Editing Tsirelson's bound (section)

==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
<math>K_G^{\mathbb R}(\lfloor r\rfloor),</math>
where
<math>r = \min \left\{m,n,-\frac12 + \sqrt{\frac14 + 2(m+n)}\right\},</math>
and <math>K_G^{\mathbb R}(d)</math> is the [[Grothendieck constant]] of order ''d''.<ref>{{cite journal |author1=Boris Tsirelson |title=Quantum analogues of the Bell inequalities. The case of two spatially separated domains |journal=Journal of Soviet Mathematics |date=1987 |volume=36 |issue=4 |pages=557–570 |doi=10.1007/BF01663472 |s2cid=119363229 |url=http://www.math.tau.ac.il/~tsirel/download/qbell87.pdf}}</ref> Note that since <math>K_G^{\mathbb R}(2) = \sqrt2</math>, 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 [[Semidefinite programming|semidefinite programs]], the NPA hierarchy, that in general does not halt.<ref>{{cite journal | last1=Navascués | first1=Miguel | last2=Pironio | first2=Stefano | last3=Acín | first3=Antonio | title=Bounding the Set of Quantum Correlations | journal=Physical Review Letters | volume=98 | issue=1 | date=2007-01-04 | issn=0031-9007 | doi=10.1103/physrevlett.98.010401 | pmid=17358458 | page=010401 | bibcode=2007PhRvL..98a0401N | arxiv=quant-ph/0607119 | s2cid=41742170 }}</ref><ref name="npa"/> The exact values are known for a few more Bell inequalities:

For the Braunstein–Caves inequalities we have that

: <math> \langle \text{BC}_n \rangle \le n \cos\left(\frac{\pi}{n}\right). </math>

For the WWŻB inequalities the Tsirelson bound is

: <math> \langle \text{WWZB}_n \rangle \le 2^{(n-1)/2}. </math>

For the <math>I_{3322}</math> inequality<ref>{{cite journal | last1=Collins | first1=Daniel | last2=Gisin | first2=Nicolas | title=A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality | journal= Journal of Physics A: Mathematical and General| volume=37 | issue=5 | pages=1775–1787 | date=2003-06-01 | doi=10.1088/0305-4470/37/5/021 | arxiv=quant-ph/0306129 | s2cid=55647659 }}</ref> the Tsirelson bound is not known exactly, but concrete realisations give a lower bound of {{val|0.250875384514}},<ref>{{cite journal |author1=K.F. Pál |author2=T. Vértesi |title=Maximal violation of the I3322 inequality using infinite dimensional quantum systems |journal=Physical Review A |date=2010 |volume=82 |pages=022116 |doi=10.1103/PhysRevA.82.022116 |arxiv=1006.3032}}</ref> and the NPA hierarchy gives an upper bound of {{val|0.2508753845139766}}.<ref>{{cite arXiv |last1=Rosset |first1=Denis |title=SymDPoly: symmetry-adapted moment relaxations for noncommutative polynomial optimization |date=2018 | eprint=1808.09598|class=quant-ph}}</ref> It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound.