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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.
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