# Sample-optimal learning of quantum states using gentle measurements

arXiv: https://arxiv.org/abs/2505.24587 · HTML: https://mlsanity.com/paper/2505.24587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 55 references that are in the map.
- First posted on arXiv 30 May 2025.

## Builds on (most cited first, 10 of 10)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Exponential separations between learning with and without quantum memory](https://mlsanity.com/paper/2111.05881/) (2021)
- [Quantum Differential Privacy: An Information Theory Perspective](https://mlsanity.com/paper/2202.10717/) (2022)
- [On quantum backpropagation, information reuse, and cheating measurement collapse](https://mlsanity.com/paper/2305.13362/) (2023)
- [Entanglement is Necessary for Optimal Quantum Property Testing](https://mlsanity.com/paper/2004.07869/) (2020)
- [Inference under Information Constraints III: Local Privacy Constraints](https://mlsanity.com/paper/2101.07981/) (2021)
- [Measuring Quantum Information Leakage Under Detection Threat](https://mlsanity.com/paper/2403.11433/) (2024)
- [Multivariate Mean Comparison under Differential Privacy](https://mlsanity.com/paper/2110.07996/) (2021)

## Abstract (arXiv)

Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance α from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of α-locally-gentle measurements (α-LGM) on a finite dimensional quantum system which are product measurements on product states and prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved relation between gentleness and quantum differential privacy. We further show a gentle quantum Neyman-Pearson lemma which implies that our qDPI is asymptotically optimal (for small α). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy ε is of order 1/(ε2 α2) for both quantum tomography and quantum state certification. Finally, we propose an α-LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an α-LGM.