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) who first derived them.
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 A 0 {\displaystyle A_{0}} , A 1 {\displaystyle A_{1}} , B 0 {\displaystyle B_{0}} , B 1 {\displaystyle B_{1}} (i.e., two observables for Alice and two for Bob) with outcomes + 1 , − 1 {\displaystyle +1,-1} such that [ A i , B j ] = 0 {\displaystyle [A_{i},B_{j}]=0} for all i , j {\displaystyle i,j} , then
⟨ A 0 B 0 ⟩ + ⟨ A 0 B 1 ⟩ + ⟨ A 1 B 0 ⟩ − ⟨ A 1 B 1 ⟩ ≤ 2 2 . {\displaystyle \langle A_{0}B_{0}\rangle +\langle A_{0}B_{1}\rangle +\langle A_{1}B_{0}\rangle -\langle A_{1}B_{1}\rangle \leq 2{\sqrt {2}}.}
For comparison, in the classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of + 1 , − 1 {\displaystyle +1,-1} is allowed, it is 4. The Tsirelson bound is attained already if Alice and Bob each make 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
B = A 0 B 0 + A 0 B 1 + A 1 B 0 − A 1 B 1 , {\displaystyle {\mathcal {B}}=A_{0}B_{0}+A_{0}B_{1}+A_{1}B_{0}-A_{1}B_{1},}
and A i 2 = B j 2 = I {\displaystyle A_{i}^{2}=B_{j}^{2}=\mathbb {I} } , i.e., if the observables' outcomes are + 1 , − 1 {\displaystyle +1,-1} , then
B 2 = 4 I − [ A 0 , A 1 ] [ B 0 , B 1 ] . {\displaystyle {\mathcal {B}}^{2}=4\mathbb {I} -[A_{0},A_{1}][B_{0},B_{1}].}
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