Quantum parameter estimation theory is an important component of quantum information theory and provides the statistical foundation that underpins important topics such as quantum system identification and quantum waveform estimation. When there is more than one parameter the ultimate precision in the mean square error given by the quantum Cram\'er-Rao bound is not necessarily achievable. For non-full rank quantum states, it was not known when this bound can be saturated (achieved) when only a single copy of the quantum state encoding the unknown parameters is available. This single-copy scenario is important because of its experimental/practical tractability. Recently, necessary and sufficient conditions for saturability of the quantum Cram\'er-Rao bound in the multiparameter single-copy scenario have been established in terms of i) the commutativity of a set of projected symmetric logarithmic derivatives and ii) the existence of a unitary solution to a system of coupled nonlinear partial differential equations. New sufficient conditions were also obtained that only depend on properties of the symmetric logarithmic derivatives. In this paper, key structural properties of optimal measurements that saturate the quantum Cram\'er-Rao bound are illuminated. These properties are exploited to i) show that the sufficient conditions are in fact necessary and sufficient for an optimal measurement to be projective, ii) give an alternative proof of previously established necessary conditions, and iii) describe general POVMs, not necessarily projective, that saturate the multiparameter QCRB. Examples are given where a unitary solution to the system of nonlinear partial differential equations can be explicitly calculated when the required conditions are fulfilled.

翻译：量子参数估计理论是量子信息论的重要组成部分，为量子系统辨识和量子波形估计等重要课题提供了统计基础。当存在多个参数时，由量子Cramér-Rao界给出的均方误差终极精度不一定可实现。对于非满秩量子态，当仅能获取编码未知参数的量子态单个副本时，该界能否被饱和（实现）此前尚不明确。单副本场景因其实验/实践可操作性而具有重要意义。近期，针对多参数单副本场景中量子Cramér-Rao界饱和性的充分必要条件已被建立，条件涉及：(i) 一组投影对称对数导数的对易性；(ii) 耦合非线性偏微分方程组是否存在酉解。同时，仅依赖于对称对数导数性质的新型充分条件也已获得。本文揭示了能够饱和量子Cramér-Rao界的最优测量的关键结构性质，并利用这些性质：(i) 证明该充分条件实质上是实现最优测量为投影测量的充要条件；(ii) 给出已有必要条件的替代性证明；(iii) 描述能够饱和多参数QCRB的一般性POVM（不必为投影测量）。当所需条件满足时，本文给出了可显式计算耦合非线性偏微分方程组酉解的实例。