Editing Tsirelson's bound

{{short description|Theoretical upper limit to non-local correlations in quantum mechanics}}
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 Tsirelson|Boris S. Tsirelson]] (or Cirel'son, in a different [[Romanization of Russian|transliteration]]), the author of the article<ref>{{cite journal | last=Cirel'son | first=B. S. | title=Quantum generalizations of Bell's inequality | journal=Letters in Mathematical Physics | volume=4 | issue=2 | year=1980 | issn=0377-9017 | doi=10.1007/bf00417500 | pages=93–100|url=http://www.tau.ac.il/~tsirel/download/qbell80.html| bibcode=1980LMaPh...4...93C | s2cid=120680226 | url-access=subscription }}</ref> in which the first one was derived.

==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 ([[Self-adjoint operator|Hermitian]]) dichotomic observables <math>A_0</math>, <math>A_1</math>, <math>B_0</math>, <math>B_1</math> (i.e., two observables for [[Alice and Bob|Alice]] and two for [[Alice and Bob|Bob]]) with outcomes <math>+1, -1</math> such that <math>[A_i, B_j] = 0</math> for all <math>i, j</math>, then

: <math> \langle A_0 B_0 \rangle + \langle A_0 B_1 \rangle + \langle A_1 B_0 \rangle - \langle A_1 B_1 \rangle \le 2\sqrt{2}.</math>

For comparison, in the classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of <math>+1, -1</math> 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

: <math> \mathcal{B} = A_0 B_0  +  A_0 B_1  +  A_1 B_0  -  A_1 B_1, </math>

and <math>A_i^2 = B_j^2 = \mathbb{I}</math>, i.e., if the observables' outcomes are <math>+1, -1</math>, then

: <math> \mathcal{B}^2 = 4 \mathbb{I} - [A_0, A_1] [B_0, B_1]. </math>

If <math>[A_0, A_1] = 0</math> or <math>[B_0, B_1] = 0</math>, which can be regarded as the classical case, it already follows that <math>\langle \mathcal{B} \rangle \le 2</math>. In the quantum case, we need only notice that <math>\big\|[A_0, A_1]\big\| \le 2 \|A_0\| \|A_1\| \le 2</math>, and the Tsirelson bound <math>\langle \mathcal{B} \rangle \le 2\sqrt{2}</math> follows.

==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.

==Derivation 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,<ref>{{cite journal | last1=Linden | first1=Noah | last2=Popescu | first2=Sandu | last3=Short | first3=Anthony J. | last4=Winter | first4=Andreas |author-link4=Andreas Winter| title=Quantum Nonlocality and Beyond: Limits from Nonlocal Computation | journal=Physical Review Letters| volume=99 | issue=18 | date=2007-10-30 | issn=0031-9007 | doi=10.1103/physrevlett.99.180502 | pmid=17995388 | page=180502| bibcode=2007PhRvL..99r0502L | arxiv=quant-ph/0610097 }}</ref> [[information causality]]<ref>{{cite journal | last1=Pawłowski | first1=Marcin | last2=Paterek | first2=Tomasz | last3=Kaszlikowski | first3=Dagomir | last4=Scarani | first4=Valerio | last5=Winter | first5=Andreas |author-link5=Andreas Winter| last6=Żukowski | first6=Marek | title=Information causality as a physical principle | journal=Nature | volume=461 | issue=7267 | year=2009 | issn=0028-0836 | doi=10.1038/nature08400 | pmid=19847260 | pages=1101–1104| bibcode=2009Natur.461.1101P | arxiv=0905.2292 | s2cid=4428663 }}</ref> and macroscopic locality.<ref>{{cite journal | last1=Navascués | first1=Miguel | last2=Wunderlich | first2=Harald | title=A glance beyond the quantum model | journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume=466 | issue=2115 | date=2009-11-11 | issn=1364-5021 | doi=10.1098/rspa.2009.0453 | pages=881–890| doi-access=free | arxiv=0907.0372 }}</ref> 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.<ref>{{cite journal | last1=Craig | first1=David | last2=Dowker | first2=Fay |author-link2=Fay Dowker| last3=Henson | first3=Joe | last4=Major | first4=Seth | last5=Rideout | first5=David | last6=Sorkin | first6=Rafael D. |author-link6=Rafael Sorkin | title=A Bell inequality analog in quantum measure theory | journal=Journal of Physics A: Mathematical and Theoretical | volume=40 | issue=3 | year=2007 | issn=1751-8113 | doi=10.1088/1751-8113/40/3/010 | pages=501–523 | arxiv=quant-ph/0605008 | bibcode=2007JPhA...40..501C | s2cid=8706909 }}</ref>

==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>

==See also==
* [[Quantum nonlocality]]
* [[Bell's theorem]]
* [[EPR paradox]]
* [[CHSH inequality]]
* [[Quantum pseudo-telepathy]]

== References ==
<references/>

{{DEFAULTSORT:Tsirelson's Bound}}
[[Category:Quantum measurement]]