Sample-optimal learning of quantum states using gentle measurements

TL;DR

This paper introduces a new class of gentle measurements for quantum states, establishing optimal bounds on the number of states needed for accurate quantum learning tasks like tomography and certification.

Contribution

It defines $ ext{alpha}$-locally-gentle measurements, proves a strong data-processing inequality, and presents an implementable method achieving optimal bounds for quantum state learning.

Findings

Established an order of $1/( ext{epsilon}^2 ext{alpha}^2)$ for the number of states needed.

Proved the asymptotic optimality of the quantum data-processing inequality.

Introduced the quantum Label Switch method for practical implementation.

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 $\alpha$ from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of $\alpha-$locally-gentle measurements ($\alpha-$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 $\alpha$). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy $\epsilon$ is of order $1/(\epsilon^2 \alpha^2)$ for both quantum tomography and quantum state certification. Finally, we propose an $\alpha-$LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an $\alpha-$LGM.