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，而是提供与初始态在指定迹距离$α$内的测量后状态，同时产生一个用于量子学习初始态的随机变量。我们在有限维量子系统上引入$α-$局部温和测量（$α-$LGM）类，该类作用于乘积态时具有乘积测量特性，并通过改进温和性与量子差分隐私之间的关系，证明了该类测量满足强量子数据处理不等式（qDPI）。进一步，我们证明了一个温和的量子内曼-皮尔逊引理，该引理表明我们的qDPI在渐近意义下（当$α$较小时）是最优的。利用该不等式，我们证明在量子层析成像和量子态认证中，达到给定精度$ε$所需的量子态数量级为$1/(ε^2 α^2)$。最后，我们提出一种称为量子标签交换（quantum Label Switch）的$α-$LGM方法，该方法能实现这些界，是一种将任意二元测量转化为$α-$LGM的通用可实施方法。