Quantum chi-squared tomography and mutual information testing

Abstract

For quantum state tomography on rank-r dimension-d states, we show that O~(r.5d1.5/ϵ)≤O~(d2/ϵ) copies suffice for accuracy ϵ with respect to (Bures) χ2-divergence, and O~(rd/ϵ) copies suffice for accuracy ϵ with respect to quantum relative entropy. The best previous bound was O~(rd/ϵ)≤O~(d2/ϵ) with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the χ2-divergence. For algorithms that are required to use single-copy measurements, we show that O~(r1.5d1.5/ϵ)≤O~(d3/ϵ) copies suffice for χ2-divergence, and O~(r2d/ϵ) suffice for relative entropy.Using this tomography algorithm, we show that O~(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 O~(d/ϵ).