A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates Bell inequalities (i.e., it cannot be described by a local hidden-variable theory), a natural question to ask is how large can the violation be. The answer is precisely the Tsirelson bound for the particular Bell inequality in question. In general, this bound is lower than the bound that would be obtained if more general theories, only constrained by "no-signalling" (i.e., that they do not permit communication faster than light), were considered, and much research has been dedicated to the question of why this is the case. The Tsirelson bounds are named after Boris S. Tsirelson (or Cirel'son, in a different transliteration), the author of the article in which the first one was derived. 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 case (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. Several proofs of this bound exist, but perhaps the most enlightening one is based on the Khalfin–Tsirelson–Landau identity. If we define an observable and , i.e., if the observables' outcomes are , 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. Other Bell inequalities Tsirelson also showed that for any bipartite full-correlation Bell inequality with m inputs for Alice and n inputs for Bob, the ratio between the Tsirelson bound and the local bound is at most where and is the Grothendieck constant of order d. Note that since , this bound implies the above result about the CHSH inequality. In general, obtaining a Tsirelson bound for a given Bell inequality is a hard problem that has to be solved on a case-by-case basis. It is not even known to be decidable. The best known computational method for upperbounding it is a convergent hierarchy of