Universally Fisher-Symmetric Informationally Complete Measurements

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.

Figure 1

Scaled MSE (left plot) and scaled MSB (right plot) achieved by universally Fisher-symmetric measurements (UFS), including the collective SIC, in qubit state tomography. Here $s$ is the length of the Bloch vector. The performances of SIC, MUB, and covariant measurements (averaged over states with the same purity) are shown for comparison (reproduced from Ref. [22]).