Talk:Tsirelson's bound

message: 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:
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)
thread_title:  Following text is by the late Tsirelson (added the header) 

message: 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)
replies: text: 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)
thread_title:  Proposed change 

message: * 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)
thread_title:  Cleanup 

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

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Upper bound