Indexed on: 18 Sep '17Published on: 18 Sep '17Published in: arXiv - Quantum Physics
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.