Quantum chi-squared tomography and mutual information testing

For quantum state tomography on rank-r dimension-d states, we show that widetildeO(r.5d1.5/ε) ≤ widetildeO(d2/ε) copies suffice for accuracy~ε with respect to (Bures) chi2-divergence, and widetildeO(rd/ε) copies suffice for accuracy~ε with respect to quantum relative entropy. The best previous bound was widetildeO(rd/ε) ≤ widetildeO(d2/ε) with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the chi2-divergence. For algorithms that are required to use single-copy measurements, we show that widetildeO(r1.5 d1.5/ε) ≤ widetildeO(d3/ε) copies suffice for chi2-divergence, and widetildeO(r2 d/ε) suffice for relative entropy. Using this tomography algorithm, we show that widetildeO(d2.5/ε) copies of a d× 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 classical version of mutual information testing to widetildeO(d/ε).