Quantum chi-squared tomography and mutual information testing

Abstract

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

34 pages