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[1] in which the first one was derived.
It states that if we have four (Hermitian) dichotomic observables
, then For comparison, in the classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of
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.
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
, 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.
The best known computational method for upperbounding it is a convergent hierarchy of semidefinite programs, the NPA hierarchy, that in general does not halt.
[7] It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound.
Significant research has been dedicated to finding a physical principle that explains why quantum correlations go only up to the Tsirelson bound and nothing more.
Three such principles have been found: no-advantage for non-local computation,[8] information causality[9] and macroscopic locality.
[10] That is to say, if one could achieve a CHSH correlation exceeding Tsirelson's bound, all such principles would be violated.
Tsirelson's bound also follows if the Bell experiment admits a strongly positive quantal measure.
[11] There are two different ways of defining the Tsirelson bound of a Bell expression.
Tsirelson's problem is the question of whether these two definitions are equivalent.
The tensor product Tsirelson bound is then the supremum of the value attained in this Bell expression by making measurements
: The commuting Tsirelson bound is the supremum of the value attained in this Bell expression by making measurements
In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras,[12] so only for infinite dimensions it is possible that
Tsirelson's problem is the question of whether for all Bell expressions
This question was first considered by Boris Tsirelson in 1993, where he asserted without proof that
[13] Upon being asked for a proof by Antonio Acín in 2006, he realized that the one he had in mind didn't work, and issued the question as an open problem.
[14] Together with Miguel Navascués and Stefano Pironio, Antonio Acín had developed an hierarchy of semidefinite programs, the NPA hierarchy, that converged to the commuting Tsirelson bound
from above,[4] and wanted to know whether it also converged to the tensor product Tsirelson bound
, then this procedure can be combined with the NPA hierarchy to produce a halting algorithm to compute the Tsirelson bound, making it a computable number (note that in isolation neither procedure halts in general).
In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that