Consequences and Limits of Nonlocal Strategies

0.22 MB·

by  Richard Cleve

Consequences and Limits of Nonlocal Strategies
Richard Cleve Peter Høyer† BenToner‡ John Watrous§
∗
January 11,2010
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2 Abstract

n
Thispaperinvestigatesthepowersandlimitationsofquantumentanglementinthecontext
a
J of cooperative games of incomplete information. We give several examples of such nonlocal
1 gameswherestrategiesthatmakeuseofentanglementoutperformallpossibleclassicalstrate-
1 gies. Oneimplicationoftheseexamplesisthatentanglementcanprofoundlyaffectthesound-
nesspropertyoftwo-proverinteractiveproofsystems. Wethenestablishlimitsontheproba-

2
bility with which strategiesmaking use of entanglement canwin restrictedtypesof nonlocal
v
games. TheseupperboundsmayberegardedasgeneralizationsofTsirelson-typeinequalities,
6
7 which place bounds on the extent to which quantum information canallow for the violation
0 ofBellinequalities. Wealsoinvestigatetheamountofentanglementrequiredbyoptimaland
4 nearlyoptimalquantumstrategiesforsomegames.
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1 Introduction
/
h
p
This paper studies the implications of quantum entanglement to nonlocal games, which are coop-
-
nt erative games of incomplete information. Thereare twocooperating playersin a nonlocal game:
a Alice and Bob. A referee, also sometimes called a verifier in this context, determines the game as
u
follows. First, the referee randomly chooses questions for the players, drawn from finite sets
q
: according to some fixed probability distribution, and sends each player their question. Without
v
i communicating,andthereforewithoutknowing(ingeneral)whatquestionwasaskedoftheother
X
player,AliceandBobmusteachrespondtotherefereewithananswer. Therefereethenevaluates
r
a some predicate on the questions asked of the players together with their answers, to determine
whethertheywinorlose.
Whenanalyzingagivennonlocalgame,weassumethatAliceandBobhavecompleteknowl-
edgeoftheprobabilitydistributionthatdeterminesthereferee’squestions,aswellasthepredicate
that determines whether they win or lose. Based on this information they can agree before the
game starts on a joint strategy. A quantum strategy allows Alice and Bobto make use of a shared
entangled state, while a classical strategy doesnot. For some nonlocal games, quantum strategies
can allow the players to win with a higher probability than is possible for any classical strategy,
due to the fact that measurements of an entangled state can result in correlations that are not
achievable withouttheuseofentanglement.
∗InstituteforQuantumComputingandSchoolofComputerScience,UniversityofWaterloo,Canada
†InstituteforQuantumInformationScienceandDepartmentofComputerScience,UniversityofCalgary,Canada
‡BQPSolutionsPtyLtd,Elsternwick,Australia
§InstituteforQuantumComputingandSchoolofComputerScience,UniversityofWaterloo,Canada
1
Nonlocalgamesarenaturalabstractionsofmulti-proverinteractive proofsystems(consisting
of a single round of interaction with two provers), and have previously been considered in this
context in purely classical terms [19, 13, 45]. They also provide a natural setting in which the
quantum-physicalnotionofnonlocalitycanbecast. Inquantuminformationtheory,aBellinequality
is analogous to an upperbound on theprobability with which Alice and Bobcan win a nonlocal
game using a classical strategy, and a violation of a Bell inequality is analogous to the situation
in which a quantum strategy wins a particular nonlocal game with a higher probability than is
possible for any classical strategy. Tsirelson-type inequalities are bounds on the amount by which
quantuminformation can allow for theviolation ofBellinequalities, andthereforeare analogous
to upper bounds on the probabilities with which nonlocal games can be won using quantum
strategies.
We will begin our study of nonlocal games in Section 2, where we give precise definitions of
nonlocalgamesandtheclassical andquantumvaluesofthesegames. Thesevaluesrepresentthe
optimal probabilities with which Alice and Bob can win the games using classical and quantum
strategies,respectively.
Then,inSection3,wepresentfourexamplesofnonlocalgamesforwhichquantumstrategies
outperform classical strategies, including nonlocal games for which there exist perfect quantum
strategies(meaningthatthestrategieswinwithprobability one),butforwhichtheredonotexist
perfectclassicalstrategies. Theseexamplesarenotnew,butforthemostparthavebeenpresented
in the theoretical physics literature as hypothetical thought experiments, and their connections
with nonlocal games and multi-prover interactive proofs are obscure. The simplicity of some of
ourpresentationsmayhelptoelucidatesomeofthefeaturesofnonlocality.
Next,inSection4,wediscusstheimplicationsofquantumstrategiesfornonlocalgamestothe
study of multi-prover interactive proof systems. In particular, we exhibit two natural examples
oftwo-proverinteractiveproofsystemsthatareclassically sound,butbecomeunsoundwhenthe
proversmayemployquantumstrategies.
Finally, in Section 5, we provide the beginnings of a systematic understanding of the limits
of nonlocal strategies for two restricted classes of games: binary games and XOR games. Binary
gamesarenonlocalgamesinwhichAliceandBobeachrespondwithasinglebit,andXORgames
are binary games for which the referee’s predicate that determines whether Alice and Bob win
or lose depends only on the parity (or exclusive-OR) of their answer bits. The results proved in
this section include generalizations ofTsirelson’s inequality. We also prove upper boundson the
amountofentanglementneededtoplayXORgamesoptimallyornearlyoptimally.
Remarkon a previousversion of thispaper
A preliminary version of this paper appeared at the IEEE Conference on Computational Complexity
in 2004, and was postedto thearXiv.org e-printserverin Aprilof2004. This preliminary version
contains an error: oneofthe technical lemmas (numberedLemma 5.4 in theproceedingsversion
andLemma5.5inthearXiv.orgversion)isfalse. Thepresentversioncorrectsthismistake,which
requiresanewproofofTheorem3below.
Aside from this correction, the presentversion of this paper is similar to the preliminary ver-
sion. Whilesomerevisionstothewriting,notation,andorganizationofthepaperhavebeenmade,
andafewotherminorpointshavebeenadded,thereis(forthemostpart)nodiscussionofnewer
work that was donesubsequentto the writing ofthe preliminary version. (Readersinterestedin
furtherdevelopmentsonnonlocalgamesmay,forinstance,consult[17,29,32,33].)
2
Alice Bob
a b
s t
Referee
Figure1: Thecommunicationstructureofanonlocalgame.
2 Definitions
Nonlocal games
Letπ beaprobability distributiononS T,andletV beapredicateonS T A B,forfinite,
× × × ×
non-empty sets S, T, A, and B. Then V and π define a nonlocal game G = G(V,π) as follows.
A pair of questions (s,t) S T is randomly chosen according to the distribution π, and s is
∈ ×
sent to Alice and t is sent to Bob. Alice must respond with an answer a A and Bob with an
∈
answer b B. Alice and Bob are not permitted to communicate after receiving s and t, but they
∈
may agreeonwhateversortofstrategytheylike priortoreceiving theirquestions. TheywinifV
evaluates to 1 on (s,t,a,b) and lose otherwise. To stress the fact that (a,b) is correct or incorrect
givenquestions(s,t) wewilldenotethevalueofthepredicateV on(s,t,a,b) asV(a,b s,t).
|
Classical strategiesand classical valuesof nonlocal games
Theclassical value ofagame G is themaximum probability with which Alice andBobcan win G,
rangingoverallpurelyclassicalstrategies. TheclassicalvalueofagameGwillbedenotedω (G).
c
A deterministic strategyis a restricted typeof classical strategyin which a and b are simply func-
tionsof s and t, respectively. Theclassical value ofagameisalways obtainedbysomedetermin-
istic strategy given that any probabilistic strategy can be expressed as a convex combination of
deterministicstrategies. Inotherwords,itholdsthat
ω (G(V,π)) = max∑π(s,t)V(a(s),b(t) s,t),
c
a,b |
s,t
wherethemaximumisoverallfunctionsa : S Aandb : T B.
→ →
Quantumstrategies andquantumvaluesof nonlocal games
Wewillassumeforthisdiscussionandthroughouttherestofthepaperthatthereaderisfamiliar
with the basics of quantum information, which is discussed(for instance) in the books [43], [35],
and[31].
A quantumstrategyfor a game G consistsofan initial bipartite state ψ sharedby Alice and
| i
Bob, a quantum measurement for Alice for each s S, and a quantum measurement for Bobfor
∈
each t T. Oninput(s,t), Aliceperformshermeasurementcorrespondingtosonherportionof
∈
ψ , yielding an outcome a. Similarly, Bob performs his measurement corresponding to t on his
| i
portionof ψ ,yieldingoutcomeb. Theresultsaandbaresentbacktothereferee.
| i
3
In more descriptive mathematical terms, a quantum strategy is determined by the following
items.
1. Apositiveintegernandaunitvector ψ ,for and isomorphiccopiesofthevector
| i ∈ A⊗B A B
spaceCn. Thespace representsAlice’spartof ψ andthespace representsBob’spart.
A | i B
2. Twocollectionsofpositivesemidefiniten nmatrices
×
Xa : s S, a A and Yb : t T, b B
s ∈ ∈ t ∈ ∈
n o n o
satisfying
∑ Xa = 1 and ∑ Yb = 1
s t
a A b B
∈ ∈
foreverychoiceofs S andt T,where1 denotesthen nidentitymatrix. Foreach s S,
∈ ∈ × ∈
the collection Xa : a A describes the measurement performed by Alice on her part of
{ s ∈ }
ψ when she receives the question s. Likewise, for each t T, the collection Yb : b B
| i ∈ { t ∈ }
describesBob’smeasurementgiventhequestiont.
Given aquestions S forAlice and aquestiont T for Bob,such astrategycausesAlice to
∈ ∈
answerwitha AandBobtoanswerwithb Bwithprobability ψ Xa Yb ψ . Theprobability
∈ ∈ h | s ⊗ t| i
thatAliceandBobwinthegame Gusingsuchastrategyisthereforegivenby
∑ π(s,t) ∑ ψ Xa Yb ψ V(a,b s,t).
h | s ⊗ t| i |
(s,t) S T (a,b) A B
∈ × ∈ ×
Thequantumvalueofagame G,denotedω (G),isthesupremumofthewinningprobabilities
q
over all quantum strategies of Alice and Bob. It is not known if the quantum value is always
achievedbysomestrategy,duetothefactthatthenumbernisnotaprioribounded. Forinstance,
it has not been ruled out that there could sometimes exist a sequence of strategies requiring in-
creasing values of n that win a game G with probabilities converging to, but never reaching, its
quantum value. (As is shown in [40], this phenomenon indeed occurs in an extended form of
nonlocalgamesinwhichtherefereecansend,receive,andprocessquantuminformation.)
Observables
When we refer to an observable in this paper, we are referring to a Hermitian matrix that one as-
sociates with a projective measurement whose measurement outcomes are real numbers. More
precisely, suppose that we have a measurement described by a collection of projection matrices
Π ,...,Π forwhichΠ + +Π = 1,andsupposefurtherthatweassociatetheoutcomesof
1 k 1 k
{ } ···
themeasurementwithacollectionofrealnumbers λ ,...,λ . Thentheobservablecorrespond-
1 k
{ }
ingtothismeasurementisgivenby
k
A = ∑λ Π . (1)
j j
j=1
Given such a matrix A, one may determine the corresponding projective measurement by com-
putingthespectraldecompositionof A.
We will primarily use observables when discussing strategies for games that require Alice
andBobtoperformtwo-outcomemeasurements,especiallyinthecaseofbinarygames(whichare
the main topic of Section 5). In this setting it is convenient to associate real numbers in the set
4
+1, 1 with the binary values 0,1 using the correspondence 0 +1 and 1 1 (as is
{ − } { } 7→ 7→ −
commonwhenusingdiscreteFourieranalysisofBooleanfunctions). Whenusingthisconvention,
theobservable A correspondingtoabinary-valued projectivemeasurement Π ,Π isgivenby
0 1
{ }
A = Π Π .
0 1
−
3 Examples of nonlocal games
The fact that entanglement can cause non-classical correlations is a familiar idea in quantum
physics, introduced in a seminal 1964 paper by Bell [5]. In the following subsections, we give
fourexamplesofthisphenomenon.
3.1 The CHSH game
Our first example of a game for which a quantum strategy outperforms any classical strategy is
a well-known example based on the CHSH inequality, named for its discoverers Clauser, Horne,
Shimony, and Holt [14]. Rephrased in terms of nonlocal games, the example is as follows. Let
S = T = A = B = 0,1 ,letπ betheuniformdistributiononS T,andletV bethepredicate
{ } ×
1 if a b = s t
V(a,b s,t) = ⊕ ∧
| 0 otherwise.
(cid:26)
The classical value of the game G = G(V,π) is ω (G) = 3/4, which is easily verified by con-
c
sidering all deterministic strategies. Using a quantum strategy, however, Alice and Bob can win
this game with probability cos2(π/8) 0.85. This probability is optimal, so we have ω (G) =
q
≈
cos2(π/8). A descriptionofa quantumstrategythat achieves thisprobability ofsuccessfollows,
andthefactthatitisoptimalfollowsfromTsirelson’sInequality[34,47].
First,lettheentangledstatesharedbyAliceandBobbe ψ = ( 00 + 11 )/√2,define
| i | i | i
φ (θ) = cos(θ) 0 +sin(θ) 1 ,
0
| i | i | i
φ (θ) = sin(θ) 0 +cos(θ) 1 ,
1
| i − | i | i
andletAliceandBob’smeasurementsbegivenas
Xa = φ (0) φ (0) ,
0 | a ih a |
Xa = φ (π/4) φ (π/4) ,
1 | a ih a |
Yb = φ (π/8) φ (π/8) ,
0 | b ih b |
Yb = φ ( π/8) φ ( π/8)
1 | b − ih b − |
for a,b 0,1 . Each of these matrices is a rank-one projection matrix, so the measurements
∈ { }
Alice and Bobare making are examplesofprojective measurements. Given our particular choice
of ψ , we have ψ X Y ψ = 1 Tr XTY for arbitrary matrices X and Y. Thus, as each of the
| i h | ⊗ | i 2
matrices Xa and Yb is real and symmetric, the probability that Alice and Bob answer (s,t) with
s t (cid:0) (cid:1)
(a,b) is 1Tr XaYb . Itisnowroutinetocheckthatineverycase,thecorrectanswerisgivenwith
2 s t
probabilitycos2(π/8)andtheincorrectanswerwithprobabilitysin2(π/8).
(cid:0) (cid:1)
5
3.2 The Odd Cycle game
Forthefollowinggame,imaginethatAliceandBobaretryingtoconvincetherefereethatanodd
cycleoflengthnis2-colorable(whichitisnot,asnisodd). Therefereesendsthenameofavertex
toeachofAliceandBobsuchthatthetwoverticesareeitherthesameoradjacent. AliceandBob
eachsendoneoftwocolorsbacktothereferee. Thereferee’srequirementisthat,whenthevertices
are the same, the two colors should agree, and when the vertices are adjacent, the colors should
bedifferent.
Formally, let n 3 be an odd integer, let S = T = Z , and let A = B = 0,1 . Take π to be
n
≥ { }
theuniformdistributionovertheset (s,t) Z Z : s = tors+1 t(mod n) andletV be
n n
{ ∈ × ≡ }
definedas
1 if a b = [s+1 t(mod n)]
V(a,b s,t) = ⊕ ≡
| 0 otherwise.
(cid:26)
(Here we have written [s+1 t(mod n)] to denote the Boolean value representing the truth
≡
or falsehood of the congruence s+1 t(mod n).) This is a variation on a game based on the
≡
chainedBellinequalitiesofBraunsteinandCaves[10]thatgeneralizetheCHSHinequality. Itisalso
discussedbyVaidman[49].
It is easy to see that ω (G) = 1 1/2n for this game. Any deterministic strategy must fail
c
−
for at least one of the possible pairs (s,t), as an odd cycle cannot be 2-colored, while a strategy
achievingsuccessprobability1 1/2n isthatAliceandBobleta = s mod 2andb = t mod 2.
−
On the other hand, a quantum strategy can attain a success probability quadratically closer
to1. Thefollowingquantumstrategy[10]winswithprobability cos2(π/4n) 1 (π/4n)2. The
≥ −
sharedstateisthesameasfortheCHSHgame:
1
ψ = ( 00 + 11 ).
| i √2 | i | i
Thistimewedefine
Xa = φ (α ) φ (α ) ,
s | a s ih a s |
Yb = φ (β ) φ (β ) ,
t | b t ih b t |
where
π π π
α = s+ ,
s
2 − 2n 4n
(cid:16)π π (cid:17)
β = t,
t
2 − 2n
(cid:16) (cid:17)
and where φ (θ) and φ (θ) are as defined for the CHSH game. Given questions (s,t), the
0 1
| i | i
probability that Alice and Bob answer the same bit may be calculated to be cos2(α β ), which
s t
−
impliestheyanswerdifferentbitswithprobabilitysin2(α β ). Incases = twehave
s t
−
π
α β = ,
s t
− 4n
so they answer correctly (i.e., with a = b) with probability cos2(π/4n). If s+1 t (modn), on
≡
theotherhand,wehave
π π
α β = ,
s t
− 2 − 4n
so they answer correctly (i.e., with a = b) with probability sin2(π/2 π/4n) = cos2(π/4n).
6 −
Thereforethisstrategyanswerscorrectlywithprobability cos2(π/4n) oneverypairofquestions.
Thisquantumstrategyisoptimal,asweshallshowinCorollary9laterinthepaper.
6
3.3 The MagicSquaregame
Thenextgameweconsiderisbasedonthefactthattheredoesnotexista3 3binarymatrixwith
×
thepropertythateachrowhasevenparityandeachcolumnhasoddparity. Itisaslightvariation
of an example presented by Aravind [3], which builds on work by Mermin [41, 42]. The idea is
toaskAlicetofillinthevaluesineitheraroworacolumnofthematrix (randomlyselected)and
to ask Bob to fill in a single entry of the matrix, that is randomly chosen among the three entries
giventoAlice. TherequirementisthattheparityconditionsaremetbyAlice’sanswers(evenfor
rows,oddforcolumns)andthatBob’sanswerisconsistentwithAlice’sanswers.
Formally,let S = Z indexthesixpossiblequestionstoAlice (threerowsplusthreecolumns)
6
and let T = Z index the nine possible questions to Bob (one for each entry of the matrix). Let
9
A = 0,1 3 andB = 0,1 . ThepredicateV(a,b s,t)isdefinedtotakevalue1ifandonlyifahas
{ } { } |
the appropriate parity (0 for a row and 1 for a column) and theentry of a correspondingto t has
value b. Thedistribution π istheuniformdistributionover (s,t) S T : entrytisintriples .
{ ∈ × }
It is not hard to see that ω (G) = 17/18 for this game. It should be noted that, although it is
c
convenient to set A = 0,1 3 for this game, we could take A = 0,1 2, because the third bit of
{ } { }
Alice’soutputisdeterminedbythefirsttwobitsandtheparityconstraints.
Remarkably, there exists a perfect quantum strategy for this game, meaning that the strategy
winswithprobability1. Theparticularstrategywewilldescribeisderivedfrom[3],whereaslight
variantofthisgameispresented.
This strategy is best described using the notion of observables, which were discussed at the
end of Section 2. Consider the following 3 3 matrix of observables, each corresponding to a
×
measurementoftwoqubits:
σ σ σ σ σ σ
x y y x z z
⊗ ⊗ ⊗
σ σ σ σ σ σ .
 y z z y x x
⊗ ⊗ ⊗
σ σ σ σ σ σ
z x x z y y
 ⊗ ⊗ ⊗ 
 
Here,thematricesσ ,σ ,andσ arethe(non-identity)Paulimatrices:
x y z
0 1 0 i 1 0
σ = , σ = − , σ = .
x 1 0 y i 0 z 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
AliceandBobwillsharetwocopiesofthestate
1 1
ψ = 01 10 .
−
| i √2 | i− √2 | i
Toanswer any questiontheyreceive about the 3 3 square, Alice and Bobsimply measure their
×
partsofthetwosharedstateswith respecttothecorrespondingobservablesin theabove matrix.
For example, if Alice is asked to assign values to the first row of the square, she measures her
twoqubitswithrespecttotheobservableσ σ ,thenwithrespecttotheobservableσ σ ,and
x y y x
⊗ ⊗
finallywithrespecttoσ σ ,eachtimeobtaininga 1outcomecorrespondingtoabinaryvalue.
z z
⊗ ±
Similarly, if Bob is asked to assign a value to the (1,2) entry of the square, he measures his two
qubitswithrespecttotheobservableσ σ .
y x
⊗
To understand why this strategy works perfectly, a few basic observations are required. The
first observation is that ψ is a 1 eigenvector of each of the operators σ σ , σ σ , and
− x x y y
| i − ⊗ ⊗
σ σ ,whichimplies
z z
⊗
ψ− σx σx ψ− = ψ− σy σy ψ− = ψ− σz σz ψ− = 1. (2)
h | ⊗ | i h | ⊗ | i h | ⊗ | i −
7
This implies that Alice and Bob’s answers are always in agreement, as the product of their mea-
surement outcomes (considered as 1 values) always equals 1. For instance, if Alice and Bob
±
bothmeasuretheirtwocopiesof ψ withrespecttotheobservableσ σ ,theproductoftheir
− y x
| i ⊗
outcomeswillbe
ψ− σy σy ψ− ψ− σx σx ψ− = ( 1)( 1) = 1.
h | ⊗ | ih | ⊗ | i − −
ThesecondobservationisthatthePaulimatricesanti-commuteinpairs:
σ σ = σ σ , σ σ = σ σ , σ σ = σ σ . (3)
x y y x x z z x y z z y
− − −
This implies that the observables commute within any row or column. For instance, σ σ and
y x
⊗
σ σ bothbelongtothefirstrow,andwehave
z z
⊗
(σ σ )(σ σ ) = (σ σ ) (σ σ ) = ( σ σ ) ( σ σ ) = (σ σ ) (σ σ ) = (σ σ )(σ σ ).
y x z z y z x z z y z x z y z x z z y x
⊗ ⊗ ⊗ − ⊗ − ⊗ ⊗ ⊗
This implies that Alice can simultaneously measure the three observables within whichever row
or column she was asked—or equivalently that the outcomes she obtains do not depend on the
orderinwhichsheperformsthesemeasurements.
Thefinalobservationthatisrequiredisthattheproductoftheobservablesineachrowisequal
to1 1, while theproductoftheobservablesin each column is 1 1. Thisimplies that Alice’s
⊗ − ⊗
parityrequirementsarealwaysmet.
3.4 The Kochen-Speckergame
ThisgameisbasedontheKochen-SpeckerTheorem,whichcanbestatedasfollows.
Theorem1(KochenandSpecker[37]). Thereexistsanexplicitsetofvectors v ,..., v inR3
0 m 1
{| i | − i}
thatcannotbe 0,1 -colored sothatbothofthefollowingconditionshold:
{ }
1. Foreveryorthogonalpairofvectors v and v ,theyarenotbothcolored1.
i j
| i | i
2. Foreverymutuallyorthogonal tripleofvectors v , v ,and v ,atleastoneofthemiscolored 1.
i j k
| i | i | i
The original theorem in [37] used 117 vectors, but this has subsequently been reduced to 31
vectors[44]. Wewillassumethateveryorthogonalpairofvectorsinthesetispartofanorthogonal
triple—whichiseasilyachievedbyaddingafewmorevectorstotheset—andthatthevectorsare
normalized. ConnectionsbetweentheKochen-SpeckerTheoremandnonlocalityhavepreviously
beenmadein[28].
The Kochen-Specker game is defined relative to the above set of vectors. Alice receives a
randomtripleoforthogonalvectorsasherinputandBobreceivesasinglevectorrandomlychosen
from the triple as his input. Alice outputsa trit indicating which of her three vectors is assigned
color1(implicitly, theothertwovectorsareassignedcolor0). Boboutputsabitassigningacolor
to his vector. Therequirement is that Alice and Bobassign thesame color tothe vector that they
receiveincommon.
It is straightforward to show that the existence of a perfect classical strategy for this game
would violate the Kochen-Specker Theorem, so ω (G) < 1 for this game. On the other hand
c
there is a perfect quantum strategy, using entanglement ψ = ( 00 + 11 + 22 )/√3. Alice’s
| i | i | i | i
projectors(forinput v , v , v )are v v , v v , v v ,andBob’sprojectors(forinput
i j k i i j j k k
| i | i | i | ih | | ih | | ih |
v )are v v and1 v v .
l l l l l
| i | ih | −| ih |
8
4 Connections with multi-prover interactive proof systems
The two-prover interactive proof systemmodel was defined by Ben-Or, Goldwasser, Kilian, and
Wigderson [7], and has been the focus of a great deal of study. Babai, Fortnow, and Lund [4]
proved that every language in NEXP has a two-prover interactive proof system. Several refine-
ments to this result were made [12, 18, 39], leading to a proof by Feige and Lova´sz [19] that a
languageisin NEXPifandonlyifithasatwo-proverone-roundproofsystemwithperfectcom-
pletenessandexponentiallysmallsoundnesserror.
In mostworkon multi-prover interactive proofsystems,theproversare computationally un-
bounded, subject to the restriction that they cannot communicate with each other during the
courseoftheprotocol. Becausethespiritoftheinteractiveproofsystemparadigmistoboundthe
capabilities oftheverifierratherthantheprover(s),it isnaturaltoconsiderproverstrategiesthat
entail sharing entangled quantum information prior to the execution of the proof system. Note
thatsuchastrategydoesnotnecessarilyrequirethecomputationallyboundedverifiertomanipu-
late(orknowanythingabout)quantuminformation. However,muchofthestudyofmulti-prover
interactive proof systems occurred prior to the mid 1990s, when quantum information was not
well-known within the theoretical computer science community, and quantum strategies were
generally not considered. In fact, the methodologies for analyzing these proof systems usually
maketheimplicit assumptionthatproversarerestrictedtoclassicalstrategies.
In this section, we consider what happens when the provers can employ quantumstrategies.
Wedonotmakeanychangetotheverifier,whoremainsclassical,andallcommunicationbetween
theverifierandtheproversremainsclassical.1 Anaturalquestionis: Whatistheexpressivepower
ofsuchproofsystems?
Let us write MIP and MIP to distinguish between the cases of no shared entanglement and
∗
shared entanglement, respectively. That is, MIP denotes the class of languages recognized by
multi-proverinteractiveproofsystemswhereallcommunicationbetweentheproversandverifier
isclassicalandtheproversdonotshareentanglement(ashasbeenimplicitlyassumedinprevious
contexts). The definition of MIP∗ is identical to that of MIP, except that the provers may share
an arbitrary entangled quantum state at the beginning of the protocol. Furthermore, let MIP[k]
and MIP∗[k] denote the same classes, but with the number of provers set to k. It is known that
MIP = MIP[2] = NEXP. We do not know any relationships between MIP∗, MIP∗[2] and NEXP,
exceptthetrivialcontainmentMIP∗[2] MIP∗.
⊆
A one-round two-prover interactive proof system is one where the interaction is restricted to
two stages: a query stage where the verifier sends information to the provers, and a response
stagewheretheproverssendinformationtotheverifier. Notethatsuchaproofsystemassociates
anonlocalgame G toeachstring x withthefollowingproperty. Forallyes-inputsx,thevalueof
x
ω (G )isclosetoone,and,forallno-inputsx,thevalueofω (G )isclosetozero.
q x q x
We give two examples of natural two-prover one-round proof systems that are classically
sound,but become unsoundwhen theproversusequantumstrategies: one is for languagesthat
express graph chromatic numbers and the other is for 3-SAT. These examples are related to the
examplesinSection3. WealsoexplainwhytheexistingproofsthatequateMIPwithNEXPbreak
downintermsoftheirmethodologyinthecaseofMIP∗. Resultsin[36]implythat,iftheamount
ofentanglementbetweentheproversispolynomiallybounded,thenanylanguagerecognizedby
an MIP∗ proof system is contained in NEXP; but without this polynomial restriction, we do not
knowifthisholds.
1KobayashiandMatsumoto[36]considerarelatedbutdifferentmodel,wheretheproversandtheverifiermanipu-
latequantuminformationandquantumcommunicationoccursbetweentheverifierandtheprovers.
9
4.1 GraphColoring proof system
The Odd Cycle game in Section 3.2 can be regarded as a protocol where two provers are trying
toconvince averifier thataparticular graph is two-colorable. Thisideageneralizes toany graph
G and number of colors k. The verifier asks each prover for thecolor (among k possibilities) of a
vertex and requires that the colors be the same whenever each prover gets the same vertex and
differentwhenevertheproversgetadjacentvertices. Formally,thegameforGandkisasfollows.
LetS = T = V(G),let A = B = Z ,let
k
1 if(s = tand a = b)or((s,t) E(G)and a = b)
V(a,b s,t) = ∈ 6
| 0 otherwise,
(cid:26)
andπ betheuniformdistributionon (s,s) : s V(G) (s,t) E(G) .
{ ∈ }∪{ ∈ }
If G is k-colorable then the provers can satisfy V by basing their answers on a valid coloring
ofG. Therefore,thevalueoftheassociatedgameis1. IfGisnotk-colorable then,foranyclassical
strategyonthepart oftheprovers,theremustbe an inconsistencyfor somevalue of(s,t), sothe
classicalvalueoftheassociatedgameisatmost1 1/( V(G) + E(G) ). Theverifiercanamplify
− | | | |
the difference between the two cases (k-colorable and not k-colorable) by repeating this game a
polynomialnumberoftimes(inparallel[45]). Thusthisisaclassicaltwo-proverinteractiveproof
systemforthelanguageconsistingofallk-colorable graphs.
This proof system breaks down in the case of entangled provers. Based on a protocol in [9],
there exists a sequence of graphs G (where n ranges over all powers of two) with the following
n
properties. First,foranyn,thereisaperfectquantumstrategyfortheGraphColoringproofsystem
withgraphG andk = ncolors. Second,forsufficientlylargen,G isnotn-colorable.
n n
Forany n, G is simpleto describe: it has vertices 0,1 n andtwo verticesare adjacent if and
n
{ }
onlyiftheHamming distancebetweenthemis n/2. However,resultsin[9]showthatthere exists
an nsuchthat G is notn-colorable, withoutgivingan explicit nforwhich thisholds. (Theproof
n
isbasedonarelatedresultin[11],whichmakesuseofacombinatorialresultin[23].) Theresultis
madeexplicit in[24], whereitisshownthat G isnot16-colorable. Thus,theresultinggraphfor
16
whichtheGraphColoringproofsystembreaksdownhas216 vertices,anditcanbesimplifiedby
takingonlyhalfofitsvertices,resultinginagraphof32,768vertices.
4.2 3-SATproof system
Nextweconsiderasimpleandnaturaltwo-proverinteractiveproofsystemforprovingthat3-CNF
formulas aresatisfiable. Theverifiersendsthefirstprover(Alice)aclauseandthesecondprover
(Bob) a variable from that clause. Alice must provide a satisfying assignment to the variables in
thatclauseandBobmustassignavalueforthevariable hereceivesthatisconsistentwithAlice’s
assignment.
This proof system associates a nonlocal game G with every 3-CNF Boolean formula f over
f
variables x ,...,x withmclausesc ,...,c . Specifically,wetake
0 n 1 0 m 1
− −
S = Z , T = Z , A = 0,1 3, and B = 0,1 .
m n
{ } { }
Foreachclause,everya 0,1 3 inducesanassignmenttoeachvariablethatoccursintheclause
∈ { }
in a natural way. The predicate V(a,b s,t) takes the value 1 if and only if the assignment for
|
the variables in c induced by a satisfies c and is consistentwith the assignment x = b, and the
s s t
distributionπ maybetakentobeuniformon
(s,t) S T : clausec containsvariable x .
s t
{ ∈ × }
10