- Generalized Estimators, Slope, Efficiency, and Fisher Information Bounds(arXiv)
Author : Paul W. Vos
Abstract : Point estimators may not exist, need not be unique, and their distributions are not parameter invariant. Generalized estimators provide distributions that are parameter invariant, unique, and exist when point estimates do not. Comparing point estimators using variance is less useful when estimators are biased. A squared slope Λ is defined that can be used to compare both point and generalized estimators and is unaffected by bias. Fisher information I and variance are fundamentally different quantities: the latter is defined at a distribution that need not belong to a family, while the former cannot be defined without a family of distributions, M. Fisher information and Λ are similar quantities as both are defined on the tangent bundle TM and I provides an upper bound, Λ≤I, that holds for all sample sizes — asymptotics are not required. Comparing estimators using Λ rather than variance supports Fisher’s claim that I provides a bound even in small samples. Λ-efficiency is defined that extends the efficiency of unbiased estimators based on variance. While defined by the slope, Λ-efficiency is simply ρ2, the square of the correlation between estimator and score function
2. Direct measurement of quantum Fisher information(arXiv)
Abstract : In the adiabatic perturbation theory, Berry curvature is related to the generalized force, and the quantum metric tensor is linked with energy fluctuation. While the former is tested with numerous numerical results and experimental realizations, the latter is less considered. Quantum Fisher information, key to quantum precision measurement, is four times quantum metric tensor. It is difficult to relate the quantum Fisher information with some physical observable. One interesting candidate is square of the symmetric logarithmic derivative, which is usually tough to obtain both theoretically and experimentally. The adiabatic perturbation theory enlightens us to measure the energy fluctuation to directly extract the quantum Fisher information. In this article, we first adopt an alternative way to derive the link of energy fluctuation to the quantum Fisher information. Then we numerically testify the direct extraction of the quantum Fisher information based on adiabatic perturbation in two-level systems and simulate the experimental realization in nitrogen-vacancy center with experimentally practical parameters. Statistical models such as transverse field Ising model and Heisenberg spin chains are also discussed to compare with the analytical result and show the level crossing respectively. Our discussion will provide a new practical scheme to measure the quantum Fisher information, and will also benefit the quantum precision measurement and the understand of the quantum Fisher information.
3. Fisher information of a Black Hole Spacetime(arXiv)
Abstract : Relativistic quantum metrology is the study of optimal measurement procedures within systems that have both quantum and relativistic components. Here we use Unruh-DeWitt detectors coupled to a massless scalar field as probes of thermal parameters in different spacetimes via a relativistic quantum metrology analysis. We consider both 3-dimensional Anti-de Sitter and BTZ black hole spacetimes. We compute the Fisher information to identify characteristics of the black hole spacetime and to compare it to a uniformly accelerating detector in Anti-de Sitter space. We find the dependence of the Fisher information on temperature, detector energy gap, black hole mass, interaction time, and the initial state of the detector. We identify strategies that maximize the Fisher information and therefore the precision of estimation.
4.A Fisher information-based incompatibility criterion for quantum channels (arXiv)
Abstract : We introduce a new incompatibility criterion for quantum channels, based on the notion of (quantum) Fisher information. Our construction is based on a similar criterion for quantum measurements put forward by H.~Zhu. We then study the power of the incompatibility criterion in different scenarios. Firstly, we prove the first analytical conditions for the incompatibility of two Schur channels. Then, we study the incompatibility structure of a tuple of depolarizing channels, comparing the newly introduced criterion with the known results from asymmetric quantum cloning.