# Quantum state certification

arXiv: https://arxiv.org/abs/1708.06002 · HTML: https://mlsanity.com/paper/1708.06002/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2017.

## Built on by (most cited first, 13 of 13)

- [Exponential separations between learning with and without quantum memory](https://mlsanity.com/paper/2111.05881/) (2021)
- [Toward Instance-Optimal State Certification With Incoherent Measurements](https://mlsanity.com/paper/2102.13098/) (2021)
- [When Does Adaptivity Help for Quantum State Learning?](https://mlsanity.com/paper/2206.05265/) (2022)
- [Tight Bounds for Quantum State Certification with Incoherent Measurements](https://mlsanity.com/paper/2204.07155/) (2022)
- [Certifying almost all quantum states with few single-qubit measurements](https://mlsanity.com/paper/2404.07281/) (2024)
- [Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation](https://mlsanity.com/paper/2408.06967/) (2024)
- [On the sample complexity of purity and inner product estimation](https://mlsanity.com/paper/2410.12712/) (2024)
- [Estimating quantum relative entropies on quantum computers](https://mlsanity.com/paper/2501.07292/) (2025)
- [Sampling and Identity-Testing Without Approximate Tensorization of Entropy](https://mlsanity.com/paper/2506.23456/) (2025)
- [Instance-Optimal Quantum State Certification with Entangled Measurements](https://mlsanity.com/paper/2507.06010/) (2025)
- [Multi-Armed Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2603.17925/) (2026)
- [New Bounds on Quantum Sample Complexity of Measurement Classes](https://mlsanity.com/paper/2408.12683/) (2024)
- [Non-iid hypothesis testing: from classical to quantum](https://mlsanity.com/paper/2510.06147/) (2025)

## Abstract (arXiv)

We consider the problem of quantum state certification, where one is given n copies of an unknown d-dimensional quantum mixed state ρ, and one wants to test whether ρ is equal to some known mixed state σ or else is ε-far from σ. The goal is to use notably fewer copies than the Omega(d2) needed for full tomography on ρ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using n = O(d/ε) copies, and one with respect to trace distance using n = O(d/ε2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.