Short answer: The Tsirelson bound is the largest score quantum theory allows on the CHSH test, 2√2, against a bound of 2 under local hidden variables. A formalized proof summary in the Archive of Formal Proofs states it, a 2006 Physical Review A paper by Stephanie Wehner proves bounds for generalized CHSH inequalities, and a 2017 arXiv preprint computes exact quantum violations for other low-dimensional Bell inequalities. These are mathematical results.
The CHSH score
The Archive of Formal Proofs abstract describes the CHSH inequality, named after Clauser, Horne, Shimony and Holt, as based on an experiment in which two separate parties perform joint measurements several times and a score is derived from the runs. If local hidden variables were correct, the score would be bounded by 2. A suitable choice of observables in a quantum setting violates this when measuring the Bell state. The abstract credits Alain Aspect with obtaining this result experimentally.
The Tsirelson bound
The same abstract says Tsirelson answered how large the violation could be, by proving that in the quantum setting 2√2 is the highest score obtainable. The authors formalize this result and summarize the main results on the CHSH score: it is bounded by 2 under local hidden variables, the bound cannot be violated for a separable density matrix, it remains valid when one party uses commuting observables, and otherwise the bound is 2√2 whatever the observables and quantum state. It is reached for a suitable choice of observables on the Bell state.
Generalized inequalities
Wehner's 2006 Physical Review A paper considers generalized CHSH inequalities with many measurement settings and two outcomes each. It shows how to prove Tsirelson bounds for any such inequality using semidefinite programming. As an example, for any shared entangled state and observables with eigenvalues plus or minus 1, the chain of correlations X1Y1 + X2Y1 + X2Y2 + X3Y2 + ... + XnYn - X1Yn is bounded in absolute value by 2n cos(π/(2n)). The abstract notes that observables achieving equality are known to exist and shows they are optimal, and says the approach generalizes to other inequalities for such observables.
Low-dimensional Bell inequalities
The 2017 arXiv preprint (1712.08506) says the CHSH inequality certifies a quantum violation by a factor of √2 in the simplest nontrivial setup, two systems with two dichotomic measurements each, and that this is the largest possible violation there, known as the Tsirelson bound. The authors calculate exact values of quantum violations for the other Bell correlation inequalities in setups with up to four measurements, finding them all smaller than √2, using an analytic approach where earlier work used numerical methods. They also note that for bipartite systems and dichotomic measurements the gap between classical and quantum correlations cannot exceed the Grothendieck constant.
Limits
The bound is specific to the setups described. The 2017 work is an arXiv preprint, and the Archive of Formal Proofs text is a formalization summary. None of the three works reports an engineering application.
FAQ
What is the Tsirelson bound for CHSH?
According to the Archive of Formal Proofs abstract, 2√2 is the highest score obtainable in the quantum setting, compared with 2 under local hidden variables.
Is the bound reached?
The same abstract says it is reached for a suitable choice of observables when measuring the Bell state.
How are bounds for generalized CHSH inequalities proven?
Wehner's 2006 paper uses semidefinite programming for inequalities with many settings and two outcomes each.
Do other low-dimensional Bell inequalities exceed this?
The 2017 preprint reports that, for setups with up to four measurements, the other Bell correlation inequalities have exact quantum violations smaller than √2.
What limits the gap between classical and quantum correlations?
The 2017 preprint says that for bipartite systems and dichotomic measurements the discrepancy cannot exceed the Grothendieck constant.
All field notes