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Quantum chi-squared tomography and mutual information testing

Steven T. Flammia, Ryan O'Donnell

TL;DR

The best known sample complexity for the classical version of mutual information testing to $\widetilde{O}(d/\epsilon)$ is improved and the results are an improvement since.

Abstract

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 $\widetilde{O}(d/ε)$.

Quantum chi-squared tomography and mutual information testing

TL;DR

The best known sample complexity for the classical version of mutual information testing to is improved and the results are an improvement since.

Abstract

For quantum state tomography on rank- dimension- states, we show that copies suffice for accuracy~ with respect to (Bures) -divergence, and copies suffice for accuracy~ with respect to quantum relative entropy. The best previous bound was with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the -divergence. For algorithms that are required to use single-copy measurements, we show that copies suffice for -divergence, and suffice for relative entropy. Using this tomography algorithm, we show that copies of a -dimensional bipartite state suffice to test if it has quantum mutual information~ or at least~. As a corollary, we also improve the best known sample complexity for the \emph{classical} version of mutual information testing to .
Paper Structure (19 sections, 47 theorems, 134 equations)

This paper contains 19 sections, 47 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Different quantum divergences, and prior work
  3. Our results
  4. Basic results on distances and divergences
  5. Classical distances and divergences
  6. Quantum distances and divergences
  7. Quantum Tomography with Quantum Relative Entropy Loss
  8. Quantum state tomography
  9. Qubit tomography with single-copy measurements
  10. Upgrading Frobenius-squared tomography algorithms
  11. The plan for learning in chi squared: refining diagonal estimates on submatrices
  12. The central estimation algorithm
  13. Conclusions from the central estimation algorithm
  14. A simple Frobenius-squared estimator
  15. Testing zero mutual information
  16. ...and 4 more sections

Key Result

theorem 1.1

(KUENG201788.) There is a state tomography algorithm using nonadaptive single-copy measurements achieving expected Frobenius-squared error $O(rd/n)$ on $d$-dimensional states of rank at most $r$. Hence $n = O(rd/\epsilon)$ samples suffice to getWith probability at least $.99$, say, by Markov's inequ

Theorems & Definitions (110)

  • theorem 1.1
  • remark 1.2
  • theorem 1.3
  • theorem 1.4
  • remark 1.5
  • theorem 1.6
  • corollary 1.7
  • corollary 1.8
  • corollary 1.9
  • corollary 1.10
  • ...and 100 more