Is this new version of the CHSH inequality valid?

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Is this new version of the CHSH inequality by Richard Gill valid? Please see paper linked.

Yes
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No
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Total votes : 9

Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Tue Feb 04, 2014 5:53 pm

"Statistics, Causality and Bell's Theorem" is the title of a paper by Richard Gill which can be found here.

http://arxiv.org/abs/1207.5103

In section 2 of this paper, Gill establishes a new version of the CHSH inequality which he claims is equally valid as the original version basically. See eq. (2) and eq. (4) of the paper. Then he further claims that quantum theory can violate this inequality; see eq. (6) in section 3. It seems to me that with A, A', B and B' restricted to values of +/- 1, that it is impossible for anything including quantum theory to violate that inequality. So eq. (6) is false.

I think Gill is mixing up polarizer settings, a, a', b and b' with results. For one, why is there even a A' and B' when there is only detection stations A and B in an EPR-Bohm scenario.

Your comments are appreciated.
Last edited by FrediFizzx on Fri Feb 07, 2014 11:40 pm, edited 1 time in total.
Reason: Messing around with Poll Creation
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Tue Feb 04, 2014 11:37 pm

I think that Fred has not understood the transition from Section 2 to Section 3 of my paper.

Section 2 is about some elementary facts concerning an Nx4 spreadsheet of numbers +/-1. The columns are labelled A, A', B, B'. In each row, it is easy to check that AB + AB' + A'B - A'B' can only equal -2, 0 or +2. The average of (AB + AB' + A'B - A'B') over the rows therefore lies between -2 and +2. But of course it also equals ave(AB) + ave(AB') + ave(A'B) - ave(A'B'). Now suppose that for each row, all independently of one another, we completely randomly pick just one of the four products AB, AB', A'B, A'B'. Average the values of AB over the rows for which we chose AB, and so on. We'll get four new averages, each based on a random selection of about a quarter of the whole table. Each of these four averages will be close to the corresponding averages over the whole table. Hence with large probability, sample_ave(AB) + sample_ave(AB') + sample_ave(A'B) - sample_ave(A'B') will lie within a slightly larger interval from just below -2 to just about +2.

So the question is, does this have anything to do with quantum mechanics?

The answer is: not directly, but it would apply to a rigorous event-based local-realistic computer simulation of a Bell-CHSH experiment.

What do I mean by that?

There are going to be N runs: N times a pair of particles is created, sent to two distant locations. (N is, say, 10 000). At those two locations two observers randomly and independently choose a setting (angle) on a measurement device. Alice chooses every time randomly between settings a and a'. Bob between b and b'. The settings go into the measurement device, the particle arrives at the measurement device, and out comes an outcome +/-1.

Simulate this on a network of computers. N times, the following happens. Hidden variables from the source arrive at Alice's measurement computer, Alice doesn't see them but hits button 1 or button 2 to choose between a and a', i.e., to choose between observing A and A'. Similarly Bob does the same thing, far away (hits button 1 or button 2 to choose between b and b', i.e., to choose between observing B and B'. The computers print out the setting and the outcome. (Alice and Bob choose their settings by tossing fair coins.)

The point is that although Alice only gets to make one choice in each run, we can perfectly well define both outcomes which she would have observed, had she hit either button 1 or 2. They are just the result of running a certain computer problem with inputs 1, hidden variables from source, or with inputs 2, same hidden variables from source. Thus both A and A' "exist" - she just chooses randomly which one to actually observe.

If we suppose that the computer simulation does not exploit the memory loophole, thus each run can be considered completely separately from the previous runs, we see that we can use the computer programs to generate the Nx4 spreadsheet which I talked about before. (In other work I used martingale theory to extend the results from memory-less systems to systems with memory).

From a suite of computer programs simulating a Bell-CHSH experiment one generalizes to the metaphysical concept of a local realistic physical theory underlying the QM of that experiment. Or if you like, to the concept of a local hidden variables model underlying QM.

One may now ask, how come Joy Christian now claims that his model has been implemented in Python, Java, Mathematica ... and allows a perfect local-realistic event-based (i.e., particle by particle) simulation which violates the CHSH inequality?

If anyone is interested I can answer that question too, but maybe we should start a new thread for that topic.
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Re: Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Wed Feb 05, 2014 5:52 pm

Richard, you are side-stepping the question that I asked which is the topic of this thread. I did not mis-understand any transition from sect. 2 to sect. 3. In sect. 3 you state, "...certain models in quantum physics predict,

<AB>_lim + <AB'>_lim + <A'B>_lim - <A'B'>_lim = "

I contend that expression is impossible so your eq. 6 is false. Since nothing can violate your eq. (2) and (4), it is meaningless to use it for any kind of computer simulation network to test hidden variables.

Those that wish to discuss Joy Christian's model, please start a new thread.
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Re: Is this new version of the CHSH inequality valid?

Postby Joy Christian » Wed Feb 05, 2014 10:22 pm

Gill's error lies in this innocent-looking statement: "The point is that although Alice only gets to make one choice in each run, we can perfectly well define both outcomes which she would have observed, had she hit either button 1 or 2." This is the assumption of "counterfactual definiteness." Had Alice hit button 2 instead of 1, she would have observed outcome A' instead of A." This is a highly nontrivial assumption, both philosophically and mathematically. And physically of course it is *impossible* for Alice to observe both A and A' at the same time. Therefore Gill's treatment of this assumption is extraordinarily naïve, to say the least.

Let us first think about this philosophically. Do we sometimes wonder in real life like: "Had I taken a left turn instead of a right turn on the Baker street, I would have been enjoying doughnuts instead of getting mugged"? Yes we do. But it is getting mugged is what is actually happening to you, and enjoying doughnuts is just your fantasy. Thus one has to be extremely careful in using the assumption of counterfactual definiteness, especially in the EPR-Bohm type experiments. For what is involved in such experiments are events occurring in space and time, or at least in space at a given time. Therefore some knowledge and understanding of the actual physics of the general relativistic notions of space and time is absolutely essential. Any argument without this understanding is pure fantasy, like enjoying doughnuts.

So what, then, is the correct mathematical treatment of using counterfactual definiteness in analysing such experiments? Well, we must first ask: In which solution of Einstein's field equation the events A and A' which could have been observed by Alice could be actually occurring? Now there are compelling physical reasons to believe that they are in fact occurring in a parallelized 3-sphere [or in the topological space of the group SU(2)]. They are most certainly NOT occurring in an Nx4 spreadsheet. Nx4 spreadsheet is a fantasy, whereas parallelized 3-sphere is the reality. But once we recognize this, then it is clear that the counterfactually occurring outcomes like A and A' cannot be represented in a flat spreadsheet. That would be simply an incorrect representation of the actual events A, A' etc. In other words, what Gill is doing is simply incorrect. The only correct way to represent counterfactually occurring events is by representing them on the space of a unit parallelized 3-sphere. But once this is done, all quantum mechanical predictions for the experiment are easily reproduced, in a strictly local, realistic, and deterministic fashion.
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Re: Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Thu Feb 06, 2014 12:25 am

gill1109 reply to this was moved to a new topic.

viewtopic.php?f=6&t=11

Please stay on topic for this thread.
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Thu Feb 06, 2014 12:42 am

Joy hits the nail on the head. Yes, that assumption is the assumption of counterfactual definiteness, sometimes called briefly "realism".

Bell's theorem says that quantum mechanics is in conflict with locality + realism + freedom. To prove the theorem you assume the latter three, derive CHSH, and remark that QM violates CHSH.

If we believe quantum mechanics and we believe the proof of Bell's theorem, one must relinquish one of the three: "locality", "realism", "freedom". My personal preference is to relinquish "realism". Seems that Joy is on my side here.

Fred started this thread with a discussion of a new proof of Bell's theorem in a recent paper of mine. One of the themes of that paper is how this proof applies to computer simulations of violations of Bell's theorem. That issue is continued on another thread

viewtopic.php?f=6&t=11
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Thu Feb 06, 2014 1:00 am

Fred, I don't see your problem with the 2 sqrt 2 statement. This is the prediction of QM. It is experimentally confirmed (Aspect, Weihs, ...). Joy even believes it can be observed with classical systems (his colourful exploding balls experiment, which goes back to Peres - Peres of course arguing that we would not see a violation of CHSH in that context).
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Thu Feb 06, 2014 5:03 am

By the way my paper does not give a "new version of the CHSH inequality". It gives a proof of a sharper form of the standard CHSH inequality, under standard assumptions. It is sharper in the sense that it is finite N inequality. By letting N go to infinity, one recovers the standard CHSH inequality, proven under the standard conditions.

As Joy Christian perceptively wrote, one cannot contradict a true theorem, but one can hope circumvent it. He did this in his earlier works by computing a different correlation to the "usual" one (by "usual", I refer to the one which is conventionally used both by experimental and by theoretical physicists in this field). I think this is the main reason why his ideas did not gain much of a following.
Last edited by gill1109 on Thu Feb 06, 2014 6:52 am, edited 1 time in total.
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Re: Is this new version of the CHSH inequality valid?

Postby Joy Christian » Thu Feb 06, 2014 5:21 am

gill1109 wrote:Joy hits the nail on the head. Yes, that assumption is the assumption of counterfactual definiteness, sometimes called briefly "realism".


Partly correct, but one must not confuse "counterfactual definiteness" with "realism." A philosopher would call such a mixing a "category error."

gill1109 wrote:Bell's theorem says that quantum mechanics is in conflict with locality + realism + freedom. To prove the theorem you assume the latter three, derive CHSH, and remark that QM violates CHSH.


Again, partly correct. Bell assumed locality, counterfactual definiteness, and experimenter's free will. But he implemented counterfactual definiteness *incorrectly* by assuming an *incorrect* topology (that of R instead of S^3) in the *codomain* of his proposed measurement functions. Bell thus made a mathematical error and confused Boole's well known inequality with Einstein's concerns of locality and realism. By assuming the correct codomain (S^3 instead of R) in his own measurement functions one can derive a CHSH-like inequality, but bounded by 2\/2 instead of 2. In doing so no compromise with "locality", "realism", or "freedom" is needed.

gill1109 wrote:If we believe quantum mechanics and we believe the proof of Bell's theorem, one must relinquish one of the three: "locality", "realism", "freedom". My personal preference is to relinquish "realism". Seems that Joy is on my side here.


I believe, as did Einstein, that orthodox quantum mechanics makes correct predictions. I believe Bell's mathematical theorem is simply wrong (in the sense mentioned above). It simply does not follow from Bell's argument that we must relinquish "locality", or "realism", or "freedom." There is absolutely no reason to relinquish any of the three. More specifically, choosing the codomain S^3 in Bell's measurement functions does not compromise realism in anyway, because the actual measurement results (i.e., the image points within the codomain S^3) are still +1 or -1, just as demanded by Bell. So I am strictly on Einstein's side, and definitely not on Bell's side.
Last edited by Joy Christian on Thu Feb 06, 2014 5:50 am, edited 2 times in total.
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Re: Is this new version of the CHSH inequality valid?

Postby Joy Christian » Thu Feb 06, 2014 5:46 am

gill1109 wrote:As Joy Christian perceptively wrote, one cannot contradict a true theorem, but one can hope to get around it. He did this in his earlier works by computing a different correlation to the "usual" one (by "usual", I refer to the one which is conventionally used both by experimental and by theoretical physicists in this field). I think this is the main reason why his ideas did not gain much of a following.


This is completely incorrect. I DID NOT compute a different correlation to the usual one. I derived correlation E(a, b) = -a.b between measurement results A(a, L) = +1 or -1 and B(b, L) = +1 or -1, in a completely standard manner.

The reason my ideas allegedly did not gain much following is because many people did not understand the mathematical concepts I used; and some people even deliberately set out to misrepresent my work in a systematic manner. The standard strategy of such a misrepresentation followed an elementary logical fallacy, known as a straw-man fallacy. What the diehard believers in Bell's argument typically did was to erect a grotesque straw-man of my work, and then triumphantly knocked down their own distortion. In other words, they replaced my manifestly local, realistic, and deterministic model X with their own distorted version Y, triumphantly knocked down Y, and then wrongly proclaimed that they have in fact knocked down X. We often see this type of disingenuous strategy being played out in politics. But here it has been played out over and over again against my scientifically impeccable work on quantum correlations. No wonder my ideas did not gain much following.
Last edited by Joy Christian on Thu Feb 06, 2014 6:48 am, edited 1 time in total.
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Thu Feb 06, 2014 6:30 am

Dear Joy

"Locality", "realism" and "freedom" are for me abbreviations (convenient code-words) for "counterfactual definiteness", "relativistic local causality", and "no-conspiracy" - as my article makes clear. I followed an excellent discussion due to Boris Tsirelson on Citizendium, http://en.citizendium.org/wiki/entanglement_(physics).

Regarding the definition of correlation: you are welcome to your own interpretation of what you did. I write according to my understanding of what you wrote. I base myself on your written words and your written formulas.

I later discussed this issue with Azhar Iqbal, Lucien Hardy, Manfried Faber and others... they all told me that they had lost interest in your work on account of your new definition of correlation. And this they had done, *long before* I contacted them. So I hardly think that *the new definition of correlation* is a straw-man erected by your enemies. I recall hearing you speak at a work-shop in Berlin many years ago and you explained very clearly how you were computing correlations. I discussed it at the time with Reinhard Werner and others. If you want to circumvent Bell, it seems almost necessary to re-define one of the core concepts. The onus is on you to explain to the world why the correlations which people calculated before were the wrong ones.

This reminds me of Bryan Sanctuary who has his own local realistic model for the singlet correlations, unfortunately the correlation he finds is half the cosine, not the whole cosine. Unfortunately it has been well known for a long time how to generate half cosine theta in a completely local realistic way. He seems to want all the experimenters to divide their correlations by 2 so that they will fit to his model. I am afraid he is not getting much of a following.

Richard
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Re: Is this new version of the CHSH inequality valid?

Postby Joy Christian » Thu Feb 06, 2014 6:42 am

gill1109 wrote:I later discussed this issue with Azhar Iqbal, Lucien Hardy, Manfried Faber and others... they all told me that they had lost interest in your work on account of your new definition of correlation. And this they had done, *long before* I contacted them.


I do not believe a single word of this for a second. My work and my impeccably local-realistic model speak for themselves: http://libertesphilosophica.info/blog/.

In this world many people believe in many things. If someone wants to believe in the magic of non-locality and/or non-reality, then they are free to do so.
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Re: Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Thu Feb 06, 2014 12:12 pm

gill1109 wrote:Fred, I don't see your problem with the 2 sqrt 2 statement. This is the prediction of QM. It is experimentally confirmed (Aspect, Weihs, ...). Joy even believes it can be observed with classical systems (his colourful exploding balls experiment, which goes back to Peres - Peres of course arguing that we would not see a violation of CHSH in that context).


Joy does not believe your eq. (2) or (4) can be violated by his experiment. You lost the bet about that on FQXi, remember. You have mangled the true expression for CHSH. In CHSH, there is no A' and B' and there is no A' and B' in the experiments either. If you think your eq. (6) is true, you should be able to come up with a table of say N = 100 for A, A', B, B' that would violate it. You can't. It is impossible.

Let's stay on topic for this thread which is about the question I asked as a topic. Thanks.
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Fri Feb 07, 2014 2:49 am

I don't recall losing any bet whatsoever on FQXi! I do recall, Fred, that you and I had great difficulty communicating with one another.

In my paper, equations (2) and (4) are about certain averages derived from a given Nx4 table of numbers +/-1 with columns labelled A, A', B, B'. These represent the counterfactual outcomes of both possible measurements (with settings a, a'; and b, b') on both particles.

<AB> is defined to be the average of the product of the numbers in columns A and B, taken over all N rows.

<AB>_{obs} is defined to be the average of the product of the numbers in columns A and B of the Nx4 table, taken over a certain random subsample of rows. Each row, independently of the others, has probability 1/4 to be included.

<AB>_{lim} in equation (4) is defined as the limit of <AB>_{obs} in equation (3), if we think of the given Nx4 table as consisting of the first N rows of an infinitely long table.

Similarly, <AB'> etc.

Equation (6) is about data coming from experiments. Such data might also be represented as an Nx4 table, but now with columns labelled "setting Alice", "outcome Alice", "setting Bob", "outcome Bob". The "settings" columns contain the numbers 1 and 2 (referring to a and a', or to b and b', as appropriate); the "outcomes" columns contain the numbers +/-1.

If you like I can easily create the following: an Nx6 table with columns labelled by A, A', B, B', setting Alice, setting Bob.

The first four columns contain +/-1's. The last two columns contain 1's and 2's. One can extract from this an experimental data table with columns "setting Alice", "outcome Alice", "setting Bob", "outcome Bob". The settings are just copied over; and the outcomes are taken, for Alice, by choosing A or A' according to whether Alice's setting is 1 or 2, and, for Bob, B or B' according to whether Bob's setting is 1 or 2.

So you want to see such data which violates CHSH, giving approximately 2 sqrt 2? No problem, it's an easy job.
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Fri Feb 07, 2014 11:25 am

Fred, is this the spreadsheet you wanted?

http://www.math.leidenuniv.nl/~gill/fred.csv

It was created (using the R statistical language) with

http://www.math.leidenuniv.nl/~gill/fred.R

Note, the spreadsheet has 6 columns: setting Alice = SA, setting Bob = SB, A = A1, A' = A2, B = B1, B' = B2. If you proceed to select A or A' according to setting Alice = 1 or 2, and B or B' according to setting Bob = 1 or 2, you'll be able to create a real-life experimental data set:

setting Alice, outcome Alice, setting Bob, outcome Bob

You should observe a resounding violation of CHSH
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Re: Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Fri Feb 07, 2014 12:14 pm

That is not what your eq. (6) says. I don't think you are fooling anyone. I will try A1, A2, B1, and B2 for your eq. (6) after work.
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Fri Feb 07, 2014 12:58 pm

Equation (6) talks about what we would see if N goes to infinity. It refers to quantities determined from a real data set: ie a set of N quadruples "setting Alice, outcome Alice, setting Bob, outcome Bob". I've explained how, for N=100, you can extract the experimental data from the spread sheet which you asked me to prepare for you.

If you like, I can also post the experimental data.

BTW I am certainly not trying to fool anyone. It is quite unwaranted to suggest otherwise. Please try to behave according to the rules of this forum.
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Re: Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Fri Feb 07, 2014 9:49 pm

Here is an Excel file that computes the result for N = 100 for A = A1, A' = A2, B = B1 and B' = B2.

download/Nequal100GillCHSH.xlsx

And the result is 0.72 for <AB> + <AB'> + <A'B> - <A'B'> which is not even close to violation. You don't really need to try any more as even for the limit of N goes to infinity you won't be able to get a solution greater than 2. Quantum theory cannot violate your expression because nothing can. Not sure why you can't see that as you basically prove it before eq. (2). ??? So eq. (6) is false.

BTW: Sorry, but if you are not fooling anyone then you must be fooling yourself. There is nothing impolite about that. It just means you are wrong.
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Re: Is this new version of the CHSH inequality valid?

Postby gill1109 » Sat Feb 08, 2014 4:10 am

Dear Fred

Let's assume that no-one is trying to fool anyone else. Maybe even no-one is wrong - maybe we just don't understand one another?

So thanks for the new spreadsheet, which illustrates exactly this point. You forgot that in each of the N = 100 runs, one only gets to observe one of A and A', and only one of B and B'. You must first extract the observed data from the spreadsheet I sent you, and then compute CHSH for the "experimental data" which results.

You have to use the columns "setting Alice" and "setting Bob" to select one of A and A', and one of B and B'. Please try that. The four correlations are each to be computed on a different subset of the data. I have indicated which subsets to take.

Richard

PS to help you, I have uploaded a version of "fred.csv" from which I have deleted all non observed outcomes. You can find it at

http://www.math.leidenuniv.nl/~gill/fred2.csv
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Re: Is this new version of the CHSH inequality valid?

Postby FrediFizzx » Sat Feb 08, 2014 12:55 pm

But that is not what your eq. (6) is saying. If you want eqs. (2), (4) and (6) to say that, then formulate them to say that. Oh, guess what? It has already been done by CHSH. Listen, I am just trying to be helpful here with some constructive criticism. You should fix your mistakes in that paper before someone important notices your errors. I am not sure if you are doing the mistakes intentionally to mislead people or that you have somehow tricked yourself into believing that it is true. I hope it is the latter case as it explains why you are having such problems seeing that Joy's model does give the predictions of quantum theory in a local realistic way.
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