Quantum chi-squared tomography and mutual information testing

Abstract: For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $$\widetilde{O}(r^{{.5}d^{{1.5}/\epsilon)}} \leq \widetilde{O}(d<sup>2/\epsilon)$$ copies suffice for accuracy~$\epsilon$ with respect to (Bures) $\chi<sup>2$-divergence, and $\widetilde{O}(rd/\epsilon)$ copies suffice for accuracy~$\epsilon$ with respect to quantum relative entropy. The best previous bound was $$\widetilde{O}(rd/\epsilon) \leq \widetilde{O}(d<sup>2/\epsilon)$$ with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the $\chi<sup>2$-divergence. For algorithms that are required to use single-copy measurements, we show that $$\widetilde{O}(r^{1.5} d^{{1.5}/\epsilon)} \leq \widetilde{O}(d<sup>3/\epsilon)$$ copies suffice for $\chi<sup>2$-divergence, and $\widetilde{O}(r^{2} d/\epsilon)$ suffice for relative entropy. Using this tomography algorithm, we show that $\widetilde{O}(d<sup>{2.5}/\epsilon)$ copies of a $d\times d$-dimensional bipartite state suffice to test if it has quantum mutual information~$0$ or at least~$\epsilon$. As a corollary, we also improve the best known sample complexity for the \emph{classical} version of mutual information testing to $\widetilde{O}(d/\epsilon)$.