Quantum chi-squared tomography and mutual information testing

TL;DR

This paper develops efficient quantum state tomography methods based on $oldsymbol{ ext{chi}^2}$-divergence and quantum relative entropy, improving sample complexity bounds and enabling mutual information testing with fewer copies.

Contribution

It introduces new quantum tomography algorithms with improved sample complexity bounds for $oldsymbol{ ext{chi}^2}$-divergence and relative entropy, and applies these to mutual information testing.

Findings

Reduced sample complexity for quantum state tomography.

Achieved mutual information testing with fewer copies of bipartite states.

Improved classical mutual information testing sample complexity.

Abstract

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 $d\times d$-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 \emph{classical} version of mutual information testing to $\widetilde{O}(d/\epsilon)$.