Nonlocal correlations: Fair and Unfair Strategies in Bayesian Game

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by  Arup Roy

Nonlocal correlations: Fair and Unfair Strategies in Bayesian Game
1 1 1 2 1 2
ArupRoy, AmitMukherjee, TamalGuha, SibasishGhosh, SomeSankarBhattacharya, andManikBanik
1
PhysicsandAppliedMathematicsUnit, IndianStatisticalInstitute, 203B.T.Road, Kolkata700108, India.
2
Optics&QuantumInformationGroup,TheInstituteofMathematicalSciences,C.I.TCampus,Tharamani,Chennai600113,India.
Interestingconnectionhasbeenestablishedbetweentwoapparentlyunrelatedconcepts,namely,
quantumnonlocalityandBayesiangametheory. Ithasbeenshownthatnonlocalcorrelationsinthe
formofadvicecanoutperformclassicalequilibriumstrategiesincommoninterestBayesiangames
andalsoinconflictinginterestgames. However,classicalequilibriumstrategiescanbeoftwotypes,
fairandunfair. Whereasinfairequilibriumpayoffsofdifferentplayersaresame,inunfaircasethey
differ. Advantageofnonlocalcorrelationhasbeendemonstratedoverfairstrategies. Inthisworkwe
showthatquantumstrategiescanoutperformeventheunfairclassicalequilibriumstrategies. Forthis
purposeweconsideraclassoftwoplayersgameswhichasaspecialcaseincludestheconflicting
6
gameproposedin[Phys. Rev. Lett. 114,020401(2015)]. Thesegamescanhavebothfairandunfair
1
classicalequilibriaandalsocanhaveonlyunfairones. Weprovideasimpleanalyticcharacterization
0
2 ofthenonlocalcorrelationsthatareadvantageousovertheclassicalequilibriumstrategiesinthese
games.
r
a
M
Undoubtedlyoneofthemostfundamentalcontradic- inputs and produces some outputs. No communica-

9 tionsofQuantummechanics(QM)withclassicalphysics tionisallowedamongthepartiesduringthegamebut,
getsmanifestedinitsnonlocalbehavior. Thisbizarrefea- they can share correlations. The aim is to cooperat-
]
h tureofQMwasfirstestablishedintheseminalworkofJ. ivelyoptimizesomepayofffunction. Ifthecorrelation
p S.Bell[1],wherehehasshownthatQMisincompatible sharedamongthemisclassicalone(ormorepreciselyto
-
t with the local-realistic world view of classical physics. saylocalrealisticone)thenthepayoffisupperbounded
n
Moreprecisely,Bellshowedthatmeasurementstatistics by some threshold value. However it may be possible
a
u ofmultipartiteentangledquantumsystemscanviolate thatusingcorrelationofentangledquantumstatesthese
q anempiricallytestablelocalrealisticinequality(ingen- optimalclassicalboundscanbesuperseded,whiches-
[ eralcalledBelltypeinequalities)whichestablishesthe tablishesnonlocalbehaviorofthesharedentangledstate.

3 denialoflocalrealismunderlyingQM.SinceBell’swork, ThecanonicalexampleofsuchgameisClauser-Horne-
v nonlocality remains at the center of quantum founda- Shimony-Holt(CHSH)game[8]involvingtwoparties,
9 tional research (see [2] and references therein) and it sayAliceandBob,eachwithbinaryinputsandbinary
4
hasbeenverifiedinnumeroussuccessfulexperiments, outputs. Whereas payoff of this game in local realistic
3
2 startingfromthefamousAspect’sexperiment[3]tovery theoryisupperboundedby3/4√,inquantumtheoryone
0 recentloop-holefreetests[4]. Apartfromfoundational canachieveitupto1/2(1+1/ 2).
.
1 interestquantumnonlocalcorrelationshavebeenproved Game theoretic formulation of Bell type inequalit-
0 to be the key resource in various device-independent ies prompted Brunner et al. to explore the connection
6 protocols[5]. VeryrecentlyBrunnerandLindenhavees- betweenBellnonlocalityandBayesiangametheory[6].
1
: tablishedthatBellnonlocalityhasinterestingconnection ThetheoryofBayesiangamewasdiscoveredbyHarsa-
v
withaseeminglydifferentareaofresearch,namely,the- nyi and the framework is developed for games with
i
X oryofBayesiangame[6]. ABayesiangamecanbeplayed incomplete information [9] (see also [10]). In such a
r underclassicalequilibriumstrategieswhichareoftwo game,eachplayermayhavesomeprivateinformation
a
types,fairequilibriaandunfairequilibria. Whereasin unknowntootherplayers; ontheotherhandtheplay-
fair equilibria payoffs of different parties are same, in ers may have a common piece of advice and thus can
unfair equilibria they differ. It has been shown that followcorrelatedstrategies,givingrisetotheconceptof
QMcanprovideadvantageousstrategiesoverthebest correlatedNashequilibrium[11]. Aspointedoutin[6],
classicalstrategiesincommoninterestBayesiangames the concept of private information in Bayesian games
[6] and can also outperform the fair classical equilib- is analogous to the notion of locality in Bell inequalit-
riumstrategiesinconflictinggames[7]. Theaimofthis ies (BI). And the fact that common advice in Bayesian
presentpaperistostudywhethernonlocalcorrelation games does not reveal the private information mimics
can be advantageous over the classical unfair equilib- the concept of no-signaling resources in case of BI. A
riumstrategiesinsuchagames. largerclassofsuchno-signalingcorrelations,provento
Operationally Bell type inequalities can be best un- bestrongerresourcethanlocalrealisticcorrelationsin
derstood in terms of games involving several number nonlocalgames,areavailableinQM.Naturallytheques-
ofspatiallyseparatedparties. Eachpartyreceivessome tionariseswhethersuchnonlocalquantumcorrelations
2
indeed provide advantage in Bayesian games over the
Table I. Utility table for the game G(κ,τ). Both κ and τ are
classicalstrategies. Interestingly,intheirpaperBrunner
positive.
et al. have answered affirmatively to this question [6].
The CHSH game as well as GHZ game [12], Mermin xA∧xB =0 xA∧xB =1
game[13],MagicSquareGame[14,15],HiddenMatch- yB =0 yB =1 yB =0 yB =1
ing game [16, 17] and the three games of [6] are all yA =0 (1,κ) (0,0) (0,0) (3/4,3/4)
y =1 (0,0) (1/2,τ) (3/4,3/4) (0,0)
exampleofcommoninterestgameswheretheinvolving A
partieshavetooptimizesomepayoff,cooperatively. On
theotherhandinconflictinginterestgameinterestsof
ies are given in Table-I. In accordance with the para-
the players differ, resulting to conflict in their best ac-
meter κ and τ of Table-I let us denote such a game as
tions. BattleofSexes(BoS)isaclassicexampleofsuch
G(κ,τ). Whenever κ < τ, there is a conflict between
conflicting interest game. Interestingly in [7], the au-
Alice and Bob in choosing their actions. Note that the
thors have provided an example of a conflicting game game G(1/2,1) istheconflictinggamestudiedin[7].
wherequantumstrategiescanoutperformtheclassical
fair equilibrium strategies. Furthermore, using semi-
In the case of correlated strategies, i.e., when the
definiteprogramming(SDP)theyhaveshownthatBell
parties are given some common advice, the average
statesalongwithsuitablychosenmeasurementsarethe
payoffiscalculatedas:
quantumequilibriumstrategiesforthesaidgame. Using
theinitialsoftheauthorsletcallthisPKLSZDKgame. F = ∑P(x)P(y|x)u (x,y). (1)
i i
A Baysian game can have only fair equilibria, both x,y
fair and unfair equilibria or only unfair equilibria. In
Here P(x)istheprobabilitydistributionovertheAlice’s
this present paper we address the question whether
andBob’sjointtype x ≡ (x ,x )whichisconsideredto
A B
quantumadvicecansurpasstheclassicalunfairequilib-
beuniformforthisparticulargameand P(y|x) denote
riumstrategiesinsuchgames. Interestingly,weanswer
theconditionaljointaction y ≡ (y ,y ) giventhetype
A B
affirmativelytothisquestions. Forthispurposewehave
x,i.e.,theprobabilitythatAlicetakesactiony andBob
A
constructedatwoparametricclassoftwoplayersgame,
takes action y given their joint type (x ,x ). In the
B A B
eachplayerhavingtwotypes/inputsandtwopossible
caseofsuchcorrelatedstrategies,Aumannintroduced
actions/outputs for each type. As a special case, the
theconceptofcorrelatedequilibria(CE)[11],whichhave
PKLSZDKgameisalsoincludedinthisclass. Forcertain
severalniceproperties: theyareeasiertofind[18],every
conditionsontheparametersweshowthatthesegames
Nash equilibrium is a CE and convex combinations of
canhaveonlyunfairequilibria. Toshowtheadvantages
CEareagainCE.Forplayingthegame G(κ,τ) eachof
of nonlocal correlation over the classical equilibrium
Alice and Bob can take one of the following four pure
strategies we take a very analytic approach. First we
classicalstrategies:
showthatinthe2−2−2scenario(i.e. two-party,each
with two measurements and each measurement with g1(x ) =0; g2(x ) =1; g3(x ) = x; g4(x ) = x ⊕1;
i i i i i i i i i i
two outcomes) any no signaling correlation can be ex-
pressedinacanonicalform. Usingthiscanonicalform where g1(x ) =0meansthatith partytakestheaction0
i i
wecompletelycharacterizetheno-signalingcorrelations whateverbethetypeisandsimilarlytheothercases; ⊕
providingadvantagesovertheclassicalfairandunfair denotesmodulo2sum. For16possiblepurestrategies
strategiesinthesegames. Oneinterestingconsequence ofAliceandBob(together)theiraveragepayoffs(F ,F )
A B
ofouranalysisisthatitprovidesasimpleexplanation are listed in Table-II. As discussed earlier, G(κ,τ) be a
(withoutusinganySDP),whythesaidstrategiesof[7] conflictinginterestgamewhen τ >κ. Inthiscasewhen
arequantumfairequilibriumone. κ < 3 the strategies (g1,g3), (g3,g4), and (g4,g2)
4 A B A B A B
A class of two players Bayesian games: Here we ad- are Nash equilibria and let’s denote the correspond-
(cid:16) (cid:17)
opt the same notations used in Ref.[7]. Let Alice and ing payoffs, (Feq1,Feq1) ≡ 11, 3 + κ , (Feq2,Feq2) ≡
A B 16 16 2 A B
Bob are two players involved in the game. Alice’s (cid:0) 9 , 3 + κ+τ(cid:1), and (Feq3,Feq3) ≡ (cid:0) 7 , 3 + τ(cid:1), respect-
and Bob’s types/inputs are denoted as x ∈ X and 16 16 4 A B 16 16 2
A A ively;andforκ > 3 thestrategies(g1,g1),(g3,g4),and
x ∈ X ,respectively. Foreachtypestheytakesomeac- 4 A B A B
B B (g4,g2) areNashequilibriawithcorrespondingpayoffs
tions/outputsdenotedas y ∈ Y and y ∈ Y andac- A B
cordinglytheyaregivenpayAoffs/uAtilitiesBdenotBedas u as (Feq1(cid:48),Feq1(cid:48)) ≡ (cid:0)3,3κ(cid:1), (Feq2,Feq2) ≡ (cid:0) 9 , 3 + κ+τ(cid:1),
A A B 4 4 A B 16 16 4
and u ,respectively,where u : X ×X ×Y ×Y → and (Feq3,Feq3) ≡ (cid:0) 7 , 3 + τ(cid:1), respectively. For the
B i A B A B A B 16 16 2
R , for i ∈ {A,B}. For the class of games considered parameter value τ > κ > 1, all three equilibrium are
here, X = X = Y = Y = {0,1} and the utilit- unfair and in every case Bob’s payoff is greater than
A B A B
3
g1 g2 g3 g4 where (cid:104)ij(cid:105) := P(++|ij)− P(+−|ij)− P(−+|ij)+
B B B B
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) P(−−|ij). Ano-signalingprobabilitydistributionhas
g1 3,3κ 3, 3 11, 3 + κ 1,κ
A (cid:16)4 4 (cid:17) (cid:16)16 16(cid:17) 16(cid:16)16 (cid:17)2 (cid:16) 4 4 (cid:17) alocalrealisticdescriptionifandonlyifitsatisfiesthe
g2A 136,136 38,34τ 18,τ4 176,136 + τ2 Bell-CHSH inequality, i.e., iff |B| ≤ 2 [20]. In terms of
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
g3 11, 3 + κ 1,τ 1,κ 9, 3 + κ+τ probabilities,theBell-CHSHexpressionbecomes,
A 16 16 2 8 4 4 4 16 16 4
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
g4A 14,κ4 176,136 + τ2 196,136 + κ+4τ 81,τ4 B=2+4(c +c +c −c )−4(m +n ). (3)
00 01 10 11 0 0
Table II. (Color on-line) Alice’s and Bob’s average payoffs Collectionofallsuchno-signalingcorrelationsNS form
(F ,F )fordifferentpurestrategiesinthegameG(κ,τ). For
A B a 8 dimensional polytope with 24 vertices and the fa-
conflictinginterestcase(τ>κ)therearethreeequilibriawhen
κ < 3 (redandblue)andwhenκ > 3 (greenandblue),and cetsaredeterminedbypositivityandno-signalingcon-
4 4
forτ<κthereisoneequilibrium: redwhenκ< 3 andgreen straints. Collection of all classical (local realistic) cor-
whenκ> 3. 4 relations L andallquantumcorrelations Q areconvex
4 setslyingwithin NS. While L isagainapolytopewith
trivialfacetsgivenbypositivityandthenontrivialfacets
Alice’s. Note that in these cases even no fair correl- determinedbyBell-CHSHinequalities,theset Q iscon-
atedequilibriumstrategyispossible. Thecaseswhere vexbutnotapolytope[21]. Anycorrelationoutside L
κ+τ = 1.5 give a fair equilibria strategy as occurred is called nonlocal and the strict set inclusion relations
in the conflicting game of [7]. When κ > τ it turns L ⊂ Q ⊂ NS reflectthefactthatquantumtheorycon-
outtobeagamewithonlyoneequilibrium; forκ < 3 tainsnonlocalcorrelations,butnonlocalityofquantum
4
the strategy (g1A,g3B) is Nash equilibrium and the cor- theory is restricted compared to gener√al no-signaling
responding payoff, (Feq1,Feq1) ≡ (cid:16)11, 3 + κ(cid:17), and for correlations. While B canbe atmost 2 2 in Q [22], in
A B 16 16 2 NS itcangoupto4[23].
κ > 3 the Nash equilibrium is the strategy (g1,g1)
4 A B Nonlocal advantage over classical strategies: In the
with (Feq1(cid:48),Feq1(cid:48)) ≡ (cid:0)3,3κ(cid:1). Since any classical (local Bayesian game described above, the two players can
A B 4 4
realistic) advice can be written as P(y ,y |x ,x ) = becommonlyadvisedbyageneralno-signalingcorrela-
A B A b
(cid:82)
dλP(y |x ,λ)P(y |x ,λ), with λ being a local vari- tion. Givensuchanadvice NS ∈ NS Alice’sandBob’s
A A B B
able(alsocalledhiddenvariablebythequantumfound- averagepayoffs,respectively,reads:
ationcommunity),soconvexityensuresthatusingany
1
such advice it is not possible to overcome the equilib- FNS = [3+3/2B+2(m +n )+(m +n )], (4)
A 16 0 0 1 1
rium payoffs. However in quantum world there are
1
no-signaling correlations that are not in this local real- FNS = [(10τ−2κ)+(τ+κ)B+4(κ−τ)(m +n )
B 16 0 0
istic form (thus called nonlocal) and hence there may
+(3−4τ)(m +n )+4(κ+τ−3/2)c ]. (5)
be a possibility to overcome the classical equilibrium 1 1 11
payoffs. Ano-signalingnonlocaladviceoutperformssomeclas-
2−2−2no-signalingcorrelations: Forthetwo-partyscen- sical equilibrium payoff (FAeq,FBeq) if FiNS > Fieq, for
ariowithtwotwo-outcomemeasurementsforeachparty i = A,B. As an explicit example in the following we
let denote the joint probability distribution as P(ab|ij), studythePKLSZDKgame,i.e.,thegame G(1/2,1). For
(cid:16) (cid:17)
wheretheoutcomesa,b ∈ {+,−}andthemeasurement this game it becomes that F +F = 3 1+ 1B , i.e.,
A B 4 4
settings i,j ∈ {0,1}. We can express the joint distribu- thesumofAlice’sandBob’saveragepayoffsinclassical,
tionas[19]: quantum,a√ndno-signalingtheoriescanbeatmost9/8,
3/4(1+1/ 2),and3/2,respectively.
P(ab|ij) ≡ (c ,m −c ,n −c ,1−n −m +c ), (2)
ij ij ij ij ij ij ij ij (a) Fair strategy: For fair strategy the average equi-
librium payoffs of Alice and Bob are identical, which
with outcomes in the order (++,+−,−+,−−). Here
givesthecondition,2(m +n )+m +n =3andboth
m and n denote the corresponding marginal prob- 0 0 1 1
ij ij Alice’sandBob’spayoffsbecome3/8(1+1/4B). Thus
abilities of Alice and Bob, with positivity impos-
anyno-signalingadvicewillbeadvantageousoverthe
ing the restrictions, max{0,m + n − 1} ≤ c ≤
ij ij ij
bestfairclassicalstrategyif,
min{m ,n }∀ij. Accordingtono-signalingAlice’smar-
ij ij
ginaloutcomeprobabilityshouldnotdependonBob’s (cid:18) (cid:19)
3 1 9
measurementsettingsandviceversa,whichcanbeex- 1+ B > ⇒ B>2. (6)
8 4 16
pressed as m = m := m , m = m := m , n =
00 01 0 10 11 1 00
n := n , n = n := n . The celebrated Bell-CHSH As described in [7], a quantum strategy is specified
10 0 01 11 1
expression is given by,B = (cid:104)00(cid:105)+(cid:104)01(cid:105)+(cid:104)10(cid:105)−(cid:104)11(cid:105), by the triple {ρ ,(M0,M1),(M0,M1)} [7], where
AB A A B B
4
ρ issomebipartitequantumstateprovidedtoAlice theadvice. Forthesharpaswellasunsharp[26]meas-
AB
andBobasadviceand Mk aretwooutcomespositive- urementsperformedbyAliceorBobontheirrespective
i
operator-valued-measures (POVM) for k ∈ {0,1}, i ∈ partsthemarginalprobabilitiesarecompletelyrandom
{A,B}. For example, two qubit Werner class of andhenceitisnotpossibletofulfillthecondition(7)and
states W = p|ψ−(cid:105)(cid:104)ψ−|+(1−p)I ⊗ I, where |ψ−(cid:105) = (8)simultaneously. Thoughmoregeneraltwooutcome
p 2 2
√1 (|0(cid:105)⊗|1(cid:105)−|1(cid:105)⊗|0(cid:105)) isthesingletstateandIbethe POVMcangivebiasedmarginalbutstilltheyarenotuse-
2 ful(see[27]fordetail). Sosingletstateisnogoodinthis
identityoperator,satisfiesthefirstconditionforproject-
purposeandsimilarargumentappliesforanybipartite
ive measurements performed by Alice and Bob. And
state with completely mixed marginals. Naturally we
forsuitablychosenmeasurementtheysatisfythesecond
conditionwhenever p > √1 . MoreovertheoptimalBell- thenconsiderapurestate |ψ(cid:105)AB = a|00(cid:105)+b|11(cid:105) asthe
2 √ advice,withAlice’sandBob’smeasurementdirections
CHSHviolationinquantumtheory,i.e.,2 2isuniquely (cid:16) (cid:17)
specifiedas,Mk ≡ sin θkcos φk,sin θksin φk,cos θk ,
(up to localunitary freedom) achieved by singlet state i i i i i i
with suitably chosen measurements [24]. This implies where k ∈ {0,1} and i ∈ {A,B}. Interestingly, we find
that such advice can fulfill the requirement. For ex-
thatthesingletstatewithcorrespondingmeasurement
ample,ifwetakeanadvicewith a =0.9andthemeas-
settingsdescribeafairequilibriumstrategyinquantum
urement directions θ0 = −θ0 = −π, φ0 = −φ0 = π,
theory. Itisnoteworthythatmoregeneralno-signaling A B 15 A B 2
θ1 = π, and φ1 = π, then their average payoffs be-
advicescanoutperformbestquantumstrategy. i 3 i 2
an(au)nUfanifrai(rtostrBaotebg)yc:lFarsosimcaTlaebqlue-ilIiIbnrioutemthsatrta(tgeg1Ay,gw3B)itihs c[2o7m])e. FHAo=we0v.e7r0,6t6h(i>s s11t16ra)taengyd iFsBn=ot0a.51q6u3a(n>tum176)e(qsueei-
Alice’sandBob’spayoffs11/16and7/16,respectively librium strategy. Because, given the same advice and
for the game G(1/2,1). So a no-signaling advice will same measurement settings for Alice, Bob can fix his
be considered advantageous over this strategy if F > measurementdirectionsasθ0(new) =0.451517, θ1(new) =
A B B
11/16and FB >7/16orequivalentlytosay: 1.25911, φ0(new) = −1.5708, φ1(new) =1.5708sothathis
B B
new payoff becomes Fnew = 0.5213 > F > 7 . But for
3B+2(m0+n0)+(m1+n1) >8, and (7) thesenewmeasuremenBtchoicesAlice’spBayof1f6modifies
2
3B−2(m +n )−(m +n ) > −2. (8) to 1116 < FAnew = 0.6981 < 0.7066 = FA. Even if the
0 0 1 1 partiesconsidertwooutcomePOVMs,theequilibrium
2
willnotbeachieved.
From conditions (7) and (8) it is clear that the no-
At this point one can think of a stronger refinement
signaling correlations need to be nonlocal to provide
of the equilibrium concept, known as social optim-
advantageovertheclassicalunfairequilibriumstrategy.
ality [28]. It is a choice of strategies, one by each
If we consider the following one parameter family of
player, that maximizes the sum of the players’ pay-
no-signalingcorrelations:
offs. Of course, maximization of the sum of play-
A = qPR+(1−q)D, (9) ers’ payoff may not necessarily lead to the satisfac-
tion of all the participating players. Given the advice,
(cid:112)
where 0 ≤ q ≤ 1 and PR = {PPR(ab|ij)} and D = FA+FB = 34(1+ 12 1+4a2(1−a2)) =1.2266,andone
{PD(ab|ij) with, PPR(ab|ij) = 1/2 if α⊕β = ij and canfindastrategy(see[27])suchthat FA∗ =0.6978> 1161
0 otherwise, and PD(ab|ij) = 1 if α = β = 1 and 0 andFB∗ =0.5288> 176. Clearlythis‘∗(cid:48) strategyisasocial
otherwise,thenstraightforwardcalculationgives, optimalone,whiletheprevioustwoarenot. Itisquite
mentioning that quantum strategies also outperform
11 q 7 5q
F = + , F = + , (10) the other classical unfair (unfair to Alice) equilibrium
A B
16 16 16 16
strategy. Theanalysisremainssameifamixedentangled
establishing advantage over the unfair classical equi- adviceisgiveninsteadofpureone.
librium strategy for nonzero value of q. However this Discussions: Nonlocality,arguablyoneofthemostcon-
exampledoesnotsolvethequestionwhetherquantum troversialissuesinquantumfoundations,hasacquired
theoryprovidesadvantageovertheclassicalstrategies. lotsofresearchattentionofinformationtheoristsduring
ThisisbecauseinRef.[25]ithasbeenshownthatinthe last two decades due to its practical usefulness in sev-
asymptoticlimitthecorrelation A ofEq.(9)canbedis- eraldeviceindependentinformationtheoreticprotocols.
tilledtomaximallynonlocal PR correlationandhence Veryrecentlyinterestingconnectionofthispeculiarre-
quantumrealizationofthiscorrelationisnotpossible. sourcehasbeenestablishedinanotherveryimportant
Tofindaquantumstrategybetterthanclassicalunfair branch of study, namely Bayesian game theory which
equilibriumstrategywefirstconsiderthesingletstateas hasseveralapplicationsineconomics,socialandpolit-
5
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1.0
APPENDIX 0.5
0.0
Singletstateisnogoodforunfairstrategy
(cid:45)2 (cid:45)1 0 1 2 3
As we have discussed in the manuscript, a strategy Α
will be advantageous over classical unfair equilibrium
oneif, Figure2. (Coloron-line)BS =−2.82. Redcurveisfor(19)and
blueisfor(20). Allowedvaluesof(α,µ)satisfyingcondition
3B+K>8, and (11) (15)lieinthegreenregion. Allowedvaluesof(α,µ)satisfying
2 boththeconditions(19)and(20)liesintheyellowregions.
3
B−K> −2, (12)
2
√
whereK=2(m +n )+m +n ,withm (n)beingthe atmost3(1+ 2) <8. Sosingletstatealongwithsharp
0 0 1 1 i i
marginalof + outcomeoftheith measurementofAlice orunsharpmeasurementsisnogoodforthepurpose.
(Bob). Forsingletstatethemarginalstateofeachparty However considering a more general two outcome
iscompletelymixedstate. Projectivemeasurement M qubit measurement one can get the value of K very
aˆ
(λ) close to 6 and hence having a possibility of satisfying
andunsharpmeasurement M alongsomedirection aˆ,
aˆ the condition (11). At the same time the setting must
isgivenby,
have satisfy the condition (12). In the following we
M = {1(I+aˆ.(cid:126)σ),1(I−aˆ.(cid:126)σ)}, (13) studythesepossibilities. Themostgeneraltwooutcome
aˆ
2 2 qubitmeasurement(POVM)along mˆ canbeexpressed
M(λ) = {1(I+λaˆ.(cid:126)σ),1(I−λaˆ.(cid:126)σ)}, (14) as {Emˆ(α,µ),I−Emˆ(α,µ)} [26],where
aˆ 2 2
where 12(I±nˆ.(cid:126)σ) are projectors corresponding to up Emˆ(α,µ) = 1(αI+µmˆ.(cid:126)σ), with
and down eigenstates, and 1(I±λnˆ.(cid:126)σ) being the cor- 2
responding unsharp effect w2ith 0 < λ ≤ 1. For such 0< α ≤2, and0≤ µ ≤min{α,2−α}. (15)
measurementsperformedontheeachpartofthesing-
let state the marginal statistics is completely random. If Alice performs the POVM M ≡ {E (α,µ),I−
mˆ
Hencethevalue√ofKisalways3. SincethevalueofB Emˆ(α,µ)} and Bob performs the POVM N ≡
canbeatmost2 2,thelefthandsideofEq.(11)canbe {E (α,µ),I−E (α,µ)} on their respective part of the
nˆ nˆ
7
singletstate,thejointconditionalprobabilitiesread: With B taking +ve and −ve values, respectively, the
S
equality of the conditions (19)-(20)√represent ellipses
1 andhyperbolas. For |B | =2.82(cid:39)2 2corresponding
P(++) = [α2−µ2mˆ.nˆ], S
4 ellipses and hyperbolas are plotted in Fig.1 and Fig.2,
P(+−) = 1[α(2−α)+µ2mˆ.nˆ], respectively.
4 Achoiceof (α,µ) willviolateboththeconditions(19)
P(−+) = 1[(2−α)α+µ2mˆ.nˆ], and (20) if it lies outside both the ellipse (hyperbolas),
4 i.e., in the yellow region in Fig.1 (Fig.2). However, the
P(−−) = 1[α2−µ2mˆ.nˆ]. (16) allowedvaluesof(α,µ)[compatiblewithcondition(15)]
4 areinthegreenregion,whichhasnooverlapwiththe
yellow region. Of course there are (α,µ) outside one
Themarginalprobabilitiesaretherefore, curve but inside other. Changing the value of B , we
S
findthatthereisnoallowed(α,µ)whichcanfulfillboth
Pmˆ(+) := P(++)+P(+−) = α/2, the requirements imposed by the conditions (19) and
Pnˆ(+) := P(−+)+P(−−) = α/2, (20).
Therefore singlet state, even with most general two
andtheexpectation (cid:104)MN(cid:105) becomes, outcomePOVM,isnotanusefuladviceovertheunfair
classical equilibrium strategy in the PKLSZDK game.
(cid:104)MN(cid:105) := P(++)−P(+−)−P(−+)+P(−−) Similarkindofargumentisalsotrueforanytwoqubit
= (α−1)2−µ2mˆ.nˆ. statewithboththemarginalstatescompletelymixed.
Generalstrategywithnon-maximallypureentangled
TheBell-CHSHquantityandKaretherefore, advice
B = 2(α−1)2−µ2(mˆ .nˆ +mˆ .nˆ +mˆ .nˆ −mˆ .nˆ ) Foranarbitrarypurenon-maximallyentangledstate
0 0 0 1 1 0 1 1 |ψ(cid:105) = a|00(cid:105)+b|11(cid:105) andtheprojectivemeasurements
:=2(α−1)2+µ2B , and (17) AB
S performedbyAliceandBobforarbitrarydirections,say,
K = 3α, (18)
M0 ≡ (p ,p ,p ); M1 ≡ (q ,q ,q ), (21)
A 1 2 3 A 1 2 3
with BS be the CHSH value for sharp measurement M0B ≡ (r1,r2,r3); M1B ≡ (s1,s2,s3), (22)
taking both +ve and −ve values. And the conditions
the input-output probability distribution is listed in
(11)-(12)look:
Table.III. Also one can express the measurement dir-
ectionsinpolarcoordinate,i.e.,
(α−1/2)2 µ2
+ >1, (19)
23/12 23/6BS Mk ≡ (cid:16)sin θkcos φk,sin θksin φk,cos θk(cid:17). (23)
i i i i i i
(α−3/2)2 µ2
+ >1. (20)
7/12 7/6B
S
(cid:20)
Givensuchastrategy {|ψ(cid:105) ,(M0,M1),(M0,M1)}, 1 3
AB A A B b FB = 3+ ab(r1(p1+q1)+s1(p1−q1)−r2(p2+q2)
whereMk betheprojectivemeasurementalong Mk,the 8 2
i i
averagepayoffsofAliceandBobcanbecalculatedfrom 3
−s (p −q ))+ (r +s )(p −q )
theTable.III,usingtheexpressionofEqs.(3),(4),and(5), 2 2 2 4 3 3 3 3
(cid:21)
andtheexpressionsbecome: −1(a2−b2)(q +s +2(p +r )) . (25)
3 3 3 3
4
(cid:20) Theconditionforfairstrategiesthusreads:
1 3
F = 3+ ab(r (p +q )+s (p −q )−r (p +q )
A 1 1 1 1 1 1 2 2 2
8 2 (a2−b2)(q +s +2(p +r ) =0, (26)
3 3 3 3
3
−s (p −q ))+ (r +s )(p −q )
2 2 2 4 3 3 3 3 whichcanbesatisfiedbynon-maximallypureentangled
1 (cid:21) state(i.e. a (cid:54)= b)withappropriatelychosenmeasurement
+ (a2−b2)(q +s +2(p +r )) , (24)
4 3 3 3 3 settings(i.e. q3+s3+2p3+2r3 =0). However,herewe
8
TableIII.Input-outputprobabilitydistributionforprojectivemeasurementsperformedbyAliceandBobonthestate|ψ(cid:105) =
AB
a|00(cid:105)+b|11(cid:105).
++ +− −+ −−
M0M0 1{2ab(r p − r p ) + 1{2ab(−r p + r p ) − 1{2ab(−r p + r p ) − 1{2ab(r p − r p ) +
A B 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2
b2(−1 + r )(p − 1) +b2(1+r )(p −1)−a2(r − a2(1+r )(p −1)−b2(r − a2(−1 + r )(p − 1) +
3 3 3 3 3 3 3 3 3 3
a2(r +1)(p +1)} 1)(p +1)} 1)(p +1)} b2(r +1)(p +1)}
3 3 3 3 3 3
M0M1 1{2ab(s p − s p ) + 1{2ab(−s p + s p ) − 1{2ab(−s p + s p ) − 1{2ab(s p − s p ) +
A B 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2
b2(−1 + s )(p − 1) +b2(1+s )(p −1)−a2(s − a2(1+s )(p −1)−b2(s − a2(−1 + s )(p − 1) +
3 3 3 3 3 3 3 3 3 3
a2(s +1)(p +1)} 1)(p +1)} 1)(p +1)} b2(s +1)(p +1)}
3 3 3 3 3 3
M1M0 1{2ab(r q − r q ) + 1{2ab(−r q + r q ) − 1{2ab(−r q + r q ) − 1{2ab(r q − r q ) +
A B 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2
b2(−1 + r )(q − 1) +b2(1+r )(q −1)−a2(r − a2(1+r )(q −1)−b2(r − a2(−1 + r )(q − 1) +
3 3 3 3 3 3 3 3 3 3
a2(r +1)(q +1)} 1)(q +1)} 1)(q +1)} b2(r +1)(q +1)}
3 3 3 3 3 3
M1M1 1{2ab(s q − s q ) + 1{2ab(−s q + s q ) − 1{2ab(−s q + s q ) − 1{2ab(s q − s q ) +
A B 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2 4 1 1 2 2
b2(−1 + s )(q − 1) +b2(1+s )(q −1)−a2(s − a2(1+s )(q −1)−b2(s − a2(−1 + s )(q − 1) +
3 3 3 3 3 3 3 3 3 3
a2(s +1)(q +1)} 1)(q +1)} 1)(q +1)} b2(s +1)(q +1)}
3 3 3 3 3 3
are interested whether they can be useful for unfair
0.70
strategies. Given the above expressions for F and F
A B
theaimistosearchforstrategiessurpassingtheclassical
0.65
unfair equilibrium one. And it can be done efficiently
byusinglinearprogramming. A
F
Examplesgiveninmanuscript: Foranarbitrary|ψ(cid:105) = 0.60
AB
a|00(cid:105)+b|11(cid:105),ifAliceandBobchoosetheirmeasurement
directions as θ0 = −π , φ0 = π, θ1 = π , φ1 = π then 0.55
i 15 i 2 i 3 i 2
theiraveragepayoffslook:
F = 1(cid:20)(cid:18)2a2− 1(cid:19)cos (cid:16)π (cid:17)+ 3cos2 (cid:16)π (cid:17) 0.500.0 0.2 0.4 0.6 0.8 1.0
A 8 4 15 4 15 a
√
(cid:112)
+3a 3(1−a2)sin (cid:16)π (cid:17)− 3a 1−a2 sin2(cid:16)π (cid:17) Figure 3. (Color on-line) Blue curve denotes the value of
2 15 2 15 F [Eq.(31)] and the red line is the value 11/16. For every
√ (cid:35) A
45 9a 1−a2 2a2−1 a∈(0,1),FA islessthan11/16.
+ + + , (27)
16 8 4
∼ ∼
we get,F = 0.6978, F = 0.5288, giving a social op-
1(cid:20)(cid:18)7 (cid:19) (cid:16)π (cid:17) 3 (cid:16)π (cid:17) A B
F = −2a2 cos + cos2 timal strategy. At this point please note that, not all
B
8 4 15 4 15 measurementsettingsgivingoptimalBforaquantum
√
+3a(cid:112)3(1−a2)sin (cid:16)π (cid:17)− 3a 1−a2 sin2(cid:16)π (cid:17) advice will not give FA > 1116 and FB > 176. As
2 15 2 15 for example, for the state |ψ(cid:105)AB = a|01(cid:105)+b|10(cid:105) the
45 9a√1−a2 2a2−1(cid:35) measurements choices M0A ≡ (0,0,1), M1A ≡ (1,0,0),
+ + − . (28) M0 ≡ (sinβ,0,cosβ), and M1 ≡ (sinβ(cid:48),0,cosβ(cid:48)), with
16 8 4 B √ B
cosβ = −c√osβ(cid:48) = 1/ 1+4a2b2, give the optimal Bell
For a = 0.9, we get F = 0.7066(> 11) and F = violation2 1+4a2b2 [29]. Howeverforthissettingswe
A 16 B
0.5163(> 7 ). get,
16
Social optimality strategy: For the following measure-
mentsettingsofAliceandBob: F = 1 (cid:18) 7√+22a2−24a4 +2a2+5(cid:19), (31)
A
θ0 =0, θ1 ∼= −1.5708, 16 2 1+4a2−4a4
A A 1 (cid:18) 5+26a2−24a4 (cid:19)
φ0A ∼= −2.3636, φ1A ∼=0.777996, (29) FB = 16 2√1+4a2−4a4 −2a2+7 . (32)
θB0 ∼= −0.6653, θB1 ∼=0.6653, For any value of a ∈ (0,1), FA does not satisfy the
φ0 ∼= −0.7780, φ1 ∼= −0.7780, (30) requiredcondition(seeFig.3).
B B