Parallel Repetition for k-Player XOR Games

In a $k$-XOR game $\mathrm{\Psi }$ over ${\mathbb{F}}_p$, a verifier samples challenges $(x_1,\dots ,x_k)\sim \mu$ where $\mu$ is a distribution over ${\mathrm{\Sigma }}_1\times \dots \times {\mathrm{\Sigma }}_k$, sends $x_i$ to player $i$ and receives an answer $f_i(x_i)\in {\mathbb{F}}_p$, and accepts if and only if $f_1(x_1)+\dots +f_k(x_k)=t(x_1,\dots ,x_k)$, where $t\text{\hspace{0.17em}⁣}:{\mathrm{\Sigma }}_1\times \dots \times {\mathrm{\Sigma }}_k\to {\mathbb{F}}_p$ is a fixed map. We show that if $\mathrm{\Psi }$ has value strictly less than $1$, then the value of the $n$-fold repeated game ${\mathrm{\Psi }}^{\otimes n}$ is at most $2^{-cn}$ where $c=c(\mathrm{\Psi })>0$ is a fixed constant. This extends a previous result of [Bhangale-Braverman-Khot-Liu-Minzer] showing exponential decay for some $3$-XOR games. Our proof uses a relaxation of the value of games studied in the quantum setting, called the commuting value of a game. We show a transformation that maps a $k$-player game $\mathrm{\Psi }$ with value strictly less than $1$ to a $2k$-player game ${\mathrm{\Psi }}^{\mathrm{\prime }}$ with commuting value less than $1$, such that $\mathsf{v}\mathsf{a}\mathsf{l}({\mathrm{\Psi }}^{\otimes n})\le \mathsf{v}\mathsf{a}\mathsf{l}(({\mathrm{\Psi }}^{\mathrm{\prime }}{)}^{\otimes n})$. We prove that $\mathsf{v}\mathsf{a}\mathsf{l}({\mathrm{\Psi }}^{\mathrm{\prime }})<1$ has a direct proof using nothing more than the Cauchy-Schwarz inequality, which we are able to tensorize to get that $\mathsf{v}\mathsf{a}\mathsf{l}(({\mathrm{\Psi }}^{\mathrm{\prime }}{)}^{\otimes n})$ vanishes exponentially fast in $n$.