Seeking the Analytic Quaternion by Colin Walker

Colin Walker

Thank you, Christian.

Best wishes for you, your team and your research.

Cheers, Colin

Colin Walker

Thank you, Vladimir.

I will comment soon on your blog after taking a better look at your essay.

Best to you,

Colin

Jonathan Dickau

Hello Colin,

I had to go at this 3 times to read it through. And I did a little research in between. But once I did, I found it easy to follow the Maths used in support of your thesis. I am not certain, but I think you are discovering here that there is an interior and exterior quaternion derivative component (perhaps inner and outer would be better terminology). I am no expert, but I suspect that you are re-discovering things known in the field of exterior derivatives and differential forms.

However; this could be a step forward for Physics folks, in terms of having a missing piece of our understanding filled in. You appear to be revealing that there are symmetric and anti-symmetric components - under the Cayley-Dickson construction. An interesting romp through some fun Maths. It was not all obviously. or completely, pertinent to the assigned topic throughout, but was largely germane considering the result and its interpretation. So I give you kudos and high marks.

All the Best,

Jonathan

Jonathan Dickau

I'm probably bearing old news but...

The quaternions are isomorphic to parallelized S3, so examples can be geometrized and algebratized as is needed. Sometimes; it is easier to see contrasting cases as geometric examples, but I am not sure what the exact equivalency is here, in the case of the two analytic derivatives. So I would hesitate, and need to think further, before making an association.

Regards,

Jonathan

Colin Walker

Hi Jonathan,

I am glad you put in the effort to get a good understanding. Getting to a symmetric balance in the derivatives is the main thing. All else falls from that. This is quite obscure stuff which can get complicated, but ultimately it comes down to ordinary real derivatives.

I am fairly sure this is a new take on quaternion derivatives and analyticity, but it would not surprise me to have reinvented the wheel in some other context. Actually, it would be great if a correspondence to the quaternion derivatives could be found in some existing formulation, possibly linking the two.

Your point about geometric examples is a good one. There must be a way to interpret the derivatives in terms of well-known geometric algebra, but I will have to set this project aside for a while.

Another project has captured my attention. Rob McEachern put forward the idea (with a computer program as a demonstration) that quantum correlations are a result of sampling a process that can provide, at most, one bit of information per sample. I stumbled upon a faster way to produce the correlations using a similar Monte Carlo method, which also requires a large number of trials to accumulate statistics. It is easily verified that McEachern's one-bit criterion is satisfied. Bell correlations are produced matching the theoretical sinusoid to about 1% accuracy. Second paper will be posted on vixra hopefully by the end of the month. It turns out that there is a way to speed the procedure up by replacing Monte Carlo statistics by easily calculated probabilities. The CHSH test verifies that the procedure is classical, but there is a simple way to nearly reproduce the quantum expectation for CHSH. Here is my original paper on the subject, which includes links to Rob's work. I have a feeling that that this is fundamentally related to QM, but it is also interesting for its potential in simulations.

You have an impressive base of knowledge, Jonathan, and I really appreciate your feedback.

All the best to you,

Colin

Jonathan Dickau

Wow Colin!

Your idea above sounds impressive actually. Quantum information theory asserts that simplicity of calculation is what sets the likelihood for a process to run or outcome to occur. I'll have more to say, and hope we can stay in touch after.

All the Best,

Jonathan

[deleted]

Colin,

You might find this paper on Classical entanglement to be of some interest.

It is worth noting the discussion (section C) on detection efficiency. My vixra paper reports the double detection efficiency. But, since that cannot be directly measured experimentally, most other papers report only a conditional detection efficiency. The latter is equal to the square root of the former (sqrt(0.72) = 0.85) for the model in my paper.

Rob McEachern

Colin Walker

Hi Rob and Jonathan,

Thanks for the link to Danforth's paper on classical entanglement.

A draft of my second paper on quantum correlations has been posted at sites.google.com/site/quantcorr. There is also a file with C functions for calculating correlations based on the "geometric probability" of crossing a threshold. The detection rate is mentioned briefly, but I will have to look into Danforth's discussion.

I am just starting to get back to work on CHSH, after being away from it for some time. It seems like just a matter of using the results of Bell-like correlations. Might as well look at GHZ test as well.

Colin

Jonathan Dickau

Thanks Colin..

I'll check that out. And Rob's recommendation too.

JJD

Robert McEachern

Colin,

Here is another idea that you might like to consider. I mentioned to you previously that the Fourier transform of the triangular classical curve, consists of odd harmonics, in which the fundamental is a scaled version of the quantum correlation curve. I also mentioned that instead of assigning the same value to all the polarity pixels on a half-coin, a more sophisticated coding could be used. Now imagine combining those two ideas: for example, a coin with alternating values every 30 degrees i.e. the third harmonic, and another alternating every 18 degrees (the fifth harmonic) etc. Producing a coded coin as a weighted sum of those harmonic coins (with appropriate phase-shifts; i.e. rotations), might enable a better match between the classical and quantum curves, with a higher detection efficiency, by building the harmonic relationship between the classical and quantum correlation curves, into the coding of a coin's polarity. Comparing measured polarities (matched filter outputs) from different sectors might even enable a crude form of bit-error detection.

Rob McEachern

Robert McEachern

Colin,

By the way, here is the reason I mentioned detector efficiency in my earlier post:

In A strong loophole-free test of local realism on page 4, the authors claim that:

"Alice and Bob have system detection efficiencies of 74.7±0.3% and 75.6±0.3%, respectively... These background counts in our system raise the efficiency needed to violate a Bell inequality from 2/3 to 72.5%."

In other words, the authors claim to have eliminated the possibility of a detection loophole, even though their detection efficiency is well below the conditional detection efficiency obtainable via the single-bit-of-information, classical model.

Rob McEachern

Colin Walker

Hi Rob. There is certainly a difference in detector efficiency between the coin and vector models. I just checked that the detection efficiency for the noisy vector models is always less than 70%, whether using linear or cosine projection - compared to 72% for your coin model. I would suppose that the two-dimensionality of the surface of the coin must make the difference, somehow.

It is still not clear how to calculate signal to noise ratio for the coin. The vector model essentially takes all the noise samples from the coin added together, as the final position of a random walk, resulting in a single random sample. Perhaps that is an approach worth considering for the coin's SNR, even though it seems to go against dimensionality making the difference.

Your last idea, about getting a better fit to the theoretical quantum correlation by noise-coding the coins (or vectors), is intriguing. Posing the problem as minimization of error to determine operational parameters could be challenging, so a computerized search would likely be involved. - Colin

Robert McEachern

Colin,

Note that the matched filter detection implemented in my model, is far from an optimal detector. If there was any external noise in the system, in addition to the intrinsic noise of the coin, the matched filter would increase the noise power without any corresponding increase in signal power, since it extends far beyond the edges of the coins. An actual matched filter would duplicate the filter used within the coins, to eliminate this problem. I deliberately did not do this in the paper, since it would make the paper and figure harder to understand by the physics community, that has little or no intuitive understanding of such issues.

Another issue is aliasing. Since the image extends only a limited distance beyond the edge of the coin, high frequencies fold-back into the image without being properly attenuated by the filter. These types of issues, along with many others, could easily contribute a 1% error, such as you have observed.

By the way, we ought to move this discussion to a more appropriate web-page. On FQXi, a better choice might be the newly created Quickfire Quantum Qs

I will post a figure and comment there, pertaining to the odd harmonic series of the classical correlation curve as compared to the quantum correlation curve, and a possible connection with detector efficiencies.

Rob McEachern

Colin Walker

Hi Rob. I will keep an eye on the QQQ page. The connection between quantum correlations and a 1-bit process should attract interest. Trying to replicate the perfect sinusoidal quantum correlation might be a little obscure as an introduction, but there are many related questions to consider. - Colin

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