Sample-optimal learning of quantum states using gentle measurements

Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of locally-gentle measurements (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 ). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy is of order for both quantum tomography and quantum state certification. Finally, we propose an LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an LGM.
View on arXiv@article{butucea2025_2505.24587, title={ Sample-optimal learning of quantum states using gentle measurements }, author={ Cristina Butucea and Jan Johannes and Henning Stein }, journal={arXiv preprint arXiv:2505.24587}, year={ 2025 } }