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.

翻译：量子态的温和测量不会完全坍缩初始态，而是在指定迹距离 $α$ 下提供一个与初始态接近的测量后态，以及一个用于量子学习初始态的随机变量。本文针对有限维量子系统引入了 $α$-局部温和测量（$α$-LGM）类，该类对乘积态执行乘积测量，并利用改进的温和性与量子差分隐私关系证明了该测量类上的强量子数据处理不等式（qDPI）。进一步，我们提出了温和量子奈曼-皮尔逊引理，表明该 qDPI 在渐近意义下（对较小 $α$）是最优的。基于该不等式，我们证明了在量子层析成像和量子态验证任务中，达到指定精度 $ε$ 所需的量子态样本量级为 $1/(ε^2 α^2)$。最后，我们提出了一种称为量子标签交换的 $α$-LGM 方案，该方案可达到上述理论界限，为将任意二输出测量转化为 $α$-LGM 提供了一种通用的可实施方案。