Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

Abstract

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 rho such that Vert cdot Vert _{Xotimes _pi Y}leqslant rho ,Vert cdot Vert _{Xotimes _varepsilon Y} holds for all Banach spaces of dimensions dim X=n and dim Y=m, where Xotimes _pi Y and X otimes _varepsilon Y are the projective and injective tensor products. By requiring that X=Y, one obtains a symmetrised version of the above ratio, denoted by r_s(n). We prove that r(n,m)geqslant 19/18 for all n,mgeqslant 2, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and r_s(n), showing that, up to log factors: r_s(n) is of the order sqrt{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 ‘ell _1/ell _2/ell _{infty } 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.