Geometry In Quantum Metrology And Measurement Theory Of Fisher Information

With the development of experimental technology,it is becoming more and more possible to use quantum precision measurement to break the limit of classical measurement accuracy.Related theoretical and experimental research is also widely carried out in the world,and the contents of quantum metrology is increasingly abundant.Quantum Fisher information is the most important concept in quantum metrology,and it gives the parameter’s information contained in a quantum state when it is used to measure a specific parameter,i.e.,the sensitivity of the parameter measured by this state.The main content of this paper is the geometric representation in the quantum metrology and the measurement theory of quantum Fisher information.Quantum Fisher information has the same form with the real part of quantum geometric tensor,i.e.,quantum metric tensor.We have explored the geometric representation of quantum geometric tensor.In addition,we theoretically discuss how to measure quantum Fisher information by direct and indirect methods.The main contents of this thesis are as follows:1.In Chapter 2,we first review the definition,relationship and basic calculation methods of quantum Fisher information and classical Fisher information.We also discuss the singleparameter quantum Fisher information and multi-parameter quantum Fisher information matrix.Among the calculation methods of quantum Fisher information,the unitary parametrization process is particularly important,and we will use this method in Chapter 3.In addition,we discuss the quantum geometric tensor,and illustrate the relation between quantum geometric tensor and quantum Fisher information.The real part of quantum geometric tensor is called quantum metric tensor,which differs from the quantum Fisher information of pure state only by a coefficient.It can be seen that the distance of quantum states,fidelity and quantum Fisher information are all connected,and this has a profound meaning on the measurement theory of quantum Fisher information in Chapter 5.2.In Chapter 3,we discuss a problem in quantum optics,namely the effect of kinetic energy term in the light-matter interaction Hamiltonian on the metrology of Rabi frequency.This is because we often ignore the kinetic energy term in the theoretical calculation,however,when the kinetic energy term is considered,the external motion of the atom also has an influence on the Rabi frequency measuring accuracy.We will explore the importance of this influence in some parameter regions.We first carry out an unitary transformation to eliminate the position operator in the original Hamiltonian,so that a su(2)-type Hamiltonian can be obtained.For this unitary parametrization process,the final quantum Fisher information is known.We also calculate the classical Fisher information obtained from the actual measurement of some operators.When the Hamiltonian contains kinetic energy term,population difference measurement is no longer suitable for measuring Rabi frequency and the joint measurement of momentum and population difference is needed.3.In Chapter 4,we discuss Majorana’s stellar representation of the quantum geometric tensor.We give the results of fully symmetric state up to spin 3/2.As for arbitrary spin cases,we list the Majorana’s stellar representations of some special quantum states.These special states play an important role in many scenarios,and their Majorana’s stellar representations for quantum geometric tensor can be seen directly from our conclusion.4.In Chapter 5,we discuss the direct measurement scheme of quantum Fisher information,which uses adiabatic perturbation to relate the quantum Fisher information(matrix)of the ground state of an interested Hamiltonian to the energy fluctuations,and the quantum Fisher information is directly given by measuring the energy fluctuations.It avoids the concern about the symmetric logarithmic derivative(SLD),which is also directly related to quantum Fisher information but is difficult to be obtained both theoretically and experimentally.We verify the feasibility of this scheme through a two-level system,which can extract not only single-parameter quantum Fisher information,but also multi-parameter quantum Fisher information matrix.We also simulate the extraction process of quantum Fisher information using the actual physical parameters in the nitrogen-vacancy color center.In addition,the extracted quantum Fisher information can also show the physical process of phase transitions and energy-level crossing.5.In Chapter 6,we discuss the indirect measurement scheme of quantum Fisher information,which is mainly elucidated by our proposed instantaneous indirect measurement principle.In the short-time limit,through introducing a known reference system we can measure the physical quantities of one subsystem of the target system and obtain the information of other physical quantities of another subsystem,which can indirectly help the measurement of quantum Fisher information.As a theoretical demonstration,in the atom-photon interacting systems,by measuring the change of the atomic energy,we can obtain the p-order correlation functions of an unknown light field,and similarly,measuring the average photon number of an unknown light field can obtain the atomic state’s information.Due to the measuring time required is very short,the atom-light entanglement is still weak,and hence the effect of measuring one subsystem on the other is very small,resulting in the high fidelity of the other system.6.In Chapter 7,we have given summaries and outlook of this thesis.The appendix shows the detailed calculation for the omission in Chapter 3 and Chapter 5.