Universal gaps for XOR games from estimates on tensor norm ratios

Understand

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.