Universal gaps for XOR games from estimates on tensor norm ratios

TL;DR

This paper investigates the fundamental differences between local and global strategies in XOR games through tensor norm ratios, revealing universal gaps that grow with system size and applying these findings to quantum data hiding efficiency.

Contribution

It introduces the projective-injective ratio for tensor norms, establishes lower bounds and asymptotic growth rates, and connects these mathematical results to operational advantages in XOR games and quantum data hiding.

Findings

The ratio r(n,m) is at least 19/18 for all n,m≥2.

r_s(n) scales as the square root of n, up to log factors.

The gap between local and global strategies grows polynomially with system dimension.

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 $\|\cdot\|_{X\otimes_\pi Y}\leq\rho\,\|\cdot\|_{X\otimes_\varepsilon Y}$ holds for all Banach spaces of dimensions $\dim X=n$ and $\dim Y=m$, where $X\otimes_\pi Y$ 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 $r(n,m)\geq 19/18$ for all $n,m\geq 2$, 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.