Sample-optimal learning of quantum states using gentle measurements

• URL: http://arxiv.org/abs/2505.24587v1
• Date: Fri, 30 May 2025 13:34:11 GMT
• Title: Sample-optimal learning of quantum states using gentle measurements
• Authors: Cristina Butucea, Jan Johannes, Henning Stein,
• Abstract summary: We introduce here the class of $alpha-$locally-gentle measurements ($alpha-$LGM) on a finite dimensional quantum system.<n>We prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved gentleness between relation and quantum differential privacy.<n>We propose an $alpha-$LGM called quantum Label Switch that attains these bounds.
• Score: 2.867517731896504
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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