Quantum chi-squared tomography and mutual information testing

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