Sample-optimal learning of quantum states using gentle measurements
Cristina Butucea, Jan Johannes, Henning Stein
TL;DR
We address learning quantum states under the constraint of gentle measurements that only slightly disturb the system. Our core approach introduces $α$-locally-gentle measurements (α-LGM) and proves a strong quantum Data-Processing Inequality (qDPI) linking information gain to gentleness via quantum differential privacy, along with a gentle quantum Neyman-Pearson lemma. We show that both quantum tomography and quantum state certification require $n=Θ\left(\frac{1}{ε^2 α^2}\right)$ copies, and we provide an implementable α-LGM, the quantum Label Switch (qLS), that attains these bounds for qubits. These results illuminate a privacy–utility trade-off in quantum learning and yield practical protocols for gentle quantum state estimation and testing, with extensions to higher dimensions discussed for future work.
Abstract
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.