# Optimality of the CHSH strategy

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.

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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? – Piotr Migdal Dec 4 '11 at 1:50
@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. – Charles Dec 4 '11 at 5:45
I found the link with 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). – Piotr Migdal Dec 4 '11 at 15:29
Have you even read Tsirelson's paper? 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. – Mateus Araújo Dec 6 '11 at 1:42

## 1 Answer

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.)

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