# Talk:CHSH inequality

WikiProject Physics (Rated B-class, Mid-importance)

## Flawed Argument

The sentence The mathematical formalism of quantum mechanics predicts a maximum value for S of 2√2 (Tsirelson's bound),[4] which is greater than 2, and CHSH violations are therefore predicted by the theory of quantum mechanics. uses a flawed argument. The therefore is wrong. One must show that Tsirelson's bound is the BEST possible maximum or, by the same token, that the maximum of the Tsirelson bound is ASSUMED in a number of cases. It is not sufficient to merely demonstrate one bound. Both is, of course, true and can be done. So the result is fine but the line of reasoning does not live up to the standards of precision which are possible.

## Preceding Discussion

Hi,

May I suggest also to include criticism on the completeness of the CHSH. To be found at: link http://dx.doi.org/10.1016/j.rinp.2014.06.002.

If Alice and Bob are allowed additional random elements i.e. a four sided dice and an additional coin and a third person Carrol is allowed to randomly draw LHV models from a "model pool"(an urn with names of models for instance) then the CHSH contains a statistical loophole for LHV models violating S=E(1,1)-E(1,2)-E(2,1)-E(2,2), and Probability{|S| > 2 | using LHV} is nonzero for quartets (1,1),(1,2),(2,1) and (2,2).

Regards. — Preceding unsigned comment added by Hgeurdes (talkcontribs) 10:14, 18 August 2014 (UTC)

## Suggestion for re-write of whole article

Hi Franck I've just been reading Be bold and realised that I have to re-write the whole article! What we need is basically:

What we need is basically:

• A few words of introduction.
(Frank W ~@) R 03:41, 2 Aug 2004 (UTC)) Yes, a few words of introduction seem appropriate, such as those in the present article.
• Reproduce the approved "derivation", which is not the original CHSH 1969 one but Bell's 1971 version (reproduced in Clauser and Shimony's 1978 report, pp 1892-1893, or p 37 or p 156 of Speakable and Unspeakable).
I agree that the derivation(s) presented in the article may draw on several sources (of course in compliance with the GNU Free Documentation License);
more details on different derivations may be spelt out in separate dedicated articles.
In trying to reproduce any derivation (as far as is indeed reproducible at all) what we basically need (and have already) are careful definitions of the symbols used, of quantities to be measured experimentally, of distinctions to be denoted; together with the complete statement of the conditions required to derive (or, indeed, prove) the inequality under consideration.
As part of a whole series of articles on related topics, within an encyclopedia, the article should use notation distinctly, consistenly and carefully; e. g. (ideally)
* reserving letters A and B (etc.) to denote and reference distinct detector systems (a.k.a. "Alice" and "Bob");
* denoting counts (natural numbers) with n or N;
* avoiding the use of operator symbols (such as + and -) or number values (such as 1 and 2) as mere distinct indices or labels.
• Mention the fact that it can alternatively be derived from the CH74 inequality.
• Quote some of what Clauser et al actually wrote in 1969, 1974 and 1978 re use of the inequality, making it clear that in 1969 what they in fact recommended was the CH74 inequality, only with a rather clumsy derivation that implied the latter needed assumptions that are in fact irrelevant.
• Mention some of the experiments in which the CHSH inequality has been used, and the typical reservations regarding the fair sampling/detection/efficiency loophole.
* To point out consequences of the derivation (and the corresponding thought experiment) itself, i. e. in particular the class(es) of "objective local theories (OLT)" which don't satisfy the conditions under which the inequality can be derived; and
* To point out the advance of CHSH, especially over Bell, Physics1 (1964), eq. (15), e. g. by obtaining the latter as a special case of the former.

What we don't need are false statements such as:

It is named for the authors, John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, of the paper [1.] in which this inequality was first derived, and where its utility for experimental test was emphasized. [My italics]

[It was not recommended for practical use, perhaps because at the time only single-channel experiments could be done, but also because of the difficulty of non-detections.]

The indicated emphasis is evident in title of reference: "Proposed Experiment to Test ...", PRL23, 880 (1969), as well as in statements therein, such as
"The aim of this paper is to propose explicitly such an experiment." or
"[...] the condition for violation of Inequality (2b) is [...] (4). This is the essential requirement on the design of a decisive experiment."
If such emphasis and suggested decisiveness was modified in later publications, then this ought to be mentioned, of course; thus approaching a NPOV.
Also, whether "non-detections" and associated assumptions have any relevance at all, depends on the definition of the quantities in terms of which the CHSH inequality is derived.

We don't need any of the first page: it is absurd to introduce something that was designed to test hidden variables in any other way than in terms of hidden variables!

On the contrary: without careful definition in terms of experimental procedures and values (here: counts) we have no experimental test at all; nor any scientific/falsifiable theory to begin with.

We don't need to even define ρ(j), since it is assumed that the distribution of ρ is the same for all subexperiments. The ρ that we are interested in is a function of the hidden variable, λ.

That would be neglecting the important class of OLTs for which the hidden variable does vary trial by trial, for which the trial index j consequently is appropriate to denote the system state, of which the assumption of distribution ρ being the same for all subexperiments cannot be made, and for which the CHSH inequality (among others) can consequently not be derived.

We don't need functions A(aA, λ) that take values +1 or -1 but Bell's 1971 "A bar", the mean value, allowing for non-detections.

We don't need ... to be limited to just one derivation; inequalities which are known by distinct names and/or which were published by distinct sets of authors generally deserve separate articles.
But whoever lets statements about "non-detections" appear in articles, such as Bell's 1971 eq. (6), needs to be very careful to point out all related assumptions, such as Bell's 1971 eq. (10), and to avoid generally false assertions such as Bell's 1971 eq. (12).

We don't need even to mention the Tsirelson Inequality, which I for one have never heard of and which has never been applied in any experiment.

The Tsirelson Inequality surely deserves its own article. Its relevance here is of course that it is of a similar form as the CHSH inequality, but weak enough to be satisfied by all OLTs. This can be referenced for instance per M. A. Nielsen, I. L. Chuang, "Quantum Computation and Quantum Information", Cambridge UP (2000).

Caroline Thompson 21:36, 29 Jul 2004 (UTC)

Regards, Frank W ~@) R 03:41, 2 Aug 2004 (UTC)

Caroline Thompson 18:44, 2 Aug 2004 (UTC)
Frank, I agree with many of your points. Exceptions are:
Re notation: I find all the little arrows make for far poorer legibility than the traditional + and -.

[FW] ... point out consequences of the derivation (and the corresponding thought experiment) itself, i. e. in particular the class(es) of "objective local theories (OLT)" which don't satisfy the conditions under which the inequality can be derived;

There must be an infinite number of thought-experiments that don't match the conditions! CHSH, though, were concerned with the *real* possibilities. I think that on this page we should restrict ourselves to just these, vis the ideal CHSH inequality as expressed in (1a) of the CHSH 1969 paper, or the equivalent more symmetrical form -2 <= S <= 2, but in the more general form proved by Bell, 1971, in which non-detections are allowed for. Since CHSH did not in fact recommend this test for practical use (see below), but instead the CH74 test we can scarcely avoid mentioning the latter.

[CHT] [It was not recommended for practical use, perhaps because at the time only single-channel experiments could be done, but also because of the difficulty of non-detections.]

[FW] The indicated emphasis is evident in title of reference: "Proposed Experiment to Test ...", PRL23, 880 (1969), as well as in statements therein, such as "The aim of this paper is to propose explicitly such an experiment." or "[...] the condition for violation of Inequality (2b) is [...] (4).

Yes, but inequality (2b) is the CH74 test, involving single-channel polarisers and subexperiments with polarisers removed. They said quite clearly that inequalities (1) (the CHSH inequality) could not be used in optical experiments ( the only kind envisaged) because of the low efficiency of photodetectors. There follows a rather indirect derivation of the CH74 inequality, using distincly confusing notation. '-' now means not the '-' channel of a two-channel analyser but non-detection in the single channel of a one-channel one, i.e. it would correspond to -1 or 0 if we had two channels.

[CHT] We don't need any of the first page: it is absurd to introduce something that was designed to test hidden variables in any other way than in terms of hidden variables!

[FW]On the contrary: without careful definition in terms of experimental procedures and values (here: counts) we have no experimental test at all; nor any scientific/falsifiable theory to begin with.

But what that first page is really saying is something very simple: that we look for + and - outputs on both sides and clock up the appropriate "coincidence" counter when there are non-null observations. And I find the notation makes for very poor legibility. In any event, we can surely drop most of the subscripts.

[CHT] We don't need to even define ρ(j), since it is assumed that the distribution of ρ is the same for all subexperiments. The ρ that we are interested in is a function of the hidden variable, λ.

[FW] That would be neglecting the important class of OLTs for which the hidden variable does vary trial by trial, for which the trial index j consequently is appropriate to denote the system state, of which the assumption of distribution ρ being the same for all subexperiments cannot be made, and for which the CHSH inequality (among others) can consequently not be derived.

If such OLT's exist they need to be on a separate page.
(Frank W ~@) R 06:05, 4 Aug 2004 (UTC)) Surely eventually. More immediately, the article on OLTs ought to define them more explicitly and separately.
What we need here is to follow as succinctly as possible the evolution of the CHSH test as used in practice.
No: what's needed here (according to the title of the article is most of all an adequately self-contained derivation of the CHSH inequality; with applications and relation to other articles and topics mentioned and linked, for instance (why not?!), according to your suggestions below, to CHSH Bell test.
In actual experiments
(... at least in some, which may be referenced ...)
they assume the source always produces the same distribution of possible states,
(... presumably in "sufficiently large ensembles" or "over sufficiently (though necessarily finitely) many trials" ...)
... where the diversity of (all) "possible states" is taken to include any hidden diversity; i. e. if "λs" denotes the (complete) system state in (at least) one particular trial of set J, then it is assumed that in (at least) one trial of set K (disjoint from set J) a completely equal system state occured, which is equally to be denoted by "λs".
Yes, this assumption is not only relevant in some actual experiments, but it is essential for deriving the CHSH inequality itself in the first place.
Consequently the separate consideration of OLTs which don't satisfy this assumption.
and the probability of the state λ can be specified by a single function ρ(λ) that does not vary between subexperiments.
Under the described assumption, yes. ρ( j ) appears of course in the definition of "P" as experimental quantity, as the simple 1/({ j = first of J Σ last of J} 1).

[FW]... But whoever lets statements about "non-detections" appear in articles, such as Bell's 1971 eq. (6), needs to be very careful to point out all related assumptions, such as Bell's 1971 eq. (10), and to avoid generally false assertions such as Bell's 1971 eq. (12).

Non-detections have been there from the start. CHSH 1969 paper, last para of page 881:
"Unfortunately, if the particles are optical photons (as in the experiment proposed below) no practical tests of [the CHSH inequality] (1) can presently be performed in this way, because available photoelectric efficiencies are rather small ..."
correct, FWIW, and followed by
"We shall therefore henceforth interpret A( a ) = +/- 1 and B( b ) = +/- 1 to mean emergence or nonemergence of the photons from the respective filters [... instead of] detection or nondetection [in a single-channel detector ...]"
Note that this is subsequent to the derivation of the CHSH inequality (1b). The possible consideration of (non-unity) "efficiencies" and "non-detections" is contingent on interpretation of what's to be counted (and perhaps appropriately addressed in CHSH Bell test); it's not part of the derivation of (1b) itself.
Also, there are two complicating issues:
1. CHSH seem to go on to consider accumulative "rates" rather can sets of distinct, separate, single-photon-pair trials; and
2. "non-detection with a single-channel detector" is a notion quite different from "non-detection with a two-channel detector" ...
I was under the impression that our discussion so far only dealt with "non-detections" in the latter sense, i. e. with trials in which
"nj (A↑) = 0" as well as "nj (A↓) = 0" were counted ...
Incidentally, Bell's eq. (12) of his 1971 paper is the QM prediction.
I have to be more precise: In question is the statement immediately preceding (12) that "P( a, b )) is given" by (12) even if the definition of "P( a, b )" involves "non-detections" (in the latter sense).
The whole purpose of Bell's theorem is to show that the local realist prediction differs from it.

[FW] The Tsirelson Inequality surely deserves its own article. Its relevance here is of course that it is of a similar form as the CHSH inequality, but weak enough to be satisfied by all OLTs. This can be referenced for instance per M. A. Nielsen, I. L. Chuang, "Quantum Computation and Quantum Information", Cambridge UP (2000).

But the assumptions behind it have no physical meaning! If it's not used anywhere surely it does not deserve a mention?
I strongly disagree; however, the derivation as sketched presently is quite simplistic ... although, thereby, it may not violate any copyright per Lett. Math. Phys. 4, p. 93, (1980) ...
Other inequalities, such as the trivial one variously attributed to D'Espagnat, Wigner or whoever, discussed in Bell's "Bertlmann's socks" article (Speakable and Unspeakable, pp 139-158) could be mentioned somewhere in wikipedia
Obviously as Wigner-d'Espagnat inequality, as named in Bell's article ...
but there's no point in mentioning everything.
(Which would be an awful lot more than what's being menitioned ...)

Anyway, what I think we need to do is create a completely new page, perhaps with title "CHSH Bell test"
Certainly.
then if this looks an improvement the old CHSH inequality page can be dropped.
I strongly disagree; those topics are quite distinct, though closely related, of course.

Is there some way of creating a temporary page to develop it on?
(By wikifying the term as article name; as I did above.)
Caroline Thompson 18:50, 2 Aug 2004 (UTC)
Regards, Frank W ~@) R 06:05, 4 Aug 2004 (UTC)

I'm working on a "CHSH Bell test" page, but have come across a difficulty! As far as I can see, Shimony's derivation (which I am considering including) does not really need the assumption that you can multiply independent probabilities at all. All it needs is that the state of the complete system be unaffected by the polariser settings. This is generally the case in real experiments other than ones such as Rowe's "trapped ion" one, where everything, including "detector settings" could interact.

BTW you've completely misrepresented Malus! I've put a comment on the discussion page, though wikipedia doesn't seem to think the article exists. Also I find your notation in the Wigner-d'Espagnat inequality article absolutely stultifying. Where did you get your ideas about Malus' from? I've recently been reading about his true work in Mach's "Principles of Optics" and you are definitely misrepresenting him. You are also, I think, misrepresenting John Bell if you associate his harmless discussion in the Bertlmann's socks article with your calculated cosines.

Caroline Thompson 10:09, 4 Aug 2004 (UTC)

I've just checked Shimony's article: his derivation does not seem to be correct. A pity. I'll revert to using John Bell's 1971 one unless I can find out how to correct Shimony's. Caroline Thompson 10:13, 4 Aug 2004 (UTC)

## What CHSH really proved (and more on the Chaotic Ball geometry)

Hello Franck

I fear this problem with geometry is mainly a red herring. It would help if you went back and re-read my Chaotic Ball model paper, this time forgetting all preconceptions. The ball is assumed to move randomly about a fixed centre. The assistants stand and look at it. The directions of their gazes are the "detector settings" and are quite unambiguous. This is only an analogy so accuracy does not matter. If you look at something you will be facing a particular direction. It is surely not beyond the wit of man to fix the two angles of gaze with respect to some fixed base line -- a wall of the room or something?

To take (some of) your points in turn:

[FW:] ... under which circumstances do you suppose "the system state" could undergo change, if not in general from one trial of the sequence to the next trial ?

Yes, the state (the orientation of the ball) is assumed to change from one observation to the next, but randomly. The "trial index" has no physical significance. As I explained, in a real experiment one organises things to ensure that the source is always producing the same random set of states. It makes sense to define one distribution function ρ(λ) to give the probability of emissions being in the state λ, and this probability distribution stays the same for all individual trials and all sub-experiments (groups of trials with the same fixed detector settings).

[CHT:] But what is it we are really trying to reach agreement on? The major issue, I thought, was whether or not the CHSH test as currently applied, using the sum of coincidence rates as denominator, is valid?

[FW:] I tried to address this already above: that's certainly valid, especially as being free of any particular assumptions which would have to be made in order to consider "non-detections".

The CHSH inequality has only been proved true if the terms involved are all built from sums and differences of probabilities. You are not going to be able to reliably get an unbiased estimate of a probability if you divide by the number of observed coincidences, since the observed coincidences are not the set you started with. You have to divide by N, the number of emitted pairs.

[FW:] ... neither do they appear decisive about how to obtain the number "N", the "number of emissions" itself.

Read on and you will find that they realise that N is not in practice known and therefore recommend a different test, in fact the CH74 one. Later discussion by Bell of possible use of the CHSH inequality was always on the assumption that there would be an "event-ready" detector. The number registered by this would be used as N. No such experiment has even been done, though. Even in pulsed laser experiments, where the number of pulses ought, one would have thought, to be known, this number does not seem to be used.

[FW:] ... it is at least one sensible approach, as well as the simplest, to define and count the "number of emissions", "N" as "number of detections" which is directly available.

My [chaotic ball model] is sufficient to illustrate the fact that the inequality if used with the number of detections as denominator cannot serve its original purpose. Local realist models can infringe it if there are in fact some non-detections. Clauser et al were aware of this.

[FW:] [Re the geometry of the ball model] ... Could you for starters please point out just where the "index" of a protractor would have to be placed, in the presumably "precise diagram" (fig. 2)? My above guess, "(on) the center C of the ball" may have been a misunderstanding, of course. (I also note that the "Vectors a and b" of fig. 2 have no intersection drawn at all ...)

As explained above, there is absolutely no problem here. And just why you think angles have to be defined in terms of distances I have no idea. Can't you talk about the direction of the sun without any reference to distance?

[CHT:] But perhaps more importantly, if you think you can estimate the angle from the observed counts (assuming Malus' Law) this is entirely begging the issue!

[FW:] it certainly underscores the importance of CHSH's suggestions as thought experiments ...

The CHSH 1969 paper is not about thought experiment but real ones. The results of the first real one using ideas from the paper (though not the "CHSH test" but the CH74 single-channel one) were published in 1972.

[CHT:] I have seen the formula you use ("ArcCos[ (NN + SS - NS - SN) / (NN + SS + NS + SN) ]") elsewhere, but that does not make it meaningful.

[FW:] The fact that this formula represents a mathematically unambiguous expression in terms of actual experimental counts (based on your own definition) does make it meaningful; a physical quantity to be measured ...

OK

[FW:] Since the days of Malus, this quantity even has a name: the "orientation angle", e. g. between a pair of "polarizer axes" (or in the later version: between a pair of "analyzer axes").

You're putting the cart before the horse, Franck! Malus would never have dreamed to using the term "orientation angle" to mean anything other than the geometrical angle.

Yours, Caroline Thompson 09:22, 29 Jul 2004 (UTC)

I've replaced the whole page as promised. It's rather different! I cover all the functionality of the original and more, except that I've left out all mention of the Tsirelson Inequality, which is both irrelevant and based on nonsensical assumptions. Caroline Thompson 09:17, 14 Aug 2004 (UTC)

## Is wikipedia's policy of a "Neutral point of view" suppose to mean the blatant suppression of a minority one?

Dr Chinese, your wholesale elimination of refs to my Chaotic Ball paper and pruning out of links to the Bell test loopholes page has surely overstepped the mark! Can't we please be neutral, i.e. not biased towards the supposedly "accepted" POV? It's not as if I was the only local realist in the world, or the only person to have realised the relevance of the fair sampling loophole to the validity of actual Bell test experiments.

As you will see, I've replaced the ref to my own paper by one to Gisin's. Wikipedai is the worse for this, since my paper is considerably more comprehensible, giving an intuitive analogy that correctly covers the main logic of the fair sampling loophole. Caroline Thompson 21:45, 31 Jan 2005 (UTC)

All other than Caroline: the substance of the debate on this is at talk:Bell's theorem, and is not duplicated on related articles. Caroline: stop promoting yourself. Your POV deserves the same relative attention as it would get in a QM textbook... so quit wasting our time with coy arguments about fair representation. You have it with the Bell test loopholes article.--DrChinese 22:27, 31 Jan 2005 (UTC)

## Robot spelling "correction"

Perhaps I just have to give up: the robot thinks the word "oriented" is preferable to "orientated", which it does not seem to have heard of. My "Word" spellchecker accepts both. Is there any way of persuading the robot I'm right? Despite what "Word" says, "oriented" is not part of my language. Caroline Thompson 5 July 2005 09:10 (UTC)

## List of inequalities

I just added this article to the list of inequalities, where, strangely, it did not appear earlier. If anyone knows of others that should be there and are not, could they please add those too? Michael Hardy (talk) 14:13, 11 December 2008 (UTC)

## odd paragraph

Is it ok to remove the following paragraph from the article? It doesn't seem to make sense - how can something be assumed to be a hidden variable *in an experiment*, rather than in a theory?

"Note that in all actual Bell test experiments it is assumed that the source stays essentially constant, being characterised at any given instant by a state ("hidden variable") λ that has a constant distribution ρ(λ) and is unaffected by the choice of detector setting."

Nathaniel Virgo (talk) 15:42, 30 May 2011 (UTC)

I agree. It's nonsense. The article generally is very poor. Experimental and theoretical issues are muddled and mixed. It's also very outdated in tone. The issues discussed here were big things in the 70's but there has been enormous progress both in experiment and theory since then. Richard Gill (talk) 19:08, 5 June 2012 (UTC)

## ..The constraint but can on the other hand be infringed..

'...The constraint but can on the other hand be infringed by quantum mechanics....' . . The intended meaning of this sentence is elusive. . I know 'butt-can' wasn't intended or any better, but it is the only thing that comes close to making sense. 70.185.109.98 (talk) 17:54, 4 February 2013 (UTC)

## The experimental estimate for ${\displaystyle {\displaystyle E(a,b)}E(a,b)}$ is then calculated as: (see Eq. 3)

I'm not clear from reading this. Should incidences be counted as +1, -1 or should they be counted as +1, 0?

103.118.46.247 (talk) 07:45, 3 November 2020 (UTC)