Quantum chi-squared tomography and mutual information testing

For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $\widetilde{O}(r^{{.5}d{1.5}/\epsilon)} \leq \widetilde{O}(d^2/\epsilon)$ copies suffice for accuracy $\epsilon$ with respect to (Bures) $\chi^{2$-divergence,} and $\widetilde{O}(rd/\epsilon)$ copies suffice for accuracy $\epsilon$ with respect to quantum relative entropy. The best previous bound was $\widetilde{O}(rd/\epsilon) \leq \widetilde{O}(d^2/\epsilon)$ with respect to infidelity; our results are an improvement since [ \text{infidelity} \leq \text{relative entropy} \leq \text{$\chi^{2$-divergence}.]} For algorithms that are required to use single-copy measurements, we show that $\widetilde{O}(r^{1.5} d^{{1.5}/\epsilon)} \leq \widetilde{O}(d^3/\epsilon)$ copies suffice for $\chi^{2$-divergence,} and $\widetilde{O}(r^{2} d/\epsilon)$ suffice for relative entropy. Using this tomography algorithm, we show that $\widetilde{O}(d^{{2.5}/\epsilon)$} 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 classical version of mutual information testing to $\widetilde{O}(d/\epsilon)$.