Zhu Huangjun, Hayashi Masahito
Institute for Theoretical Physics, University of Cologne, Cologne 50937, Germany.
Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China.
Phys Rev Lett. 2018 Jan 19;120(3):030404. doi: 10.1103/PhysRevLett.120.030404.
A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.
如果量子测量能为表征感兴趣的量子态的所有参数提供均匀且最大的信息，那么该量子测量就是费希尔对称的。利用（复射影）2 - 设计，我们构建了对一对相同制备的量子态的测量，这些测量对于所有纯态都是费希尔对称的。这种测量在不进行自适应测量的情况下，能以最优方式实现最小统计误差。然后，我们确定了一对量子态上对于完全混合态和所有纯态同时都是费希尔对称的所有集体测量。对于一个量子比特，这些测量对于所有态都是费希尔对称的。最小最优测量与难以捉摸的对称信息完备测量相关联，这反映了局部对称性和全局对称性之间的深刻联系。在这项研究中，我们推导了对一对量子态上任何集体测量的费希尔信息矩阵的一个基本约束，这为表征集体测量的层析成像效率提供了一个有用的工具。
