Quantum estimation for quantum technology

TL;DR

This paper reviews quantum estimation theory, providing formulas and tools to quantify quantum noise and optimize measurements of nonlinear quantum properties crucial for quantum technology.

Contribution

It offers explicit formulas for quantum Fisher information and links estimation optimization to quantum statistical geometry, advancing quantum measurement techniques.

Findings

Derived formulas for symmetric logarithmic derivative and quantum Fisher information.

Quantified quantum noise in measurements of non-observable quantities.

Connected estimation optimization with quantum statistical model geometry.

Abstract

Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology.