Quantum chi-squared tomography and mutual information testing

Quantum 8, 1381 (2024).

https://doi.org/10.22331/q-2024-06-20-1381

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 infidelity is bounded above by both the relative entropy and the $chi^2$-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 $dtimes d$-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 $widetilde{O}(d/epsilon)$.