Tsirelson's bound

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 $A\_0$, $A\_1$, $B\_0$, $B\_1$ (i.e., two observables for Alice and two for Bob) with outcomes $+1,\; -1$ such that $[A\_i,\; B\_j]\; =\; 0$ for all $i,\; j$, then

: $\backslash langle\; A\_0\; B\_0\; \backslash rangle\; +\; \backslash langle\; A\_0\; B\_1\; \backslash rangle\; +\; \backslash langle\; A\_1\; B\_0\; \backslash rangle\; -\; \backslash langle\; A\_1\; B\_1\; \backslash rangle\; \backslash le\; 2\backslash sqrt\{2\}.$

For comparison, in the classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of $+1,\; -1$ 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

: $\backslash mathcal\{B\}\; =\; A\_0\; B\_0\; +\; A\_0\; B\_1\; +\; A\_1\; B\_0\; -\; A\_1\; B\_1,$

and $A\_i^2\; =\; B\_j^2\; =\; \backslash mathbb\{I\}$, i.e., if the observables' outcomes are $+1,\; -1$, then

: $\backslash mathcal\{B\}^2\; =\; 4\; \backslash mathbb\{I\}\; -\; [A\_0,\; A\_1]\; [B\_0,\; B\_1].$

If $[A\_0,\; A\_1]\; =\; 0$ or $[B\_0,\; B\_1]\; =\; 0$, which can be regarded as the classical case, it already follows that $\backslash langle\; \backslash mathcal\{B\}\; \backslash rangle\; \backslash le\; 2$. In the quantum case, we need only notice that $\backslash big\backslash |[A\_0,\; A\_1]\backslash big\backslash |\; \backslash le\; 2\; \backslash |A\_0\backslash |\; \backslash |A\_1\backslash |\; \backslash le\; 2$, and the Tsirelson bound $\backslash langle\; \backslash mathcal\{B\}\; \backslash rangle\; \backslash le\; 2\backslash sqrt\{2\}$ follows.

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

$K\_G^\{\backslash mathbb\; R\}(\backslash lfloor\; r\backslash rfloor),$

where

$r\; =\; \backslash min\; \backslash left\backslash \{m,n,-\backslash frac12\; +\; \backslash sqrt\{\backslash frac14\; +\; 2(m+n)\}\backslash right\backslash \},$

and $K\_G^\{\backslash mathbb\; R\}(d)$ is the Grothendieck constant of order d.{{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}} Note that since $K\_G^\{\backslash mathbb\; R\}(2)\; =\; \backslash sqrt2$, 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 programs, the NPA hierarchy, that in general does not halt.{{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 }} The exact values are known for a few more Bell inequalities:

For the Braunstein–Caves inequalities we have that

: $\backslash langle\; \backslash text\{BC\}\_n\; \backslash rangle\; \backslash le\; n\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{n\}\backslash right).$

For the WWŻB inequalities the Tsirelson bound is

: $\backslash langle\; \backslash text\{WWZB\}\_n\; \backslash rangle\; \backslash le\; 2^\{(n-1)/2\}.$

For the $I\_\{3322\}$ inequality{{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 }} the Tsirelson bound is not known exactly, but concrete realisations give a lower bound of {{val|0.250875384514}},{{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}} and the NPA hierarchy gives an upper bound of {{val|0.2508753845139766}}.{{cite arXiv |last1=Rosset |first1=Denis |title=SymDPoly: symmetry-adapted moment relaxations for noncommutative polynomial optimization |date=2018 | eprint=1808.09598|class=quant-ph}} 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,{{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 }} information causality{{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 }} and macroscopic locality.{{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 }} 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.{{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 }}

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

: $B\; =\; \backslash sum\_\{abxy\}\; \backslash mu\_\{abxy\}\; p(ab|xy)$

be a Bell expression, where $p(ab|xy)$ is the probability of obtaining outcomes $a,\; b$ with the settings $x,\; y$. The tensor product Tsirelson bound is then the supremum of the value attained in this Bell expression by making measurements $A^a\_x\; :\; \backslash mathcal\{H\}\_A\; \backslash to\; \backslash mathcal\{H\}\_A$ and $B^b\_y\; :\; \backslash mathcal\{H\}\_B\; \backslash to\; \backslash mathcal\{H\}\_B$ on a quantum state $|\backslash psi\backslash rangle\; \backslash in\; \backslash mathcal\{H\}\_A\; \backslash otimes\; \backslash mathcal\{H\}\_B$:

: $T\_t\; =\; \backslash sup\_\{|\backslash psi\backslash rangle,\; A^a\_x,\; B^b\_y\}\; \backslash sum\_\{abxy\}\; \backslash mu\_\{abxy\}\; \backslash langle\; \backslash psi\; |\; A^a\_x\; \backslash otimes\; B^b\_y\; |\backslash psi\backslash rangle.$

The commuting Tsirelson bound is the supremum of the value attained in this Bell expression by making measurements $A^a\_x\; :\; \backslash mathcal\{H\}\; \backslash to\; \backslash mathcal\{H\}$ and $B^b\_y\; :\; \backslash mathcal\{H\}\; \backslash to\; \backslash mathcal\{H\}$ such that $\backslash forall\; a,\; b,\; x,\; y;\; [A^a\_x,\; B^b\_y]\; =\; 0$ on a quantum state $|\backslash psi\backslash rangle\; \backslash in\; \backslash mathcal\{H\}$:

: $T\_c\; =\; \backslash sup\_\{|\backslash psi\backslash rangle,\; A^a\_x,\; B^b\_y\}\; \backslash sum\_\{abxy\}\; \backslash mu\_\{abxy\}\; \backslash langle\; \backslash psi\; |\; A^a\_x\; B^b\_y\; |\backslash psi\backslash rangle.$

Since tensor product algebras in particular commute, $T\_t\; \backslash le\; T\_c$. In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras,{{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}} so only for infinite dimensions it is possible that $T\_t\; \backslash neq\; T\_c$. Tsirelson's problem is the question of whether for all Bell expressions $T\_t\; =\; T\_c$.

This question was first considered by Boris Tsirelson in 1993, where he asserted without proof that $T\_t\; =\; T\_c$.{{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}} 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.{{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}} 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 $T\_c$ from above,{{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 }} and wanted to know whether it also converged to the tensor product Tsirelson bound $T\_t$, the most physically relevant one.

Since one can produce a converging sequencing of approximations to $T\_t$ from below by considering finite-dimensional states and observables, if $T\_t\; =\; T\_c$, 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 $T\_t$ is not computable, then $T\_t\; \backslash neq\; T\_c$. In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that $T\_t$ is not computable, thus solving Tsirelson's problem in the negative.{{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}} Tsirelson's problem has been shown to be equivalent to Connes' embedding problem,{{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 }} so the same proof also implies that the Connes embedding problem is false.{{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}}

• Quantum nonlocality
• Bell's theorem
• EPR paradox
• CHSH inequality
• Quantum pseudo-telepathy

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Category:Quantum measurement