Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called ‘projective–injective ratio’. This is defined as the minimal constant ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} such that ‖·‖X⊗πY⩽ρ‖·‖X⊗εY\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _{X\otimes _\pi Y}\leqslant \rho \,\Vert \cdot \Vert _{X\otimes _\varepsilon Y}$$\end{document} holds for all Banach spaces of dimensions dimX=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim X=n$$\end{document} and dimY=m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim Y=m$$\end{document}, where X⊗πY\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\otimes _\pi Y$$\end{document} and X⊗εY\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \otimes _\varepsilon Y$$\end{document} are the projective and injective tensor products. By requiring that X=Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=Y$$\end{document}, one obtains a symmetrised version of the above ratio, denoted by rs(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_s(n)$$\end{document}. We prove that r(n,m)⩾19/18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(n,m)\geqslant 19/18$$\end{document} for all n,m⩾2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n,m\geqslant 2$$\end{document}, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and rs(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_s(n)$$\end{document}, showing that, up to log factors: rs(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_s(n)$$\end{document} is of the order n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}$$\end{document} (which is sharp); r(n, n) is at least of the order n1/6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{1/6}$$\end{document}; and r(n, m) grows at least as min{n,m}1/8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{n,m\}^{1/8}$$\end{document}. These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ‘ℓ1/ℓ2/ℓ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1/\ell _2/\ell _{\infty }$$\end{document} trichotomy theorem’ based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements.The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively.The intrinsic (i.e.model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called 'projective-injective ratio'.This is defined as the minimal constant ρ such that • X⊗π Y ρ • X⊗εY holds for all Banach spaces of dimensions dim X = n and dim Y = m, where X ⊗π Y and X ⊗ε Y are the projective and injective tensor products.By requiring that X = Y , one obtains a symmetrised version of the above ratio, denoted by rs(n).We prove that r(n, m) 19/18 for all n, m 2, implying that injective and projective tensor products are never isometric.We then study the asymptotic behaviour of r(n, m) and rs(n), showing that, up to log factors: rs(n) is of the order √ n (which is sharp); r(n, n) is at least of the order n 1/6 ; and r(n, m) grows at least as min{n, m} 1/8 .These results constitute our main contribution to the theory of tensor norms.In our proof, a crucial role is played by an ' 1/ 2/ ∞ trichotomy theorem' based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann.The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension.In the quantum case, we are able to narrow this gap up to universal constants.As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.arXiv:1809.10616v1[quant-ph] 27 Sep 2018

I. INTRODUCTION
One of the most prominent conceptual contributions of the celebrated 1964 paper by J.S. Bell [1] is to point out that the implications of the quantum mechanical predictions extend far beyond the very same formalism that is used to deduce them, and shed light on some of the deepest secrets of Nature.As is well known, the scenario considered by Bell features two distant parties that share a quantum entangled state and make local quantum measurements, but the conclusions of the argument, far from depending on the details of the involved physics, rely only upon the correctness of its experimental predictions, which are expressed in terms of entirely classical input-output relations.In the same spirit, we deem it important to understand what features of information processing in composite systems are truly intrinsic, meaning that they are common to all conceivable physical theories.
A suitable way to formalise the concept of a physical theory in this context is provided by the mathematical machinery of general probabilistic theories (GPTs) [2][3][4].It is sometimes convenient to think of GPTs as generalisations of finite-dimensional quantum mechanics, where the set of unnormalised states is not assumed to be the cone of positive semidefinite matrices, but it is rather taken to be an arbitrary convex cone in a finite-dimensional real vector space.As is well known, a GPT makes the host vector space a Banach space in a canonical way, by equipping it with a so-called base norm.
The starting point of our investigation is the study of XOR games in the rich landscape of GPTs.We remind the reader that any XOR game can be equivalently cast in terms of state discrimination queries subjected to locality constraints, so that our analysis applies equally well to these problems.XOR games are arguably the simplest examples of two-prover one-round games, and feature two players Alice and Bob who are asked questions by a referee in the form of bipartite states of some physical system modelled by a GPT.The correctness of the one-bit answers they provide upon measuring the state depends only on their parity.According to whether Alice and Bob are allowed to carry out product or global measurements, one talks about local or global strategies.In general, the winning probability can be significantly larger in the latter than in the former case.
While the quantitative details of this phenomenon will in general depend on the particular physical system modelling the questions, our work is instead motivated by the wish to understand which behaviours are universal, and thus pertain to the intrinsic nature of XOR games.This line of investigation brings us to develop an extensive connection with the theory of tensor norms on finite-dimensional Banach spaces, which has already proved to be instrumental in the study of classical and quantum XOR games [5].While in these more standard settings one deals with specific examples of tensor norms, the analysis of games played over arbitrary GPT models requires a systematic understanding of general tensor norms.
The main problem we investigate here asks for the maximal gap that can be guaranteed to exist between the winning probabilities associated with global and local strategies in XOR games played over GPTs of fixed local dimensions.In analogy with the classical case, we show that such winning probabilities are given by simple expressions involving respectively the projective and injective tensor norms induced by the local GPTs through their native Banach space structures.Comparing them in a model-independent fashion prompts us to investigate the least constant of domination of the injective over the projective tensor norm over all pairs of normed spaces of fixed dimensions.
We call this function 'projective/injective ratio', or π/ε ratio for short.When seen from the point of view of pure mathematics, this universal function over pairs of positive integers encodes some information regarding Grothendieck's theory of tensor products of Banach spaces.At the same time, the operational interpretation we construct here guarantees that the same object captures some intrinsic feature of general XOR games.
Our main result is that the π/ε ratio associated with two n-dimensional Banach spaces scales at least as n 1/6 (up to logarithmic factors), implying that global strategies for XOR games are intrinsically much more effective than local ones in a precise asymptotic sense.We ask the question whether this scaling can be improved up to n 1/2 , and bolster this hypothesis by showing that it holds true (again, up to log factors) when two copies of the same space are considered.Interestingly, this question is intimately connected to the problem of estimating the radius of the weak Banach-Mazur compactum, which has also been conjectured to be of order n 1/2 [6,7].We also consider the problem of establishing universal lower bounds for the π/ε ratio.We prove that for all pairs of Banach spaces X, Y there is a tensor in X ⊗ Y whose projective norm is at least 19/18 times its injective norm.In particular, these norms are always different.This should be compared with the famous construction by Pisier [8] of an infinite-dimensional Banach space X such that the injective and projective norms on X ⊗ X are equivalent.Finally, we solve the problem of computing the π/ε ratio for some specific examples of physically relevant Banach spaces.Most notably, we establish that it is of the order min{n, m} 3/2 for S n,sa

1
, where S k,sa 1 stands for the space of k × k Hermitian matrices endowed with the trace norm.The importance of this special case stems from the fact that S k,sa 1 is the natural Banach space associated with a k-level quantum system.The rest of the paper is structured as follows.Throughout this section, we provide very brief introductions to the GPT formalism (Subsection I A), to the theory of tensor norms (Subsection I B), and to XOR games (Subsection I C).In Section II, we state our main results and broadly discuss some of proof techniques we developed.Section III presents some general properties of the π/ε ratio, connecting it with other concepts in functional analysis.There, we find the universal lower bound of 19/18, and solve the quantum mechanical case up to multiplicative constants.Section IV deals with the problem of determining the asymptotic scaling of the π/ε ratio, either for two copies of the same space, or in the fully general setting of two normed spaces of different dimensions.Extra information can be found in Appendices.Appendix A investigates how our operational interpretation of the injective tensor norm is affected by the introduction of a bounded amount of two-way communication, while Appendix B provides a proof of the useful fact that any normed space is 2-isomorphic to a base norm space.Finally, Appendix C gathers the functional-analytic background that is used throughout the paper and which may be unfamiliar to a non-specialist reader.

A. General probabilistic theories
The origins of the formalism of general probabilistic theories (GPTs) lie in the attempt to axiomatise quantum mechanics, rebuilding it upon operationally motivated postulates rather than upon more evasive concepts such as 'wave function' and 'microscopic system'.Although these ideas can be found already in some antecedent works [9,10], the first major contributions were made by the 'Marburg school' led by Ludwig [11,12].This resulted in an intense debate around the nascent GPT formalism, which took place in a series of papers published in Communications in Mathematical Physics [13][14][15][16][17][18].For an account of the early development of the field, we refer the interested reader to [2].A more modern point of view can be found in [4,Chapter 1].This foundationally motivated interest has seen a revival in the last two decades, with much effort being focused on attempts to 'reconstruct' quantum mechanics starting from first principles [19][20][21][22][23].At the same time, GPTs have become central to quantum information science, as they provide indispensable tools to analyse information processing beyond classical theories, see for instance [24][25][26][27][28][29][30][31][32].An introduction to the GPT framework that will suffice for our purposes can be found in [4, Chapter 2] (see also [32,Section 2]).Throughout this subsection, we limit ourselves to recalling the basics and to fixing the notation.Definition 1.A general probabilistic theory is a triple (V, C, u), where: (i) V is a finitedimensional real vector space; (ii) C ⊂ V is a closed, convex, salient and generating cone; and (iii) u, called the order unit or the unit effect, is a functional in the interior of the dual cone C * . .= {x * ∈ V * : x * (x) 0 ∀ x ∈ C}.GPTs will be denoted by capital letters such as A, B etc., which -with a slight abuse of notation -identify also the underlying physical systems.We call dim V the dimension of the GPT.
On the mathematical level, we can think of (V, C) as an ordered vector space, the ordering being given by x y ⇔ y − x ∈ C. Also the dual vector space can be thought of as ordered by the dual cone C * .In this language, the functional u is said to be strictly positive, since x 0 and u(x) = 0 implies x = 0.The states of the physical system modelled by (V, C, u) are represented by vectors in C ∩ u −1 (1) =: Ω.The compact convex set Ω is called the state space of the GPT, and accordingly we will sometimes refer to C as the cone of unnormalised states.Convexity here plays an operationally relevant role, as the process of preparing a system in a state ω 0 with probability p and ω 1 with probability 1−p, and later forgetting the value of the binary random variable associated with its preparation, leaves the system in the state pω 0 + (1 − p)ω 1 .
The GPT formalism allows us to make probabilistic predictions of the outcomes of measurements performed on a certain state.In this context, a measurement is a finite collection (e i ) i∈I of functionals in the order interval [0, u] (generically called effects) that add up to the order unit, i.e. such that i e i = u (normalisation).The probability of obtaining the outcome i upon measuring the state ω ∈ Ω is evaluated as e i (ω).Throughout this paper, we will always make the so-called norestriction hypothesis, which guarantees that all normalised collections of effects identify a physically legitimate measurement [33].We denote by M the set of all measurements associated with a certain GPT, adding a subscript to identify it if needed.
Equipping a vector space V with a GPT structure entails selecting a special positive functional on it, i.e. the unit effect u.In turn, this special functional can be used to define a norm on the dual space V * .By definition, the unit ball of this norm is the interval [−u, u], and for x * ∈ V * one has This choice makes V * a so-called order unit space [34,35].The corresponding Banach space structure induced on V is that of a base norm space [36].The norm on V is given by any of the two expressions The equivalence is an easy consequence of the strong duality of conic optimisation programs [37]; alternatively, it can be established by checking that the convex body K . .= conv(Ω ∪ −Ω) ⊂ V is the unit ball for the base norm, while x * = sup{|x * (x)| : x ∈ K} for any x * ∈ V * .This more or less exhausts the description of single systems within the GPT framework.Note that dynamics is not part of this very basic picture, which is limited to so-called 'prepare-andmeasure procedures'.Time evolution can be accounted for within this formalism, but this goes beyond the scope of the present paper.What we will need here is instead the extension of the formalism to the case of composite systems.We will be mainly concerned with the simplest case of a bipartite system formed by two subsystems A and B described by local GPTs (V A , C A , u A ) and (V B , C B , u B ).The theory modelling the composite AB will be denoted by AB = (V AB , C AB , u AB ).Under very reasonable assumptions [38,39], the main one being that bipartite states are always uniquely determined by the statistics produced by local measurements (a principle that goes under the name of local tomography), one can identify V AB with the tensor product of the local vector spaces, i.e.V AB V A ⊗ V B .When this identification is made, one obtains also u AB = u A ⊗ u B .To fully specify the joint system, one still needs to identify the cone of unnormalised states C AB .It turns out that such a choice cannot be made a priori on the ground of some indisputable axiom, but has to be based on some information regarding the actual physics of the system.However, the operational interpretation of the theory puts some nontrivial constraints on C AB , in the form of a lower and upper bound with respect to the inclusion relation.Namely, we have where The two constructions ( 5) and ( 6) are called minimal and maximal tensor product, respectively.In (4), the lower bound comes from the fact that any tensor product of local states must represent a valid state, while -dually -the fact that any tensor product of local effects must be an effect of the joint system leads to the upper bound.In (5) we used the notation In what follows, we will call admissible any composite AB whose associated cone C AB satisfies (4).Also, we will denote by A ⊗ min B and A ⊗ max B those corresponding to the choices (5) and (6) for C AB .
We conclude this brief presentation of the GPT formalism by discussing the two physically most relevant examples, i.e. classical probability theory and quantum mechanics.Classical probability theory can be viewed as the GPT where R d + is the cone of entrywise non-negative vectors, and u(x) . .= d i=1 x i for all x ∈ R d .The induced base norm coincides with the 1 -norm Composing classical systems is easy, for when either C A or C B is simplicial (i.e. a linear image of R d + ) minimal and maximal tensor product coincide.
An n-level quantum mechanical system is modelled by the GPT where M sa n is the space of n×n Hermitian matrices, PSD n the cone of positive semidefinite matrices, and tr represents the trace functional.The quantum mechanical base norm is the appropriate noncommutative generalisation of the 1 -norm, i.e. the trace norm X 1 . .= tr √ X † X.The base norm space of n × n Hermitian matrices endowed with the trace norm will be denoted by S n,sa 1 .In contrast with the classical case, for quantum mechanics composition rules become an issue.In fact, the standard quantum mechanical composition rule dictates that if A = QM n and B = QM m then AB = QM nm .The corresponding cone C AB is well known to make both inclusions in (4) strict.

B. Tensor norms
We start by recalling the basic theory of tensor products of normed spaces.In what follows B X . .= {x ∈ X : x 1} will denote the unit ball of a Banach space X.There are at least two canonical ways in which one can construct a norm on a generic tensor product X ⊗ Y of finitedimensional real Banach spaces [40,41].The injective norm of a tensor z ∈ X ⊗ Y is defined by the expression while its projective norm is given by In Subsection I A, we learnt that the vector space associated with a GPT carries a natural norm, i.e. the base norm given by (3).Since a joint system AB lives on the tensor product V A ⊗ V B of the local vector spaces, it is natural to ask whether any of the above tensor norms admits an operational interpretation in this context.Indeed, it turns out that [32, Proposition 22] for all admissible composites AB.The last equality tells us that the projective norm corresponds to the base norm associated with the minimal tensor product of the two theories.One may thus be led to conjecture that an analogous identity exists between • V A ⊗εV B and • A ⊗ max B , but the example of two classical probability theories reveals that this is not the case.We will find an adequate operational interpretation for the injective tensor norm in the forthcoming Subsection I C.
The most elementary properties of injective and projective norms is perhaps the inequality valid for all X, Y .Moreover, since the space X ⊗ Y is of finite dimension, these two norms will always be equivalent, i.e. there will exist an optimal constant 1 ρ(X, Y ) < ∞ such that It is straightforward to verify that the above constant is formally given by the following optimisation: For reasons that will soon become clear, in this paper we are interested in studying the range of values of the function ρ(X, Y ) across all pairs of spaces of fixed dimensions.
Definition 2. The projective/injective ratio, or π/ε ratio for short, is the following universal function over pairs of integers n, m 2: where the optimisation is understood to be over all pairs of finite-dimensional Banach spaces X, Y of fixed dimensions n, m.A slight modification of the above function (15) yields the symmetric projective/injective ratio: where the infimum is taken over all Banach spaces of dimension n 2.
One could equally well investigate analogous quantities where the infimum in the above optimisations is replaced by a supremum, however it turns out that these can be evaluated exactly.In fact, it has been shown that [32,Proposition 21] In light of this, in the rest of the paper we shall be concerned with the π/ε ratios as constructed in Definition 2. By taking as ansatzes X = n 1 and Y = m 2 in (15) (and assuming without loss of generality that n m), one can see that For an explicit proof, see the discussion before (56).To upper bound the symmetrised ratio one can take as ansatzes two copies of n 1 , which yields the slightly worse estimate [32, Example 29] Note that although r(n, n) r s (n) for all n, it may happen that r(n, n) < r s (n), i.e. the infimum in (15) does not need to be achieved on two copies of the same space even when n = m.The above inequalities exhaust the elementary properties of the π/ε ratios, and leave open many interesting questions, whose thorough investigation constitutes our main contribution.For a summary of the results we obtain on these quantities, we refer the reader to Section II.

C. XOR games
A simple but extremely useful setting where different physical models can be studied and compared from the point of view of information processing is that defined by XOR games.In these games, a referee interacts with two players Alice and Bob, who can cooperate with each other in order to maximise their winning probability.In the classical setting, the referee chooses a pair of questions according to a publicly known distribution and sends one question to each player.Then, the players are requested to provide a one-bit answer each, and the winning condition of the game, for a given pair of questions, only depends on the parity of the answers.In the basic local setting, the players can agree in advance on a strategy for their answers but they are not allowed to communicate with each other once the game has started.
These games are arguably central in theoretical computer science, mainly because of their simplicity and broad applicability to different topics such as interactive proof systems, hardness of approximation and the PCP theorem.In addition, XOR games have played a major role in quantum information theory since they were first considered in [42].In fact, these games had already been implicitly considered in the context of the study of quantum nonlocality [1,43], by means of their equivalent formulation in terms of correlation Bell inequalities.Their systematic study was initiated by Tsirelson [44].Far from being purely theoretical objects, in the last years these games have been crucial in the development of device-independent quantum cryptography and random numbers generators.
Motivated by their relevance for theory and applications, and drawing from previous works that put forth suitable quantum generalisations [45,46], in this paper we introduce XOR games in the context of GPTs.In this more general setting, the two players' system will be described by some bipartite GPT AB = (V A ⊗ V B , C AB , u A ⊗ u B ).The referee samples the questions from a finite alphabet I, the probability of drawing i being denoted by p i .The answers are represented by a collection of bits (c i ) i∈I ∈ {0, 1} |I| , while the questions are described by states Upon being asked the question ω i , the players output answers a ∈ {0, 1} and b ∈ {0, 1}, respectively, and the winning condition takes the form a ⊕ b = c i .The players' behaviour can be modelled by a suitable measurement M = (g ab ) ab∈{00,01,10,11} ∈ M AB over AB, with g ab (ω i ) representing the probability that the answers a, b are given when the question ω i has been asked.We can then formalise the notion of XOR game as follows.
Definition 3.An XOR game G is a quadruple (AB, ω, p, c), where: (i) AB is a bipartite GPT; (ii) ω = (ω i ) i∈I is a discrete collection of states over AB; (iii) p is a probability distribution over the set I; and (iv) c = (c i ) i∈I is a set of bits.A strategy for the players is a measurement M = (g ab ) ab∈{00,01,10,11} over AB.
The prototypical (and simplest) example of a strategy, called a local strategy, consists in the players performing a product measurement (e a ⊗ f b ) a,b∈{0,1} ∈ M A ⊗ M B .The opposite case is naturally that of a global strategy, corresponding to the case of Alice and Bob having access to global measurements (g ab ) a,b∈{0,1} ∈ M AB , but one can equally well consider some intermediate scenarios where the players are allowed a bounded amount of communication before they are required to output the answers.Some of these variations on the theme are examined in Appendix A. From the above picture it follows that XOR games can be equivalently formulated as instances of state discrimination problems, possibly subjected to some special constraints dictated by locality.
Before we delve into the study of XOR games over GPTs, let us point out some caveats in the terminology.According to the standard conventions, a classical XOR game is more than an XOR game played over the composite AB = Cl nm formed by two classical GPTs A = Cl n and B = Cl m defined by (7).In fact, it is usually understood that in this case the questions are taken from the standard basis of V AB = R nm , i.e. ω xy . .= v x ⊗ v y , where 1 x n and 1 y m are integers, and v k is the k-th vector in the standard basis of R k .In view of the perfect local distinguishability of the questions, one usually refers directly to the labels x and y as the questions.In compliance with the established conventions, from now on we will stick to the above definition of a classical XOR game.
We now flesh out the theory of XOR games in the GPT setting.One is usually interested in maximising the success probability of the players given a certain set of measurements they have access to.Since an XOR game can always be won with probability 1/2 by just answering randomly, it is standard to quantify the effectiveness of the players' strategy by introducing the bias β(G) of the game G: It is understood that the bias depends also on the strategy adopted by the players, which we specify with a subscript.The following result yields explicit expressions for the bias corresponding to local and global strategies.
Then the biases corresponding to local and global strategies evaluate to respectively.Here, • AB is the base norm associated with the GPT AB.In particular, we obtain β ALL (G) z G V A ⊗πV B for all composites AB, and the bound is achieved when AB = A ⊗ min B. Proof.It is not difficult to realise that when the players adopt global strategies, their task is equivalent to a state discrimination problem with all measurements on the composite system being available.Hence, (22) follows from [32, Lemma 7].As for (21), note that a pair of local measurements That the optimisation over all local measurements yields ( 21) is a consequence of the elementary fact that and analogously for system B. The last claim follows from (11).
Remark.The value of the bias under local strategies depends only on the local structure of the GPTs A and B, and is thus independent of the particular rule we chose to compose them.
Theorem 1 generalises the mathematical description of the non-entangled bias of classical and quantum XOR games [5,46], and yields the operational interpretation of the injective norm we were seeking.In Appendix A we show that this interpretation is 'robust', in the sense that even allowing the players a bounded amount of communication before they output the answers does not increase the bias by more than a factor equal to the product of the dimensions of the GPTs used to carry the messages.The same type of argument also shows that not much can be gained if the players have access to a pre-shared physical system of bounded dimension.
For classical XOR games more is true: namely, Tsirelson's theorem [44] states that even assistance by entangled quantum states of arbitrarily large dimension does not allow for a significant improvement over product strategies.However, this has to be regarded as a peculiar feature of quantum systems, deeply linked with the underlying Hilbert space structure through Grothendieck's inequality.It is therefore not surprising that it does not carry over to the general GPT setting.In fact, it turns out that a classical XOR game can always be won with probability 1 if one allows for assistance from a well-chosen set of non-signalling correlations1 (remember that non-signalling correlations can be viewed as GPTs [4,Section 2.3.4]).
We now illustrate the main motivation behind our investigation.In the scenario we consider, the referee's goal is to minimise the bias (equivalently, the winning probability) corresponding to local strategies for a fixed value of that corresponding to global strategies, effectively testing Alice and Bob's ability to perform joint measurements on their systems.On the other hand, the two players aim to win the game with high probability while employing only local strategies.They are free to choose the local systems A and B of fixed local dimensions that better suit their needs, as this choice may be dictated -among other things -by their experimental capabilities.However, they have no control over the composite AB the referee will be able to access nor over the questions they will be asked.Given these constraints, how large is the global/local bias ratio the referee can hope to achieve?
Having specified the setting, we now proceed to perform the quantitative analysis that will ultimately lead us to identify our main subject of study.Start by observing that (3) implies that every element z ∈ V A ⊗ V B such that z AB 1 for some legitimate composite AB is such that z = z G for some game G on AB (with just two questions).Hence, the maximal increase in the bias of an XOR game on AB when passing from the local to the global strategies is given by where the supremum on the l.h.s. is over all games on a fixed composite AB.Since the referee is also free to choose the optimal composite AB for given local systems A and B, we can also optimise over all composition rules.This is easily done by means of (11), and yields where the last step is an application of the definition (14).The above equation ( 25) is important because it connects the theory of XOR games over GPTs to that of tensor norms, and it can be used as a starting point to investigate some intrinsic aspects of the behaviour of information processing in bipartite systems.For instance, in [32] the optimal performance of data hiding against 'locally constrained sets of measurements' is connected to the quantity sup A,B ρ(V A , V B ), the supremum running over all GPTs A and B of fixed dimensions2 .In the setting we study here, instead, Alice and Bob's goal is to minimise the global/local bias ratio in (25) by making a clever choice of A and B in the first place.Hence, the relevant quantity is the infimum running over all GPTs A, B (equivalently, over all base norm spaces V A , V B ) of fixed dimensions.If we find that r bn (n, m) > 1 for all n, m, then there is a point in claiming that global strategies for XOR games perform better than local ones, independently of the underlying physical theories.If we manage to determine the asymptotic scaling of the quantity (26), we will even be able to make this statement quantitative.Comparing ( 26) with (15), it is elementary to observe that for all n, m 2, as the infimum that defines the r.h.s. is over all pairs of Banach spaces, while that on the l.h.s.includes only base norm spaces.In spite of this, thanks to the result of Appendix B we know that the two sides of ( 27) are in fact comparable.Namely, Lemma B.2 tells us that for all n, m 2 one has In light of the above equivalences, in the rest of the paper we shall study the function r instead of r bn .This simplifies our investigation considerably.Before we present our results, let us comment on the optimisation over the composition rules performed in (25).While we argued above that this may not be the most natural choice, it is conceivable to consider a modified scenario where Alice and Bob fix not only the systems A, B, but also the composite AB.Instead of (25), one should rather compute inf AB sup G β ALL (G)/β LO (G), and then take also the infimum over A and B of fixed dimensions.Remarkably, the choice of the composite is irrelevant when either A or B are classical theories.In this case, we are also able to give sharp estimates of the r.h.s. of (25) (see Remarks after Proposition 12 and Lemma 16).

II. MAIN RESULTS
Throughout this section, we present our main results on the universal functions r(n, m) and r s (n) introduced in ( 15) and ( 16), respectively.The discussion of the elementary properties of these objects, as conducted in Section I B, left many natural questions open.For instance, is it the case that r(n, m) c > 1 for all n, m 2 and for some universal constant c?In other words, is it true that for any pair of Banach spaces X, Y of dimension at least 2 there is a tensor z ∈ X ⊗ Y for which z X⊗πY c z X⊗εY ?Looking only at large enough dimensions, how do r(n, m) and r s (n) behave asymptotically in n, m?Some insight into this latter question was provided by Pisier [47], who showed that r(n, m) → ∞ when min{n, m} → ∞, but with no asymptotic growth explicitly stated.
Our first result answers the first of the above questions in the affirmative.
Theorem 2. For any pair of Banach spaces X, Y with dim X 2 and dim Y 2, we have ρ(X, Y ) 19/18.Equivalently, there exists a nonzero tensor z ∈ X ⊗ Y such that Consequently, the function r(n, m) defined in (15) satisfies r(n, m) Theorem 2 also applies to infinite-dimensional spaces and shows that the injective and projective tensor products cannot be isometric.Our proof of Theorem 2 requires a variation on Auerbach's lemma that is susceptible to an intuitive geometrical interpretation ( Lemma 13).
The following open problem asks whether the injective and projective tensor norms are always √ 2 apart.The value of √ 2 would be optimal, since ρ( We now move on to the analysis of the asymptotic behaviour of the function r s (n).Developing functional analysis techniques from [48], we arrive at a lower estimate of ρ(X, X) involving the Banach-Mazur distance of X from the Hilbert space of the same dimension (Theorem 14).Since by John's theorem this cannot exceed the square root of the dimension, we deduce the following estimate, which is sharp up to logarithmic factors (compare with (19)).
Theorem 4. The function r s defined in (16) satisfies for some universal constant c > 0.
Note.Throughout the paper we will denote universal (always strictly positive) constants by c, C, C etc. Unless explicitly indicated, these symbols do not necessarily refer to the same numerical values when they appear in different formulae.
The investigation of the function r(n, m) poses more substantial technical hurdles.For simplicity, we start by looking at r(n, n).Our main result is the following.
Theorem 5.The function r defined in (15) satisfies for some universal constant c > 0.
The above estimate shows in particular that r(n, n) grows at least as a power of n.In turn, this implies that global strategies for XOR games are intrinsically asymptotically (much) more efficient than local ones, which was one of our main claims.Our proof of Theorem 5 rests upon two main ingredients: (1) a lower estimate for ρ ( n 1 , X), ρ ( n ∞ , X) and ρ ( n 2 , X) when dim X n (Lemma 16), in turn based on the 'proportional Dvoretzky-Rogers factorisation' of [49], which we state as Theorem C.9; and (2) a 'trichotomy theorem' inspired by previous results in [7,50], which states that every Banach space hosts sufficiently well-behaved subspaces on which the norm is similar enough to either an 1 -norm, or an ∞ -norm, or a Euclidean norm (Theorem 17).
Due to the technical complexity of managing many different estimates simultaneously, for the case n = m we could not obtain an exponent as good as 1/6.However, we were nevertheless able to ensure that there is power law scaling in min{n, m}.
for some universal constant c > 0.
The exponents we obtained in Theorems 5 and 6 are unlikely to optimal.We present the following problem concerning the scaling of the projective/injective ratio.Problem 7. Does there exist a universal constant c > 0 such that r(n, m) c min{n, m} 1/2 (35) for all positive integers n, m?In other words, for all pairs of finite-dimensional Banach spaces X, Y , does there exist a nonzero tensor z ∈ X ⊗ Y such that While the value of r(n, m) grows with the dimension, new phenomena appear when considering infinite-dimensional spaces.Indeed, a famous construction by Pisier [8], solving negatively a conjecture by Grothendieck, entails that there exists an infinite-dimensional Banach space X such that ρ(X, X) < ∞.Using the information from Lemma B.1, we conclude that the same behaviour occurs in the realm of GPTs: there exist infinite-dimensional GPTs A, B such that local and global strategies are equivalent up to a universal constant in any possible composite AB.
The study of asymptotic behaviours should not distract us from the fact that certain GPT models are of prime importance because of their compliance with known physics.Therefore, one of our results is the determination of the quantity ρ(X, Y ) when X, Y are the base norm spaces corresponding to quantum mechanical systems, i.e. the Banach spaces S k,sa 1 .The following constitutes a notable improvement over [51,Lemma 20], [52,Theorem 15] and [32, Eq. ( 72)], as detailed below.
for some constants C, c > 0.More precisely, we have the following estimate for the upper bound in the above relation: Note.When comparing (37) with the other estimates on r(n, m) that we presented throughout this section, one should remember that the dimension of the space S n,sa 1 is n 2 rather than n.Hence, curiously, for X = S n,sa 1 the quantity ρ(X, X) ≈ n 3/2 is of the same order as the geometric mean between the theoretical minimum r s (n 2 ) ≈ n and the absolute maximum R(n 2 , n 2 ) = n 2 .Although we will not consider complex spaces in this work, it is worth mentioning that the same estimate holds (with a slight modification in the constants) if the real space S k,sa 1 is replaced by the complex space S k 1 of k × k matrices endowed with the trace norm.Among other things, the above Theorem 8 enables us to give a new upper bound on the maximal efficiency of quantum mechanical data hiding against local measurements, quantitatively encoded in the data hiding ratio function R LO (n, m) defined in [32].
Corollary 9.For all Hermitian matrices Z acting on a bipartite system C n ⊗ C m , we have where • LO is the distinguishability norm under local measurements [52].In other words, the data hiding ratio against local measurements can be upper bounded as The above result improves upon several previously known estimates.In [51, Lemma 20], an inequality analogous to (40) was shown, that featured an exponent 2 on the r.h.s.; moreover, the relation proven in [52,Theorem 15] implies that R LO (n, m) √ 153 nm, which can be worse than (40) e.g. when n 2 m or m 2 n.

A. Some notions of functional analysis
We start by reminding the reader of some facts in elementary linear algebra.Given a pair of finite-dimensional vector spaces X, Y , there is a canonical isomorphism between the tensor product space X ⊗ Y and the space of linear maps X * → Y .We will write this correspondence as Note that one has F (z) = z * , where on the l.h.s.we have the flip operator F : X ⊗ Y → Y ⊗ X defined by F (x ⊗ y) . .= y ⊗ x, while on the r.h.s.(•) * stands for the adjoint (transposition) operation L(X * , Y ) → L(Y * , X).It is also easy to see that one has where tr denotes the trace.
In this paper we are interested in tensor products of finite-dimensional real Banach spaces, so from now on X and Y will denote a pair of such objects.We already encountered the concepts of injective and projective tensor products (see ( 9) and ( 10)).Below we discuss some elementary properties of these constructions.For a start, injective and projective norm are dual to each other, in the sense that By means of the correspondence (41), it is possible to translate tensor norms into the language of operators.One has where • N is the so-called nuclear norm [53, Section 8].
The Banach-Mazur distance between two normed spaces X, Y with the same finite dimension is defined as [53] where the infimum is taken over invertible linear maps u from X to Y .It satisfies the multiplicative version of the triangle inequality, i.e. d(X, The Banach-Mazur distance only makes sense for a pair of spaces of equal dimension.When X, Y are normed spaces such that dim X dim Y , one may define as a substitute the factorisation constant of X through Y as Clearly, f(X, Y ) is finite if and only if dim X dim Y (otherwise the above infimum is over an empty set).In order to circumvent this restriction, it is sometimes relevant to consider a relaxed version of the above quantity, where we allow the factorisation to be realised only after averaging.This leads to the definition of the weak factorisation constant as where now the infimum is taken over pairs of operator-valued random variables (u, v).We get the inequality wf(X, Y ) f(X, Y ) by restricting the infimum to constant random variables.Note also that the quantity wf(X, Y ) is well-defined without any restriction on the dimensions of X and Y .It is easy to verify that the factorisation constants dualise, in the sense that We may also consider a symmetric variant of the weak factorisation constant, called the weak Banach-Mazur distance and defined as Let us review some classical facts about the geometry of the Banach-Mazur compactum, i.e. the space of all normed spaces of fixed dimension, up to isometries.For a more complete introduction, we refer the reader to the excellent monograph [53].A general upper bound valid for any ndimensional normed space is the estimate following from John's theorem (Theorem C.7) As a consequence, for any pair of n-dimensional spaces X, Y , we have This bound is essentially sharp: Gluskin proved, via a random construction, the existence of spaces X, Y such that d(X, Y ) cn for some c > 0 [54].However, the estimate ( 52) can be improved in many specific cases.In particular, a question that is of relevance to us is that of the distance between a space and its dual: it was proved by Bourgain and Milman [48] that whenever dim(X) = n, we have In a similar vein, (52) can be improved if we switch to the weak Banach-Mazur distance: a result by Rudelson [7] asserts that wd(X, Y ) Cn 13/14 log C n (54) whenever dim(X) = dim(Y ) = n.We also point out that the exact diameter of the Banach-Mazur compactum is known in dimension 2: namely, equality being achieved when the unit balls of X and Y are respectively a square and a regular hexagon [55].

B. Basic properties
We start the investigation of the elementary properties of the parameter ρ(X, Y ) by presenting a simple calculation in the case X = n 1 , Y = m 2 , where n m.For a matrix z ∈ n 1 ⊗ m 2 , we write where to obtain the first inequality we randomised over σ, assuming that σ 1 , . . ., σ n are i.i.d.±1 Bernoulli random variables.The above computation shows that z n √ n.That this upper bound is in fact tight can be seen by considering the matrix z with entries z ij = δ i,j (i.e. the identity if m = n).We conclude that entailing the upper bound (18) on r(n, m).
We now move on to investigating some more general properties of the function ρ(X, Y ).We start by stating a very useful reformulation in terms of operators rather than tensors.
Lemma 10.For any pair of finite-dimensional normed spaces X, Y , we have Proof.The statement follows from the properties of injective and projective norms under the cor-respondence (41).We write The above passages can be justified as follows: 1: we used the duality relation (43); 2: we applied ( 42), ( 44) and ( 45); 3: we just renamed v . .= ( z) * and u . .= w.
The following proposition gathers several estimates of the function ρ(X, Y ).
In particular, when dim(X) = dim(Y ) = n we have Proof.The identities (58) can be obtained for instance from (57) by exchanging the role of u and v and/or taking their duals u * , v * .Remember that one has tr(uv) = tr(vu) = tr(u * v * ) = tr(v * u * ).Also ( 59) is elementary, and follows by taking u = v = Id X in (57).
To show (60), pick some operators u : X → Y * and v : Y * → X of norm no larger than 1, and consider random variables u : X → X and v : X → X such that E[v u ] = Id X .Since for all realisations of u and v the operator uv v : X → Y * has norm no larger than 1, and an analogous reasoning holds for u v u : Y * → X, using (57) we deduce that Then we can write Taking the supremum over u, v and the infimum over u , v subjected to the above constraints, and using ( 48) and ( 57), we finally obtain (60).Since wf(X , X) f(X , X), (61) follows as well.To prove (62), note that d(X, X ) = f(X, X ) whenever dim(X) = dim(X ).Note that (63) also follows immediately: where: 1: follows from ( 59) and ( 58); 2: is an application of (60); and 3: is again a consequence of ( 58).The estimate (64) was proved in [32, Proposition 21] (compare with ( 17)) as a consequence of Auerbach's lemma [56,Vol. I,Sec. 1.c.3].An alternative proof can be given using the following fact from linear algebra (left as an exercise for the reader): every linear map w on R n that is a contraction for some norm satisfies tr(w) rank(w).To recover (64), apply this observation to the composite map w = vu, with u, v as in (57).
Remark.It follows in particular from (61) that any upper bound on ρ(X, Y ) is also valid for 1complemented subspaces of X and Y .Recall that a subspace X ⊆ X is λ-complemented if there is a projection P : X → X with P λ.The complementation hypothesis cannot be omitted.To give a concrete example, consider a cn-dimensional subspace X ⊆ n 1 with d(X, cn 2 ) C (existence of such a subspace is well known and follows for example from Dvoretzky-Milman theorem We point out that weaker versions of Theorems 4 and 5 follow easily by combining Proposition 11 and 'off the shelf' results.More precisely, the lower bound r s (n) cn 1/6 /(log n) C is an immediate consequence of (65) and ( 53), and the lower bound r(n, n) cn 1/14 /(log n) C follows from ( 63) and (54).Interestingly, the special case n = m of Problem 7 would follow from (65) if one could prove Rudelson's conjecture that wf(X, Y ) C √ n for all n-dimensional Banach spaces X, Y .

C. Universal lower bounds
In this section we prove that the injective and projective tensor products of any two Banach spaces cannot be isometric, unless one of them is 1-dimensional.That this is the case when both dimensions are equal to 2 can be seen by combining (55) (61), it suffices to consider n = 2.In that case, there are explicit formulas for both the projective and injective norms: for any z = It remains to justify that any Banach space Y contains two vectors y 1 and y 2 such that This follows from some properties of the so-called modulus of uniform convexity of Y , a real function defined for ε ∈ [0, 2] by It is known [57] that for any Banach space Y and any ε ∈ [0, 2], we have δ Y (ε) 1 − 1 − ε 2 /4 (the value obtained for a Euclidean space).Applying this inequality with ε = √ 2 show the existence of unit vectors y 1 , y 2 such that y 1 − y 2 = √ 2 and y 1 + y 2 √ 2, and therefore (69) is satisfied.
Remark.An immediate consequence of Proposition 12 is that there is always a gap at least as large as √ 2 between local and global bias for XOR games played over a system AB in which either A or B is a classical theory (7).
We now proceed to establish the universal lower bound r(n, m) 19/18, formalised earlier as Theorem 2. Our main technical tool is the following variant of Auerbach's lemma, illustrated in Figure 1.
Lemma 13.Let X be a Banach space of dimension at least 2. Then there exist vectors e 1 , e 2 ∈ X, e * 1 , e * 2 ∈ X * such that for any i, j ∈ {1, 2} we have e i X = e * j X * = 1, e * j (e i ) = δ i,j , and moreover Proof.It is enough to prove the lemma when dim X = 2, since the general case follows by considering any 2-dimensional subspace Y ⊆ X and extending the linear forms.Suppose now dim X = 2. Without loss of generality we may assume that X = (R 2 , • X ), and identify as well X * with R 2 .By applying a suitable linear transformation, we may assume that the variational problem max{| det(f, g)| : f, g ∈ B X * } is achieved when (f, g) = (e 1 , e 2 ), the canonical basis.It is clear that e 1 X * = e 2 X * = 1, and one checks that e 1 X = e 2 X = 1.Let us show this explicitly for e 1 .On the one hand, 1 = e 1 , e 1 e 1 X * e 1 X = e 1 X .On the other hand, if e 1 X > 1 one could find a vector f ∈ B X * such that f , e 1 > 1, which would imply that | det(f , e 2 )| > 1 = | det(e 1 , e 2 )|, in contradiction with the assumption that the pair (e 1 , e 2 ) achieves the maximum in the above variational expression.
Proof of Theorem 2. Apply Lemma 13 to both X and Y , and consider the tensor Consider also Since the linear forms e * i are bounded in absolute value by 1 on the unit ball, an argument following closely the proof of the CHSH inequality [43] shows that w * X * ⊗εY * Both pairs (a, b) and (c, d) belong to the hexagon Under these constraints it can be proved that 5ac + 5ad + 5bc − 4bd 9 .
Indeed, it suffices to verify this inequality when (a, b) and (c, d) are extreme points of H; this yields a total of 36 different combinations to check.Finally, we have Combining ( 70) with (71) gives z X⊗πY 19 18 z X⊗εY , as needed.Remark.The proof of Lemma 13 gives more information, namely that e 1 + e 2 X α, e 1 − e 2 X β for some real numbers α, β ∈ [1,2] such that α + β 3.This extra information can presumably be used to improve the lower bound in Theorem 2 to z X⊗πY 8 7 z X⊗εY for an appropriate choice of z depending on α, β.However, since our arguments for that would rely heavily on computer assistance (and since the bound 8/7 is unlikely to be optimal), we do not present them.

D. An important special case: quantum theory
In this section we will study the case where both parties are described by a quantum model.Before we start, let us expound some notation that we already partially introduced.We denote by M sa k the real vector space of k × k Hermitian matrices.By equipping it with a Schatten norm • p , defined by z p . .= (tr |z| p ) 1/p , we can make such space a Banach space, which we denote by S k,sa p .In what follows, we will be interested in the two particular cases p = 1 and p = ∞, whose corresponding norms are the trace norm and the operator norm, respectively.For simplicity, we will make the canonical identification (S k,sa 1 ) * = S k,sa ∞ .Accordingly, the action of (S k,sa 1 ) * on S k,sa 1 is given simply by the Hilbert-Schmidt inner product, i.e. y(x) = tr[xy] for x ∈ S k,sa 1 and y ∈ S k,sa ∞ .As for the tensor product, remember that M sa n ⊗ M sa m = M sa nm canonically.We now proceed to prove Theorem 8, hence determining the scaling of the function ρ(S n,sa 1 , S m,sa

1
) with respect to n and m.The proof of the Corollary 9 appears at the end of the present section.
Note.From now on, in some of the proofs we will find it convenient to adopt Dirac's notation for vectors and functionals in (or acting on) R n and C n .This will be done without further comments.
Proof of Theorem 8.In order to establish (37), we have to show the existence of two constants c, C > 0 such that c min{n, m} 3/2 sup We break down the argument to prove (72) into two parts.
Step 1: lower bound.We assume without loss of generality that n m, and consider two Hilbert-Schmidt orthonormal bases (x i ) 1 i n 2 and (y j ) 1 j m 2 of M sa n and M sa m , respectively.We form the random tensor where (g ij ) are independent N (0, 1) Gaussian random variables.Let us observe that the distribution of z does not depend on the choice of the local orthonormal bases.We use the results from Corollary C.4: By duality, (75) implies a lower bound on the projective norm of z in S n,sa . More precisely, using the duality between S n,sa ∞ together with the Cauchy-Schwartz inequality, we obtain Since the l.h.s. of (76) is of order nm, combining (76) and (75) yields the lower bound Using the above relation together with (74), we see that the random variable has a nonnegative expectation.In particular, the event {U 0} is nonempty, from which it follows that ρ(S n,sa Step 2: upper bound.As before, we assume, without loss of generality, that n m.Let us consider an element z ∈ M sa n ⊗ M sa m such that z S n,sa for every x ∈ M sa n (the notation z was introduced in ( 41)).In Dirac's notation, the spectral decomposition of ϕ reads

A. A lower bound for two copies of the same theory
In this section we prove that r s (n) c √ n/ log 3 n for a certain universal constant c > 0, which is the technically challenging part of the statement of Theorem 4. In fact, remember that the example of n 1 shows that r s (n) √ 2n (as reported in ( 19)), hence the aforementioned result is optimal up to logarithmic factors.For an n-dimensional Banach space we denote d X = d(X, n 2 ), where d is the Banach-Mazur distance.
Theorem 14.There exists a universal constant c > 0 such that for every Banach space X of dimension n, we have ρ(X, X) cn d X log 3 n .
In particular, r s (n) c √ n/ log 3 n.
Proof.The lower bound on r s (n) follows immediately by combining ( 14) and the well-known estimate d X √ n in (51).We now set out to prove (14).We may assume that X is equal to Our main tool is the following lemma, whose proof we postpone.It is based on ideas from [48] (see also [53,Lemma 46.2] and comments below it).
Lemma 15.Consider a Banach space X = (R n , • X ), and assume that 1 Here, c and C are universal constants and (g i ) n i=1 is a sequence of independent N (0, 1) Gaussian random variables.Lemma 15, and form the random tensors

Let us consider the vectors
where (g ij ) are independent N (0, 1) Gaussian random variables.It is clear that On the other hand, according to Chevet's inequality (Theorem C.3), we have where Id K denotes the identity map restricted to K . .= span{|f i : 1 i k}.We can then write The justification of the above steps is as follows: 1: we used the duality of injective and projective norm (43); 2: follows by definition of ρ(X, X); 3: is an application of the Cauchy-Schwarz inequality; 4: is the p = 2 case of the Khintchine-Kahane inequality (Theorem C.2); 5: derives from ( 83) and (84); 6: can be derived from (80), using the fact that orthogonal projections onto subspaces of Hilbert spaces have norm 1; 7: is the statement of Lemma 15.Combining the above estimate with the lower bound in (82), we deduce that ρ(X, X) c n d X log 3 n , which concludes the proof.
Proof of Lemma 15.According to the M M * -estimate (Theorem C.1), there exists an isomorphism T : n 2 → X such that Moreover, since X (T ) = X (T • U ) for every unitary U , it can be assumed that T is positive definite.By the spectral theorem, T can be written as for some positive numbers λ i and (|f i ) n i=1 an orthonormal basis of R n .Then, inequality (85) implies that Using the inequalities |f i X 1/ √ n and f j | X * 1 together with Jensen inequality (or Lemma C.5), we see that λ Cn log n , completing the argument.

B. A lower bound for any pair of theories
The aim of this section is to prove Theorems 5 and 6, which provide general lower bounds on ρ(X, Y ) as functions of the dimensions n, m only.As discussed in Section II, our strategy requires two preliminary results that allow to reduce the problem to the more manageable special case where either X or Y is one of the 'classical' spaces n 1 , n 2 or n ∞ .We start by presenting the solution to these special cases.
Case (ii) : λ n/3 > A/ √ n.Let F = span{e i : i > n/3} and denote by Id F : F → R n the identity map restricted to F .We have To see the previous bound, just write We apply the same dichotomy to X * .If case (i) occurs for either X or X * , we are done.It remains to consider the situation when case (ii) occurs for both.This means that there exist subspaces F 1 and F 2 of dimension 2n/3 such that Consider the subspace F = F 1 ∩ F 2 (note that dim F n/3).We are going to apply the Dvoretzky-Milman theorem (Theorem C.6) to both X ∩ F and X * ∩ F (that is, to the space F with the norms inherited from X and from X * respectively).The corresponding Dvoretzky dimensions are On the other hand, if we consider the random vector g = dim F i=1 g i f i , where (g i ) are independent N (0, 1) Gaussian random variables and (f i ) is an orthonormal basis in F , we compute where 1. follows from Cauchy-Schwarz inequality together with the inequality |g| 2 g X g X * , and 2. from Khintchine-Kahane inequalities (Theorem C.2). Since X * (P F ) X * (Id) C √ n log n, we have X (P F ) c n/ log n, and similarly X * (P F ) c n/ log n.It follows that k * (X ∩ F ) cA 2 / log n and k * (X * ∩ F ) cA 2 / log n.
By Dvoretzky-Milman theorem (Theorem C.6) and the remark following it, there is a subspace E ⊆ F of dimension cA 2 / log n such that both X ∩ E and X * ∩ E are 4-Euclidean.Moreover, using the extra information given by Theorem C.6, we have and therefore E is a C log n-complemented 4-Euclidean subspace of X.
Proof of Theorem 5.As usual, the difficult part is to establish the lower bound on r(n, n), while the upper bound is reported in (18) (set n = m).Let X and Y be n-dimensional normed spaces, and A > 1 be a number whose value will be optimised later.Theorem 17 implies in particular that at least one of the following occurs (here we use the classical fact that a subspace isometric to m ∞ is automatically 1-complemented): If (i) holds, then by Lemma 16 and ( 61) we obtain A similar estimate applies when X satisfies (ii), or when Y satisfies (i) or (ii).The only uncovered case is when X and Y both satisfy (iii), and we have then The optimal choice is A = n 1/12 (log n) 5/6 , which gives the announced lower bound.
Proof of Theorem 6.The upper bound on r(n, m) follows again from ( 18), so we focus on the lower bound.Let X and Y be normed spaces of respective dimensions n and m (with n m), and A > 1 be a number whose value will be optimised later.As is the previous proof, we combine Theorem 17 (applied only to X) and Lemma 16.In case (i), we have Case (ii) is similar by duality.In case (iii), we have The optimal choice A = n 1/8 √ log n always gives the lower bound ρ(X, Y ) cn 1/8 / log n, concluding the proof.

V. CONCLUSIONS
In this paper, we have defined and investigated XOR games from the foundational standpoint provided by general probabilistic theories.This has led us to identify a deep connection between the minimum relative increase in the bias when global strategies displace local ones on the one hand, and the so-called projective/injective ratio on the other.The existence of such a connection is made possible by the fact that all norms on a given vector space can be well approximated by suitable base norms induced by GPTs.
The projective/injective ratio r(n, m) is a universal function over pairs of integers that encodes some structural information about the theory of Banach spaces.For instance, we have shown that n/r(n, n) provides a lower bound on the diameter of the Banach-Mazur compactum in dimension n as measured by the weak distance [6,7].We have also proved that r(n, m) is always lower bounded by a universal constant strictly larger than 1.This shows the remarkable fact that injective and projective tensor product can never be isometric, even though Pisier's celebrated construction [8] demonstrates that they can be isomorphic in the case where the spaces have infinite dimension.Along the way, we developed an Auerbach-type lemma that may be of independent interest.
The main results we have presented concern the asymptotic behaviour of the ratio r(n, m) and of its symmetrised version r s (n).In this context, we were able to show that, up to logarithmic factors, r s (n) is of the order √ n.We showed that r(n, m) grows at least as min{n, m} 1/8 , and that one can improve the exponent to 1/6 if n = m.The proofs of these latter results follow by putting together an understanding of the projective/injective ratio in tensor products of the form d p ⊗ X, with p = 1, 2, ∞, and a 'trichotomy theorem' that identifies in any normed space a sufficiently large subspace that is close in the Banach-Mazur distance to either (a The main technical hurdle consists in establishing the additional requirement that in case (b) the chosen subspace is also well-complemented.As we have discussed, our findings draw on previous ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann.
Finally, although our primary subject of study is the intrinsic theory of XOR games played over general physical systems, it would be futile to deny that quantum systems hold great importance, due to their omnipresence in Nature as we currently understand it.In this spirit, we determined the exact scaling of the maximal global/local bias ratio in quantum XOR games, finding that it is of the order min{n, m} 3/2 , with n, m being the local Hilbert space dimensions.Interestingly, this implies a new bound on the maximal strength of quantum mechanical data hiding against local operations.
Our work leaves a number of open problems that we believe are worth investigating.Let us recall briefly some of them.First, it would be interesting to compute exactly the absolute minimum of r(n, m) across all pairs of integers, which one may conjecture to be equal to √ 2. A perhaps more profound question is to determine the best exponent γ opt such that r(n, m) c min{n, m} γopt for all n, m.We ask whether γ opt = 1/2.As we have seen, the simplified statement with n = m would follow from Rudelson's conjecture [7] that the Banach-Mazur compactum in dimension n has a diameter of the order √ n with respect to the weak distance.

B. EVERY NORMED SPACE IS 2-ISOMORPHIC TO A BASE NORM SPACE
In this appendix we justify our choice of characterising the intrinsic difference between global and local strategies in XOR games by means of the projective/injective ratio as defined by (15) instead of ( 26), as discussed in Subsection I C.This corresponds to letting the optimisation run over all pairs of Banach spaces of fixed dimensions instead of restricting it to the base norm spaces alone, and does not lead to a significant loss of information because of the inequalities ( 28) and ( 29), whose proof we present here.Let us start with a preliminary result.
Lemma B.1.Every Banach space (possibly infinite-dimensional) is 2-isomorphic to a base norm space.
Proof.Let X be a Banach space.Pick a unit vector x ∈ X such that x = 1, and consider the associated norming functional x * ∈ X * , which satisfies x * = 1 and x * (x) = 1.Calling B X the unit ball of X, construct the set F . .= (x * ) −1 (1/2) ∩ B X , and then set B . .= cl conv (F ∪ (−F )), where the closure is possibly needed only in the infinite-dimensional case.It is not difficult to verify that B is the unit ball of the base norm space induced on X by the positive cone x /2} and the unit functional u . .= 2x * [2, p. 26].Since B ⊆ B X , it suffices to check that B X ⊆ 2B to establish the claim.To this end, we pick y ∈ B X and we check that y 2 ∈ B. We can assume without loss of generality that x * (y) 0, while |x * (y)| 1 holds by construction.We now distinguish two cases.
Observe that z + lies at the intersection of the segment joining y and x with the plane x * = 1/2.Analogously, z − lies at the intersection of the segment joining y and −x with the plane x * = −1/2.In particular, z ± ∈ B. We now try to obtain a multiple of y by taking a convex combination of z + and z − .Setting By rescaling the vector on the l.h.s.we see that y/2 ∈ B.
This concludes the proof.We are now ready to prove the inequalities ( 28) and ( 29) discussed in the main text.Proof.Lemma B.1 proves that for all Banach spaces X there is a base norm Banach space X such that d(X, X ) 2, where d is the Banach-Mazur distance (46).We apply this to a pair of finite-dimensional Banach spaces X, Y , with dim X = n and dim Y = m, obtaining two base norm Banach spaces X and Y of the same dimension that are 2-isomorphic to X and Y , respectively.We find that r bn (n, m) ρ(X , Y ) d(X , X)ρ(X, Y ) d(X , X) d(Y , Y )ρ(X, Y ) 4ρ(X, Y ) , where we used (62) twice, once for each of the two arguments of ρ (this is possible as ρ is symmetric, see (58)).Taking the infimum over all pairs X, Y yields (28).We now move on to proving the second inequality (29).The main idea of the argument is to construct, given a pair of Banach spaces X, Y of dimensions n − 1, m − 1, another pair of base norm spaces X , Y of dimensions n, m such that ρ(X , Y ) ≈ ρ(X, Y ).This can be done by setting X . .= X ⊕ ∞ R, where (x, a) X = max{ x X , |a|} for all x ∈ X and a ∈ R, and analogously for Y .It is not difficult to check that • X is in fact the base norm induced by the cone C . .= {(x, a) : a x X } and the order unit u X ∈ (X ) * defined by u X (x, a) = a for all x ∈ X and a ∈ R. Thus, X and Y are base norm spaces.Incidentally, this is a systematic way of associating 'centrally symmetric' GPTs to Banach spaces, see [32,Section 6.1].We now proceed to show that ρ(X , Y ) 2 + ρ(X, Y ) , (B1) using a similar technique to that employed in the proof of [32,Proposition 26].Take where z ∈ X ⊗ Y , s ∈ X, t ∈ Y , and a ∈ R. Using the fact that a unit functional ϕ ∈ B (X ) * acts as ϕ(x, a) = px * (x) ± (1 − p)a, for some x * ∈ B X * and p ∈ [0, 1], it is not difficult to show that z X ⊗εY = max { z X⊗εY , s , t , |a|} .(B2) We now give an upper estimate of the corresponding projective norm.Taking vectors x i ∈ X and y i ∈ Y such that z = i x i ⊗ y i and z X⊗πY = i x i y i , and for an arbitrary p ∈ [0, 1], we consider the decomposition Optimising over all z ∈ X ⊗ Y and using (14) gives the estimate in (B1).We can now write This concludes the proof.
Note that, given a Banach space Z and (z i ) n i=1 ⊂ Z, we have Here T is the linear map defined by T |e i = z i for every i = 1, • • • , n, where (|e i ) i is an orthonormal basis of n 2 .Here is a typical application of Chevet's inequality.Fix integers m, n, and consider (x i ) 1 i m 2 and (y j ) 1 j n 2 orthonormal bases of M sa m and M sa n respectively, with respect to the Hilbert-Schmidt inner product.We form the random tensor where (g ij ) are independent N (0, 1) Gaussian random variables.Proof.In both cases we apply Theorem C.3 and need to estimate all quantities appearing on the r.h.s.The random matrix G m = m j=1 g j x j is distributed according to the Gaussian Unitary Ensemble.It is well known (see for example [60,Proposition 6.24]) that as m tends to infinity, On the other hand, it is also well known (and easy to check) that hence the result follows.
Finally, we will use the following lemma, whose proof is elementary.
Lemma C.5 (Contraction principle).Let (α i ) n i=1 and (β i ) n i=1 be two sequences of numbers with 0 α i β i for every i.Let (g i ) n i=1 be a sequence of independent N (0, 1) Gaussian random variables.Then, for every Banach space X and every x 1 , • • • , x n ∈ X, we have

Theorem 8 . 1 ⊗ S m,sa 1 satisfiesc 1 , S m,sa 1 )
Denoting by S k,sa 1 the space of k ×k Hermitian matrices endowed with the trace norm, the best constant of domination of • X⊗εY over • X⊗πY on S n,sa min{n, m} 3/2 ρ (S n,sa C min{n, m}3/2

i λ − 1 jλ j m 2 s
Cn 3/2 log n holds for any indices i, j.Let us denote m = min{λ k : 1 k n} and M = max{λ k , 1 k n}.It follows that M m Cn 3/2 log n .(86) Now, (86) implies that the sets A s = 1 j n : 2 s−1 with s = 1, • • • , r, define a partition of {1, • • • , n} for a certain r C log n.By the pigeonhole principle, one can immediately deduce the existence of a set A s0 such that |A s0 | n C log n .Now, consider the set of orthonormal vectors {|f j : j ∈ A s0 }.Applying Lemma C.5, we see that

FIG. 2 .
FIG.2.Geometric intuition behind the proof of Lemma B.2, to which we refer for notation.The plane depicted is that spanned by the vectors x and y.From now on, all sets we consider are understood to be intersected with this plane.The curved solid and dashed line represents part of the boundary of the unit ball BX (we set y = 1 for simplicity).The points a and b are the intersections of the line x * = 1/2 with the boundary of BX .The associated parallelogram B is depicted in dark grey colour, while 2B is in light grey colour.Then, an informal version of the argument is as follows.Since between x and b the boundary of BX lies above the dash-dotted line, convexity implies that it must lie below that same line between b and y.This ensures that y belongs to the dash-dotted tilted parallelogram, which is readily verified to fit inside 2B.

Corollary C. 4 . 1 ⊗εS
Let z be defined as in (C2).Remember that we denote by S n,sa p the space of n × n Hermitian matrices equipped with the Schatten norm • p .Then E z S m,sa

||Id : S m,sa 2 → 2 →Since S m,sa 2 is isometric to m 2 2 , 1 ⊗εS
S m,sa ∞ || 1, ||Id : S m,sa we can apply Theorem C.3 to conclude that E z S m,sa Is it true that for any pair of Banach spaces X, Y with dim X, dim Y 2, we have ρ(X, Y ) √ 2? In other words, does there always exist a nonzero tensor z ∈ X ⊗ Y such that Problem 3 asks whether the injective and projective norms are always at least √ 2 apart.As a partial answer, we present a proof in the special case when one of the spaces is n ∞ (Proposition 12).Then, we solve a weaker version of Problem 3, with √ 2 replaced by the value 19/18 (Theorem 2).