# Online Learning of Pure States is as Hard as Mixed States

arXiv: https://arxiv.org/abs/2502.00823 · HTML: https://mlsanity.com/paper/2502.00823/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 2 Feb 2025.

## Builds on (most cited first, 11 of 11)

- [Smoothed Analysis with Adaptive Adversaries](https://mlsanity.com/paper/2102.08446/) (2021)
- [Smoothed Online Learning is as Easy as Statistical Learning](https://mlsanity.com/paper/2202.04690/) (2022)
- [Smoothed Analysis of Online and Differentially Private Learning](https://mlsanity.com/paper/2006.10129/) (2020)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Generalization Bounds for Data-Driven Numerical Linear Algebra](https://mlsanity.com/paper/2206.07886/) (2022)
- [Efficient estimation of Pauli observables by derandomization](https://mlsanity.com/paper/2103.07510/) (2021)
- [Adaptive Online Learning of Quantum States](https://mlsanity.com/paper/2206.00220/) (2022)
- [When Does Adaptivity Help for Quantum State Learning?](https://mlsanity.com/paper/2206.05265/) (2022)
- [Learning pure quantum states (almost) without regret](https://mlsanity.com/paper/2406.18370/) (2024)
- [Online learning of a panoply of quantum objects](https://mlsanity.com/paper/2406.04245/) (2024)
- [Provable learning of quantum states with graphical models](https://mlsanity.com/paper/2309.09235/) (2023)

## Built on by (most cited first, 1 of 1)

- [Sequential Capacity of Quantum Processes with Finite Memory](https://mlsanity.com/paper/2610.02068/) (2026)

## Abstract (arXiv)

Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling. We also generalize previous results on full quantum state tomography in the online setting to (i) the ε-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.