Quantum chi-squared tomography and mutual information testing
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For quantum state tomography on rank-r dimension-d states, we show that O(r^.5d^1.5/ϵ) ≤O(d^2/ϵ) copies suffice for accuracy ϵ with respect to (Bures) χ^2-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(d^2/ϵ) with respect to infidelity; our results are an improvement since infidelity≤relative entropy≤χ^2-divergence. For algorithms that are required to use single-copy measurements, we show that O(r^1.5 d^1.5/ϵ) ≤O(d^3/ϵ) copies suffice for χ^2-divergence, and O(r^2 d/ϵ) suffice for relative entropy. Using this tomography algorithm, we show that O(d^2.5/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).
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