A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics is non-local, i.e., that quantum mechanical correlations violate Bell inequalities, a natural question to ask is "how non-local can quantum mechanics be?", or, more precisely, by how much can the Bell inequality be violated. The answer is precisely the Tsirelson bound for the particular Bell inequality in question. In general this bound is lower than what would be algebraically possible, and much research has been dedicated to the question of why this is the case.
Tsirelson bound for the CHSH inequality
The first Tsirelson bound was derived as an upper bound on the correlations measured in the CHSH inequality. It states that if we have four (Hermitian) dichotomic observables , , , (i.e., two observables for Alice and two for Bob) with outcomes such that for all , then
For comparison, in the classical (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of is allowed it is 4. The Tsirelson bound is attained already if Alice and Bob each makes measurements on a qubit, the simplest non-trivial quantum system.
Lots of proofs have been developed for this bound, but perhaps the most enlightening one is based on the Khalfin-Tsirelson-Landau identity. If we define an observable
and , i.e., if the outcomes of the observables are associated to projective measurements, then
If or , which can be regarded as the classical case, it already follows that . In the quantum case, we need only notice that and the Tsirelson bound follows.
Tsirelson bounds for other Bell inequalities
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Obtaining a Tsirelson bound for a given Bell inequality is in general a hard problem that has to be solved on a case-by-case basis, although there are numerical algorithms that can upperbound it. The exact values are known for a few more Bell inequalities:
For the Braunstein-Caves inequalities we have that
For the WWŻB inequalities the Tsirelson bound is
Finding the Tsirelson bound for the inequality is a notorious open problem in quantum information theory.
Tsirelson bounds from physical principles
Lots of research have been dedicated to find a physical principle that explains why quantum correlations go only up to the Tsirelson bound and nothing more. Two such principles have been found, the non-triviality of communication complexity and information causality. That is to say, if one could achieve a CHSH correlation of any more than Tsirelson's bound, such measurements can be chained together to create a PR Box. Armed with PR boxes, it is possible for two observers to calculate any function with one bit of output, sharing only one bit of communication between them. For example, the problem in which each party has access to a large bit string and be asked whether or not the bit strings are identical, can be solved with one party transmitting only a single bit to the other.    
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