Continuous input nonlocal games
aa r X i v : . [ qu a n t - ph ] M a r Continuous Input Nonlocal Games
N. Aharon, S. Machnes, B. Reznik, J. Silman, and L. Vaidman School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel
We present a family of nonlocal games in which the inputs the players receive arecontinuous. We study three representative members of the family. For the first twoa team sharing quantum correlations (entanglement) has an advantage over any teamrestricted to classical correlations. We conjecture that this is true for the third memberof the family as well.
The nonlocal nature of quantum mechanics, as man-ifested in Bell inequalities violation [1, 2], has recentlybeen highlighted in a number of games [4–10]. Termednonlocal [9], these are cooperative games with incom-plete information for a team of remote players. Each ofthe players is assigned by a verifier an input generatedaccording to a known joint probability distribution. Theplayers must then send an output to the verifier, whocarries a truth table dictating for each combination ofinputs, which combinations of outputs result in a win.The players may coordinate a joint strategy prior toreceiving their input, but cannot communicate with oneanother subsequently. A team sharing quantum corre-lations (entanglement) is said to employ a “quantumstrategy,” while a team restricted to sharing classicalcorrelations is said to employ a “classical strategy.”In this paper we analyze three representative membersof a novel family of nonlocal games, which differ fromother nonlocal games in the literature in that the inputsets are continuous rather than discrete and finite.Moreover, most nonlocal games include a “promise”regarding the allowed input combinations and theirfrequency. This means that the joint probability dis-tribution governing the assignment of combinations ofinputs is not uniform. This restriction is especiallytailored to guarantee a maximum quantum advantage,and can make the rules of the game complex. In thegames that we analyze there is no such promise. Thejoint probability distribution governing the assignmentof inputs is uniform, and the rules are simple. Neverthe-less, a non-negligible quantum advantage obtains.In the first game two remote players A and B receivea uniformly generated input a ∈ [0 ,
1] and b ∈ [0 , o i ∈ { , − } ( i = A, B ) to the verifier. The game is considered to havebeen won if o A · o B = +1 , a + b < − , a + b ≥ . (1)The game, therefore, amounts to the problem of return-ing a positive (negative) product of outputs when thesum of the inputs is less than (greater than or equal to) 1.In the following we show that a team employing a quan- FIG. 1: Game 1 - the classical strategy. The lower (upper)big triangle is the region where identical (opposite) outputsare required to win. Given the choice of outputs regions inwhich the game is won (lost) are colored in green (red). Itis easy to see that the green regions add up to of the totalarea of the square. tum strategy can achieve a higher probability for winningthe game than a team restricted to classical strategies.We begin by presenting the optimal classical strategy.It is easy to show that it is deterministic, i.e. the outputis a single-valued function of the input, and is given forexample by o A = 1 , o B = +1 , b < − , b ≥ . (2)The winning probability then equals 75% (see Fig. 1).This may be verified by noting that the game can be castas the continuum limit of a family of Bell inequalities,first discovered by Gisin [11], for which Tsirelson provedboth the classical and quantum bounds [12]. For moredetails see [13].In the quantum strategy we present the players sharea two qubit singlet state | ψ s i = 1 √ |↑↓i − |↓↑i ) . (3)Having beforehand agreed on a coordinate system, theplayers then measure the spin component of their qubits FIG. 2: Game 1 - the quantum strategy. θ A and θ B denotethe angles at which players A and B , respectively, measurethe spin of their qubit. The dotted and dashed arcs denotethe range of θ A and θ B . along different axes in the xy -plane. The choice of axesis dictated by the inputs as follows: A measures alongan axis spanning an angle of θ A ( a ) from the negative x -axis, while B measures along an axis spanning an an-gle of θ B ( b ) from the negative y -axis (see Fig. 2). Theplayers then send the results of the measurements to theverifier. For a + b ≥ a + b < a and b the probability for identical resultsis sin ( ∆2 ), where ∆ ≡ π − θ A ( a ) − θ B ( b ) is the angle be-tween the axes of measurement. The winning probabilityis therefore given by P W = Z da Z db [Θ( a + b −
1) cos ( ∆2 )+Θ(1 − a − b ) sin ( ∆2 )] , (4)where Θ is the unit step function (Θ(0) = 1). To max-imize P W we look for θ A ( a ) and θ B ( b ) such that when a + b ≥ a + b <
1) ∆ is small (large). A most naturalchoice is θ A ( a ) = πa , θ B ( b ) = πb , (5)as is evident from Fig. 2. The integral then equals + π corresponding to a winning probability of ≈ .
8% andsaturating the Tsirelson bound of the corresponding Bellinequality [12]. This gives an advantage of ≈ .
8% to ateam making use of quantum correlations over a teamlimited to classical correlations.The above game is a special case of a more generaljoint task in which A and B are assigned the uniformlygenerated inputs a ∈ [0 , m ] and b ∈ [0 , n ], respectively,and must return correlated (anticorrelated) outputswhen a + b < n + m ( a + b ≥ n + m ). Note that by setting FIG. 3: Game 2 - the classical strategy. The two small tri-angles and the strip between the two middle dashed lines areregions where identical outputs are required to win. Giventhe choice of outputs regions where the game is won (lost)regions are colored in green (red). The green regions add upto of the total area of the square. n = − m and defining ˜ b ≡ − b , the task reduces to havingto return identical outputs when a < ˜ b and oppositeotherwise.The second game is identical to the first in all but thewinning conditions. The game is now considered to havebeen won if o A · o B = +1 , | b − a | mod 3 > − , | b − a | mod 3 ≤ . (6)That is, the players must return correlated outputs ifthe absolute value of the their inputs’ difference is in theinterval (cid:2) , (cid:3) , otherwise they must return anticorrelatedoutputs.A possible realization of the optimal classical strategyis o A = +1 , a ≤ − , a > , o B = − , b ≤ +1 , b > . (7)The winning probability equals 75%, as in the first game(see Fig. 3). To see that this is the maximum, considerFig. 3. If we cyclically shift the input of one of theplayers by , then the regions that require correlated oranticorrelated outputs within each quadrant correspondto the first game [14]. Therefore, if the game admitteda strategy with a winning probability greater than 75%in any of the quadrants, so would the first game. Thequantum strategy we present differs from that of the firstgame only in the choice of axes A and B measure along.The winning probability now equals P W = Z da Z db [Θ(4 | b − a | mod 3 −
1) cos ( ∆2 )+Θ(1 − | b − a | mod 3) sin ( ∆2 )] , (8)Here ∆ ≡ θ A ( a ) − θ B ( b ) with both angles now spanningfrom the y -axis in the xy -plane. The maximum obtainsfor θ A ( a ) = 2 πa , θ B ( b ) = 2 πb , (9)giving the same winning probability as in the first game,i.e. ≈ . π . The question arises as tohow the quantum advantage changes when playing thegame in three dimensions. More specifically, two remoteplayers are each assigned a pair of angles 0 ≤ θ i ≤ π ,0 ≤ ϕ i < π , designating a three dimensional unit vectorˆ r i ( i = A, B ). The game is considered to have been wonif o A · o B = +1 , ˆ r A · ˆ r B < − , ˆ r A · ˆ r B ≥ . (10)The joint probability distribution governing the assign-ment of angles is a product ρ A ( θ A , ϕ A ) · ρ B ( θ B , ϕ B ) with ρ i ( θ i , ϕ i ) = sin θ i , (11)guaranteeing isotropy [15]. The classical strategy that wepresent is an extension of the optimal classical strategyof the second game, where in the geometric description A ( B ) returns an output equal to 1 ( − π . Otherwise, A ( B ) returns − A ( B ) return 1 ( −
1) when θ A ≤ π ( θ B ≤ π ),independent of ϕ A ( ϕ B ), and − ≈ .
2% (1 − π ) probability of winning. It seemslikely that this strategy is the optimal.As in the other games, in the quantum strategy thatwe consider, A and B share a singlet state of two qubitsand measure along axes dictated by their inputs, ˆ n A (ˆ r A ) and ˆ n B (ˆ r B ). The probability for winning is then givenby P W = Z Ω A d Ω A Z Ω B d Ω B [Θ(ˆ r A · ˆ r B ) cos ( ∆2 )+Θ( − ˆ r A · ˆ r B ) sin ( ∆2 )] , (12)with ∆ ≡ arccos(ˆ n A (ˆ r A ) · ˆ n B (ˆ r B )), and maximizes forˆ n A (ˆ r A ) = ˆ r A , ˆ n B (ˆ r B ) = ˆ r B . (13)The probability of winning than equals 75%. Numericalevidence obtained using semi-definite programming(SDP) indicates that this stratgey is optimal. Inter-estingly, the quantum advantage remains unchangedequaling ≈ . A and B each receive the coordinates of a ran-domly generated three dimensional vector r A and r B ,respectively. Then by a suitable choice of the joint prob-ability distribution governing the assignment of the vec-tors, each of the games translates to a question about thequantity ξ ≡ | r B − r A | = q r B − r B · r A + r A . (14)The third game obtains if we restrict the vectors to unitmagnitude. Actually, it is enough to require that thevectors be nonvanishing so long as they are generatedisotropically. We then ask whether ξ < p r B + r A . Thesecond game is identical except that we further restrictthe vectors to lie on the same plane. In the first game weabolish isotropy altogether. The vectors are generatedanitparallel to one another, with their magnitudesuniformly distributed between 0 and 1. ξ then equals r A + r B , and the players must decide whether ξ > Acknowledgments
We acknowledge support fromthe Israeli Science Foundation (grants no. 784/06 and990/06), and from the European Commission under theIntegrated Project Qubit Applications (QAP) funded bythe IST Directorate (contract no. 015848). [1] J.S. Bell, Physics , 195 (1964).[2] J.F. Clauser, R.A. Holt, M.A. Horne, and A. Shimony,Phys. Rev. Lett. , 880 (1969).[3] D.M. Greenberger, M.A. Horne, and A. Zeilinger, in Bells Theorem, Quantum Theory and Conceptions of the Uni-verse, edited by M. Kafatos (Springer, 1988), pg. 69.[4] B. Tsirelson, Lecture Notes in Quantum InformationProcessing, Tel-Aviv University (1996). [5] L. Vaidman, Found. Phys. , 615 (1999).[6] L. Vaidman, Phys. Lett. A , 241 (2001). See alsoL. Vaidman, in Quantum [Un]speakables: From Bell toQuantum Information, edited by R. Bertlmann and A.Zeilinger (Springer, 2002), pg. 221.[7] A. Cabello, Phys. Rev. A , 042104 (2003).[8] P.K. Aravind, Am. J. Phys. , 1303 (2004).[9] R. Cleve, P. Høyer, B. Toner, and J. Watrous, Pro-ceedings of the 19th IEEE Conference on ComputationalComplexity, (2004), pg. 236.[10] G. Brassard, A. Broadbent, and A. Tapp, Found. Phys. , 1877 (2005). [11] N. Gisin, Phys. Lett. A , 1 (1999).[12] B. Tsirelson, arXiv:0706.1091 [quant-ph].[13] J. Silman, S. Machnes, and N. Aharon, Phys. Lett. A , 3796 (2008).[14] To be more precise, each of the quadrants corresponds tothe truth table of the a < b or a > b formulation of thefirst game.[15] The differential of a solid angle, Ω, in spherical coordi-nates is proportional to sin θ . This introduces a weightfunction when integrating over θ and ϕϕ