TY - JOUR

T1 - Multiplicative Bell inequalities

AU - Te'Eni, Amit

AU - Peled, Bar Y.

AU - Cohen, Eliahu

AU - Carmi, Avishy

N1 - Funding Information:
We thank Pawel Horodecki for insightful discussions. A.C. acknowledges support from the Israel Science Foundation, Grant No. 1723/16. E.C. was supported by the Canada Research Chairs (CRC) Program and by the Engineering Faculty in Bar Ilan University.
Publisher Copyright:
© 2019 American Physical Society.

PY - 2019/4/26

Y1 - 2019/4/26

N2 - Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a unique coordination game. In the simplest case, Alice and Bob, each having two random variables, attempt to maximize the area of a rectangle and the rectangle's area is represented by a certain parameter. This parameter, which is a function of the correlations between their random variables, is shown to be a Bell parameter, i.e., the achievable bound using only classical correlations is strictly smaller than the achievable bound using nonlocal quantum correlations We continue by generalizing to the case in which Alice and Bob, each having now n random variables, wish to maximize a certain volume in n-dimensional space. We term this parameter a multiplicative Bell parameter and prove its Tsirelson bound. Finally, we investigate the case of local hidden variables and show that for any deterministic strategy of one of the players, the Bell parameter is a harmonic function whose maximum approaches the Tsirelson bound as the number of measurement devices increases. Some theoretical and experimental implications of these results are discussed.

AB - Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a unique coordination game. In the simplest case, Alice and Bob, each having two random variables, attempt to maximize the area of a rectangle and the rectangle's area is represented by a certain parameter. This parameter, which is a function of the correlations between their random variables, is shown to be a Bell parameter, i.e., the achievable bound using only classical correlations is strictly smaller than the achievable bound using nonlocal quantum correlations We continue by generalizing to the case in which Alice and Bob, each having now n random variables, wish to maximize a certain volume in n-dimensional space. We term this parameter a multiplicative Bell parameter and prove its Tsirelson bound. Finally, we investigate the case of local hidden variables and show that for any deterministic strategy of one of the players, the Bell parameter is a harmonic function whose maximum approaches the Tsirelson bound as the number of measurement devices increases. Some theoretical and experimental implications of these results are discussed.

UR - http://www.scopus.com/inward/record.url?scp=85065307292&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.99.040102

DO - 10.1103/PhysRevA.99.040102

M3 - Article

AN - SCOPUS:85065307292

VL - 99

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 4

M1 - 040102

ER -