# Quantum chi-squared tomography and mutual information testing

arXiv: https://arxiv.org/abs/2305.18519 · HTML: https://mlsanity.com/paper/2305.18519/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2023.

## Built on by (most cited first, 7 of 7)

- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Phase Transitions in the Output Distribution of Large Language Models](https://mlsanity.com/paper/2405.17088/) (2024)
- [Quantum state-agnostic work extraction (almost) without dissipation](https://mlsanity.com/paper/2505.09456/) (2025)
- [Quantum Natural Stochastic Pairwise Coordinate Descent](https://mlsanity.com/paper/2407.13858/) (2024)
- [Sampling and Identity-Testing Without Approximate Tensorization of Entropy](https://mlsanity.com/paper/2506.23456/) (2025)
- [Neural Total Variation Distance Estimators for Changepoint Detection in News Data](https://mlsanity.com/paper/2506.18764/) (2025)
- [Non-iid hypothesis testing: from classical to quantum](https://mlsanity.com/paper/2510.06147/) (2025)

## Abstract (arXiv)

For quantum state tomography on rank-r dimension-d states, we show that widetildeO(r.5d1.5/ε) ≤ widetildeO(d2/ε) copies suffice for accuracy~ε with respect to (Bures) chi2-divergence, and widetildeO(rd/ε) copies suffice for accuracy~ε with respect to quantum relative entropy. The best previous bound was widetildeO(rd/ε) ≤ widetildeO(d2/ε) with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the chi2-divergence. For algorithms that are required to use single-copy measurements, we show that widetildeO(r1.5 d1.5/ε) ≤ widetildeO(d3/ε) copies suffice for chi2-divergence, and widetildeO(r2 d/ε) suffice for relative entropy. Using this tomography algorithm, we show that widetildeO(d2.5/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 widetildeO(d/ε).