We revisit the problem of estimating an unknown parameter of a pure quantum state, and investigate `null-measurement' strategies in which the experimenter aims to measure in a basis that contains a vector close to the true system state. Such strategies are known to approach the quantum Fisher information for models where the quantum Cram\'{e}r-Rao bound is achievable but a detailed adaptive strategy for achieving the bound in the multi-copy setting has been lacking. We first show that the following naive null-measurement implementation fails to attain even the standard estimation scaling: estimate the parameter on a small sub-sample, and apply the null-measurement corresponding to the estimated value on the rest of the systems. This is due to non-identifiability issues specific to null-measurements, which arise when the true and reference parameters are close to each other. To avoid this, we propose the alternative displaced-null measurement strategy in which the reference parameter is altered by a small amount which is sufficient to ensure parameter identifiability. We use this strategy to devise asymptotically optimal measurements for models where the quantum Cram\'{e}r-Rao bound is achievable. More generally, we extend the method to arbitrary multi-parameter models and prove the asymptotic achievability of the the Holevo bound. An important tool in our analysis is the theory of quantum local asymptotic normality which provides a clear intuition about the design of the proposed estimators, and shows that they have asymptotically normal distributions.

翻译：我们重新审视了纯量子态未知参数估计问题，并研究了“零测量”策略——实验者旨在通过与真实系统状态相近的基向量进行测量。此类策略在量子克拉美-罗界可实现时能接近量子费舍信息，但多副本场景下实现该界的详细自适应策略尚不完善。我们首先证明以下朴素零测量实现方式无法达到标准估计标度：基于小子样本估计参数，并对剩余系统施加对应估计值的零测量。这是由于零测量特有的非可识别性问题——当真实参数与参考参数接近时产生。为避免该问题，我们提出替代性位移零测量策略：对参考参数施加微小偏移，该偏移足以确保参数可识别性。我们利用该策略为量子克拉美-罗界可实现的模型设计了渐近最优测量。更一般地，我们将该方法推广至任意多参数模型，并证明霍列沃界的渐近可达性。本分析的重要工具是量子局部渐近正态理论，该理论为所提估计量的设计提供了清晰直观解释，并表明其具有渐近正态分布特性。