Study On Quantum Metrology And Quantum Fisher Information

Over the past several decades, quantum metrology has become a fast developing discipline, which is closely related to practical applications, such as atomic clocks, quan-tum imaging, and gravitational wave detection etc. Quantum metrology is to estimate accurately the value of an unknown parameter with the assistance of the effect of quan-tization. The lower bound of the error of estimation is determined by the inverse of quantum Fisher information. Hence, how to attain this precision limit is the essential task in quantum metrology. In the realistic experiments, the achievable accuracy of parameter estimation is dependent on four aspects:probe states, parameterization pro-cess, measurements and data processing. This dissertation focus on two aspects:the parameterization in the presence of decoherence and the optimal measurement to sat-urate the sensitivity bound of parameter estimation. The structure of this dissertation is arranged as follows.Chapter1reviews the research background of quantum parameter estimation, and introduces some models of interferometer experiment.Chapter2first reivews quantum Crmamer-Rao theorem in quantum estimation theory, and then introduces the definition of the quantum Fisher information. Furhter-more, we especially introduce the properties of the quantum Fisher information, as well as its expressions in the diagonalization representation and Bloch representation.In Chapter3, we investigate the time evolution of the quantum Fisher information and its preserving in noisy quantum systems. We start by introducing the methods to describle the dynamical process of the quantum open systems. Then we investigate the dynamics of quantum Fisher information under typical decoherences from a geomet-rical point of view. We consider three different decoherence channels:phase-damping channel, depolarzing channel, amplitude damping channel. We obtain analytical results under three different decoherence channels, which are expressed as affine transformation matrices. Using the hierarchy equation method, we numerically study the dynamics of quantum Fisher information in a dissipative model and compare the numerical results with the analytical ones obtained by applying the rotating-wave approximation. We further investigate the dynmaics of the quantum Fisher information in the collevtive dephasing model. We find that the collevtive dephasing significantly diminishes the precision of the phase parameter with the Ramsey interferometry. Finally, we propose a scheme to protect quantm Fisher information with repect to phase paramters from phase noises.In Chapter4, we address the optimization problem of measurements for achieving the ultimating sensitivity deterimind by the quantum Cramer-Rao bound from error propagation formula. Propagation of error is a widely used estimation tool in experi-ments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the esti-mation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramer-Rao theorem in parameter estimation. By invoking the Schrodinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for both single and multi-parameter estimates. For single parameter estimation, applying the theo-retic condition to GHZ states, we give the general expression of the optimal observables for local measurements to achieve the Heisenberg-limit precision. For multi-parameter estimation, we examine the optimal condition in the estimation of two phase shifts in optical interferometry, and find that the photon-counting based detection can always achieve the maximal sensitivity for all path-symmetric pure states.Chapter5denotes the conclusion and prospect.