Talk:Tsirelson's bound

Following text is by the late Tsirelson (added the header)

1. Thank you for your interesting article on `my bound'.

2. I am astonished by the order `proof - formulation - interpretation'. In journal articles, it is usually `introduction - formulation - proof'. For an encyclopedia it is even more important (so I think), to start with an informal explanation (what is it good for), then give an exact formulation, and then (if at all) a proof.

3. The notion `operator' could be more clear. Many people believe that operator is usually a linear transformation of a Hilbert space (I mean, in the context of quantum theory; in general, the space may be Banach etc, and the operator may be nonlinear). I guess that your notion of operator is sometimes (or always?) different; probably you mean an element of some algebra (associative? ordered? C-star algebra?) or even someting more general (what exactly?). Could you please be more explicit?

4. More than 80 relevant references are collected on my site, here: [1] Maybe, an external link to it would be appropriate.

--Boris Tsirelson

Maybe I shouldn't throw any stones, but it seems to me this article could be much clearer, for instance following the exposition given in Asher Peres' book p 174 which is self-contained and lucid. CSTAR 20:40, 15 Feb 2005 (UTC)

--

Sorry, but I was shocked by the poor quality of the article. The article is not readable even for those familiar with the result.

My opinion: All the details should go except for the simple one for the CHSH inequality in Tsirelson's original paper. Afaik the bound is nowadays understood as a bound on the correlations that one can obtain in the setup of an injective C* algebra. I am not aware of any usage of the bound in the setup of hidden variables models. Rather, the reason for the peak in the citations around 2006 is the advent of Popescu-Rohrlich-Boxes (these Boxes are hypothetical but not yet unreasonable devices which violate the bound). --13:04, 10 April 2010 (UTC) —Preceding unsigned comment added by IXhdBAH (talk • contribs)

Proposed change

Suppose

$X_{A},Y_{A},X_{B},Y_{B}\,$

are self-adjoint operators such that

$X_{A}^{2}=Y_{A}^{2}=X_{B}^{2}=Y_{B}^{2}=1\,$

and the A operators commute with the B operators, that is

$[X_{A},X_{B}]=[X_{A},Y_{B}]=[Y_{A},X_{B}]=[Y_{A},Y_{B}]=0\,$

Then for any state φ,

$\langle X_{A}X_{B}\rangle _{\phi }+\langle X_{A}Y_{B}\rangle _{\phi }+\langle Y_{A}X_{B}\rangle _{\phi }-\langle Y_{A}Y_{B}\rangle _{\phi }\leq 2{\sqrt {2}}\,$

and the upper bound is achieved by the Pauli spin matrices as in the Bell theorem article. The proof considers

$C=X_{A}X_{B}+X_{A}Y_{B}+Y_{A}X_{B}-Y_{A}Y_{B}$

and computes C².

This would mean deleting most of what is written in the article currently.CSTAR 06:21, 19 Feb 2005 (UTC)

My knee-jerk reaction is that rewriting in this way would be a good idea. It would also be nice to have a statement of the theorem before launching into its proof. I might give this a shot on some rainy day. linas 22:22, 18 July 2005 (UTC)

Please fix the encoding of the proof starting F • U + F • V + U • G - V • G on the article page, in my browser the bold font renders the minus sign like a smaller dot so that last term looks like U • G • V • G.

Formatting

Is there a particular reason that the equations and inequalities show up in text form and not in .png form with all the nice and tidy formating?

Cleanup

• In non-TeX notation, one italicizes variables but NOT parentheses and NOT digits.
• In non-TeX notation, a minus sign is much longer than a mere hyphen. Thus:

3 − 5,

not

3 - 5.

• "Inline" TeX can be obnoxious by "displayed" math is better in TeX than in non-TeX notation.

See Wikipedia:Manual of Style (mathematics). All this was neglected in three sections of this article. Michael Hardy (talk) 00:12, 20 October 2008 (UTC)

...continuing with the above, I suspect that "≤" was intended where "=<" appears. And lots of similar problems. Michael Hardy (talk) 17:44, 20 August 2009 (UTC)

Major rewrite

Thanks to Mateus Araújo it is now considerably better than before. Boris Tsirelson (talk) 13:31, 11 December 2012 (UTC)

Upper bound

I agree with User:Qcomp reversion of my edit. But perhaps we could make it even clearer what the actual no-signaling bound is; it seems that in a world where Alice sees +1 and -1 randomly 50/50 no matter her settings, and Bob always sees the opposite, the CHSH expression would be 4, and there would still be no way they could signal to each other. 84.211.177.136 (talk) 19:26, 9 August 2021 (UTC)

in the comment accompanying my edit there was one error: as you rightly observe, even the maximal correlations (allowed algebraically) do not allow signalling and they are, actually, what can be achieved with a Popescu-Rohrlich box, so there are only three bounds: LHV$<$quantum$<$no-signalling. I think it should not surprise that purely random events (however correlated they are), do not allow transmission of information. I don't know if there's a non-empty super-quantum but less-then-PR scenario. The framework in which this is usually discussed is that of Generalized probabilistic theorys and one could link there; I'm not sure what (if anything) should be added to this article. Maybe a brief section putting the bound in context of the other bounds? Relation to bounds for larger-dimensional systems? I would not put more in the lead (and, if possible, rather move details to that new section) --Qcomp (talk) 20:34, 9 August 2021 (UTC)

There's plenty of bounds between the quantum and no-signalling bounds. Some of them are described in Quantum nonlocality#The physics of supra-quantum correlations. Also, keep in mind that in general the signalling bound is larger than the no-signalling bound, it's mostly by chance that they coincide in the CHSH expression. For example, in the CH variant the signalling bound is larger. Tercer (talk) 15:49, 10 August 2021 (UTC)

On locality

@Tercer: sorry for the wording but I still think we should be careful when saying that quantum mechanics is non-local. As explained in principle of locality and quantum nonlocality, locality is not the only thing that is at stake here it is either locality, realism or statistical independence (superdeterminism). Forgetting the last one, I propose a new version for the lead:

A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates Bell inequalities (it cannot be described by a classical local hidden-variable theory), a natural question to ask is how much can the Bell inequality be violated within quantum mechanics.

Any other suggestion? --ReyHahn (talk) 12:11, 9 June 2022 (UTC)

I would slightly rephrase your paragraph like this

A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates Bell inequalities (i.e., it cannot be described by a local hidden-variable theory), a natural question to ask is how large can the violation be.

I think it is fine to sidestep the interpretation of Bell's theorem, as that is gigantic can of worms. It is not true, however, that the what is at stake is either "locality, realism, or superdeterminism". First of all, "realism" has nothing to do with the informal interpretation of realism, which would be the existence of an objective reality. It simply means determinism. I'm rather aghast that this terminology caught on, and led to countless people saying nonsense like "Bell's theorem proves that there's no objective reality". In any case, it is true that one can derive Bell's theorem from the assumptions of locality, determinism, and no superdeterminism. This is CHSH's 1969 theorem. However, we also have Bell's 1976 theorem, that uses the assumptions of local causality and no superdeterminism. Therefore, one can correctly say "quantum mechanics is nonlocal" referring to the latter theorem, i.e., meaning that quantum mechanics violates local causality. This got established as jargon, thought, so often you'll see "quantum mechanics is nonlocal" in paper meaning simply that it violates Bell inequalities. Tercer (talk) 13:46, 9 June 2022 (UTC)

Thanks for the rewording. I would also be careful with determinism, nobody meant any of the philosophical items in realism, but more like counterfactual definiteness and the whole idea of needing more hidden variables to explain quantum mechanics. "Determinism" per se is a whole can of worms too, consider many-worlds interpretation. I also get that "nonlocality" means violating Bell inequalities. However if somebody is reading for the first time that quantum mechanics is non-local, it is very misleading, it is like if physicist did a Bell test and all assumed that we need non-local hidden variables when in fact the most practical way to do QM is to forget about any of that and assume that is probabilistic and local (as in space-like objects cannot influence each other).--ReyHahn (talk) 17:36, 9 June 2022 (UTC)

Well, quantum mechanics does violate local causality. Local causality is defined as the probability of some event being independent of space-like separated events. So no, you can't just assume that it is probabilistic and local and be done with it. That's the problem with the CHSH version of Bell's theorem, people conclude that giving up determinism is enough to retain locality, and this is not the case. You can retain locality as defined by CHSH, which is a rather weak definition, and does not imply local causality. I think local causality captures much better what is informally thought of as locality.

Determinism, on the other hand, I think it's rather straightforward. It is meant as single-world determinism. The fact that Many-Worlds is local, deterministic, and violates Bell inequalities doesn't contradict anything. Tercer (talk) 18:54, 9 June 2022 (UTC)

Well the main debate if over. I do not think I will bite much more into the issue. Putting determinism aside, I admit that I have never understood how Bell's local causality is so different from the local realism argument. If you have any good and clear material that makes a case for the former against the latter please link it (there are a few but see the bold part) and I gladly take a look. As you say this such a can of worms in terms of definitions and terminology.--ReyHahn (talk) 19:54, 9 June 2022 (UTC)

Well, both versions are different and important on their own. The CHSH version because it rules out the sort of hidden-variable models people want to do, and the Bell version because it shows there's something nonlocal about quantum mechanics. Unfortunately the literature on the topic is a minefield, all sorts of confusion and inconsistent terminology, as well as outright mistakes. I did write a couple of blog posts to help clear things up, maybe you'll find it helpful [2]. Tercer (talk) 20:55, 9 June 2022 (UTC)