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 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/(ε^2 α^2) 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.