A201488 Decimal expansion of maximal success probability of the CHSH game.

8, 5, 3, 5, 5, 3, 3, 9, 0, 5, 9, 3, 2, 7, 3, 7, 6, 2, 2, 0, 0, 4, 2, 2, 1, 8, 1, 0, 5, 2, 4, 2, 4, 5, 1, 9, 6, 4, 2, 4, 1, 7, 9, 6, 8, 8, 4, 4, 2, 3, 7, 0, 1, 8, 2, 9, 4, 1, 6, 9, 9, 3, 4, 4, 9, 7, 6, 8, 3, 1, 1, 9, 6, 1, 5, 5, 2, 6, 7, 5, 9, 7, 1, 2, 5, 9, 6

Offset: 0

Author

Charles R Greathouse IV, Dec 02 2011

Keywords

• nonn
• cons
• nice

Comments

A referee chooses two random bits and gives one to each of two players who share an entangled quantum state but are not permitted to communicate. The players each choose a bit to send to the referee. If both of the bits from the referee are 1, then the players win if their chosen bits are different; otherwise they win if their chosen bits are the same. The best classical win probability is 3/4, but this can be improved in a quantum setting.

The optimality of this probability follows from Tsirelson's inequality and is implicit in the CHSH paper.

Ratio of leg length to base length in an isosceles triangle with the property that the areas of the two smaller excircles sum up to the area of the third excircle. - Martin Janecke, Aug 05 2012

Weight factor in Laguerre-Gauss quadrature, corresponding to the smallest root of the Laguerre polynomial of degree 2. - A.H.M. Smeets, May 15 2025

Examples

			0.853553390593273762200422181052424519642417968844237...
		

Links

• J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Phys. Rev. Lett. 23 (1969), pp. 880-884.
• Claude Crépeau, Louis Salvail, Jean-Raymond Simard, and Alain Tapp, Classical and quantum strategies for two-prover bit commitments, Ninth Workshop on Quantum Information Processing (2006).
• Erica Klarreich, The Proof in the Quantum Pudding, Quanta Magazine, 21 Aug 2013.
• Stephen J. Summers and Reinhard Werner, The vacuum violates Bell's inequalities, Physics Letters A 110:5 (1985), pp. 257-259.
• Wikipedia, CHSH inequality
• Wikipedia, Quantum pseudo-telepathy
• Wikipedia, Tsirelson's bound
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A020765, A019863.

Programs

Mathematica

RealDigits[Cos[Pi/8]^2,10,120][[1]] (* Harvey P. Dale, Jan 21 2012 *)

PARI

cos(Pi/8)^2 \\ Charles R Greathouse IV, Dec 02 2011

Formula

Equals cos^2(Pi/8) = (1 + 1/sqrt(2))/2.

Equals (theta_3(0, q^2)/theta_3(0, q))^2 where q = 1/e^Pi. - Michael Somos, Dec 02 2022