiation '**' preceding over multiplication '*', shouldn't then the formula be:

k2=ceiling(((abs(1-(c2*c2))**(d/2))*r0)/tau) ! delay time

with d=2 that would reduce simply to

k2=ceiling((abs(1-(c2*c2)))*r0/tau)

I haven't read all the other posts so maybe it was already mentioned.

 

[DrChinese]

I've never used fortran but from what I was able to google the operator precedence seems to be standard with exponentiation '**' preceding over multiplication '*', shouldn't then the formula be:

k2=ceiling(((abs(1-(c2*c2))**(d/2))*r0)/tau) ! delay time

with d=2 that would reduce simply to

k2=ceiling((abs(1-(c2*c2)))*r0/tau)

I haven't read all the other posts so maybe it was already mentioned.

Yes, that is how I read it as well. I am simulating in an Excel spreadsheet with VB programming embedded so that I can check a variety of scenarios myself. Then I use Excel to graph it too. That way I can share easily. I downloaded the EXE of the De Raedt simulation itself from a site, but so far cannot actually execute it as I need the MATLAB library (but not the latest version, which I have).

 

[DrChinese]

OK, finally, I have my spreadsheet documenting the issue above on the De Raedt model. Email me or send me a message with your address and I will send to you. I will post on my site in a bit, and will post a link when available. I will upload some screen shots so you can see some of the salient points.

There are 2 images. The first shows the De Raedt model working properly for what they call Experiment I. The second shows the De Raedt model working incorrectly for a variation on their Experiment II, which I refer to as Experiment IV (the variation, that is).

 

[DrChinese]

Here are links to the spreadsheet model (Excel 2003 is .XLS, 2007 is .XLSM):

http://drchinese.com/David/DeRaedtComputerSimulation.EPRBwithPhotons.A.xls

http://drchinese.com/David/DeRaedtComputerSimulation.EPRBwithPhotons.A.xlsm

Overview:
========

This spreadsheet is based around the computer simulations of De Raedt et al per references [1] and [2] below. Generally, the papers show that Local Realistic theories could reasonably make predictions which violate a Bell Inequality.

The attached worksheet pages cover several aspects of the model, and faithfully use the published information about the model to achieve the results. The original code was in FORTRAN. In this spreadsheet, I have implemented using Visual Basic functions (VBA) which are accessible from the Visual Basic button in the Developer tab.

Conclusion
========

The results show that the De Raedt model successfully models 2 important elements of Local Realism (LR), both based around Type II PDC entanglement.

These respect Bell by assuming that the sample observed is not a fair representation of the full universe. See the worksheets "De Raedt.TypeII.Entangled" and "EventDetail".

However, the model fails to handle the situation in which the photon pairs are correlated but not polarization entangled. See the worksheet "De Raedt.TypeII.NotEntangled". According to the De Raedt Experiment II results (not simulated explicitly here), a different result should be expected which is more in line with the function .25+(cos^2(Theta)/2).

References
========

"A computer program to simulate Einstein–Podolsky–Rosen–Bohm experiments with photons" by K. De Raedt, H. De Raedt, and K. Michielsen (2007), published in Computer Physics Communications, 28 March 2007.

"Event-by-event simulation of Einstein-Podolsky-Rosen-Bohm experiments" by S. Zhao, H. De Raedt and K. Michielsen (2007), published in the Foundations of Physics.

 

[ajw1]

It seems a bit strange that http://rugth30.phys.rug.nl/eprbdemo/simulation.php" produces results around the QM prediction, while your DeRaedt results are always between this prediction and the straight line.

I have found a small error in your code, but this doesn't produce significant different results:

  k2 = ((1 - (c2 * c2)) * r0 / tau) + 1 ' delay time - exponent removed since d/2 = 1, +1 at end is equivalent to FORTRAN ceiling function

should probably be

  k2 = Math.Round(((1 - (c2 * c2)) * r0 / tau) + 0.5) ' delay time - exponent removed since d/2 = 1, +1 at end is equivalent to FORTRAN ceiling function

 

[DrChinese]

1. It seems a bit strange that http://rugth30.phys.rug.nl/eprbdemo/simulation.php" produces results around the QM prediction, while your DeRaedt results are always between this prediction and the straight line.

2. I have found a small error in your code, but this doesn't produce significant different results:

  k2 = ((1 - (c2 * c2)) * r0 / tau) + 1 ' delay time - exponent removed since d/2 = 1, +1 at end is equivalent to FORTRAN ceiling function

should probably be

  k2 = Math.Round(((1 - (c2 * c2)) * r0 / tau) + 0.5) ' delay time - exponent removed since d/2 = 1, +1 at end is equivalent to FORTRAN ceiling function

1. This is an artifact of the angles the silmultation is run at. Theirs is run at maybe 20-30 anglles randomly selected in a 90 degree range, while mine are run at evey degree. I see their results as safely within the intended range.

2. Because k2 is an integer type, the rounding causes truncation... I added +1 to the result which has the exact same effect as what you have.

 

[ajw1]

2. Because k2 is an integer type, the rounding causes truncation... I added +1 to the result which has the exact same effect as what you have.

No, not exactly (let for example c2 be 1)

 

[ajw1]

A major difference is that De Raedt's value for k is 1 (see fortran code), whereas you're model uses 30 as default for this value.

 

[DrChinese]

No, not exactly (let for example c2 be 1)

First, thank you VERY much for looking at the code.

The time window function they use has a random value associated with it, so the result cannot be a whole number.

 

[DrChinese]

A major difference is that De Raedt's value for k is 1 (see fortran code), whereas you're model uses 30 as default for this value.

That threw me at first before I realized that k=1 as a default makes no sense. That is the lowest value their routine can take and still produce a sample. Also, when you graph it - which you can in my model by setting k to be 1 - the results are very strange because there is so little data. k=30 or 10 or 100 produces more data points.

 

[ajw1]

First, thank you VERY much for looking at the code.

The time window function they use has a random value associated with it, so the result cannot be a whole number.

When you define a k3 integer using my formula and check whether a k3<>k2 occurs (set a breakpoint in the if statement) you will see the different values will occur

 

[ajw1]

That threw me at first before I realized that k=1 as a default makes no sense. That is the lowest value their routine can take and still produce a sample. Also, when you graph it - which you can in my model by setting k to be 1 - the results are very strange because there is so little data. k=30 or 10 or 100 produces more data points.

I have checked your model using k=1 (In their text they say W=k.tau) with 500000 iterations and this produces very accurate results

 

[DrChinese]

I have checked your model using k=1 (In their text they say W=k.tau) with 500000 iterations and this produces very accurate results

You are right, looks good!

 

[DrChinese]

When you define a k3 integer using my formula and check whether a k3<>k2 occurs (set a breakpoint in the if statement) you will see the different values will occur

I will try it as you say, if it produces different results I will definitely change it... my goal is a faithful representation of their simulation.

 

[zonde]

Coincidence rate for particular angle is not calculated properly in this model.
In your code there is the line:

    MainModel = MatchesWithinWindow / WithinWindow

But you should compare coincidences with coincidences at maximum angle (theta=90deg for Type II PDC) taking into account coincidences at minimum angle. Like that:
max=result at 90deg
min=result at 0deg
X=result at x angle
coincidence rate at x angle=(X-min)/(max-min)
assuming that results at all angles have the same singlet counts (or are normalized against singlet counts).
In this particular case you can look at result that is produced like that:

    MainModel = MatchesWithinWindow / Iterations

because "Iterations" faithfully represent singlet counts in this model and min=0.

And that way the model produces results that are far off from expected result.

 

[DrChinese]

Coincidence rate for particular angle is not calculated properly in this model.
In your code there is the line:

    MainModel = MatchesWithinWindow / WithinWindow

But you should compare coincidences with coincidences at maximum angle (theta=90deg for Type II PDC) taking into account coincidences at minimum angle. Like that:
max=result at 90deg
min=result at 0deg
X=result at x angle
coincidence rate at x angle=(X-min)/(max-min)
assuming that results at all angles have the same singlet counts (or are normalized against singlet counts).
In this particular case you can look at result that is produced like that:

    MainModel = MatchesWithinWindow / Iterations

because "Iterations" faithfully represent singlet counts in this model and min=0.

And that way the model produces results that are far off from expected result.

I do not follow what you are saying. Can you help me to understand better?

My model returns results (one of the options anyway) as = MatchesWithinWindow / WithinWindow for the Alice & Bob detector angle settings being some Theta between 0 and 90 degrees. So the way to compute this is: for each trial iteration, determine per the formula whether it is within the coincidence time window (which uses k, k1 and k2). If the answer is Yes, then add 1 to WithinWindow. If it is also a Coincidence (Match), then add 1 to that counter. That is the sample coincidence rate (subset of full universe) and the same way the De Raedt's report their numbers. It is the same way most papers report as well, relative to Theta.

What is your max and min, and what do they represent? I don't see this in their calculation code.

 

[DrChinese]

For example, here are trial runs that show the model working as it should (as far as I can tell anyway). Each does 500,000 iterations by degree over the range 0 to 90 degrees. One is at k=10 and the other is at k=30. Lower k means closer fit to the QM prediction. However, technically that (close fit to the QM prediction) is not a requirement of the De Raedt model. They are instead trying to show that a Bell Inequality can be violated by a simulation which exploits the fair sampling loophole, while simultaneously showing that the full universe doe NOT. I believe these 2 graphs should suffice to demonstrate that (superficially at least) for this particular setup - they call this Experiment I.

 

[DrChinese]

Here is the diagram showing a fundamental problem with the De Raedt model. It should produce identical results for Figure A and Figure B. Per the spreadsheet referenced above, it produces dramatically different results.

 

[zonde]

I do not follow what you are saying. Can you help me to understand better?

To find out result you divide coincidences in +/+ and -/- channels of PBS (drawing analogy with real experiment) with coincidences from all channels (+/+; -/-; +/-; -/+).
But suppose you have setup with polarizers not PBSes. Then you can know only say +/+ coincidences. What you do in real experiment? You run the experiment with certain angle setting for 10 second then change angle setting and run it for other 10 seconds and so on.
If you are careful you check that singlet rate does not change from one 10 sec. period to other 10 sec. period.
So in real experiment you do not compare coincidences in one channel with coincidences in all channels. You compare coincidences for different angles directly without calculating proportions like in Raedt's programm.

My model returns results (one of the options anyway) as = MatchesWithinWindow / WithinWindow for the Alice & Bob detector angle settings being some Theta between 0 and 90 degrees. So the way to compute this is: for each trial iteration, determine per the formula whether it is within the coincidence time window (which uses k, k1 and k2). If the answer is Yes, then add 1 to WithinWindow.

That means if coincidence can be detected in any channel.

If it is also a Coincidence (Match), then add 1 to that counter.

If you mean "MatchesWithinWindow" then it's coincidences in specific channel/-s as opposed to coincidences in any achannel.

That is the sample coincidence rate (subset of full universe) and the same way the De Raedt's report their numbers. It is the same way most papers report as well, relative to Theta.

What is your max and min, and what do they represent? I don't see this in their calculation code.

Yes there are noting like that in their code but it should be there if you want to mimic experiments.

 

[DrChinese]

1. To find out result you divide coincidences in +/+ and -/- channels of PBS (drawing analogy with real experiment) with coincidences from all channels (+/+; -/-; +/-; -/+).

2. Yes there are noting like that in their code but it should be there if you want to mimic experiments.

1. This is the technique used in the De Raedt simulation... so that is why I used it. I.e. to count +/+ and -/- as coincidences and the +/+; -/-; +/-; -/+ cases as being seen in ideal polarizing beam splitters with allowance made for the hypothesized "time delay" effect they exploit.

2. I prefer to stick to the simulation, because that is where I am trying to make a point. And that point is: the De Raedt model does NOT work as claimed for unentangled photon pairs with known polarizations. And that is a hard requirement for any model, as they themselves point out.

 

[ajw1]

2. I prefer to stick to the simulation, because that is where I am trying to make a point. And that point is: the De Raedt model does NOT work as claimed for unentangled photon pairs with known polarizations. And that is a hard requirement for any model, as they themselves point out.

Have you noticed they probably use a different model for their combined type I and type II experiments? They use something called a DLM (deterministic learning machine). This is not present in the model you base your conclusion on.

The
difference between this model and the model described in Ref.,9 is the algorithm to simulate
the polarizer. In Ref.9 we used a model for the polarizers that is too simple to correctly
describe experiments of type II.

See"[URL Event-based computer simulation model of Aspect-type experiments
strictly satisfying Einstein’s locality conditions[/URL]

 

[DrChinese]

Have you noticed they probably use a different model for their combined type I and type II experiments? They use something called a DLM (deterministic learning machine). This is not present in the model you base your conclusion on.

See"[URL Event-based computer simulation model of Aspect-type experiments
strictly satisfying Einstein’s locality conditions[/URL]

That is the reference I am working with. And yes, they refer to the DLM but it is not a part of their model as far as I can see except as general justification for their program. Their formula is arrived at as follows, quoting:

"... Therefore we use simplicity as a criterion to select a specific form. By trial and error, we found that T(n − 1) = T0F(| sin 2(n − 1)|) = T0| sin 2(n − 1)|d yields useful results. Here, T0 = max T() is the maximum time delay and defines the unit of time, used in the simulation and d is a free parameter of the model. In our numerical work, we set T0 = 1. As we demonstrate later, our model reproduces the quantum results of Table I under the hypothesis that the time tags tn,1 are distributed uniformly over the interval [0, | sin 2(n − 1)|d] with d = 2. Needless to say, we do not claim that our choice is the only one that reproduces the results of quantum theory for the EPRB experiments."


And again, since I am following their model which uses Type II PDC, I am sticking with that alone for my example. And I use that for the spreadsheet. However, the same problem does exist with Type I PDC so I don't think it is necessary to further document.

 

[ajw1]

That is the reference I am working with. And yes, they refer to the DLM but it is not a part of their model as far as I can see except as general justification for their program. Their formula is arrived at as follows, quoting:

"... Therefore we use simplicity as a criterion to select a specific form. By trial and error, we found that T(n − 1) = T0F(| sin 2(n − 1)|) = T0| sin 2(n − 1)|d yields useful results. Here, T0 = max T() is the maximum time delay and defines the unit of time, used in the simulation and d is a free parameter of the model. In our numerical work, we set T0 = 1. As we demonstrate later, our model reproduces the quantum results of Table I under the hypothesis that the time tags tn,1 are distributed uniformly over the interval [0, | sin 2(n − 1)|d] with d = 2. Needless to say, we do not claim that our choice is the only one that reproduces the results of quantum theory for the EPRB experiments."

I think they do use different logic for the model in the reference cited:

We now describe the DLM that simulates the operation of a polarizer

with the logic of the DLM explained on page 12. Your quote must be about the implementation of the logic for the timewindow.

 

[DrChinese]

I think they do use different logic for the model in the reference cited:
with the logic of the DLM explained on page 12. Your quote must be about the implementation of the logic for the timewindow.

I think you are essentially correct. I do not know (nor do I need to know) what their DLM logic consists of. That is not a part of their formula for Experiment I, which is where their FORTRAN simulation comes into play. I have not seen any published code for their Experiment II. I only know that when you model their Experiment II WITHOUT their "DLM" polarizer - which is my "Experiment IV" - you get results that are materially different than they predict and what experiment and QM predicts.

In other words, they used a "trick" to get the results to look right for Experiment II. That "trick" (it is not an inappropriate trick, so that is why I put in quotes) is that they ascribe new properties to the Polarizer... and they say that somehow relates to the DLM. OK, fine. So my "trick" (also not inappropriate) is to remove the Polarizer. And guess what? Without the Polarizer (and the DLM), the results do NOT work right. So their model CANNOT be correct, even in principle.

We all know that De Raedt et al are not actually asserting their model is accurate. They are simply claiming that SOME model could hypothetically work. And I am saying: NO, it doesn't, because you must FIRST provide at least one consistent example before the point can be yielded. And this is not that example. It must work for the same scope as what they claim it works for, and I have the counterexample that shows it does not.

The fact is: No "local realistic" simulation of the quantum mechanical predictions has been provided that even in principle meets the criteria of Bell by exploiting the fair sampling loophole.

 

[zonde]

As I understand by "the criteria of Bell" in this context you mean some sort of relation like that:
P_{VV}(\alpha,\beta) = sin^{2}\alpha\, sin^{2}\beta\, cos^{2}\theta_{l} + cos^{2}\alpha\, cos^{2}\beta\, sin^{2}\theta_{l} + \frac{1}{4}sin 2\alpha\, sin 2\beta\, sin 2\theta_{l}\, cos \phi
This is equation (9) from paper - http://arxiv.org/abs/quant-ph/0205171/"

This relation produces cos^{2}(\alpha-\beta) law when \theta_{l} is Pi/4 and \phi is 0. But when for example \theta_{l} is 0 it produces sin^{2}\alpha\, sin^{2}\beta that is simply product of two probabilities from Malus law.

But interesting how do you view this PDC TypeI from QM perspective?
As I see it if we have one crystal incident photon is converted (sometimes) into two photons that go out of the crystal diverted in opposite directions from original direction. They are not polarization entangled.
However if these two photons encounter second crystal right after the first one if certain conditions are met two photons become polarization entangled.
It seems to me that pilot wave interpretation provides nice way to resolve this. Empty pilot wave of incident photon continues its way next to second crystal and gets partly downconverted there and then overlaps with downconverted photons (somehow) creating entangled state.
How would you explain creation of polarization entangled state in TypeI PDC?

 

[DrChinese]

As I understand by "the criteria of Bell" in this context you mean some sort of relation like that:
P_{VV}(\alpha,\beta) = sin^{2}\alpha\, sin^{2}\beta\, cos^{2}\theta_{l} + cos^{2}\alpha\, cos^{2}\beta\, sin^{2}\theta_{l} + \frac{1}{4}sin 2\alpha\, sin 2\beta\, sin 2\theta_{l}\, cos \phi
This is equation (9) from paper - http://arxiv.org/abs/quant-ph/0205171/"

This relation produces cos^{2}(\alpha-\beta) law when \theta_{l} is Pi/4 and \phi is 0. But when for example \theta_{l} is 0 it produces sin^{2}\alpha\, sin^{2}\beta that is simply product of two probabilities from Malus law.

But interesting how do you view this PDC TypeI from QM perspective?
As I see it if we have one crystal incident photon is converted (sometimes) into two photons that go out of the crystal diverted in opposite directions from original direction. They are not polarization entangled.
However if these two photons encounter second crystal right after the first one if certain conditions are met two photons become polarization entangled.
It seems to me that pilot wave interpretation provides nice way to resolve this. Empty pilot wave of incident photon continues its way next to second crystal and gets partly downconverted there and then overlaps with downconverted photons (somehow) creating entangled state.
How would you explain creation of polarization entangled state in TypeI PDC?

Yes, the Type I PDC is especially interesting to consider. If you think in classical terms, it is difficult to explain. For those who are unfamiliar with the PDC Types:

Both Type I and Type II use specially cut thin (typically 1mm) non-linear crystals (often made of Barium Borate or BBo) which produced pairs of correlated photons from an input laser source. The outputs are actually conic regions that vary in wavelength and intensity by angle, and the experimenter usually locates a region of the cone with the best output according to the desired characteristics. Both Type I and Type II can produce known outputs from a known input, and such outputs are NOT polarization entangled.

Type I: H> input produces VV> output when crystal is oriented correctly for H>, and produces nothing for V> input. If the crystal is turned 90 degrees and the source is also turned 90 degree, you have V> input produceing HH> output (and produces nothing for H> input).

Type II: H> input produces HV> output, V> input produces VH> output.

To get polarization entanglement, you do the following:

Type I: Use 2 crystals oriented at 90 degrees apart (0 and 90), with the source input at a 45 degree angle.

Type II: Use 1 crystal oriented at 0 degrees, with the source input at a 45 degree angle.

The question zonde is asking is: how to consider the Type I case which requires 2 crystals to achieve entanglement? After all, doesn't the light convert at either one crystal or the other? The QM explanation is as follows:

There are 2 paths the photon pair can take from the source to the target. Path 1 could have gone through the first crystal IF the input photon is H, and that yields a VV> output. Path 2 could have gone through the second crystal IF the input photon is V, and that yields an HH> output. Since we cannot know, in principle, if the input photon resolved to H or to V, then we have a superposition in the output: VV> + HH>. This superposition is polarization entangled and is rotationally invariant (i.e. Alice and Bob can rotate around 360 degrees and will still see full correlations).

The above explanation is not "realistic" because the light does not go through one crystal OR the other. Like the double slit, it somehow converts within both and collapses to one or the other (whatever that means) upon observation. If one of the crystals is removed, then the output stream is NOT polarization entangled (though it is correlated). On the other hand, there is nothing WHATSOEVER to indicate that down conversion is due to some physical interaction of the 2 crystals. After all, Type II does NOT require 2 crystals to obtain entanglement!

So the QM explanation works for both Type I and Type II because you follow the typical QM rules for when you have superpositioned states. The Bohmian explanation works because it is designed to yield equivalent results as QM. But I personally don't see how the BM interpretation allows a "better" visual by saying the particle goes one way and the pilot wave goes the other (whatever that means), and they interfere to create entanglement.

You can clearly see that both visuals (QM, BM) have fuzzy components. I like the QM explanation simply because it follows naturally from the superposition rules, while it seems to me that the BM explanation does not. In other words, the QM explanation is based on a superposition of probability amplitudes and treats those as "relatively real". The BM explanation must then treat the pilot wave as "relatively real" too. To me, "relatively real" is not realistic. So I don't see the BM interpretation as being "more" realistic than QM. They are equal. But hey, that is just my opinion and I am perfectly comfortable with others coming to a different conclusion.

 

[zonde]

There are 2 paths the photon pair can take from the source to the target. Path 1 could have gone through the first crystal IF the input photon is H, and that yields a VV> output. Path 2 could have gone through the second crystal IF the input photon is V, and that yields an HH> output. Since we cannot know, in principle, if the input photon resolved to H or to V, then we have a superposition in the output: VV> + HH>. This superposition is polarization entangled and is rotationally invariant (i.e. Alice and Bob can rotate around 360 degrees and will still see full correlations).

As I see from QM perspective it is more correct to talk about ensembles and not individual photons and from that viewpoint ensemble is really taking both paths. The question is how to treat certain moment in time where we assume quantized photon of one ensemble is there and there is no quantized photon from other ensemble.
And I prefer to think that quantization conserves energy on average across the whole ensemble but when we talk about individual photon we can not talk about strict conservation of energy without considering environment. So individual photons can interact indirectly through environment.

 

[DrChinese]

Attached is an updated version of the Experimental setup. It shows the 4 setups side by side and explains how either Entangled State or Product State statistics are obtained. Note that in Figure D, the De Raedt statistics are Entangled State but the actual observation is Product State. This also occurs when you do a similar analysis on Type I PDC.

In other words: The De Raedt simulation model works for some PDC cases, but is inconsistent (and wrong) in others. See the second attachment for the graphed simultation results that demonstrate this. I use parameters k=30 and i=50000, but the results do not change much regardless of parameter selection; and they never look like the observed/theoretical Product State statistics.

----------------------------

You may also find it helpful to look at a good exposition on the Type II PDC setup: http://www.ino.it/~azavatta/References/JMO48p1997.pdf

Generation of correlated photon pairs in type-II parametric down conversion revisited
(2001) by CHRISTIAN KURTSIEFER, MARKUS OBERPARLEITER and HARALD WEINFURTER

See especially Figures 1 and 5.

 

[DrChinese]

OK, I have sent a letter to Hans De Raedt. I am curious as to response!

 

[ajw1]

OK, I have sent a letter to Hans De Raedt. I am curious as to response!

And, did you receive any response yet?

 

[DrChinese]

And, did you receive any response yet?

Yes, I received a very kind (and prompt!) response and am in the process of sending something back. I want them to have the Excel spreadsheet simulation I posted here, as they may find it useful (since more people have Excel than Fortran).

I prefer not to disclose his comments without prior OK from him, but I would characterize the response as follows (without, I believe, saying anything that he hasn't said before):

1. The De Raedt simulation does not handle the case I describe.
2. It also does not match Malus (a separate issue that I did not raise as this is a consequence of any algorithm respecting Bell).

Dr. De Raedt also provided me with additional materials which I am reviewing, and I believe they are already present in the archives. I have been out of it the past few days due to a surgery in the family.

 

[DrChinese]

I am reviving this thread in hopes that this will assist some readers in following some arguments about Bell's Theorem and Bell tests.

It has been argued that perhaps there is SOME set of hidden variables in which there may be a) a double dependency on theta (inflector); b) a common cause related to some global variable (ThomasT); c) cyclic hidden variables (billschnieder); or similar. See some of the active threads and you will see these themes.

I have repeatedly indicated that Bell is a roadmap to understanding that local realistic theories must be abandoned. This is generally accepted science.

In trying to show that there "could" be an exception to Bell, please consider the following to add to your list of tests for you candidate LHV theory:

a) You will be providing a formula which leads to a realistic dataset, say at angle settings 0/120/240 degrees (or some standard combination such as for CHSH inequality). This should be generated for a full universe, not just the observed cases.
b) The formula for the underlying relationship will be different than the QM predictions, and must respect the Bell Inequality curve. I.e. usually that means the boundary condition which is a straight line, although there are solutions which yield more radical results.
c) The relevant hidden variables/forumulae must be determined in advance, such that Alice's setting does not itself influence Bob's result - and vice versa.
d) There is a formula or set of hidden variables - they can be random - which leads to the detection or non-detection so that the Fair Sampling Assumption is shown to be violated (thus explaining how a LHV theory can reproduce the Entangled State statistics in a sampled environment).

And here is the little trick that should doom anything left standing:

e) You must be able to use the same assumptions and setup to yield Product State statistics when the photon pairs coming from the PDC crystal are NOT entangled.

See, that last one is a real trick: the only apparent different between PDC photons that are polarization entangled versus those that are not is that the H/V orientation is known for one, but not for the other. And yet, that flies completely in the face of the thinking of the LHV candidate theory. There should be NO DIFFERENCE! And yet, experimentally there is!

To recap: LHV candidate theories argue that the hidden variables are unknown but are pre-existing with definite values. These should lead to determinate outcomes (for polarization entangled pairs) that yield Entangled State stats when a subset is sampled. Yet when the same assumptions are made for non-polarization entangled pairs, the prediction should be for the same Entangled State stats. Yet experiments yield Product state stats for these! How can that be?

Good luck!

 

[DrChinese]

I have been working with the De Raedt team for several months to address the issue identified in this thread. Thanks especially to Dr. Kristel Michielsen for substantial time and effort to work with me on this.

The issue I identified was rectified very quickly using what they call their "Model 2" algorithm. My earlier analysis was using their older "Model 1" algorithm. After getting to the point where we were able to compare statistics for a jointly agreed upon group of settings, I am satisfied that they have a simulation which accomplishes - in essence - what they claim.

I am still analyzing the event by event results. I do expect to have some follow up issues. As I get some more information, I will share this too. For those of you with a computer background, I thought you might be interested in the solution:

Replace:

If c1 > 0 Then

k1 = ((1 - (c1 * c1)) * r0 / tau) + 1 ' delay time...

by:

if(c1>2*RND()-1) then

k1 = ( (1 - (c1*c1))**2 * r0 / tau) + 1 ' delay time...

This subtle change made a huge difference! I will be updating my Excel spreadsheet model and posting a link when it is ready.

I wanted to provide this update for those who follow this subject. Please keep in mind that the De Raedt model is a computer simulation which exploits the coincidence time window as a means to achieve a very interesting result: It is local realistic. Therefore, it is able to provide event by event detail for 3 (or more) suimultaneous settings (i.e. it is realistic). It does this with an algorithm which is fully independent (i.e. local/separable). It does not violate a Bell Inequality for the full universe but does (somewhat) for the sample. Its physical interpretation is something else entirely and not something which I was intending to address in this thread. Although I would be happy to discuss this too.

 

[ajw1]

I have been working with the De Raedt team for several months to address the issue identified in this thread. Thanks especially to Dr. Kristel Michielsen for substantial time and effort to work with me on this.

The issue I identified was rectified very quickly using what they call their "Model 2" algorithm. My earlier analysis was using their older "Model 1" algorithm. After getting to the point where we were able to compare statistics for a jointly agreed upon group of settings, I am satisfied that they have a simulation which accomplishes - in essence - what they claim.

I am still analyzing the event by event results. I do expect to have some follow up issues. As I get some more information, I will share this too. For those of you with a computer background, I thought you might be interested in the solution:

Replace:

If c1 > 0 Then

k1 = ((1 - (c1 * c1)) * r0 / tau) + 1 ' delay time...

by:

if(c1>2*RND()-1) then

k1 = ( (1 - (c1*c1))**2 * r0 / tau) + 1 ' delay time...

This subtle change made a huge difference! I will be updating my Excel spreadsheet model and posting a link when it is ready.

I wanted to provide this update for those who follow this subject. Please keep in mind that the De Raedt model is a computer simulation which exploits the coincidence time window as a means to achieve a very interesting result: It is local realistic. Therefore, it is able to provide event by event detail for 3 (or more) suimultaneous settings (i.e. it is realistic). It does this with an algorithm which is fully independent (i.e. local/separable). It does not violate a Bell Inequality for the full universe but does (somewhat) for the sample. Its physical interpretation is something else entirely and not something which I was intending to address in this thread. Although I would be happy to discuss this too.

The changed algorithm produces correct results for 'entangled' photons. But I don't see how you initialise not-entangled photons to obtain the classical relation between polarisator angles in the model. Or wasn't this the issue you're referring to?

Did de Raedt hint on any possible interpretation?

 

[DrChinese]

The changed algorithm produces correct results for 'entangled' photons. But I don't see how you initialise not-entangled photons to obtain the classical relation between polarisator angles in the model. Or wasn't this the issue you're referring to?

Strangely, and despite the fact that it "shouldn't" work, the results magically appeared. Keep in mind that this is for the "Unfair Sample" case - i.e. where there is a subset of the full universe. I tried for 100,000 iterations. With this coding, the full universe for both setups - entangled and unentangled - was Product State. That part almost makes sense, in fact I think it is the most reasonable point for a full universe! What doesn't make sense is the fact that you get Perfect Correlations when you have random unknown polarizations, but get Product State (less than perfect) when you have fixed polarization. That seems impossible.

However, by the rules of the simulation, it works.

Now, does this mean it is possible to violate Bell? Definitely not, and they don't claim to. What they claim is that a biased (what I call Unfair) sample can violate Bell even though the full universe does not. This particular point has not been in contention as far as I know, although I don't think anyone else has actually worked out such a model. So I think it is great work just for them to get to this point.

I am still studying the results, as there are a number of very critical issues involved in their results. For example, it is still not clear to me by how much a Bell Inequality is violated. Their Model 1 did a fine job of Entangled State, but Model 2 does not seem anywhere near as good at this. On the other hand, the Model 1 completely fails at Product State while the Model 2 does this very well. So there are trade-offs between the models. (You would expect that to a certain extent.)

(And while I haven't really looked into HOW the formula works its magic, it appears to be a function of the number of calls to a random number generator. I.e. it is almost as if 2 calls can offset each other. I guess in the right circumstances, that could happen. You need something like that to get Entangled State correlations from an otherwise Product State scenario, I think.)

 

[ajw1]

I get results that seems to be equally good as the model 1:

Maybe I have interpreted the formula differently:

            double malus = h.Malus(Particle.Polarization - this.Angle);
            if (malus > h.GetRandomPlusMin())  // sign(cos(...))
            {
                Particle.Absorbed = true; // <=> -1 event
            }
            Particle.DelayTime = Math.Ceiling(Math.Pow(Math.Pow((1 - malus * malus),2), (d / 2)) * h.GetRandom() / tau); // delay time

with h.Malues defined as:

        public static double Malus(double Angle)
        {
            return Cos(2 * Angle);
        }