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 possible without signalling fast than light, 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.
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 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
This section needs expansion. You can help by adding to it. (December 2012)
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. It is not even known to be decidable. The best known computational method for upperbounding it is a convergent hierarchy of semidefinite programs, the NPA hierarchy, that in general does not halt. 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
For the inequality the Tsirelson bound is not known exactly, but concrete realisations give a lower bound of 0.250 875 38 and the NPA hierarchy gives an upper bound of 0.250 875 39. It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound.
Tsirelson bounds from physical principles
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 , Information causality and macroscopic locality. 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 quansal measure.
There are two different ways of defining the Tsirelson bound of a Bell expression. One by demanding that the measurements are in a tensor product structure, and another by demanding only that they commute. Tsirelson's problem is the question of whether these two definitions are equivalent. More formally, let
be a Bell expression, where is the probability of obtaining outcomes with the settings . The tensor product Tsirelson bound is then the supremum of the value attained in this Bell expression by making measurements and on a quantum state :
The commuting Tsirelson bound is the supremum of the value attained in this Bell expression by making measurements and such that on a quantum state :
Since tensor product algebras in particular commute, . In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras, 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 .. 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. 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, and wanted to know whether it also converged to the tensor product Tsirelson bound , the most physically relevant one.
Since one can produce a converging sequencing of approximations to from below by considering finite-dimensional states and observables, if 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). Conversely, if is not computable, then . In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that is not computable, thus solving Tsirelson's problem.
- Cirel'son, B. S. (1980). "Quantum generalizations of Bell's inequality". Letters in Mathematical Physics. Springer Nature. 4 (2): 93–100. Bibcode:1980LMaPh...4...93C. doi:10.1007/bf00417500. ISSN 0377-9017.
- Navascués, Miguel; Pironio, Stefano; Acín, Antonio (2007-01-04). "Bounding the Set of Quantum Correlations". Physical Review Letters. 98 (1): 010401. arXiv:quant-ph/0607119. Bibcode:2007PhRvL..98a0401N. doi:10.1103/physrevlett.98.010401. ISSN 0031-9007. PMID 17358458.
- M. Navascués; S. Pironio; A. Acín (2008). "A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations". New Journal of Physics. 10: 073013. arXiv:0803.4290. doi:10.1088/1367-2630/10/7/073013.
- Collins, Daniel; Gisin, Nicolas (2003-06-01). "A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality". Journal of Physics A: Mathematical and General. 37 (5): 1775–1787. arXiv:quant-ph/0306129. doi:10.1088/0305-4470/37/5/021.
- K.F. Pál; T. Vértesi (2010). "Maximal violation of the I3322 inequality using infinite dimensional quantum systems". Physical Review A. 82: 022116. arXiv:1006.3032. doi:10.1103/PhysRevA.82.022116.
- Linden, Noah; Popescu, Sandu; Short, Anthony J.; Winter, Andreas (2007-10-30). "Quantum Nonlocality and Beyond: Limits from Nonlocal Computation". Physical Review Letters. American Physical Society (APS). 99 (18): 180502. arXiv:quant-ph/0610097. Bibcode:2007PhRvL..99r0502L. doi:10.1103/physrevlett.99.180502. ISSN 0031-9007. PMID 17995388.
- Pawłowski, Marcin; Paterek, Tomasz; Kaszlikowski, Dagomir; Scarani, Valerio; Winter, Andreas; Żukowski, Marek (2009). "Information causality as a physical principle". Nature. 461 (7267): 1101–1104. arXiv:0905.2292. Bibcode:2009Natur.461.1101P. doi:10.1038/nature08400. ISSN 0028-0836. PMID 19847260.
- Navascués, Miguel; Wunderlich, Harald (2009-11-11). "A glance beyond the quantum model". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. The Royal Society. 466 (2115): 881–890. doi:10.1098/rspa.2009.0453. ISSN 1364-5021.
- Craig, David; Dowker, Fay; Henson, Joe; Major, Seth; Rideout, David; Sorkin, Rafael D (2006-12-20). "A Bell inequality analog in quantum measure theory". Journal of Physics A: Mathematical and Theoretical. IOP Publishing. 40 (3): 501–523. arXiv:quant-ph/0605008. doi:10.1088/1751-8113/40/3/010. ISSN 1751-8113.
- Tsirelson, B.S. (1993). "Some results and problems on quantum Bell-type inequalities" (PDF). Hadronic Journal Supplement. 8: 329–345.
- Tsirelson, B. "Bell inequalities and operator algebras". Retrieved 20 January 2020.
- Tsirelson, B. "Bell inequalities and operator algebras" (PDF). Retrieved 20 January 2020.
- Z. Ji; A. Natarajan; T. Vidick; J. Wright; H. Yuen (2020). "MIP*=RE". arXiv:2001.04383. Cite journal requires
- M. Junge; M. Navascués; C. Palazuelos; D. Pérez-García; V. B. Scholz; R. F. Werner (2011). "Connes' embedding problem and Tsirelson's problem". Journal of Mathematical Physics. 52 (1): 012102. arXiv:1008.1142. doi:10.1063/1.3514538.