Quantum chi-squared tomography and mutual information testing

Abstract

For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $\tilde{O}(r^{.5}{d}^{1.5}/\epsilon )\le \tilde{O}(d^2/\epsilon )$ copies suffice for accuracy~$\epsilon$ with respect to (Bures) ${\chi }^2$-divergence, and $\tilde{O}(rd/\epsilon )$ copies suffice for accuracy~$\epsilon$ with respect to quantum relative entropy. The best previous bound was $\tilde{O}(rd/\epsilon )\le \tilde{O}(d^2/\epsilon )$ 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 $\tilde{O}(r^{1.5}{d}^{1.5}/\epsilon )\le \tilde{O}(d^3/\epsilon )$ copies suffice for ${\chi }^2$-divergence, and $\tilde{O}(r^{2}d/\epsilon )$ suffice for relative entropy. Using this tomography algorithm, we show that $\tilde{O}(d^{2.5}/\epsilon )$ 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 $\tilde{O}(d/\epsilon )$.

34 pages

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Cited by in corpus (6)

• Learning Properties of Quantum States Without the I.I.D. Assumption
• Lower Bounds for Learning Quantum States with Single-Copy Measurements
• Measuring quantum relative entropy with finite-size effect
• Quantum state testing beyond the polarizing regime and quantum triangular discrimination
• Sample optimal tomography of quantum Markov chains
• Sample Optimal and Memory Efficient Quantum State Tomography