QUANTUM ESTIMATION FOR QUANTUM TECHNOLOGY

TLDR

Quantum information quantities such as entanglement and purity are nonlinear in the density matrix and cannot be measured directly, so estimating them requires indirect measurements that form a parameter estimation problem involving optimization to find the most precise estimator. The paper reviews local quantum estimation theory and derives explicit formulas for the symmetric logarithmic derivative and quantum Fisher information for relevant quantum state families. The authors define parameter estimability via the quantum signal‑to‑noise ratio and required measurement counts, and discuss how the optimization relates to the geometry of quantum statistical models. The analysis quantifies quantum noise in measuring non‑observable quantities and offers tools for characterizing signals and devices in quantum technology.

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.