Quantum chi-squared tomography and mutual information testing
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914848373211136 |
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| author | Flammia, Steven T. O'Donnell, Ryan |
| author_facet | Flammia, Steven T. O'Donnell, Ryan |
| contents | For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $\widetilde{O}(r^{.5}d^{1.5}/ε) \leq \widetilde{O}(d^2/ε)$ copies suffice for accuracy~$ε$ with respect to (Bures) $χ^2$-divergence, and $\widetilde{O}(rd/ε)$ copies suffice for accuracy~$ε$ with respect to quantum relative entropy. The best previous bound was $\widetilde{O}(rd/ε) \leq \widetilde{O}(d^2/ε)$ with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the $χ^2$-divergence. For algorithms that are required to use single-copy measurements, we show that $\widetilde{O}(r^{1.5} d^{1.5}/ε) \leq \widetilde{O}(d^3/ε)$ copies suffice for $χ^2$-divergence, and $\widetilde{O}(r^{2} d/ε)$ suffice for relative entropy.
Using this tomography algorithm, we show that $\widetilde{O}(d^{2.5}/ε)$ copies of a $d\times d$-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 \emph{classical} version of mutual information testing to $\widetilde{O}(d/ε)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18519 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum chi-squared tomography and mutual information testing Flammia, Steven T. O'Donnell, Ryan Quantum Physics Data Structures and Algorithms For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $\widetilde{O}(r^{.5}d^{1.5}/ε) \leq \widetilde{O}(d^2/ε)$ copies suffice for accuracy~$ε$ with respect to (Bures) $χ^2$-divergence, and $\widetilde{O}(rd/ε)$ copies suffice for accuracy~$ε$ with respect to quantum relative entropy. The best previous bound was $\widetilde{O}(rd/ε) \leq \widetilde{O}(d^2/ε)$ with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the $χ^2$-divergence. For algorithms that are required to use single-copy measurements, we show that $\widetilde{O}(r^{1.5} d^{1.5}/ε) \leq \widetilde{O}(d^3/ε)$ copies suffice for $χ^2$-divergence, and $\widetilde{O}(r^{2} d/ε)$ suffice for relative entropy. Using this tomography algorithm, we show that $\widetilde{O}(d^{2.5}/ε)$ copies of a $d\times d$-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 \emph{classical} version of mutual information testing to $\widetilde{O}(d/ε)$. |
| title | Quantum chi-squared tomography and mutual information testing |
| topic | Quantum Physics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2305.18519 |