Deciding the existence of perfect entangled strategies for nonlocal games
aa r X i v : . [ qu a n t - ph ] J un Deciding the existence of perfect entangled strategies for nonlocalgames
Laura Manˇcinska ∗ , David E. Roberson † , and Antonios Varvitsiotis ‡ Centre for Quantum Technologies, National University of Singapore, 117543 Singapore School of Physical and Mathematical Sciences, Nanyang Technological University, 50Nanyang Avenue, 637371 SingaporeAugust 14, 2018
Abstract
First, we consider the problem of deciding whether a nonlocal game admits a perfect entangled strategythat uses projective measurements on a maximally entangled shared state. Via a polynomial-time Karpreduction, we show that independent set games are the hardest instances of this problem. Secondly, weshow that if every independent set game whose entangled value is equal to one admits a perfect entangledstrategy, then the same holds for all symmetric synchronous games. Finally, we identify combinatoriallower bounds on the classical and entangled values of synchronous games in terms of variants of theindependence number of appropriate graphs. Our results suggest that independent set games might berepresentative of all nonlocal games when dealing with questions concerning perfect entangled strategies.
Keywords: nonlocal game, entangled value, quantum independence number, perfect entangled strategies
Entanglement plays a central role in quantum information processing and is increasingly seen as a valuableresource for distributed tasks such as unconditionally secure cryptography [Eke91], randomness certifica-tion [Col06, PAM +
10] and expansion [VV12, MH13]. Given such a scenario it is interesting to understandhow much and what kind of entanglement needs to be employed in an optimal entangled strategy. As iscommonly done, we study these questions within the framework of nonlocal games. In the computer sciencecommunity nonlocal games arise as one-round interactive proof systems, while in the physics communitythey are known as Bell inequalities [BCP + nonlocal game is specified by four finite sets A, B, Q, R , a probability distribution π on Q × R and aBoolean predicate V : A × B × Q × R → { , } . The game proceeds as follows: Using π the verifier samplesa pair ( q, r ) ∈ Q × R and sends q to Alice and r to Bob. Upon receiving their questions the players respondwith a ∈ A and b ∈ B , respectively. The players have knowledge of the distribution π and the predicate V and can agree on a common strategy before the start of the game, but they are not allowed to communicateafter they receive their questions. We say the players win the game if V ( a, b | q, r ) = 1. A strategy is called perfect if it allows the players to win the game with probability one.The goal of Alice and Bob is to maximize their probability of winning the game. The classical value ofa game G , denoted ω ( G ), is the maximum expected winning probability when the players use deterministicstrategies. An entangled strategy for a nonlocal game allows the players to determine their answers byperforming joint measurements on a shared finite-dimensional entangled state. The entangled value of a game ∗ [email protected] † [email protected] ‡ [email protected] , denoted ω ∗ ( G ), is the supremum expected winning probability the players can achieve using entangledstrategies.Despite significant efforts, many fundamental questions concerning the properties of the entangled valuehave so far remained beyond reach:(i) The computability question:
Determine (or upper bound) the computational complexity of ω ∗ ( G ).(ii) The attainability question:
Determine if ω ∗ ( G ) can always be attained.(iii) The resources question:
How much and what kind of entanglement is needed to achieve ω ∗ ( G ).The above questions are understood only for some very special classes of games. One notable exampleis the class of XOR games; for these games the answer sets, A and B , are binary and the verificationpredicate only depends on the XOR of the player’s answers. For XOR games the entangled value can beformulated as a semidefinite program which can be approximated within arbitrary precision in polynomialtime. Furthermore, the entangled value of an XOR game is always attained by a maximally entangledstate [CHTW04].At present, there has only been sporadic progress for other classes of nonlocal games. Some positiveapproximability results have been derived for the class of unique nonlocal games [KRT10]. For generalnonlocal games a hierarchy of semidefinite programming upper bounds for the entangled value was identifiedin [NPA07]. Unfortunately, the quality of the approximation at each level of the hierarchy is not understood.Given the lack of progress in addressing these questions there has been increasing interest in the study ofrestricted variants of the above problems. A decision problem that has gained some attention is the following: PERFECT
Instance:
A nonlocal game G . Question:
Does G admit a perfect entangled strategy?It follows from recent work of Ji that PERFECT is NP-hard [Ji03]. On the other hand, despite significantefforts
PERFECT is currently not known to be decidable. Some partial progress has been documented con-cerning the decidability of
PERFECT for special classes of nonlocal games. Specifically, Cleve and Mittal haveshown that for BCS games, deciding the existence of a perfect entangled strategy can be reduced to decidingthe existence of a self-adjoint operator solution to a polynomial system in non-commuting variables [CM14].This reduction does not imply decidability since no algorithms are currently known for deciding the existenceof operator solutions to non-commutative polynomial systems. In a follow up work Arkhipov studies parityBCS games with the additional requirement that each variable appears in exactly two clauses. To any suchgame he associates an undirected graph and shows that the game has a perfect entangled strategy if andonly if the corresponding graph is non-planar [Ark12]. Since non-planarity can be decided in linear time[HT74] this shows that
PERFECT can also be decided in linear time for this special subclass of BCS games.
Motivation, results, and discussion.
In view of the limited progress in understanding the computabilityand attainability questions and with the hope to gain new insights, in this work we study the decision problem
PERFECT where we impose additional operational restrictions on the set of allowed strategies. Specifically, ourgoal is to decide whether a given nonlocal game G admits a perfect entangled strategy where the players areonly allowed to apply projective measurements on a shared maximally entangled state (hereafter abbreviatedas PME strategies). Formally, we focus on the following decision problem:
PERFECT-PME
Instance:
A nonlocal game G . Question:
Does G admit a perfect PME strategy?The study of
PERFECT-PME is motivated by the following considerations. Firstly, given the impasse on thegeneral question of deciding whether a nonlocal game admits a perfect strategy,
PERFECT-PME can be viewedas an even further restricted variant of
PERFECT that can hopefully provide useful insights into the generalproblem. Secondly, to the best of our knowledge, there are no known examples of nonlocal games that admitperfect strategies but cannot be won perfectly using
PME strategies. Thus,
PME strategies might even besufficient to reach success probability one (whenever this can be done using some quantum strategy) whichwould imply that
PERFECT is in fact equivalent to
PERFECT-PME .2e note that the situation is quite different when one considers non-perfect strategies. Specifically, thereare examples of nonlocal games whose entangled value is strictly smaller than one and for which maximallyentangled states do not suffice to achieve the optimal success probability, e.g., [JP11, VW11, LVB11, Reg12].The decision problem
PERFECT-PME has also been considered by Ji [Ji03]. Similarly to [CM14], in thiswork Ji shows how one can associate to any nonlocal game G a polynomial system in non-commuting operatorvariables with the property that G admits a perfect PME strategy if and only if the corresponding system hasa solution in self-adjoint operator variables. Using this reduction he proceeds to show that
PERFECT-PME isNP-hard already when the input is restricted to be the BCS game corresponding to the 3-SAT problem.Our main result in this work is given in Theorem 5.3 where we identify independent set games as beingthe hardest instances of PERFECT-PME . In the (
X, t ) -independent set game the players aim to convince averifier that a graph X contains an independent set of size t ( i.e. , a set of t pairwise nonadjacent vertices).To play the game the verifier selects uniformly at random a pair of indices ( i, j ) ∈ [ t ] × [ t ] and sends i to Aliceand j to Bob. The players respond with vertices u, v ∈ V ( X ) respectively. In order to win, the players needto respond with the same vertex of X whenever they receive the same index. Furthermore, if they receive i = j ∈ [ t ] they need to respond with nonadjacent (and distinct) vertices of X . The second decision problemrelevant to this work is PERFECT where the input is restricted to be an independent set game.
Q-INDEPENDENCE
Instance:
An (
X, t )-independent set game.
Question:
Does the game admit a perfect entangled strategy?In our main result given in Theorem 5.3 we show that any instance G of PERFECT-PME can be transformedin polynomial-time to an instance G ′ of Q-INDEPENDENCE with the property that G admits a perfect PME strategy if and only if G ′ admits a perfect strategy. Formally: Result 1:
PERFECT-PME is polynomial-time (Karp) reducible to
Q-INDEPENDENCE . It is known that
PME strategies suffice to win independent set games perfectly (whenever this can be doneusing some quantum strategy) [RM14]. As a result, all instances of
Q-INDEPENDENCE can be identified withinstances of
PERFECT-PME and thus our first result can be understood as identifying
Q-INDEPENDENCE to beamong the most expressive subproblems of
PERFECT-PME .As an immediate consequence of our first result and the discussion in the previous paragraph it followsthat
PERFECT-PME is decidable if and only if
Q-INDEPENDENCE is decidable. Currently, it is not known whether
Q-INDEPENDENCE is decidable. Nevertheless, reducing the decidability question from arbitrary games to thespecial class of independent set games allows to narrow down our focus to this specific class of games forwhich it might be easier to make progress on the decidability question.The proof of Result 1 consists of two steps which we now briefly describe. We first need to introducesome definitions. A nonlocal game is called synchronous if it satisfies the following three requirements: (i)
Alice and Bob share the same question set Q and answer set A , (ii) π ( q, q ) > q ∈ Q , and (iii) V ( a, b | q, q ) = 0 for all q ∈ Q and all a = b . Notice that the ( X, t )-independent set game defined above is anexample of a synchronous nonlocal game. The second decision problem of interest in this paper is a variationof
PERFECT where the input is restricted to be a synchronous game:
PERFECT-SYN
Instance:
A synchronous nonlocal game G . Question:
Does G admit a perfect quantum strategy?In Lemma 3.2 we show that any synchronous game that admits a perfect quantum strategy also has aperfect PME strategy. Notice that this implies that
PERFECT-SYN is a subproblem of
PERFECT-PME .The first step in proving Result 1 is Lemma 3.5 where we show that
PERFECT-PME is polynomial-timereducible to
PERFECT-SYN . To achieve this we extend any nonlocal game G to a synchronous game ˜ G wherewe can also ask Alice any of Bob’s questions and vice versa (see Definition 3.4). The extended game ˜ G hasthe property that G has a perfect PME strategy if and only if ˜ G has a perfect strategy.The second step in proving Result 1 is Lemma 5.2 where we show that PERFECT-SYN is polynomial-time reducible to
Q-INDEPENDENCE . To achieve this, to any synchronous game G we associate an undirectedgraph X ( G ) (see Definition 4.1) and show that G admits a perfect entangled strategy if and only if the( X ( G ) , | Q | )-independent set game has a perfect strategy (where Q denotes the question set of G ).3e note that following the completion of this work it was communicated to us by Ji that building on hisrecent results in [Ji03] he has independently obtained Result 1. The proof of this fact has not been publishedbut can be derived by appropriately combining the results in [Ji03] together with two additional reductions(that are not stated in [Ji03]). Furthermore, in contrast to [Ji03] our approach is constructive and the finalinstance of Q-INDEPENDENCE is given explicitly in terms of the instance of
PERFECT-PME .As was already mentioned it is currently not known whether the entangled value of a nonlocal game (withfinite question and answer sets) is always attained by some entangled strategy. In fact, there is evidencethat this might not be true. The first example of a nonlocal game with answer sets of infinite cardinality forwhich the entangled value is only attained in the limit was identified recently in [MV14].In this work we consider the attainability question restricted to perfect strategies for symmetric syn-chronous nonlocal games. A synchronous game is called symmetric if interchanging the roles of the playersdoes not affect the value of the Boolean predicate ( cf.
Definition 5.4). We note that all games of relevanceto this work (e.g. homomorphism) are in fact symmetric. In Theorem 5.5 we show that independent setgames again capture the hardness of the attainability question for symmetric games.
Result 2:
Suppose that every independent set game G satisfying ω ∗ ( G ) = 1 admits a perfect entangledstrategy. Then the same holds for all symmetric synchronous nonlocal games. To obtain our second result we show that vanishing-error strategies for a symmetric synchronous game G give rise to vanishing-error strategies for an appropriate independent set game defined in terms of the gamegraph of G . Notice that since independent set games are synchronous, our second result can be understoodas identifying a class of synchronous games which captures the hardness of the attainability question forperfect strategies for the entire class of symmetric synchronous nonlocal games. Nevertheless, we note thatpresently we do not know whether independent set games satisfy the assumption of Result 2.A number of interesting results have been derived recently concerning the interplay between the theory ofgraphs and nonlocal games, e.g. [CSW14, RM14, CMSS14, Ark12]. In Section 4 we take a similar approachand associate an undirected graph, called its game graph , to an arbitrary synchronous nonlocal game. Asalready mentioned this is an essential ingredient in showing that PERFECT-SYN is polynomial-time reducibleto
Q-INDEPENDENCE . In Theorem 4.6 we identify combinatorial lower bounds on the classical and entangledvalue of synchronous nonlocal games in terms of their corresponding game graphs.
Result 3:
Let G be a synchronous game with question set Q and uniform distribution of questions. If X = X ( G ) is the game graph of G then ω ( G ) ≥ (cid:0) α ( X ) / | Q | (cid:1) , and ω ∗ ( G ) ≥ (cid:0) α p ( X ) / | Q | (cid:1) , where α ( X ) denotes the independence number of X and α p ( X ) the projective packing number of X (cf. Definition 4.2). We denote the set of d × d Hermitian operators by S d . Throughout this work we equip S d with the Hilbert-Schmidt inner product h X, Y i = Tr( XY ∗ ). An operator X ∈ S d is called positive , denoted by X (cid:23)
0, if ψ ∗ Xψ ≥ ψ ∈ C d . The set of d × d positive operators is denoted by S d + . We use the notation X (cid:23) Y to indicate that X − Y (cid:23)
0. An operator X is called an (orthogonal) projector if it satisfies X = X ∗ = X .The support of an operator X , denoted supp( X ), is defined as the projector on the range of X . The canonicalorthonormal basis of C d is denoted by { e i : i ∈ [ d ] } , where [ d ] := { , . . . , d } . Classical strategies and value.
A deterministic strategy for a nonlocal game G ( π, V ) consists of a pairof functions, f A : Q → A and f B : R → B , which the players use in order to determine their answers. The classical value of the game G , denoted by ω ( G ), is equal to the maximum expected probability with whichthe players can win the game using deterministic strategies. Specifically, ω ( G ) := max X q ∈ Q,r ∈ R π ( q, r ) V (cid:0) f A ( q ) , f B ( r ) | q, r (cid:1) , (1)where the maximization ranges over all deterministic strategies.4 uantum strategies and value. In this section we briefly introduce those concepts from quantum in-formation theory that are of relevance to this work. Readers without the required background are referredto [NC] for a comprehensive introduction.To any quantum system S we associate a complex inner product space C d , for some d ≥
1. The statespace of the system S is defined as the set of unit vectors in C d . The most basic way one can extract classicalinformation from a quantum system S is by measuring it. For the purposes of this paper, the most relevantmathematical formalism of the concept of a measurement is given by a Positive Operator-Valued Measure(POVM). A POVM is defined in terms of a family of positive operators M = ( M i ∈ S d + : i ∈ [ m ]) that sumup to the identity operator, i.e. , P i ∈ [ m ] M i = I d . According to the axioms of quantum mechanics, if themeasurement M is performed on a quantum system whose state is given by ψ ∈ C d then the probability thatthe i -th outcome occurs is given by ψ ∗ M i ψ . We say that a measurement M is projective if all the POVMelements M i are orthogonal projectors.Consider two quantum systems S and S with corresponding state spaces C d and C d respectively. Thestate space of the joint system (S , S ) is given by C d ⊗ C d . Moreover, if S is in state ψ ∈ C d and S isin state ψ ∈ C d then the joint system is in state ψ ⊗ ψ ∈ C d ⊗ C d . Lastly, if ( M i ∈ S d + : i ∈ [ m ]) and( N j ∈ S d + : j ∈ [ m ]) define measurements on the individual systems S and S then the family of operators( M i ⊗ N j ∈ S d d + : i ∈ [ m ] , j ∈ [ m ]) defines a product measurement on the joint system (S , S ).Given any bipartite quantum state ψ ∈ C d ⊗ C d , it is possible to choose two orthonormal basis { α i : i ∈ [ d ] } and { β i : i ∈ [ d ] } so that ψ = P di =1 λ i α i ⊗ β i and λ i ≥ i ∈ [ d ]. This is known as the Schmidtdecomposition of ψ and we refer to the λ i as the Schmidt coefficients of ψ . We say that ψ has full Schmidtrank, if all its Schmidt coefficients are positive. We say that ψ is maximally entangled if all its Schmidtcoefficients are the same. Throughout this paper we use φ to denote the canonical maximally entangledstate √ d P di =1 e i ⊗ e i and we make repeated use of the fact that φ ∗ ( A ⊗ B ) φ = d Tr( AB T ) for any operators A, B ∈ C d × d .Consider a nonlocal game G = ( V, π ) with question sets
Q, R and answer sets
A, B respectively. An entangled strategy for G consists of a bipartite state ψ ∈ C d ⊗ C d , a POVM M q = ( M aq ∈ S d + : a ∈ A )for each of Alice’s questions q ∈ Q and a POVM N r = ( N br ∈ S d + : b ∈ B ) for each of Bob’s questions r ∈ R . Upon receiving questions ( q, r ) ∈ Q × R , Alice performs measurement M q on her part of ψ and Bobperforms measurement N r on his part of ψ . The probability that upon receiving questions ( q, r ) ∈ Q × R they answer ( a, b ) ∈ A × B is equal to ψ ∗ ( M aq ⊗ N br ) ψ . The entangled value of G , denoted by ω ∗ ( G ), is thesupremum expected probability with which entangled players can win the game, i.e. , ω ∗ ( G ) := sup X q ∈ Q,r ∈ R π ( q, r ) X a ∈ A,b ∈ B V ( a, b | q, r ) ψ ∗ ( M aq ⊗ N br ) ψ, (2)where the maximization ranges over all bipartite quantum states ψ ∈ C d A ⊗ C d B and POVMs ( M q : q ∈ Q )and ( N r : r ∈ R ). A strategy for G is called projective if all the measurements M q and N r are projective.We say that a nonlocal game G admits a perfect quantum strategy if ω ∗ ( G ) = 1 and moreover, there existsa bipartite state ψ ∈ C d A ⊗ C d B and POVMs ( M q : q ∈ Q ) and ( N r : r ∈ R ) that achieve this value. Graph theory.
A graph X is given by an ordered pair of sets ( V ( X ) , E ( X )), where E ( X ) is a collectionof 2-element subsets of V ( X ). The elements of V ( X ) are called the vertices of the graph and the elementsof E ( X ) its edges . For every edge e = { u, v } ∈ E ( X ) we say that u and v are adjacent and write u ∼ X v orsimply u ∼ v if the graph is clear from the context. A set of vertices S ⊆ V ( X ) is called an independent set if no two vertices in S are adjacent. The cardinality of the largest independent set is denoted by α ( X ) andis called the independence number of X . The complement of a graph X , denoted by X , has the same vertexset as X , but u ∼ v in X if and only if u = v and u v in X . A set of vertices C ⊆ V ( X ) is called a clique in X if S is an independent set in X . In this section we introduce and study synchronous nonlocal games. We first show that synchronous gamescan always be won with perfect
PME strategies (whenever a perfect strategy exists). Our main result in this5ection is Lemma 3.5 where we show that any instance of
PERFECT-PME is polynomial-time reducible to aninstance of
PERFECT-SYN . The main ingredient in this proof is the notion of a synchronous extension of anonlocal game.
Throughout this section we focus on games where Alice and Bob share the same question and answer setsand furthermore, in order to win, they need to give the same answers upon receiving the same questions.
Definition 3.1.
A nonlocal game G = ( V, π ) is called synchronous if it satisfies the following properties:(i) A = B and Q = R ;(ii) V ( a, b | q, q ) = 0 , if a = b ;(iii) for all q ∈ Q , we have π ( q, q ) > + + a, a ′ | q, q ) = 0 whenever a = a ′ .We now study perfect entangled strategies for synchronous games and show that such strategies can,without loss of generality, be assumed to have a certain form. Lemma 3.2.
Let G be a synchronous game which admits a perfect entangled strategy. Then there also existsa perfect PME strategy for G where Bob’s projectors are the transpose of Alice’s corresponding projectors.Proof. Let G be a synchronous game with answer set A and question set Q . Consider a perfect strategy for G given by a shared state ψ ∈ C d A ⊗ C d B , a POVM M q = ( M aq : a ∈ A ) for each of Alice’s questions and aPOVM N q = ( N aq : a ∈ A ) for each of Bob’s questions. Without loss of generality, we can assume that theshared state is pure and has full Schmidt rank. Let ρ aq := Tr A (cid:0) ( M aq ⊗ I ) ψψ ∗ (cid:1) denote Bob’s residual statesafter Alice has responded a ∈ A upon receiving question q ∈ Q . We first show that h ρ aq , ρ br i = 0 , whenever V ( a, b | q, r ) = 0 . (3)For this consider a question/answer pair satisfying V ( a, b | q, r ) = 0 and assume that Bob has received question r ∈ Q . For the players to win, Bob needs to answer b ∈ Q if he holds the state ρ br since the game issynchronous. On the other hand, he cannot answer b ∈ Q if he holds the state ρ aq . Since the strategy isperfect, Bob never errs and we can use his answer to perfectly discriminate the states ρ aq and ρ br . Onlyorthogonal states can be perfectly discriminated and hence we must have that h ρ aq , ρ br i = 0.The last step is to use the support of Bob’s residual states to construct a perfect PME strategy for G . Forall a ∈ A and q ∈ Q define P qa := supp( ρ aq ). By definition of ρ aq we have that P a ∈ A ρ aq = Tr A ( ψψ ∗ ) andsince ψ has full Schmidt rank it follows that supp(Tr A ( ψψ ∗ )) = supp( P di =1 λ i e i e ∗ i ) = I d . On the other hand,since G is a synchronous game, it follows from (3) that h ρ aq , ρ a ′ q i = 0 for a = a ′ and thus supp( P a ∈ A ρ aq ) = P a ∈ A supp( ρ aq ) = P a ∈ A P aq for every q ∈ Q . Summarizing we have that P a ∈ A P aq = I d for all q ∈ Q andthus we can define projective measurements P q := ( P aq : a ∈ A ) for Alice and R q := ( P T aq : a ∈ A ) for Bob.Consider the strategy where the players share the state φ = √ d P di =1 e i ⊗ e i ∈ C d ⊗ C d , Alice uses theprojective measurement P q upon receiving question q ∈ Q and Bob uses the projective measurement R q uponreceiving q ∈ Q . To see that this strategy never errs, note that the probability to answer ( a, b ) ∈ A × A uponreceiving question pair ( q, r ) ∈ Q × Q is Pr( a, b | q, r ) = φ ∗ ( P aq ⊗ P T br ) φ = d Tr( P aq P br ) . Since the supports oforthogonal states are orthogonal it follows from (3) that Pr( a, b | q, r ) = 0 whenever V ( a, b | q, r ) = 0.This result was known for graph coloring [CMN +
07] and graph homomorphism games [RM14]. Sinceboth of these game classes are synchronous nonlocal games, Lemma 3.2 subsumes both of these results.
Remark 3.3.
Notice that the perfect strategy guaranteed by Lemma 3.2 has the property that
Pr( a, b | q, r ) =Pr( b, a | r, q ) for all a, b ∈ A and q, r ∈ Q . This observation is used in Lemma 5.2. .2 Synchronous extension In this section we introduce the notion of the synchronous extension of a nonlocal game ( cf.
Definition 3.4).We also establish that
PERFECT-PME is polynomial-time reducible to
PERFECT-SYN .In order to reduce instances of
PERFECT-PME to those of
PERFECT-SYN , to any game G we associatea synchronous game ˜ G where we can also ask Alice any of Bob’s questions and vice versa. The winningcondition in ˜ G is the same as in G if both players are asked their original questions or the other player’squestions. When both players are given the same question, we require that their answers coincide, thereforeensuring that ˜ G is synchronous. For simplicity we assume that the question sets and also the answer setsof the original game G are disjoint. Note however that this is not truly a restriction since any game can beconverted into an equivalent game with disjoint question sets and disjoint answer sets, for instance by letting Q ′ = { ( q,
0) : q ∈ Q } and R ′ = { ( r,
1) : r ∈ R } , and similarly for A and B . Definition 3.4.
Let G be a nonlocal game with disjoint question sets Q, R and disjoint answer sets
A, B .The synchronous extension of G , denoted by ˜ G , is a new synchronous game with question and answer sets˜ Q = Q ∪ R & ˜ A = A ∪ B. The probability distribution ˜ π on the question set ˜ Q × ˜ Q is any distribution of full support . Lastly theverification predicate ˜ V is given by:˜ V ( a, b | q, r ) = ˜ V ( b, a | r, q ) = V ( a, b | q, r ) , for all a ∈ A, b ∈ B, q ∈ Q, r ∈ R, (4)˜ V ( a, a ′ | q, q ) = δ aa ′ and ˜ V ( b, b ′ | r, r ) = δ bb ′ , for all q ∈ Q, r ∈ R, a, a ′ ∈ A, b, b ′ ∈ B, (5)˜ V ( y, y ′ | x, x ′ ) = 0 if either ( x, y ) or ( x ′ , y ′ ) is an element of ( R × A ) ∪ ( Q × B ) , (6)and it evaluates to one in all remaining cases. Notice that condition (4) ensures that players give correctanswers upon receiving their original questions or when their roles are reversed. Furthermore, condition (5)ensures the game is synchronous and (6) ensures that only Alice’s answers are accepted for Alice’s questionsand only Bob’s answers are accepted for Bob’s questions.Generally the synchronous extension might be harder to win than the original game. However, as we willsee in the next section, any perfect PME strategy for the game G , can be also be used to win ˜ G perfectly. PERFECT-PME to PERFECT-SYN
Using the notion of the synchronous extension we are now ready to prove the main result in this section.
Lemma 3.5.
A nonlocal game G has a perfect PME strategy if and only if its synchronous extension ˜ G hasa perfect entangled strategy. In particular, PERFECT-PME is polynomial-time reducible to
PERFECT-SYN .Proof.
First, assume that G has a perfect PME strategy using a maximally entangled state φ ∈ C d ⊗ C d andprojective measurements P q = ( P aq : a ∈ A ) and R r = ( R br : b ∈ B ) for Alice and Bob respectively. Also forall q ∈ Q and r ∈ R let P T q and R T r denote the projective measurements obtained by taking the transposeof all the projectors within the projective measurements P q and R r respectively. To play the game ˜ G theplayers use the following strategy: Alice measures her part of φ using P q upon receiving question q ∈ Q andwith R T r upon receiving question r ∈ R . In the former case she responds with some a ∈ A , while in the lattershe responds with some b ∈ B , where a and b are the respective measurement outcomes. Bob acts similarly,except that he uses his original measurements R r for a question r ∈ R and P T q for a question q ∈ Q .It remains to verify that this defines a perfect strategy for ˜ G . To do so we show that the players neverreturn answers for which ˜ V evaluates to zero. First, note that by construction both players only respondwith Alice’s answers when asked Alice’s questions and similarly for Bob’s questions and answers. Thereforethey never lose due to condition (6). Next we will show that condition (4) never causes the players to lose ˜ G . We could also allow zero probabilities for questions that correspond to zero probability questions in the original game G .
7f both players are given questions from their original question sets in G , then their strategies are exactly asthey were in G , and since their strategy for G was perfect they will win in this case. If Alice is given r ∈ R and Bob is given q ∈ Q , then they will respond with some b ∈ B and a ∈ A with probability equal to φ ∗ (cid:0) R T br ⊗ P T aq (cid:1) φ = 1 d Tr (cid:0) R T br P aq (cid:1) = 1 d Tr (cid:0) P aq R T br (cid:1) = φ ∗ ( P aq ⊗ R br ) φ. This is the probability of Alice and Bob outputting a and b respectively when receiving q and r in the originalgame G . If this probability is greater than 0, then ˜ V ( b, a | r, q ) = V ( a, b | q, r ) = 1 since they win G perfectly.Therefore condition (4) never causes Alice and Bob to lose ˜ G .Lastly, for all q ∈ Q and a = a ′ ∈ A we have thatPr( a, a ′ | q, q ) = φ ∗ (cid:0) P aq ⊗ P T a ′ q (cid:1) φ = 1 d Tr( P aq P a ′ q ) = 0 , and similarly for b = b ′ ∈ B and r ∈ R . Therefore the players always give the same answer when asked thesame question and thus they never lose ˜ G due to condition (5). Since there are no other ways for the playersto lose ˜ G , we have shown that they win this game perfectly.To show the other direction let us assume ˜ G has a perfect strategy. By construction ˜ G is synchronous,hence Lemma 3.2 allows us to conclude that there exists a perfect PME strategy for ˜ G . Since ˜ G contains theoriginal game G , any perfect strategy for ˜ G can also be used to win G perfectly.In fact, the proof of Lemma 3.5 shows that any (not necessarily perfect) PME strategy for G = ( V, π ) canbe used to win ˜ G = ( ˜ V , ˜ π ) with at least as high probability of success if ˜ π | G = π . Here, we have used ˜ π | G to refer to the distribution obtained from ˜ π by restricting to questions in G and re-normalizing. In this section we introduce the notion of the game graph of a synchronous game ( cf.
Definition 4.1). Ourmain result in this section is Theorem 4.4 where we relate the existence of perfect entangled strategies for asynchronous game to the projective packing number of its game graph. This is used in Section 5.1 to reduce
PERFECT-SYN to Q-INDEPENDENCE . Lastly, in Theorem 4.6 we identify combinatorial lower bounds on theclassical and entangled values of synchronous games in terms of their game graphs.
A nonlocal game G = ( V, π ) admits a perfect entangled strategy if there exist a quantum state ψ ∈ C d A ⊗ C d B and POVM measurements ( M qa : a ∈ A ) ⊆ S d A + and ( N rb : b ∈ B ) ⊆ S d B + such that ψ ∗ ( M qa ⊗ N rb ) ψ = 0 , when V ( a, b | q, r ) = 0 and π ( q, r ) > . (7)We have already seen in Lemma 3.2 that a synchronous game has a perfect entangled strategy if andonly if it has a perfect PME strategy. This implies that for synchronous games Condition (7) reduces to a setof orthogonality relations between the measurement operators. Next, for every synchronous nonlocal gamewe associate an undirected graph which encodes these required orthogonalities as adjacencies.
Definition 4.1.
Let G be a synchronous game with question set Q and answer set A . The game graph of G , denoted X ( G ), is the undirected graph with vertex set A × Q where ( a, q ) is adjacent to ( a ′ , q ′ ) if V ( a, a ′ | q, q ′ ) = 0 or V ( a ′ , a | q ′ , q ) = 0.An important feature of game graphs is that their vertex set admits a natural partition into cliques.Specifically, for a given question q ∈ Q of a synchronous game G , the vertices of V q := { ( a, q ) : a ∈ A } arepairwise adjacent in X ( G ). This observation will be important for the proofs in this section.8 .2 Synchronous games and the projective packing number In this section we show that a synchronous game admits a perfect entangled strategy if and only if its gamegraph has a projective packing of value | Q | ( cf. Theorem 4.4).We first recall the definition of the projective packing number of a graph [RM14, Rob13].
Definition 4.2. A d -dimensional projective packing of a graph X = ( V, E ) consists of an assignment ofprojectors P u ∈ S d + to every vertex u ∈ V such thatTr( P u P v ) = 0 , whenever u ∼ X v. (8)The value of a projective packing using projectors P u ∈ S d + is defined as1 d X u ∈ V Tr( P u ) . (9)The projective packing numbe r of a graph X , denoted α p ( X ), is defined as the supremum of the values overall projective packings of the graph X .Notice that the supremum in the definition of projective packing number is necessary because it is notclear that α p ( X ) is always attained by some projective packing of the graph X . We now give an upperbound on the projective packing number of a game graph. Lemma 4.3.
For any synchronous game G with question set Q we have that α p (cid:0) X ( G ) (cid:1) ≤ | Q | . Proof.
Let ( P aq : a ∈ A, q ∈ Q ) be a d -dimensional projective packing of X ( G ). The vertices in V q = { ( a, q ) : a ∈ A } are pairwise adjacent and thus the projectors P aq are pairwise orthogonal for every q ∈ Q . Therefore, X a ∈ A Tr( P aq ) = X a ∈ A rank( P aq ) ≤ d, where rank( M ) is the rank of matrix M . From the above inequality we further obtain that1 d X ( a,q ) ∈ A × Q Tr( P aq ) = 1 d X q ∈ Q X a ∈ A Tr( P aq ) ≤ d | Q | · d = | Q | , and thus α p (cid:0) X ( G ) (cid:1) ≤ | Q | .In view of Lemma 4.3 it is natural to ask when it is the case that α p (cid:0) X ( G ) (cid:1) = | Q | . As it turns out thishappens exactly when there exists a perfect entangled strategy for G . Theorem 4.4.
Let G be a synchronous game with question set Q . Then G has a perfect entangled strategyif and only if its game graph has a projective packing of value | Q | .Proof. Let G be a synchronous game with a perfect entangled strategy. By Lemma 3.2, there exists a perfectprojective strategy for G that uses maximally entangled state φ ∈ C d ⊗ C d , where Alice’s and Bob’s projectorsare transpose to each other. Let P aq ∈ S d + be Alice’s projector associated with question q ∈ Q and answer a ∈ A . Since this strategy is perfect we have that0 = φ ∗ ( P aq ⊗ P T a ′ q ′ ) φ = 1 d Tr( P aq P a ′ q ′ ) , (10)whenever V ( a, a ′ | q, q ′ ) = 0 or V ( a ′ , a | q ′ , q ) = 0. It follows immediately from Equation (10) that the projectors P aq form a d -dimensional projective packing of X ( G ). Since P a ∈ A P aq = I d it follows that1 d X ( a,q ) ∈ A × Q Tr( P aq ) = 1 d X q ∈ Q Tr( I d ) = | Q | , (11)and thus value of this packing is | Q | . Lastly, by Lemma 4.3 we get that α p (cid:0) X ( G ) (cid:1) = | Q | .9or the other direction, assume that X ( G ) has a d -dimensional projective packing ( P aq : a ∈ A, q ∈ Q )of value | Q | . Since G is a synchronous game we have that ( q, a ) ∼ ( q, a ′ ) for a = a ′ ∈ A and q ∈ Q . Thisimplies that P a ∈ A P aq (cid:22) I d , as the added projectors are mutually orthogonal. Furthermore, since the valueof the projective packing is | Q | , we obtain | Q | = 1 d X ( a,q ) ∈ A × Q Tr( P aq ) = X q ∈ Q (cid:16) d Tr (cid:0) X a ∈ A P aq (cid:1)(cid:17) ≤ X q ∈ Q d Tr( I d ) ≤ | Q | , (12)and thus Equation (12) holds throughout with equality. In particular, Tr (cid:0) P a ∈ A P aq (cid:1) = Tr( I d ), and since P a ∈ A P aq (cid:22) I d we conclude that P a ∈ A P aq = I d and thus P q = ( P aq : a ∈ A ) forms a valid projectivemeasurement. By the definition of the edge set of X ( G ), we see that Alice and Bob can win with probabilityone, if they measure a maximally entangled state using projective measurements P q and P T q respectively. ω ( G ) and ω ∗ ( G ) for synchronous games In this section we derive combinatorial lower bounds on the classical and entangled values of synchronousnonlocal games in terms of the independence number and the projective packing number of their game graphsrespectively ( cf.
Theorem 4.6).Our first result gives a necessary and sufficient condition for the existence of a perfect classical strategy.
Lemma 4.5.
Let G be a synchronous game with question set Q and let X := X ( G ) be the its game graph.Then, G has a perfect classical strategy if and only if α ( X ) = | Q | . Proof.
Let f A , f B : Q → A be a perfect deterministic strategy for the game G . Since G is synchronous wehave that f A = f B =: f . Set V q = { ( a, q ) : a ∈ A } and notice that { V q : q ∈ Q } forms a clique cover of X of cardinality | Q | . This shows that α ( X ) ≤ | Q | . Lastly, we show that S = { ( q, f ( q )) : q ∈ Q } is anindependent set in X . Indeed, since f is a perfect strategy, for any ( q, f ( q )) , ( r, f ( r )) ∈ S we have that V ( f ( q ) , f ( r ) | q, r ) = V ( f ( r ) , f ( q ) | r, q ) = 1. This implies that ( q, f ( q )) ( r, f ( r )).Conversely, let S be an independent set in X of cardinality | Q | . Since { V q : q ∈ Q } is a clique cover ofcardinality | Q | , for every q ∈ Q , the intersection S ∩ V q contains exactly one vertex of X which we denoteby ( q, a q ). Define f : Q → A where f ( q ) = a q for every q ∈ Q and consider the deterministic strategy for G where both players determine their answers using f . It remains to show that this is a perfect classicalstrategy. Assume for contradiction that there exist q, r ∈ Q such that V ( f ( q ) , f ( r ) | q, r ) = 0. By definitionof X this implies that ( f ( q ) , q ) ∼ ( f ( r ) , r ), contradicting the fact that S is an independent set in X .As an immediate consequence of Lemma 4.5 we recover the well-known fact that there exist a graphhomomorphism from a graph X to a graph Y if and only if α ( X ⋉ Y ) = | V ( X ) | . Here X ⋉ Y denotes the homomorphic product of X and Y whose vertex set is given by V ( X ) × V ( Y ) and ( x, y ) ∼ ( x ′ , y ′ ) if and onlyif [( x = x ′ ) and y = y ′ ] or [ x ∼ x ′ and y y ′ ]. To recover this result from Lemma 4.5 notice that the gamegraph for the ( X, Y )-homomorphism game is given precisely by X ⋉ Y (see also [RM14]).We now proceed to lower bound the classical and entangled values of synchronous games. Theorem 4.6.
Consider a synchronous game G with question set Q and uniform distribution of questions.If X = X ( G ) is the game graph of G then, ω ( G ) ≥ (cid:0) α ( X ) / | Q | (cid:1) and ω ∗ ( G ) ≥ (cid:0) α p ( X ) / | Q | (cid:1) . (13) Proof.
First, we consider the classical case. Our goal is to exhibit a deterministic strategy that wins on atleast α ( X ) out of the | Q | pairs of possible questions. Let S be an independent set in X of cardinality α ( X ). By definition of the edge set of X , for any pair ( a, q ) , ( b, r ) ∈ S we have that V ( a, b | q, r ) = 1 and V ( b, a | r, q ) = 1 . (14)Set Q ′ = { q ∈ Q : ∃ a ∈ A such that ( a, q ) ∈ S } . Since G is synchronous and S is an independent set, forevery q ∈ Q ′ there exists a unique a ∈ A such that ( a, q ) ∈ S , which we denote by f ( q ). Furthermore, noticethat | Q ′ | = α ( X ). Consider the following deterministic strategy: If a player receives as question an element10 ∈ Q ′ he responds with f ( q ) and if q Q ′ his answer is arbitrary. It follows from (14) that for q, r ∈ Q ′ ,the players win when asked ( q, r ) and ( r, q ). Since | Q ′ | = α ( X ) this strategy is correct on at least α ( X ) ofthe | Q | possible questions.Next we consider the entangled case. Let ( P aq : a ∈ A, q ∈ Q ) be a d -dimensional projective packingfor X of value γ , i.e. , γ = d P a ∈ A,q ∈ Q Tr( P aq ) . We construct an entangled strategy whose value is at least γ / | Q | . Recall that for all q ∈ Q the set V q = { ( a, q ) : a ∈ A } forms a clique in X . This implies that for fixed q ∈ Q and a = a ′ ∈ A the projectors P aq and P a ′ q are pairwise orthogonal and thus P a ∈ A P aq (cid:22) I d . Considerthe following entangled strategy for G : The players share the maximally entangled state φ ∈ C d ⊗ C d andfor every q ∈ Q Alice uses the projective measurement ( P aq : a ∈ A ) ∪ ( I − P a ∈ A P aq ) and Bob uses themeasurement ( P T aq : a ∈ A ) ∪ ( I − P a ∈ A P T aq ). Using this strategy the players win with probability at least1 | Q | X a,b,q,r φ ∗ ( P aq ⊗ P T br ) φ V ( a, b | q, r ) = 1 d | Q | X a,b,q,r Tr( P aq P br ) V ( a, b | q, r ) . (15)If V ( a, b | q, r ) = 0 then ( a, q ) ∼ ( b, r ) in the game graph and by definition of the projective packing we havethat Tr( P aq P br ) = 0. Thus (15) gives that ω ∗ ( G ) ≥ d | Q | X a,b,q,r Tr( P aq P br ) = 1 d | Q | Tr( P ) , (16)where P = P a ∈ A,q ∈ Q P aq . By the Cauchy-Schwartz inequality we get that Tr( P ) ≥ Tr( P ) /d . Finally,since γ = Tr( P ) /d, it follows from (16) that ω ∗ ( G ) ≥ γ / | Q | and the proof is completed.If a synchronous game G satisfies α p ( X ( G )) = | Q | , it follows from Theorem 4.6 that ω ∗ ( G ) = 1. Onthe other hand we have seen in Theorem 4.4 that if there exists a projective packing for the game graphwith value equal to | Q | then G has a perfect quantum strategy. Notice that these two conditions are notequivalent since we could have α p ( X ( G )) = | Q | without this value being attained. In this section we show that
PERFECT-SYN is polynomial-time reducible to
Q-INDEPENDENCE ( cf. Lemma 5.2).This fact combined with the reduction of
PERFECT-PME to PERFECT-SYN derived in Lemma 3.5 implies that
PERFECT-PME is polynomial-time reducible to
Q-INDEPENDENCE , which is the main result in this paper. Addi-tionally we consider the attainability problem for perfect strategies and synchronous games. In Theorem 5.5we show that if any independent set game whose entangled value is one also admits a perfect strategy thenthe same is true for all symmetric synchronous games.
PERFECT-PME to Q-INDEPENDENCE
Recall that in the (
X, t )-independent set game the players try to convince a verifier that the graph X containsan independent set of size t . The verifier selects uniformly at random ( i, j ) ∈ [ t ] × [ t ] and sends i to Alice and j to Bob. The players respond with vertices u, v ∈ V ( X ) respectively. The verification predicate evaluatesto zero in the following three cases: [ i = j and u = v ] or [ i = j and u = v ] or [ i = j and u ∼ X v ].The independence number of a graph X can equivalently be defined as the largest integer t ≥ X, t )-independent set game admits a perfect classical strategy. Similarly, the quantum independencenumber of a graph X , denoted by α q ( X ) is defined as the largest integer t ≥ X, t )-independentset game admits a perfect entangled strategy [RM14].It is known that the projective packing number is an upper bound to the quantum independence number.
Lemma 5.1. [Rob13, 6.11.1] Let X be a graph and k ∈ N . If α q ( X ) ≥ k then there exists a projectivepacking of X with value k . In particular, α q ( X ) ≤ α p ( X ) . We are now ready to prove the main result in this section.
Lemma 5.2.
Let G be a synchronous game with question set Q . Then G has a perfect entangled strategy ifand only if α q ( X ( G )) = | Q | . In particular,
PERFECT-SYN is polynomial-time reducible to
Q-INDEPENDENCE . roof. Assume first there exists a perfect entangled strategy for G . By Lemma 3.2 there also exists a perfect PME strategy S for G where Bob’s projectors are the transpose of Alice’s corresponding projectors. For all a, a ′ ∈ A and q, q ′ ∈ Q let Pr( a, a ′ | q, q ′ ) be the probability that Alice and Bob answer a and b respectivelyupon receiving questions q and q ′ when employing strategy S . We now construct a perfect strategy forthe ( X ( G ) , | Q | )-independent set game. Without loss of generality, we may assume that Q itself is usedas the question set. Consider the following strategy: upon receiving q and q ′ respectively, Alice and Bobuse strategy S to obtain answers a and a ′ . They then output vertices ( a, q ) and ( a ′ , q ′ ) respectively. IfPr( a, a ′ | q, q ′ ) > a ′ , a | q ′ , q ) >
0. From this we see that both V ( a, a ′ | q, q ′ ) = 1 and V ( a ′ , a | q ′ , q ) = 1, since S was perfect. This implies that ( a, q ) and ( a ′ , q ′ ) are (possiblyequal) nonadjacent vertices in X ( G ). If q = q ′ , then these two vertices are not equal and are thereforedistinct nonadjacent vertices of X ( G ), as required by the independent set game. If q = q ′ , then since G is a synchronous game and S is perfect, we have that a = a ′ and therefore the two outputted vertices areequal as required. This shows that using this strategy allows Alice and Bob to win the independent set gameperfectly and thus α q ( X ( G )) ≥ | Q | . On the other hand from Lemma 5.1 together with Lemma 4.3 it followsthat α q ( X ( G )) ≤ | Q | and thus α q ( X ( G )) = | Q | .Conversely, assume that α q ( X ( G )) = | Q | . By Lemma 5.1 and Lemma 4.3 there exists a projective packingof X ( G ) of value | Q | , and therefore by Theorem 4.4 there exists a perfect entangled strategy for G .Lastly, combining Lemma 3.5 and Lemma 5.2 directly yields the main result of this paper. Theorem 5.3.
A nonlocal game G with question sets Q and R admits a perfect PME strategy if and only if α q (cid:0) X ( ˜ G ) (cid:1) = | Q | + | R | . (17) In particular,
PERFECT-PME is polynomial-time reducible to
Q-INDEPENDENCE . In this section we focus on the the attainability problem for perfect strategies and show that the attainabilityquestion for symmetric synchronous games reduces to the attainability question for independent set games.
Definition 5.4.
A synchronous game G = ( V, π ) is called symmetric if V ( a, a ′ | q, q ′ ) = V ( a ′ , a | q ′ , q ) , for all a, a ′ ∈ A, q, q ′ ∈ Q. Notice that all synchronous games we consider in this work ( e.g. , homomorphism and coloring games)are symmetric.
Theorem 5.5.
Suppose that any independent set game G satisfying ω ∗ ( G ) = 1 admits a perfect entangledstrategy. Then the same holds for all symmetric synchronous nonlocal games.Proof. Let G = ( V, π ) be any symmetric synchronous game with question set Q and answer set A . Assumethat ω ∗ ( G ) = 1 and let X := X ( G ) be its game graph. Define G ′ = ( V ′ , π ) to be the ( X, | Q | )-independentset game with π as the distribution of questions. The crux of the proof is that from any strategy S thatsucceeds in G with probability at least 1 − ε , we can construct a strategy S ′ that wins G ′ with probabilityat least 1 − ε . Similarly to Lemma 5.2, using the strategy S for G we define the following strategy S ′ for G ′ : Upon receiving q ∈ Q , Alice uses strategy S for G and obtains an answer a ∈ A . She then replies withvertex ( q, a ) of X . Similarly, Bob, upon receiving q ′ ∈ Q he uses strategy S for G to obtain answer a ′ ∈ A .He then replies with vertex ( q ′ , a ′ ) of X . Let Pr S ( a, a ′ | q, q ′ ) denote the probability that using strategy S theplayers respond with ( a, a ′ ) ∈ A × A upon receiving questions q, q ′ ∈ Q respectively. By assumption we havethat ω ∗ ( G, S ) := X q ∈ Q,a ∈ A π ( q, q ) Pr S ( a, a | q, q )+ X q = q ′ ∈ Q π ( q, q ′ ) X a,a ′ ∈ A : V ( a,a ′ | q,q ′ )=1 Pr S ( a, a ′ | q, q ′ ) ≥ − ε. (18)Furthermore, by definition of the strategy S ′ we have thatPr S ( a, a ′ | q, q ′ ) = Pr S ′ (cid:0) ( q, a ) , ( q ′ , a ′ ) | q, q ′ (cid:1) , for all q, q ′ ∈ Q. (19)12ince G ′ is an independent set game we have V ′ (cid:0) ( q, a ) , ( q ′ , a ′ ) | q, q ′ (cid:1) = 1 if and only if[ q = q ′ and a = a ′ ] or [ q = q ′ and ( a, q ) X ( a ′ , q ′ )] . (20)Since the game G is symmetric, Condition (20) is equivalent to[ q = q ′ and a = a ′ ] or [ q = q ′ and V ( a, a ′ | q, q ′ ) = 1] . (21)Combining (19) with (21), and the fact that V ( a, a ′ | q, q ) = 1 ⇒ a = a ′ , it follows that the probability ofwinning the game G ′ using strategy S ′ is at least ω ∗ ( G, S ) ≥ − ε . Since ω ∗ ( G ) = 1 this argument can berepeated for any ε arbitrarily close to 0 which implies that the entangled value of G ′ is equal to one. Bythe assumption of the theorem, this implies that there is a perfect quantum strategy for G ′ and thus byLemma 5.2 there exists a perfect quantum strategy for G . Acknowledgements.
D. E. Roberson and A. Varvitsiotis are supported in part by the Singapore NationalResearch Foundation under NRF RF Award No. NRF-NRFF2013-13. L. Manˇcinska is supported by theSingapore Ministry of Education under the Tier 3 grant MOE2012-T3-1-009.
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