• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,720 comments
1,470 users with positive rep
818 active unimported users
More ...

  Optimality of the CHSH strategy

+ 2 like - 0 dislike

The maximum achievable probability of the Clauser-Horne-Shimony-Holt game is $\cos^2(\pi/8)\approx85.355\%,$ which can be proved with Tsirelson's inequality. But I don't imagine that this remained unknown until Tsirelson's 1980 paper. When was it first known that this constant is optimal?

I must admit—shamefully—I have not read the famous 1969 paper, so if the strategy's optimality is proved there my apologies.

This post has been migrated from (A51.SE)
asked Dec 2, 2011 in Theoretical Physics by Charles (10 points) [ no revision ]
Charles, thank you for asking. However, to be honest, the question seems to show lack of respect of other _I haven't read the paper I now I should, could you do it for me?_. Is there any particular problem preventing you from reading the 4+$\epsilon$ [page paper](http://astrophysics.fic.uni.lodz.pl/100yrs/pdf/14/022.pdf)?

This post has been migrated from (A51.SE)
@PiotrMigdal: I hadn't been able to find the paper online. Thanks for finding it for me! (I looked but found only paywalled versions.) If they do solve this in the paper then this resolves my question; if they don't, I'll edit it to clarify that they do not.

This post has been migrated from (A51.SE)
I found the link with http://scholar.google.com. Anyway, if the paper solves your question, you can edit the question _and_ answer your own (so it will be useful for other people).

This post has been migrated from (A51.SE)
Have you even read [Tsirelson's paper](http://www.tau.ac.il/~tsirel/download/qbell80.html)? He states that the bound $2\sqrt{2}$ was already known for qubits, and he proved that it holds in fact for any dimension. If you can find out who did discover it for qubits, that'd be an interesting answer.

This post has been migrated from (A51.SE)

1 Answer

+ 5 like - 0 dislike

It seems that few people were interested in this problem. In the early days, the main focus was in experimentally testing Bell inequalities. The papers I know that treat the problem abstractly are from the 80s onward.

The story goes like this:

In 1969, Clauser, Horne, Shimony and Holt publish their famous paper that introduced the CHSH inequality. In it, they stated the the maximal violation for a singlet is $2\sqrt{2}$. As they were analysing a specific experiment, they didn't bother proving it for any two-qubit state, nor even stated the bound explicitly. They were inspired by Bell (of course) and Bohm's 1957 paper with Aharonov. Amusingly, Bohm's paper displays an inequality whose quantum bound is 2.85, and the bound for some local hidden-variable models he tried is 2. Unfortunately, these numbers are just a numerical coincidence, as his inequality has nothing to do with CHSH, not being even a Bell inequality.

In 1978, Boris Tsirelson gives a seminar on Bell inequalities. The chairman (A. Vershik) asks him if it's possible to develop analogue inequalities for quantum theory, i.e., to bound the strength of quantum correlations.

In 1980, Boris Tsirelson publishes a paper that proves that in the CHSH case, it is possible to bound the correlations by $2\sqrt{2}$, for any quantum state.

In 1985, Summers and Werner rediscover Tsirelson's bound independently. (Apparently it was they who popularised it outside the Soviet Union; Landau in 1987 cites both Tsirelson and Summers.)

This post has been migrated from (A51.SE)
answered Dec 21, 2011 by Mateus Araújo (270 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights