The communication complexity of XOR games via summing operators
aa r X i v : . [ c s . CC ] A p r THE COMMUNICATION COMPLEXITY OF XOR GAMES VIASUMMING OPERATORS
CARLOS PALAZUELOS, DAVID P´EREZ-GARC´IA, AND IGNACIO VILLANUEVA
Abstract.
The discrepancy method is widely used to find lower bounds for communi-cation complexity of XOR games. It is well known that these bounds can be far fromoptimal. In this context Disjointness is usually mentioned as a case where the method failsto give good bounds, because the increment of the value of the game is linear (rather thanexponential) in the number of communicated bits. We show in this paper the existence ofXOR games where the discrepancy method yields bounds as poor as one desires. Indeed,we show the existence of such games with any previously prescribed value. Specificallywe prove the following:For any number of bits c and every 0 < δ < ǫ >
0, we show theexistence of a XOR game such that its value, both without communication or with theuse of c bits of communication, is contained in the interval ( δ − ǫ, δ + ǫ ).To prove this result we apply the theory of p -summing operators, a central topic inBanach space theory. We show in the paper other applications of this theory to the studyof the communication complexity of XOR games. Introduction
A XOR game G = ( f, π ) with N inputs on each side is defined by a function f : [ N ] × [ N ] : −→ {− , } together with a probability distribution π : [ N ] × [ N ] : −→ [0 , x, y ∈ [ N ] respectively and each of them must answer a number a, b ∈ {− , } , sothat f ( x, y ) = a · b . They can also be viewed as linear combinations of the correlationsachieved by the two parties when they are asked questions x, y .XOR games are a very natural model for the study of communication complexity incomputation as in Yao’s model ([9]). They have also been used for the study of complexityclasses ([21]), hardness of approximation ([8]), or for a better understanding of parallelrepetition results, both in the classical and the quantum contexts ([12], [6], [18], [2])In the context of quantum information, XOR games appear often with the name ofcorrelation Bell inequalities. The so-called CHSH inequality has an extrordinary relevance in this context [22]. They also provide an excellent testbed to study the relation betweenclassical computation, quantum computation and communication complexity ([19], [1]).The discrepancy method is one of the techniques most widely used to find lower boundsfor the communication complexity of a XOR game. This method is known to give poorbounds in certain cases. We show in Theorem 1.1 that these bounds can be as poor as onewants.We introduce the theory of p -summing operators with few vectors to study the commu-nication complexity of XOR games. This leads us to the use of classical tools in the localtheory of Banach spaces, like Grothendieck inequality, Chevet inequality, p -stable measuresand the concentration of measure phenomenon. We must mention here the papers [16],[15] where techniques related to ours have also been used.We say that a joint strategy between Alice and Bob γ is c -simulable if Alice and Bobcan simulate γ using c -bits of communication. We denote these strategies by S c . We saythat γ is c -simulable from Alice to Bob if they can simulate it when Alice sends c -bits ofone way communication to Bob. We denote these strategies by S c For a game G we define its values ω ( G ) , ω c ( G ) , ω c ( G ) as the maximum value that itattains on the strategies in L , S c , S c respectively.The discrepancy method, as stated in [9, Proposition 3.28] tells us that, for every game G , ω c ( G ) ≤ c ω ( G ) . Disjointness function is usually shown as an example where thediscrepancy method fails to give good lower bounds for the communication complexity,since the increment ω c ( G ) ω ( G ) is only linear in c .In our main result, we show the existence of XOR games for which the performance ofthe discrepancy method is as poor as one desires. Specifically, for any prescribed 0 < δ < c we prove the existence of a XOR game G such that both ω ( G )and ω c ( G ) are as close to δ (and hence also to each other) as we want. Theorem 1.1.
For every real number < δ < , for every c ∈ N and for every ǫ > thereexists a natural number N and a XOR game G with N inputs per player such that: ω ( G ) ∼ ǫ δ ∼ ǫ ω c ( G ) , where we use the notation a ∼ ǫ b to denote b − ǫ ≤ a ≤ b + ǫ for every a, b, ǫ > . Next, we study how sharp the bound given by the discrepancy method is, taking also intoaccount the number of inputs N . We do a full study for the case of one-way communication. HE COMMUNICATION COMPLEXITY OF XOR GAMES VIA SUMMING OPERATORS 3
Our techniques can also be applied to more general cases. We use the notation ≃ to denoteequality up to universal constants (independent of N and c ). Theorem 1.2.
For every XOR game G with N inputs per player, we have: a) ω c ( G ) ≤ K G c ω ( G ) and b) ω c ( G ) ≥ c K G √ N .These inequalities are tight in the sense that there exist games J, H with N inputsper player such that such that (1) ω c ( J ) ≃ c ω ( J ) and (2) ω c ( H ) ≃ c √ N .
Actually, in Proposition 4.1 below, we prove that, for big values of N ,“most” gamesverify conditions (1) and (2).The structure of the paper is the following: In Section 2 we introduce the notation andthe formalism we will use. Next we introduce the mathematical tools that we need ( p -summing operators with few vectors, Grothendicek inequality and Chevet inequality) andfinally we state and prove the link between communication complexity and the theory of p -summing operators with few vectors.Section 3 is devoted to the proof of Theorem 1.1, and Theorem 1.2 is proved in Section4. 2. Notation and mathematical tools
Notation.
We will need the following notations and results from tensor product the-ory. Given a normed space X , we write X ∗ for its dual space with its natural dual normand B X for its closed unit ball. Given two normed spaces X, Y , and an element u ∈ X ⊗ Y ,we define its projective norm k u k π as k u k π = inf { X i k x i kk y i k , where u = X i x i ⊗ y i } We write X ⊗ π Y for the tensor product of X and Y endowed with the projective norm.We can also define the injective norm of u as k u k ǫ = sup { X i x ∗ ( x i ) y ∗ ( y i ) , where u = X i x i ⊗ y i , x ∗ ∈ B X ∗ , y ∗ ∈ B Y ∗ } and we write X ⊗ ǫ Y for their tensor product endowed with the injective norm. C. PALAZUELOS, D. P´EREZ-GARC´IA, AND I. VILLANUEVA
In this note, X and Y will always be finite dimensional. It is well known (and not hardto see) that in that case ( X ⊗ π Y ) ∗ = X ∗ ⊗ ǫ Y ∗ and ( X ⊗ ǫ Y ) ∗ = X ∗ ⊗ π Y ∗ .In this paper we will always see games ( G x,y ) Nx,y =1 = G as elements in ( ℓ N ∞ ⊗ ℓ N ∞ ) ∗ , thealgebraic dual of ℓ N ∞ ⊗ ℓ N ∞ . We view the correlations attained by the players (or strategies)as elements ( γ x,y ) Nx,y =1 = γ in ℓ N ∞ ⊗ ℓ N ∞ . The value of the game G when the players playthe strategy γ is h G, γ i = N X x,y =1 G x,y γ x,y . For an element G ∈ ( ℓ N ∞ ⊗ ℓ N ∞ ) ∗ , we write k G k op for its norm as an element of ( ℓ N ∞ ⊗ π ℓ N ∞ ) ∗ ,which coincides with its operator norm when we identify G with the operator ˜ G : ℓ N ∞ −→ ℓ N = ( ℓ N ∞ ) ∗ defined by ˜ G ( x )( y ) = G ( x, y ). The way they are defined, XOR games arenormalized in the sense that their norm as elements of ( ℓ N ∞ ⊗ ǫ ℓ N ∞ ) ∗ is always one.When they do not communicate, Alice strategy upon receiving input x can be describedas an element α ( x, λ ) in ℓ N ∞ , where λ stands for the state of their shared randomness.Similarly, Bob’s strategy is β ( y, λ ) so that their joint strategy is an element ( γ x,y ) Nx,y =1 = γ = P i λ i α i ⊗ β i ∈ ℓ N ∞ ⊗ ℓ N ∞ such that k γ k π ≤
1. We will call these strategies local strategiesand denote it by L .2.2. Summing operators.
Following Grothendieck’s work [7], the so called local theoryof Banach spaces has been one of the cornerstones of modern functional analysis. Many ofthe main results in this theory can be expressed in terms of summing operators . We statenext the definitions and results that we use in this paper. A detailed exposition in thisarea can be read, for instance, in [5].Given a finite sequence with arbitrary length ( x i ) ni =1 in a normed space X , and a realnumber 1 ≤ p < ∞ , we define the weakly p-summing norm of ( x i ) ni =1 by k ( x i ) ni =1 k wp = sup { n X i =1 | x ∗ ( x i ) | p ! p , where x ∗ ∈ B X ∗ } . Now, given an operator T : X −→ Y between normed spaces, we define its p -summing norm as π p ( T ) = inf { C such that n X i =1 k T ( x i ) k p ! p ≤ C k ( x i ) ni =1 k wp } for every sequence ( x i ) ni =1 ⊂ X . It is well known that, for an operator T : ℓ N ∞ −→ Y , π ( T ) = P Ni =1 k T ( e i ) k . HE COMMUNICATION COMPLEXITY OF XOR GAMES VIA SUMMING OPERATORS 5
If we fix r ∈ N and restrict the previous definition to sequences ( x i ) ri =1 of maximumlength r we obtain the definition of the p -summing with r vectors norm of T , which wedenote by π rp ( T ). Summing operators with few vectors have been studied by several authors,see for instance [20] and the references therein.We will also use the following consequence of Grothendieck’s inequality. Theorem 2.1.
There exists an universal constant K G such that, for any natural numbers N, M and every operator u : ℓ N ∞ −→ ℓ M , π ( u ) ≤ K G k u k . Chevet inequality.
The following result is known as Chevet inequality. It is usuallystated for Gaussian random variables, we state it for Bernouilli random variables. See [14].
Theorem 2.2.
Given two normed spaces
E, F , there exists a universal constant b suchthat E k X x,y r x,y ϕ x ⊗ φ y k E ⊗ ǫ F ≤ b ( k ( ϕ x ) x k w E k X y r y φ y k F + k ( φ y ) y k w E k X x r x ϕ x k E ) , where r x , r y , r x,y are independent Bernouilli random variables, and ( ϕ x ) x , ( φ y ) y are finitesequences in E, F . Actually, we can take b = p π (see [3] ). We will apply Chevet inequality in the case E = F = ℓ N , and ϕ x = e x , φ y = e y ,1 ≤ x, y ≤ N . In that case, we get easily k ( e x ) Nx =1 k w = √ N and E k N X y =1 r y e y k ℓ N = N Commnunication complexity and p -summing operators. In this paper, we ap-proach the study of the different types of strategies and the corresponding value of thegames through tensor norms in ℓ N ∞ ⊗ ℓ N ∞ and its dual space. We have already mentionedthat the local strategies can be identified with the norm unit ball of ℓ N ∞ ⊗ π ℓ N ∞ . It is easy tosee that both S c and S c are symmetric convex bodies of ℓ N ∞ ⊗ ℓ N ∞ with non empty interior.Hence, they define norms on this space, and therefore the value of a game G on them canbe seen as the corresponding dual norm of the game.Lemma 2.4 below is the starting point of our approach: It identifies the value of a gameon the strategies in S c as certain operator norm. We isolate the technical parts of the proofin the following lemma. Its proof follows from [5, Proposition 2.2 and Lemma 16.13]. C. PALAZUELOS, D. P´EREZ-GARC´IA, AND I. VILLANUEVA
Lemma 2.3.
Let B = { ( α i ) ri =1 ⊂ ℓ N ∞ such that k ( α i ) ri =1 k w ≤ } . Given A ⊂ [ N ] , let α A ∈ ℓ N ∞ be the element defined by α A ( x ) = 1 if i ∈ A and α A ( x ) = 0 otherwise. Let A = { ( α A i ) ri =1 ; where A , . . . , A r is a partition of [ N ] } . Then B is the symmetric convexhull of the elements in A . Lemma 2.4.
Given a game ( G x,y ) Nx,y =1 , ω ( G ) = k ˜ G k op and ω c ( G ) = π c ( ˜ G ) .Proof. The first statement follows immediately by duality from the characterization of thelocal strategies.We prove the second statement. Let us first see that ω c ( G ) ≤ π c ( ˜ G ). We assume thatthe communication Alice sends might be dependent on a variable λ ∈ Λ. We call T ( x, λ )to the word that Alice sends when she receives the input x and the random variable takesthe value λ . We have that 1 ≤ T ( x, λ ) ≤ c . For fixed 1 ≤ i ≤ c and 1 ≤ x ≤ N , callΛ i,x = { λ ∈ Λ such that T ( x, λ ) = i } . For fixed i, λ , call X i,λ = { x such that T ( x, λ ) = i } . Calling α, β to the strategies followed by Alice and Bob, we have ω c ( G ) = sup X x,y G xy Z Λ α ( x, λ ) β ( y, λ, T ( x, λ )) dλ == sup X x,y G xy X i Z Λ i,x α ( x, λ ) β ( y, λ, T ( x, λ )) dλ = X x,y G xy X i Z Λ i,x α ( x, λ ) β ( y, λ, i ) dλ == X x,y G xy X i Z Λ i,x α ( x, λ ) β i ( y, λ ) dλ = Z Λ X i X x ∈ X i,λ X y G xy α ( x, λ ) β i ( y, λ ) dλ. For fixed λ , P i P x ∈ X i,λ P y G xy α ( x, λ ) β i ( y, λ ) is bounded above by π c ( ˜ G ) (use Lemma2.3 for this). Considering the convex hull will not change this fact.The reverse inequality follows easily from Lemma 2.3 and convexity. (cid:3) We mentioned in the Section 2 that all XOR games with N inputs per player G havenorm one considered as elements of ( ℓ N ∞ ⊗ ǫ ℓ N ∞ ) ∗ . It is well known ([5]) that this is equivalentto the condition π ( ˜ G ) = π N ( ˜ G ) = 1. In particular, if c ≥ log N , then ω c ( G ) = ω c ( G ) = 1.3. Proof of Theorem 1.1
Theorem 1.1 follows from Theorem 3.1 and Proposition 3.3 below.
HE COMMUNICATION COMPLEXITY OF XOR GAMES VIA SUMMING OPERATORS 7
Theorem 3.1.
For any real number α > , positive integer t and ǫ > , there exists anatural number N and an operator T : ℓ N ∞ −→ ℓ N such that k T k op ∼ ǫ π t ( T ) ∼ ǫ π ( T ) ∼ ǫ α The game G that we look for in Theorem 1.1 is nothing but the game whose associatedoperator is ˜ G = Tπ ( T ) , with the proper choices of ǫ , t and α .The key point of the proof of Theorem 3.1 is Levi’s embedding theorem, which says that,for every 1 < p <
2, we have an isometric embedding of ℓ p into L [0 , p -stable measures. We are interested in the (1 + ǫ )-isomorphic finite dimensionalversion of the theorem. Specifically, we use the following improvement of Levi’s embeddingtheorem due to Johnson and Schechtman. Theorem 3.2 (Theorem 1, [10]) . Let ǫ > , and suppose that < r < s < with r ≤ .Then there exists β = β ( ǫ, r, s ) > so that if m and n are positive integers with m ≤ βn ,then ℓ ms is (1 + ǫ ) -isomorphic to a subspace of ℓ nr . Note that, in the particular case of r = 1 and 1 < p <
2, Theorem 3.2 assures theexistence of β = β ( τ, , p ) > A : ℓ mp ֒ → ℓ n such that(1 − τ ) k x k ℓ mp ≤ k Ax k ℓ n ≤ (1 + τ ) k x k ℓ mp for every x ∈ ℓ mp . Proof of Theorem 3.1.
Let α, t and ǫ be as in the statement. We define: θ = log( α ), m = min { m ∈ N : t θ m < ǫ } , k = 2 m and q = m θ . Note that we can assume that 2 < q < ∞ . Indeed, if it is not, we only have to considera high enough m . Then, we define p by p + q = 1 (so 1 < p < q = p ′ . Note that, t p ′ < ǫ and k p ′ = (2 m ) θ m = 2 θ = α .We begin by considering the operator S := k − p id : ℓ k ∞ → ℓ kp . It is not difficult to check that k S k = π p ( S ) = 1 and π ( S ) = k p ′ = α (see for instance [3]). C. PALAZUELOS, D. P´EREZ-GARC´IA, AND I. VILLANUEVA
Now, we define the operator T := A ◦ S ◦ P : ℓ N ∞ → ℓ k ∞ → ℓ kp ֒ → ℓ N , where N = kβ for the β = β ( ǫ, , p ) given by Theorem 3.2, A is the associated 1 + ǫ -isomorphism given by the same theorem and P : ℓ N ∞ → ℓ k ∞ denotes the standard projection.Now, by Theorem 3.2 and the injectivity property of the p -summing operators (see forinstance [3]), we know that k T k ∼ ǫ π p ( T ) ∼ ǫ π ( T ) ∼ ǫ α . We finish the proof ifwe show that π t ( T ) ∼ ǫ x , · · · , x t ∈ ℓ N ∞ such thatsup { t X i =1 | x ∗ ( x i ) | : x ∗ ∈ B ℓ N } ≤ . Then, t X i =1 k T ( x i ) k ≤ t p ′ ( t X i =1 k T ( x i ) k p ) p ≤ (1 + ǫ ) . A suitable adjust of the ǫ ’s finishes the proof. (cid:3) This result yields immediately a “one-way communication” version of Theorem 1.1. Forthe general version, we need the following simple result.
Proposition 3.3.
Let G be a XOR game and let c be a natural number. Then ω c ( G ) ≤ c ω c ( G ) . Proof.
Applying convexity, we know that there exists a partition R , . . . , R c of [ N ] × [ N ]in rectangles and sign vectors ( α i ( x )) Nx =1 , ( β i ( y )) Ny =1 , with α i ( x ) = ± β i ( y ) for every x, y, i such that ω c ( G ) = c X i =1 X x,y ∈ R i α i ( x ) β i ( y ) M x,y . For every fixed 1 ≤ x ≤ N we define i ( x, y ) as the unique i such that ( x, y ) ∈ R i and weconsider the row of signs ( α i ( x, ( x ) β i ( x, (1) , . . . , α i ( x,N ) ( x ) β i ( x,N ) ( N )). It is easy to seethat there are at most 2 c different such rows. Clearly, 2 c bits suffice Alice to tell Bobwhich is the row associated to x . (cid:3) Proof of Theorem 1.2
Proof. a) Let G be a XOR game with N inputs per player, and let ˜ G : ℓ N ∞ → ℓ N be itsassociated operator, as in the introduction. Grothendieck’s Theorem tells us that π ( ˜ G ) ≤ HE COMMUNICATION COMPLEXITY OF XOR GAMES VIA SUMMING OPERATORS 9 K G k ˜ G k op . Now, let x , · · · , x c ∈ ℓ N ∞ be a finite sequence such that k ( x i ) c i =1 k w ≤
1. Then, ω c ( G ) = π c ( ˜ G ) ≤ c X i =1 k ˜ G ( x i ) k ≤ c ( c X i =1 k ˜ G ( x i ) k ) ≤ c K G k ˜ G k op = 2 c K G ω ( G ) . Let us see the optimality. Recall that we view games as elements in ( ℓ N ∞ ⊗ ℓ N ∞ ) ∗ = ℓ N ⊗ ℓ N .We apply Chevet inequality, to find a choice of signs ( ε x,y ) Nx,y =1 such that k c X x,y =1 ε x,y e x ⊗ e y k ℓ c ⊗ ǫ ℓ c ≤ k c X x,y =1 ε x,y e x ⊗ e y k ℓ c ⊗ π ℓ c (cid:23) √ c , where (cid:23) denotes inequality up to an universal constant. This defines an operator T : ℓ c ∞ → ℓ c such that k T k op ≤ π ( T ) = π c ( T ) (cid:23) √ c . We define T ′ = Tπ ( T ) . Letnow P : ℓ N ∞ −→ ℓ c ∞ be the canonical projection onto the first 2 c coordinates, and let ϕ : ℓ c −→ ℓ N be the canonical inclusion into the first 2 c coordinates. Then the game J defined by ˜ J : ϕ ◦ T ′ ◦ P : ℓ N ∞ → ℓ c ∞ → ℓ c → ℓ N verifies what we wanted.b)Let G be as in the hypothesiss. First we assume that N c = h ∈ N . Call A j to theisometric copy of ℓ N c ∞ contained naturally in ℓ N ∞ considering only the basis elements e i , with( j − N c < i ≤ j N c . Then1 = π ( ˜ G ) = N X i =1 k ˜ G ( e i ) k = c X j =1 N c X i =1 k ˜ G ( e ( j − N c + i ) k ≤≤ c X j =1 K G r N c k ˜ G | Ai k op ≤ K G r N c π c ( ˜ G ) = K G r N c ω c ( G ) . Now we consider the case when N c is not an integer, and we denote p the smallestnatural number such that N c ≤ p . Then again we have π ( G ) = π N ( G ) = π p c ( G ) ≤√ pK G π c ( G ) ≤ K G q N c π c ( G ) and the result follows.We see now the optimality of this result. Apply again Chevet inequality to find achoice of signs ( ε x,y ) Nx,y =1 such that k P Nx,y =1 ε x,y e x ⊗ e y k ℓ N ⊗ ǫ ℓ N ≤ k P Nx,y =1 ε x,y e x ⊗ e y k ℓ N ⊗ π ℓ N (cid:23) √ N . Let G ′ : ℓ N ∞ −→ ℓ N be its associated operator and let G be the gameassociated to G ′ π ( G ′ ) . By a), we know that ω c ( G ) (cid:22) c √ N . (cid:3) Actually, we can see that, for big values of N ,“most” games essentially attain the boundsgiven above. We write the statement for the case of games G = ( f, u ) with u the uniform distribution. Similar results can be proved for other distributions. The tool now is theConcentration of Measure Phenomenon. Proposition 4.1.
Let X N be the set of games with N inputs per player defined by G =( f, u ) , with u the uniform distribution. Consider in X N the probability µ : P ( X N ) −→ [0 , defined by µ ( A ) = Card ( A )2 N . Let r > . If m is a median of ω ( G ) under µ , then µ ( { G such that | ω ( G ) − m | ≥ r } ) ≤ e − N r . Proof.
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Quant. Inf. Comp. , (3) (2001). E-mail address : [email protected] Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
E-mail address : [email protected] Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad Complutensede Madrid, Madrid 28040, Spain
E-mail address : [email protected]@mat.ucm.es