Restrictions on Bell-type inequalities

While deriving and proving Bell-type inequalities of the form

$|E(a,b)-E(a,b')|+|E(a',b)+E(a',b')|\leq 2$

I know that the conditions on the operators $O_a$ and $O_b$ are that they must be bounded by $\pm 1$.

Joint operator $O_{ab}\equiv O_a O_b$ is consequently bounded by $\pm 1$.

However, is there any such bound on the correlation $E(a,b)$ given by operating by $O_{ab}$ on whatever state you're studying? Does $E(a,b)$ necessarily NEED to be bounded by $\pm 1$ as per the definition?

(I know that it sometimes is as a result of, say, operating on the singlet state, but is this a consequence or a condition?)

• The CHSH inequality (Clauser, Horne, Shimony, and Holt), is less severe than you write it, |E(a,b)+E(a,b′)+E(a′,b)-E(a′,b′)| ≤ 2. I attach here the address of an article of one of its parents, Abner Shimony. plato.stanford.edu/entries/bell-theorem and you go to section 2. Proof of a Theorem of Bell's Type. You can see there the equation (16), which is the form of the CHSH inequality. Nov 24 '14 at 20:31

The CHSH inequality was not proved for any state, but for the spin singlet or for the photon singlet, s.t. to your question

"is there any such bound on the correlation E(a,b) given by operating by O_{ab} on whatever state you're studying? Does E(a,b) necessarily NEED to be bounded by ±1 as per the definition?

As I say, you cannot work with whatever state. On other states there are other inequalities, you can follow in the arXiv quant-ph the works of Adan Cabello.

But WHY are you interested to break the limit +- 1 on E(a,b)? What is exactly the problem?

Good luck,

Sofia

• Ah, I get it, about the singlet state. I'm actually working with the singlet state, so it's relevant. I'm exploring different types of operators. Some of them are giving rise to correlations that exceed the $\pm 1$ bounds. The operators are normalised, though. I'm seeing violations with them, but I'm not sure if they're valid, i.e., whether I need to normalise the correlations. Nov 25 '14 at 7:37
• Before you embark on some heavy task, won't you follow my advice and look in the works of Adan Cabello? Are you familiar with the quant-ph archive? Adan Cabello is a fine physicist in this domain, and he proved all sort of inequalities, with all sort of operators. Nov 25 '14 at 13:19