The Hilbertian Tensor Norm and Entangled Two-Prover Games

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.


Introduction and Motivation
Entanglement is one of the central and most fascinating properties of quantum mechanics. The strange consequences of entangled quantum states have already puzzled Einstein, Podolsky, and Rosen [EPR35], in their seminal paper of 1935 in which they raise the issue whether quantum mechanics is complete. This leads to the question if it is possible to augment quantum mechanics with additional (yet) unknown parameters, so called local hidden variables (LHV), in order to obtain a local realistic and complete theory. It was John Bell who gave a negative answer to this question. He showed [Bel64] that there exist entangled quantum states and local measurements such that the resulting conditional probability distributions cannot be explained by a LHV theory. Consequently, such behaviours are called non-local. In the last two decades, non-locality has become an extensively studied subject within quantum information theory which has applications in subjects ranging from device independent quantum key distribution [BHK05, ABG + 07, HRW10] over questions about the foundations of quantum mechanics [BBL + 06, ABL + 09, NW10] to multi-prover games [BOGKW88, CHTW04, CSUU07, KRT08, KKM + 08, KR10].
In a two-prover game Alice and Bob, the provers, are space-like separated and receive each a classical question from a verifier. Then, each of them sends back a classical answer to the verifier. The goal of the provers is to maximize the winning probability for the predefined game. This maximal winning probability can depend on the resources Alice and Bob share. Typically it is higher if they have non-local resources at their disposal instead of only shared randomness. In order to better understand the power and limitations of quantum non-locality it is therefore of interest to investigate the question of how big the gap between the winning probabilities with quantum and classical resources can maximally be. Note that for the case of XOR games, i.e., games for which the winning condition only depends on the XOR of Alice's and Bob's answer bit, this question has been fully answered by Tsirelson [Tsi87], who showed that there is a constant gap, independent of the input alphabet sizes.
Despite the fact that non-locality has been extensively studied there are still many important open questions. One of the questions addresses the problem of deciding whether a given conditional probability distribution can be obtained by product measurements on a quantum state. There is no efficient algorithm known which decides this problem. The current state of the art is an infinite hierarchy of semi-definite programs [DLTW08,NPA08] which decides whether a system is not quantum. The drawback of this approach is that the convergence rate is, in general, not known. In order to overcome the shortage of knowledge about specific properties of the quantum set we consider a relaxation of the quantum set, obtaining a larger set of conditional probability distributions. This larger set has desirable properties while still being reasonable close to the quantum set. We will show that γ 2 can be used to define such a bigger set. In addition, by considering the dual Hilbertian tensor norm, denoted by γ * 2 , we are able to make statements about the winning probability of two-prover games with Alice and Bob having quantum systems as resources.
The first one who observed that there is a connection between tensor norms and quantum systems was Tsirelson [Tsi87]. He showed that the ratio between the maximal quantum and the maximal classical value of XOR games 1 is bounded by the Grothendieck constant 1.68 K G 1.78. Tsirelson used Grothendieck's inequality [Gro53] which establishes a connection between the Hilbertian and the projective tensor norm. Together with the fact that there is a one-to-one correspondence between quantum correlations and the Hilbertian tensor norm and between classical correlations and the projective tensor norm, the constant gap between the quantum and classical winning probabilities is implied.
Our contribution: In this paper we generalize the above mentioned argument of Tsirelson. First, we establish a connection between arbitrary quantum systems and the Hilbertian tensor norm. In particular, we prove that γ 2 evaluates to one for all quantum systems (see Proposition 2 in Section 3.2). Note however, that in contrast to the case of quantum correlations this result is not tight.
And second, we introduce a generalized Grothendieck inequality which can be applied in a setting where Alice and Bob have several possible outputs, and is therefore an extension from XOR games to arbitrary two-prover games (see Theorem 1 in Section 4). In Section 8.3 we provide a dual tensor and a matrix version of this generalized Grothendieck inequality.
Combining these two results allows us to upper bound the ratio between the maximal quantum and the maximal classical value of arbitrary two-prover games (see Theorem 4 in Section 6.2) and to improve the best known upper bound given in [DKLR09] by a square root factor.
In Section 5, we prove a direct-product theorem for the dual Hilbertian tensor norm γ * 2 (see Theorem 2). This generalizes work of [LSS08] and enables us, together with a new tight characterization of the entangled winning probability for XOR games by means of γ * 2 , to derive an alternative proof of the perfect parallel repetition theorem for entangled XOR games (see Theorem 3 in Section 6.1).
Related work: Using tools from operator space theory, Junge and Palazuelos [JP10] study large violations of Bell inequalities. In order to prove that their results are almost tight, they also provide results corresponding to our Theorem 1 (the generalized Grothendieck inequality) and Theorem 4 (the upper bound on the entangled value of two-prover games). Note that their result is more general as it holds for Bell inequalities as well. This line of research is a continuation of [JPPG + 10a, JPPG + 10b] where it is shown that operator space theory is a natural framework to study arbitrary Bell inequalities. The authors of [BRSdW10] improve the work of [JP10] by providing explicit two-prover games in order to establish near optimal lower bounds on the ratio between the quantum and classical value of Bell inequalities.
Grothendieck's inequality has been generalized in different ways before. The latest generalization can be found in [BBT09] where references to other previous generalizations [Rie74,FR94,AMMN06] are provided. Grothendieck's inequality and, in particular, the tensor norm γ 2 and its dual γ * 2 , have not only applications in quantum information theory but are also used to prove lower bounds in communication complexity [LMSS07,LS07,LSS08]. Furthermore Grothendieck's inequality serves as an inspiration to derive new semi-definite programs which can be used to approximate computationally hard problems [AN04,CW04].

Two-Prover Games
In a classical one-round two-prover cooperative game of incomplete information [BOGKW88] two classical and spatially separated provers, usually called Alice and Bob, try to win a game by interacting with a verifier. The two provers can agree on a strategy before the game. During the game the two provers are not allowed to communicate. The messages which are exchanged by the verifier and the two provers are classical bit strings. Let π : X × Y → [0, 1] be a probability distribution known by the verifier and the two provers. The verifier selects x ∈ X and y ∈ Y according to the probability distribution π and sends the value x to Alice and y to Bob. Alice and Bob send to the verifier the values s A (x) = a ∈ A and s B (y) = b ∈ B where we call the pair (s A , s B ) a strategy for the game. Note that it is sufficient to consider deterministic strategies only as the optimal (shared) randomness can be selected in advance. The provers win the game G = (π, V ) if the publicly known predicate V : A × B × X × Y → {0, 1} evaluates to 1 for the four-tuple (a, b, x, y). We consider two classes of games: Definition 1. Let G = (π, V ) be a game. Then • G is called a unique game if there exist permutations σ x,y for all inputs x ∈ X and y ∈ Y such that V (a, b, x, y) = 1 if and only if b = σ x,y (a).
• G is called an XOR game if A = B = {0, 1} with V (0, 0, x, y) = V (1, 1, x, y) and V (0, 1, x, y) = V (1, 0, x, y) for all inputs x ∈ X and y ∈ Y, i.e., the predicate V depends only on the XOR of the answers a and b.
The classical value of the protocol s A : X → A and s B : Y → B is defined by The classical value of a game, denoted by ω(G), is defined as the maximal value that can be achieved by any two strategies s A and s B for a given game G = (π, V ), i.e., We can give the two provers more power by allowing them to share entangled quantum states. Alice and Bob can then select a measurement depending on their inputs x and y, respectively, and measure the entangled state |Ψ , obtaining measurement results a and b, respectively. The entangled value of a game G = (π, V ), denoted by ω * (G), is defined as [KRT08]

Parallel Repetition of Two-Prover Games
A game G = (π, V ) can be repeated N times independently. Either the game is repeated sequentially, i.e., a full round is completed before a new round is started, or in parallel. In the latter case, N mutually independent pairs of inputs (x i , y i ) are chosen according to the distribution π and sent to the provers. The provers then compute outputs (a 1 , . . . , a N ) and (b 1 , . . . , b N ), respectively. Finally, the predicate V is evaluated for all tuples (a i , b i , x i , y i ) separately. This N -fold repetition of a game G can be seen as a new game, denoted by G ⊙N , where this new game is only won if all N rounds are won.
For sequential composition, this probability is obviously equal to the probability of winning a single game taken to the power of N . However, for parallel composition the problem gets more involved as it is generally not true that ω(G ⊙N ) is equal to ω(G) N , as shown in [For89]. The same is true for entangled games, i.e., there exist games such that ω * (G ⊙N ) > ω * (G) N [KR10]. Note that ω(G ⊙N ) ≥ ω(G) N and ω * (G ⊙N ) ≥ ω * (G) N is obviously true for all games G. Nevertheless, it can be shown that the quantity ω(G ⊙N ) decreases exponentially fast in N . A first proof of this fact, also known as the Parallel Repetition Theorem, has been given in [Raz98]. Raz's proof has been simplified in [Hol07] and extended to the case of provers using arbitrary non-signalling resources.
No such parallel repetition result is known for entangled games. However, for the special case of entangled XOR games there holds a perfect parallel repetition theorem [CSUU07]. Recently the authors of [KRT08] have shown that there is a parallel repetition theorem for entangled unique games as well. Quantitatively, it is known that if G = (π, V ) is a two-prover game then, for all N ≥ 1, it holds that

Banach Spaces
Let · X be a norm on the real finite-dimensional vector space R n . Then the tuple X := (R n , · X ) is called a Banach space. The dual space of R n , denoted by (R n ) * , is the vector space of all linear functionals from the vector space R n to the real numbers. We write G, P ∈ R for the application of the linear functional G : R n → R on the element P ∈ R n . Note that this is just the usual inner product of real vectors. The corresponding dual norm is then defined by (2.1) and the dual Banach space is given by X * := ((R n ) * , · X * ). We write f i , P , where f i ∈ (R n ) * ∼ = R n is the all-zero vector with a one at position i, to access the i'th entry of the vector P ∈ R n . And similarly, if G ∈ (R n ) * ∼ = R n we use G, e i , where e i ∈ R n is the all-zero vector with a one at position i, to access the i'th entry of G. The inner product G, P can therefore also be written as (2.2) In particular, we consider the Banach space where the ∞(1)-norm is defined as for P A ∈ R |X | ⊗ R |A| . We will also use the notation f x,a := f x ⊗ f a . See also Section 3 which gives an interpretation of the expression f x ⊗ f a , P A in the context of two-prover games. The dual space is given by (ℓ It is easy to verify that and therefore, the 1(∞)-norm is indeed the dual of the ∞(1)-norm. Note that for |A| = 1 we recover the Banach space ℓ We will use the symbols P A , P B , and P for elements in ℓ In this case they will represent (two-prover) games. Using this convention the expressions should be easier to read as we do not always have to explicitly mention the Banach space we are working on.

Connection Between Tensor Norms and Two-Prover Games
A tensor norm is a function which maps elements from tensor product spaces X ⊗ Y , where X and Y are Banach spaces, to the non-negative real numbers. Furthermore, a tensor norm inherits all properties of a regular norm and therefore fulfils the three norm-defining conditions given in Appendix A. A formal definition of tensor norms is given in Appendix B. In our particular case, we will consider the following four tensor norms (see Appendix B.1 for definitions): In the following we will show how one can represent a conditional probability distribution by a tensor P ∈ ℓ ). This will allow us to see the projective tensor norm as a map from conditional probability distributions to non-negative real numbers. On the other hand, we will show that the tensor G ∈ ℓ can be interpreted as a two-prover game and therefore the injective tensor norm assigns a non-negative real number to each game. We will see that this number is actually the classical value of a two-prover game.
Let us first give an interpretation of the term f x ⊗ f a , P A , with P A ∈ ℓ |X | ∞ (ℓ |A| 1 ), which will then lead to an explanation of the connection between tensor norms and two-prover games. First, let s A : X → A be Alice's strategy. Such a strategy can always be represented by a conditional probability distribution P A|X , with probabilities P A|X (a, x), output a, and input x. Setting for all x ∈ X , a ∈ A, defines another representation of the conditional probability distribution P A|X . Therefore, any (possibly probabilistic) strategy of Alice can conveniently be represented by a tensor P A ∈ ℓ |X | ∞ (ℓ |A| 1 ). And similarly for Bob's strategy s B : Y → B which can be represented by P B ∈ ℓ |Y| ∞ (ℓ |B| 1 ). Note that in this case we have that P A ∞(1) = P B ∞(1) = 1. Hence, any classical strategy without shared randomness can then be represented by the product representing the probability that Alice and Bob output a and b given they have inputs x and y, respectively. Entangled strategies, however, can not be represent as product tensors P A ⊗P B . Instead, they will generally be represented by non-product tensors P ∈ ℓ with P AB|XY (a, b, x, y) denoting the probability that Alice and Bob output a and b, given the inputs x and y, respectively. Any two-prover game G = (π, V ) can be interpreted as an element of the tensor product space ℓ G, e x,a ⊗ e y,b := π(x, y) · V (a, b, x, y) . (3. 2) It will be clear from the context whether G represents a pair (π, V ) or an element of a tensor product space.
The value of a protocol, which is represented as a tensor P ∈ ℓ |X | , can then be computed by where we used (2.2) in the first line and (3.1) and (3.2) in the second line.

Injective and Projective Tensor Norms
Building up on the previous section, in particular (3.3), the classical value of a two-prover game G is then given by where P A and P B are conditional probability distributions. As P A and P B represent strategies it follows that P A ∞(1) = P B ∞(1) = 1, and therefore the following upper bound on the classical value of a game is obtained: where P A and P B do not necessarily represent valid conditional probability distributions any more.
The right hand side of (3.5) is usually abbreviated by ε(G), i.e., it is an expression for the injective tensor norm of G. Note that the dual tensor norm of ε is the projective tensor norm π (see Appendix B.1.1), and therefore, (3.5) and (2.1) imply that However, as ε(G) is also a lower bound on ω(G), we obtain Proof. The statement follows from by using (3.3) in the second line and the supremum is over P A ∞(1) ≤ 1 and P B ∞(1) ≤ 1. Thus, since π(x, y) · V (a, b, x, y) ≥ 0, we have that f x,a , P A ≥ 0 and f y,b , P B ≥ 0 for the optimal case. Furthermore, it is clear that the optimum is achieved when f x,a , P A and f y,b , P B are as large as possible, meaning that a f x,a , P A = 1 and b f y,b , P B = 1 for all 1 ≤ x ≤ |X | and 1 ≤ y ≤ |Y|, respectively. But this implies that P A and P B correspond to valid (local probabilistic) strategies of Alice and Bob, respectively, and therefore the injective tensor norm of G is the same as the classical value of the game G.

Hilbertian Tensor Norm and its Dual
In the previous section we have investigated the connection between tensor norms and the classical value of two-prover games. In this section we will now establish a connection between tensor norms, in particular the dual Hilbertian tensor norm, and the entangled value of twoprover games. We will call P ∈ ℓ Note that it is no restriction to assume pure states and projective measurements (see also [NC00]). In Section 8.5 we show that the Hilbertian tensor norm has value 1 for all quantum systems.
Using this result, we can now upper bound the entangled value of an arbitrary two-prover game by the dual Hilbertian tensor norm.
Proof. The statement follows from by using (3.6) in first line, (3.3) in the second line, Proposition 2 in the third line and that γ 2 is the dual of γ * 2 in the fourth line.

Generalized Grothendieck Inequality
The previous section can be summarized by the following chain of (in)equalities which holds for any two-prover game G: . Recall that if G corresponds to a two-prover game all entries G, e x,a ⊗e y,b := π(x, y)·V (a, b, x, y) are non-negative. Let G now be an arbitrary element of the tensor product space ℓ i.e., the tensor G can have negative entries as well and therefore corresponds to a general Bell inequality (see Appendix C for a short introduction to Bell inequalities). According to Lemma 10 in Appendix B.1.1, γ * 2 (G) is still an upper bound on ε(G). However, in Section 8.3 we prove that there is an upper bound on the maximal ratio between γ * 2 (G) and ε(G).
Theorem 1 (Generalized Grothendieck Inequality in Dual Tensor Form). For any G ∈ ℓ The standard Grothendieck inequality [Gro53] (in dual tensor form) is obtained from our generalized version by setting the output alphabet sizes to 1, i.e., |A| = |B| = 1, and therefore 2) ≈ 1.78 is the best known upper bound on the Grothendieck constant K G [Kri79]. The best lower bound is K G ≥ 1.6770 due to Reeds and Davis [Dav84,Ree91].

Direct-Product Theorem
We will state a direct-product theorem for the dual Hilbertian tensor norm γ * 2 in this section. We will use this result later in the application section about parallel repetition of two-prover games. Let ∞ ) be arbitrary two-prover games between Alice, Bob and the verifier. We denote by G := G A 1 B 1 ⊙ G A 2 B 2 the composition of these two games (see also Section 2.2). Formally we have Alice and Bob will then play these two games in parallel and try two win both rounds.
To be more explicit, we define the two round game G as follows is therefore executed by Alice and Bob with systems A 1 A 2 belonging to Alice and B 1 B 2 to Bob. We say that the game is bipartite with respect to the partition A 1 A 2 : B 1 B 2 between Alice and Bob, where A i corresponds to the system ℓ 1 ) represent an arbitrary strategy of Alice and Bob for the two round game G. The winning probability of this strategy on the game G is then given by G A 1 B 1 ⊙ G A 2 B 2 , P . Note that, in general, the maximal winning probability is achieved when P is a non-product strategy, i.e., P = P A 1 B 1 ⊙ P A 2 B 2 . Therefore, in general, there exist games G A 1 B 1 and G A 2 B 2 such that with P, P A 1 B 1 , P A 2 B 2 either all classical or all entangled strategies. Hence, it is in general impossible to upper bound the classical (entangled) value of the game G A 1 B 1 ⊙G A 2 B 2 by the product of the classical (entangled) values of the individual games. On the other hand, the dual Hilbertian tensor norm has the nice property that one can actually upper bound the value of the parallel executed games by the product of the value of the single rounds.
The proof is given in Section 8.4. There we also provide additional direct product results.

Application I: Parallel Repetition of Entangled Games
In this section we will provide an alternative proof for the parallel repetition theorem for XOR games given in [CSUU07]. The proof of [CSUU07] contains two parts. In the first part, they show that the sum of XOR games obeys a perfect product rule by using semi-definite programming (SDP) techniques and then, in a second step, they use Fourier analysis to get a perfect parallel repetition theorem for XOR games. The first part corresponds to applying the direct product result for the γ * 2 tensor norm (see also the remark at the very end of Section 8.4) applied on a gameG = (π,Ṽ ), but whereṼ has now range is the quantum bias of an XOR game, denoted by ε q (G) in [CSUU07]. The second part is required because ω * (G) is a rescaling of γ * 2 (G) which is due to the fact thatṼ has range ±1.
The crucial idea in our alternative proof is to interpret the XOR game G as an element of ℓ The proof can be found in Section 8.6. We now have all the tools we need in order to give an alternative proof of the perfect parallel repetition theorem for entangled XOR games.
Proof. It is clear that ω * (G ⊙N ) ≥ ω * (G) N by executing the rounds individually. For the other direction, by Proposition 3, we have Applying the direct-product result of Theorem 2 and using Proposition 4 yields 6.2 Application II: Upper Bound on the Value of Two-Prover Games In the following we will give an upper bound on the maximal ratio between the entangled and the classical value of two-prover games, i.e., we will compute The best upper bound known so far [DKLR09] states that v ≤ O(|A| · |B|) independently of the input dimensions |X | and |Y|. If we fix the dimension of the local Hilbert spaces to d in the computation of ω * (G), it has been shown [JPPG + 10b] that v ≤ O(d), independently of the input and output dimensions. Note that these two results also hold if one considers the more general setting of Bell inequalities instead of two-prover games. We prove a result which improves the previous upper bound of [DKLR09] by a square root factor.
Theorem 4. Let the input alphabet sizes |X |, |Y| and the output alphabet sizes |A|, |B| of the two-prover games be finite.
Proof. By using Proposition 1 and Proposition 3 and the dual of the generalized Grothendieck inequality in tensor form given in Theorem 1 we get where the supremum is over two-prover games G ∈ ℓ This theorem can be seen as a generalization of Tsirelson's work in [Tsi87]. In particular, an XOR game G = (π, V ) can be interpreted as an element of ℓ |X | 1 ⊗ ℓ |Y| 1 with the respective correlation strategies P ∈ ℓ |X | ∞ ⊗ ℓ |Y| ∞ containing elements in [−1, 1] corresponding to expectation values. Proposition 1 and Proposition 3 (in this case even with equality) can also be proven for this setting of correlation strategies. Therefore, by using the standard Grothendieck inequality in dual tensor form (see also (4.1)) one obtains [Tsi87] sup for G an XOR game, independently of the input dimensions |X | and |Y|.

Concluding Remarks and Open Questions
We have investigated the Hilbertian tensor norm γ 2 and its dual γ * 2 defined over the tensor spaces ℓ . We have given an alternative proof of the perfect parallel repetition theorem for entangled XOR games using our direct-product theorems for these tensor norms. Furthermore, by applying our generalized Grothendieck inequality we could establish an upper bound on the maximal ratio between entangled and classical values of two-prover games.
As a line of possible future work, it would be interesting to investigate if our generalization of Grothendieck's inequality and our direct-product theorems have other applications in quantum information theory, communication complexity or approximation algorithms. Acknowledgements I thank Thomas Holenstein for helpful discussions and comments, Matthias Fitzi, Esther Hänggi, Marco Tomamichel and Severin Winkler for useful comments on an earlier version of this paper, the referees for constructive comments that have significantly improved the presentation of this paper and Carlos Palazuelos for pointing out an error in a previous version of Theorem 4. This research was supported by the Swiss National Science Foundation through the National Centre of Competence in Research Quantum Science and Technology.

Notation
In addition to the notation introduced in Section 2 we will use the following notation in this proof section. Let X and Y be Banach spaces and T : X → Y be a linear map. The operator norm of T is defined as Given column vectors v i,j ∈ R n with 1 ≤ i ≤ N and 1 ≤ j ≤ M , we write (v i,j ) for the n × N ·Mmatrix which has the vectors v i,j as columns. The order is such that the first M columns of (v i,j ) are given by the vectors v 1,1 , v 1,2 , ..., v 1,M . The second M columns are v 2,1 , v 2,2 , ..., v 2,M , and so forth. We write (v i,j ) T for the transposed matrix, i.e., the matrix (v i,j ) T has the vectors v T i,j as rows.
We will use (µ ij ), with 1 ≤ i ≤ n and 1 ≤ j ≤ m, to denote the n × m-matrix with entries µ ij ∈ R.
Lemma 1. Let n y,b ∈ ℓ n 2 and m x,a ∈ ℓ n 2 , with 1 ≤ n ≤ ∞, for all 1 ≤ y ≤ |Y|, 1 ≤ b ≤ |B|, 1 ≤ x ≤ |X |, and 1 ≤ a ≤ |A|. Then Proof. By using the definition of the · 1(∞)→2 -norm, we obtain As every G with G 1(∞) ≤ 1 can be written as G x,a ≡ G, e x ⊗e a = κ x ·µ x,a , with |X | x=1 |κ x | ≤ 1 and µ x,a ∈ [−1, 1], we get where we used the triangle inequality in the first line. That (m x,a ) 1(∞)→2 is greater or equal than the upper bound of (8.1) is obvious, by setting κ x = 1 for the optimal x, and hence we have equality. That the optimal vector µ in (8.1) can be chosen to consist only of +1, −1 entries follows from the convexity of norms. That (n y,b ) T 2→∞(1) = max s,y |B| b=1 s(b) · n y,b 2 holds as well follows from Lemma 9 in Appendix A.
Let us now show that (m We will now construct a function with this property. First, we can write For a = 1 we set s(1) := 1. We then set the value for s(2) which will depend on s(1). Then we set s(3) which will depend on s(1) and s(2). Hence, the value for s(a 1 ) will depend on all s(1), s(2), ..., s(a 1 − 1). In particular, we define s(a 1 ) to be s(a 1 ) := sign s(a 2 ) · m x,a 1 , m x,a 2 .
By defining the function s in this way, the right hand side of (8.2) is always non-negative which is what we wanted to prove. By the same reasoning, one can show that (n y,b ) T 2 2→∞(1) ≥ |B| b=1 n y,b 2 2 holds as well.
Note that, for |A| = |B| = 1, we have that R 2→∞ is the largest 2-norm of a row of R and S 1→2 is the largest 2-norm of a column of S.

Generalized Grothendieck Inequality
In this section, we will state and prove a generalized Grothendieck inequality. In the tensor norm picture, it is a generalization of the standard Grothendieck inequality [Gro53] in the sense that multiple outputs are allowed, or in the language of games, it generalizes from XOR games to arbitrary games. In the tensor norm language, the (generalized) Grothendieck inequality establishes a connection between the projective tensor norm π and the Hilbertian tensor norm γ 2 . The difference of our generalized Grothendieck inequality to the standard one is that the they are defined over different local Banach spaces.

Claim 1 (Generalized Grothendieck Inequality in Tensor Form). For any
Proof. Let us assume that γ 2 (P ) = 1 for some P = x,y,a,b P a,b x,y · (e x ⊗ e a ) ⊗ (e y ⊗ e b ). Showing that π(P ) ≤ π 2 ln(1+ √ 2) · |A||B| proves the claim. As γ 2 (P ) = 1, we can conclude according to Corollary 1 in Appendix B. with c = π 2 ln(1+ √ 2) . Since π is a norm, we can apply the triangle inequality and get where we also used that π is a tensor norm and therefore π(P A ⊗ P B ) = P A ∞(1) · P B ∞(1) . Furthermore, by using the definition of the ∞(1)-norm, we have that By applying Lemma 8 in Appendix A we get the following dual theorem:

Theorem 1 (Generalized Grothendieck Inequality in Dual Tensor Form). For any
As the standard Grothendieck inequality is usually stated in matrix form, we will also give a matrix representation of our generalization. As the · 2→∞(1) and · 1(∞)→2 operator norms will appear in the following claim, it might be helpful for the reader to have a look at Lemma 1 again which gives an alternative representation of these two operator norms.
The standard Grothendieck inequality in tensor as well as in matrix form are recovered from our generalized Grothendieck inequalities by setting |A| = |B| = 1.

Direct-Product Theorems
We first show a direct-product result for the γ * 2 tensor norm over Banach spaces which have the property that their norms behave "nicely" on product tensors. By behaving "nicely" we mean the following. Let X n := (R n , · Xn ), with 1 ≤ n < ∞, be a Banach space. Then the norm · Xn behaves "nicely" on product tensors if for all 1 ≤ n, m < ∞, P A ∈ R n , P B ∈ R m and with P A ⊗ P B ∈ R n·m . To shorten the notation we will usually write P A ⊗ P B X ≤ P A X · P B X .
Let us now show that the ∞(1)-norm and its dual have the property that they behave "nicely" on product tensors. First, the ∞(1)-norm can also be defined for tensor elements ). Then, we obtain: Proof. Using the definition of the ∞(1)-norm (see (8.7)) gives us Let us now prove a direct product result for the γ * 2 tensor norm. We will need the following result in order to show this result.
Lemma 4 (Bennett [Ben77]). Let A and B be n × n and m × m matrices over R, respectively. Then Lemma 5. Let X and Y be Banach spaces with the norms having the property that and let the tensor norm γ * 2 be defined over the tensor space X ⊗ Y . Then where the partition of be optimal decompositions in the definition of the γ * 2 tensor norm (see (B-3) in Appendix B.1.2). Taking the composition of these two systems gives us (see also (5.1)) We therefore get where we used Lemma 4 in the last line. The fact that the local norms behave nicely on product tensors implies immediately that The next theorem gives new direct-product results for the γ 2 and γ * 2 tensor norms over the Banach spaces ℓ , respectively. This holds since where, in the second line, we used that γ * 2 (G 1 ) ≤ 1 and γ * 2 (G 2 ) ≤ 1 imply γ * 2 (G 1 ⊙ G 2 ) ≤ 1.

Hilbertian Tensor Norm and Bipartite Quantum Systems
1 ) a quantum system if it can be obtained by measurements on a pure quantum state (see Section 3.2). On the other hand, we will call P ∈ ℓ |X | ∞ ⊗ ℓ |Y| ∞ a quantum correlation if there exists a pure quantum state |Ψ ∈ H A ⊗ H B , and observables A 1 , ..., A |X | and B 1 , ..., B |Y| on H A and H B , respectively, with eigenvalues ±1 such that Let us first focus on the case where P is a quantum correlation. In order to establish a connection to the γ 2 tensor norm, we need a theorem by Tsirelson, which says that the correlations which can be obtained by measurements on a quantum state can be represented by inner products of real unit vectors, and vice versa. More formally: Lemma 6 (Tsirelson's Theorem [Tsi80]). Let A 1 , ..., A |X | and B 1 , ..., B |Y| be observables with eigenvalues in [−1, +1]. Then for any state |Ψ ∈ H A ⊗ H B there exist real unit vectors m 1 , ..., m |X | ∈ R 2·max{|X |,|Y|} and n 1 , ..., n |Y| ∈ R 2·max{|X |,|Y|} such that for all 1 ≤ x ≤ |X | and 1 ≤ y ≤ |Y|.
Note that this is a slightly generalized version of Tsirelson's theorem where we do not need the vectors m x ∈ R N and n y ∈ R N to be unit vectors. So let us show that Lemma 6 indeed holds. In order to be allowed to apply the standard Tsirelson theorem, we need unit vectors. So let us construct them. Definem x ∈ R N +2 to be m x , e i := m x , e i for all 1 ≤ i ≤ N , m x , e N +1 := 1 − m x 2 2 and m x , e N +2 := 0. And similarly, forñ y ∈ R N +2 we set ñ y , e i := n y , e i for all 1 ≤ i ≤ N , ñ y , e N +1 := 0 and ñ y , e N +2 := 1 − n y 2 2 . We then have m x , n y = m x ,ñ y and m x 2 = 1 and ñ y 2 = 1 for all 1 ≤ x ≤ |X | and 1 ≤ y ≤ |Y|. Hence, we can apply the standard Tsirelson theorem and get m x , n y = m x ,ñ y = Ψ|A x ⊗ B y |Ψ .
Using Lemma 6, we are now ready to prove a tight connection between the γ 2 tensor norm and quantum correlations, i.e., ∞ is a quantum correlation if and only if γ 2 (P ) ≤ 1.
Proof. As P is a quantum correlation we can write it, according to Lemma 6, as with m x 2 = 1 and n x 2 = 1 for all 1 ≤ x ≤ |X | and 1 ≤ y ≤ |Y|, respectively. Furthermore, the matrices (m x ) and (n y ) T give a factorization ofP , i.e., we haveP = (n y ) T · (m x ). Using the definition of the γ 2 tensor norm given by (B-6) in Appendix B.1.2 (with |A| = |B| = 1) yields By applying Lemma 1 and using that m x 2 = 1 and n x 2 = 1, we get (n y ) T 2→∞ = (m x ) 1→2 = 1 and hence, γ 2 (P ) ≤ 1. For the converse, assume that γ 2 (P ) ≤ 1. Then, by the definition given in (B-6) we can conclude that there exist real vectors {m x } and {n y } such that (n y ) T 2→∞ ≤ 1 and (m x ) 1→2 ≤ 1 with f x ⊗ f y , P ≡ P x,y = m x , n y . Then, the second part of Lemma 1 implies that m x 2 ≤ 1 and n y 2 ≤ 1 for all 1 ≤ x ≤ |X | and 1 ≤ y ≤ |Y|. Applying the second part of Lemma 6 on the vectors {m x } and {n y } implies that P is indeed a quantum correlation.
We would also like to prove a similar result as given by Lemma 7 for quantum systems ). Unfortunately, no generalization of Tsirelson's theorem to many outputs is known to exist 2 . In particular, the second part of Tsirelson's theorem is the problem, as the first part can be generalized as will be seen in the proof of Proposition 2. We therefore get a weaker result, namely ) be a quantum system. Then γ 2 (P ) = 1. Proof. As P is a quantum system there exists a pure quantum state |Ψ ∈ H A ⊗ H B and projective measurements {M a with P = x,y,a,b P a,b x,y · e x,a ⊗ e y,b . Let {|i } i be an orthonormal basis of H A ⊗ H B . We define the complex vectorsm x,a andñ y,b bỹ 2 2 = 1 for all y ∈ {1, 2, ..., |Y|}. Furthermore, the vectors {m x,a } a are mutually orthogonal for a given x, i.e., m x,a 1 ,m x,a 2 = δ a 1 ,a 2 · m x,a 1 2 2 , with a 1 , a 2 ∈ {1, 2, ..., |A|}, and for all x ∈ {1, 2, ..., |X |}. This is the case since m x,a 1 ,m x,a 2 = Ψ|(M a 1 x · M a 2 x ⊗ id H B )|Ψ = Ψ|(M a 1 x ⊗ id H B )|Ψ · δ a 1 ,a 2 = m x,a 1 2 2 · δ a 1 ,a 2 , (8.10) as M a 1 x and M a 2 x are projectors with the property that M a 1 x ·M a 2 x = M a 1 x ·δ a 1 ,a 2 . By an analogous argument one can show that also ñ y,b 1 ,ñ y,b 2 = δ b 1 ,b 2 · ñ y,b 1 2 2 , with b 1 , b 2 ∈ {1, 2, ..., |B|}, and for all y ∈ {1, 2, ..., |Y|}. Note that the complex vectorsm x,a andñ y,b can be replaced by real vectors m x,a and n y,b of twice the length while still fulfilling (8.9) and (8.10).
By the same argument one can show that (n y,b ) T 2→∞(1) = 1 holds as well.
As Ψ|A x ⊗ B y |Ψ is the expectation value when measuring the observables A x and B y with eigenvalues ±1, we have that (8.17) By straightforward calculations, (8.16) implies (8.14), where the conditions m x,0 + m x,1 2 ≤ 1 and n y,0 + n y,1 2 ≤ 1 of (8.12) are used. And similarly for Pr[a = b|A x , B y , |Ψ ].
The algebraic tensor product of two Banach spaces X and Y is denoted by X ⊗ Y . Note that this is not yet a Banach space as we have not yet defined a norm on the tensor space X ⊗ Y . An element v ∈ X ⊗ Y can always be written as with v i A ∈ X and v i B ∈ Y , respectively. It is important to note that this decomposition is not unique as, typically, there are infinitely many such representations. The tensor product has the following properties: for all v, v 1 , v 2 ∈ X, w, w 1 , w 2 ∈ Y , and c ∈ R. Furthermore, let X and Y be Banach spaces and X * and Y * their corresponding dual Banach spaces. Then, it holds that Furthermore, if the vector spaces are isomorphic to R n and R m for some 1 ≤ n, m < ∞, and v, w ∈ R n ⊗ R m , then We will also use the notation e i,j := e i ⊗ e j and f i,j := f i ⊗ f j . The 2-norm behaves 'nicely' on product tensors, i.e., it is easy to see that for all v ∈ ℓ n 2 and w ∈ ℓ m 2 .

B Basic Properties of Tensor Norms
We will introduce the notion of tensor norms in this section. Let X and Y be arbitrary Banach spaces and X ⊗ Y the algebraic tensor product of these two spaces. We will call X and Y local spaces. A tensor norm is a norm on X ⊗ Y , which is based on the local Banach spaces X and Y with some additional special properties. See also [DF93,Rya02] which give a good introduction to the subject of tensor norms. Recall that by α * we denote the dual tensor norm of α in the sense of (2.1). The definition of tensor norms reads then as follows [Sch50]: Definition 2 (Tensor Norm). Let X and Y be finite dimensional Banach spaces. A norm α on X ⊗ Y is called a tensor norm if the following three conditions are satisfied: 1. α(P A ⊗ P B ) = P A X · P B Y for every P A ∈ X and P B ∈ Y .
If a norm on X ⊗ Y fulfils the first two conditions it is called a reasonable cross norm [Rya02]. Note that we need only the first two properties of tensor norms in this paper and that in [Sch50,Rya02] α with these three properties is called a uniform cross norm whereas in [DF93] it is called a tensor norm.

B.1 Four Different Tensor Norms
In the following we will define four different tensor norms (actually just two, but taking the duals gives us four). We will only write down the definitions for the case where the local Banach spaces X and Y are ℓ , respectively. Note, that because |X |, |Y|, |A| and |B| are finite, the resulting Banach spaces are finite dimensional as well.

B.1.1 Projective and Injective Tensor Norm
The first two tensor norms, called the projective and injective tensor norm, are the "extremal" ones., i.e., all tensor norms are larger than the injective and smaller than the projective tensor norm. The projective tensor norm of P ∈ ℓ |X | where the infimum is over all decompositions (or representations) of P . The injective tensor where the supremum is over P A ∈ ℓ |X | ∞ (ℓ |A| 1 ) and P B ∈ ℓ |Y| ∞ (ℓ |B| 1 ). One can show [Rya02] that these two norms are the dual of each other, i.e., π(P ) = sup{| G, P | : ε(G) ≤ 1} , ε(G) = sup{| G, P | : π(P ) ≤ 1} , As already state above, these two tensor norms are extremal. Formally, we have [Rya02]: Lemma 10. Let X and Y be Banach spaces. Every tensor norm α on X ⊗ Y satisfies ε(P ) ≤ α(P ) ≤ π(P ) for every P ∈ X ⊗ Y , where ε is the injective tensor norm and π is the projective tensor norm, both defined over X ⊗ Y .
The next lemma states that if P ∈ ℓ |X | ∞ (ℓ |A| 1 ) ⊗ ℓ |Y| ∞ (ℓ |B| 1 ) corresponds to an (almost) valid conditional probability distribution (we allow negative entries), then all tensor norms will assign to P a value which is at least one.
with a,b f x,a ⊗ f y,b , P = 1 for all 1 ≤ x ≤ |X | and 1 ≤ y ≤ |Y|. Then α(P ) ≥ 1 , for all tensor norms α over ℓ Proof. By the definition of the injective tensor norm we have where I A is the all-1 vector multiplied by 1/|X | and I B is the all-1 vector multiplied by 1/|Y|, where I A 1(∞) = 1 and I B 1(∞) = 1, respectively. Taking the tensor product of I A and I B yields the all-1 vector multiplied by 1/(|X ||Y|). Then, by using a,b f x,a ⊗ f y,b , P = 1, we obtain | I A ⊗ I B , P | = 1 |X ||Y| · |X ||Y| = 1 .

B.1.2 Hilbertian Tensor Norm and its Dual
In this section we introduce the Hilbertian tensor norm, denoted by γ 2 . One possible way to define it is [DF93]: where the infimum is over all decomposition On the other hand, if γ 2 is defined over ℓ where the infimum is over all decomposition . The dual of γ 2 can be represented by [DF93]: where the infimum is over all decompositions , (µ ij ) is a real n × n-matrix, and And similarly for γ * 2 over the tensor space ℓ |X | where the infimum is over all decompositions P = n i,j µ ij · P i ). There is a useful alternative representation of the Hilbertian tensor norm γ 2 whereof it actually got its name from. A tensor P ∈ ℓ |X | ) by the following identification: . Note thatP (G) does not depend on the actual decomposition of P . We are now ready to state the alternative representation of the γ 2 norm [Rya02], namely: where the infimum is over all decomposition ofP into linear operators R : ℓ 2 → ℓ |Y| ∞ (ℓ |B| 1 ) and S : ℓ |X | 1 (ℓ |A| ∞ ) → ℓ 2 . In other words,P is factored through the Hilbert space ℓ 2 . Note that this Hilbert space can be of any dimension, even infinite dimensional. By setting |A| = |B| = 1 we recover the norms used in [LMSS07,LS07,LSS08]. See Appendix D for a proof of this equivalence.
We can think of R and S being matrices of dimension |Y||B| × n and n × |X ||A|, respectively, with 1 ≤ n ≤ ∞, such that their matrix product yieldsP . Representing R as a row matrix R := (n y,b ) T and S as a column matrix S := (m x,a ) (see also Section 8.1 about the notation) yields as entries ofP = R · S the values f x,a ⊗ f y,b , P = n y,b , m x,a = m x,a , n y,b . An immediate corollary is Corollary 1. Let P ∈ ℓ |X | ∞ (ℓ |A| 1 ) ⊗ ℓ |Y| ∞ (ℓ |B| 1 ). Then γ 2 (P ) ≤ 1 if and only if there exist vectors m x,a , n y,b ∈ ℓ n 2 , with 1 ≤ n ≤ ∞, such that f x,a ⊗f y,b , P = m x,a , n y,b , and (m x,a ) 1(∞)→2 ≤ 1 and (n y,b ) T 2→∞(1) ≤ 1. where by "P is classical" we mean that P can be written as f x,a ⊗ f y,b , P = ρ(λ) · P A|XΛ (a, x, λ) · P B|Y Λ (b, y, λ)dλ with P A|XΛ (a, x, λ) ≥ 0, P B|Y Λ (b, y, λ) ≥ 0, a P A|XΛ (a, x, λ) = 1, b P B|Y Λ (b, y, λ) = 1 and ρ(λ)dλ = 1, i.e., the distribution P can be explained by a local hidden variable model, where the local hidden variable λ is selected with probability ρ(λ). Hence, by f x,a ⊗ f y,b , P we refer to the probability that the outputs are a and b, given the inputs x and y. We say that a Bell inequality G ∈ ℓ G, e x,a ⊗ e y,b · f x,a ⊗ f y,b , P > B C (G) .

C Introduction to Bell Inequalities
The most prominent example of a Bell inequality is the so-called CHSH Bell inequality [CHSH69]. Let A = B = X = Y = {0, 1}, i.e., there are only two inputs and two outputs on each side, respectively. The CHSH inequality is usually stated in the form of expectation values, but in order to fit into our presentation, we will state its equivalent "probability representation": It is not hard to show that B C (G CHSH ) = 2, where this value can be achieved for P which always "outputs" the values a = 0 and b = 0, independently of the inputs x and y.
In Section 3.1, we have shown that, for a game G ∈ ℓ |X | 1 (ℓ |A| ∞ ) ⊗ ℓ |Y| 1 (ℓ |B| ∞ ), the injective tensor norm and classical value of the game are equal, i.e., that ε(G) = ω(G) (see Proposition 1). For Bell inequalities G there is no equality relation any more. It only holds that B C (G) ≤ ε(G) for all Bell inequalities G. The reason for losing the equality stems from the fact that, in contrast to two-prover games, a Bell inequality can have negative entries. Furthermore, the fact that B C (G) is not equal to ε(G) for Bell inequalities G is the reason for our proof of Theorem 4 not going through for Bell inequalities.

D Equivalence of γ 2 Definitions
We will show the following equality: inf P =R·S R 2→Y · S X * →2 = inf w 2 (P i A ; X) · w 2 (P i B ; Y ) , (D-1) with P = n i=1 P i A ⊗ P i B ∈ X ⊗ Y for X and Y arbitrary finite dimensional Banach spaces, which implies the equivalence of (B-1) and (B-6) in Appendix B.1.2.
Let us first show that the right hand side of (D-1) is larger or equal to the left hand side. First, let P = n i=1 P i A ⊗ P i B ∈ X ⊗ Y be the optimal decomposition on the right hand side of (D-1). Then, we define R : ℓ n 2 → Y and S : X * → ℓ n 2 as follows: λ, e i · P i B , The operatorP : X * → Y corresponding to P can be represented aŝ We then get On the other hand, using the duality relation between norms, we have By setting µ, e i := G B , P i B , with µ ∈ ℓ n 2 , and using that ℓ n 2 is self dual, we get which finishes the first part of the proof.
Let us now show that the right-hand side of (D-1) is smaller or equal to the left-hand side. LetP = R · S be the optimal factorization ofP on the left-hand side of (D-1). Then there exist P i A ∈ X and P i B ∈ Y such that R(λ) = n i=1 λ, e i · P i B and S(G A ) = n i=1 e i · G A , P i A , respectively. Hence, i P i A ⊗ P i B is a valid representation of P (see also (D-2) and (D-3)). Using (D-4) and (D-5) finishes the proof.