Editing Tsirelson's bound (section)

==Tsirelson's problem==

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
: <math> B = \sum_{abxy} \mu_{abxy} p(ab|xy) </math>
be a Bell expression, where <math>p(ab|xy)</math> is the probability of obtaining outcomes <math>a, b</math> with the settings <math>x, y</math>. The tensor product Tsirelson bound is then the [[Infimum and supremum|supremum]] of the value attained in this Bell expression by making measurements <math>A^a_x : \mathcal{H}_A \to \mathcal{H}_A</math> and <math>B^b_y : \mathcal{H}_B \to \mathcal{H}_B</math> on a quantum state <math>|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B</math>:
: <math> T_t  = \sup_{|\psi\rangle, A^a_x, B^b_y} \sum_{abxy} \mu_{abxy} \langle \psi | A^a_x \otimes B^b_y |\psi\rangle.</math>
The commuting Tsirelson bound is the [[Infimum and supremum|supremum]] of the value attained in this Bell expression by making measurements <math>A^a_x : \mathcal{H} \to \mathcal{H}</math> and <math>B^b_y : \mathcal{H} \to \mathcal{H}</math> such that <math>\forall a, b, x, y; [A^a_x, B^b_y] = 0</math> on a quantum state <math>|\psi\rangle \in \mathcal{H}</math>:
: <math> T_c  = \sup_{|\psi\rangle, A^a_x, B^b_y} \sum_{abxy} \mu_{abxy} \langle \psi | A^a_x B^b_y |\psi\rangle.</math>
Since tensor product algebras in particular commute, <math>T_t \le T_c</math>. In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras,<ref>{{Cite arXiv |last1=Scholz |first1=V. B. |last2=Werner |first2=R. F. |date=2008-12-22 |title=Tsirelson's Problem |class=math-ph |eprint=0812.4305}}</ref> so only for infinite dimensions it is possible that <math>T_t \neq T_c</math>. Tsirelson's problem is the question of whether for all Bell expressions <math>T_t = T_c</math>.

This question was first considered by [[Boris Tsirelson]] in 1993, where he asserted without proof that <math>T_t = T_c</math>.<ref>{{cite journal |last1=Tsirelson |first1=B. S. |title=Some results and problems on quantum Bell-type inequalities |journal=Hadronic Journal Supplement |date=1993 |volume=8 |pages=329–345 |url=https://m.tau.ac.il/~tsirel/download/hadron.pdf}}</ref> 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.<ref>{{cite web |last1=Tsirelson |first1=B. |title=Bell inequalities and operator algebras |url=https://m.tau.ac.il/~tsirel/Research/bellopalg/main.html |accessdate=20 January 2020}}</ref> 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 <math>T_c</math> from above,<ref name="npa">{{cite journal |author1=M. Navascués |author2=S. Pironio |author3=A. Acín |title=A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations |journal=New Journal of Physics |date=2008 |volume=10 |issue=7 |pages=073013 |doi=10.1088/1367-2630/10/7/073013 |arxiv=0803.4290 |bibcode=2008NJPh...10g3013N |s2cid=1906335 }}</ref> and wanted to know whether it also converged to the tensor product Tsirelson bound <math>T_t</math>, the most physically relevant one.

Since one can produce a converging sequencing of approximations to <math>T_t</math> from below by considering finite-dimensional states and observables, if <math>T_t = T_c</math>, 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 <math>T_t</math> is not computable, then <math>T_t \neq T_c</math>. In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that <math>T_t</math> is not computable, thus solving Tsirelson's problem in the negative.<ref>{{cite arXiv |author1=Z. Ji |author2=A. Natarajan |author3=T. Vidick |author4=J. Wright |author5=H. Yuen |title=MIP* = RE |date=2020 |eprint=2001.04383 |class=quant-ph}}</ref> Tsirelson's problem has been shown to be equivalent to [[Connes' embedding problem]],<ref>{{cite journal |author1=M. Junge |author2=M. Navascués |author3=C. Palazuelos |author4=D. Pérez-García |author5=V. B. Scholz |author6=R. F. Werner |title=Connes' embedding problem and Tsirelson's problem |journal=Journal of Mathematical Physics |date=2011 |volume=52 |issue=1 |pages=012102 |doi=10.1063/1.3514538 |arxiv=1008.1142 |bibcode=2011JMP....52a2102J |s2cid=12321570 }}</ref> so the same proof also implies that the Connes embedding problem is false.<ref>{{cite web |last1=Hartnett |first1=Kevin |title=Landmark Computer Science Proof Cascades Through Physics and Math |url=https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/ |website=Quanta Magazine |language=en |date=4 March 2020}}</ref>