Universally Fisher-Symmetric Informationally Complete Measurements

TL;DR

This paper introduces Fisher-symmetric measurements using 2-designs that provide uniform information on quantum states, achieving optimal state estimation without adaptive strategies, and explores their fundamental properties and constraints.

Contribution

It constructs Fisher-symmetric measurements for pure and mixed states using 2-designs and characterizes their optimality and symmetry properties in quantum tomography.

Findings

Measurements are Fisher symmetric for all pure states.

Identifies Fisher-symmetric measurements for the completely mixed state and all pure states.

Derives a fundamental constraint on Fisher information matrices for collective measurements.

Abstract

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.