On determining which quantum measurement performs better for state estimation

TL;DR

This paper introduces a new measure called the quantum tomographic transfer function to objectively compare the effectiveness of different quantum measurements in state estimation, independent of the true state.

Contribution

It defines a physically meaningful, state-independent transfer function for assessing and comparing informationally complete quantum measurements in tomography.

Findings

The transfer function is simple for minimally complete measurements.

Explicit, computable formulas for the transfer function are provided.

Numerical simulations validate the transfer function's effectiveness and consistency with known results.

Abstract

We introduce an operational and statistically meaningful measure, the quantum tomographic transfer function, that possesses important physical invariance properties for judging whether a given informationally complete quantum measurement performs better tomographically in quantum-state estimation relative to other informationally complete measurements. This function is independent of the unknown true state of the quantum source, and is directly related to the average optimal tomographic accuracy of an unbiased state estimator for the measurement in the limit of many sampling events. For the experimentally-appealing minimally complete measurements, the transfer function is an extremely simple formula. We also give an explicit expression for this transfer function in terms of an ordered expansion that is readily computable and illustrate its usage with numerical simulations, and its consistency with some known results.