The Research On Lee-Yang Zeros And Quantum Fisher Information Matrix In Quantum Thermodynamics

The precision measurement of physical quantities plays a vital role in human development.The high-precision measurement of various physical quantities not only contributes to the development of technology and engineering but also gives a great impetus to fundamental research.Because it can break through the standard quantum limit,quantum precision measurement has become the main technology of next-generation precision measurement.As its theoretical support,quantum parameter estimation has also gained a lot of attention in recent years.Quantum Fisher information and quantum Fisher information matrix are important indicators to evaluate the potential of parameter estimation in quantum parameter estimation theory,which portrays the accuracy limit of the parameter to be estimated by the system.Meanwhile,the correlation between the quantum Fisher information matrix and other fundamental physical quantities is also gradually explored with the development of quantum parameter estimation.At present,the quantum Fisher information matrix has been widely considered a fundamental mathematical tool in quantum mechanics.Therefore,the study of the nature of the quantum Fisher information matrix itself is of great importance.In addition,most quantum precision measurement devices need to face the effect of thermal reservoirs.Therefore,the study of quantum parameter estimation in quantum thermodynamics can improve the practicality of quantum parameter estimation schemes and make them more experimentally feasible.In quantum thermodynamics,the Lee-Yang zeros are an important class of mathematical tools that describe the distribution of roots in the complex plane when the system coordination function is equal to zero.The wide application of LeeYang zeros in thermodynamics,especially its relationship with phase transition,has given it a greater interest in quantum thermodynamics.The relationship between the Lee-Yang zeros and the quantum Fisher information matrix naturally becomes an important research element when parameter estimation studies are conducted in the enthusiasm context.However,this research is still in its infancy.Therefore,this thesis will take the relationship between the Lee-Yang zeros and the quantum Fisher information matrix as an entry point for the study of parameter estimation in quantum thermodynamics.The development of quantum mechanics has promoted the study of thermodynamics in the quantum system,especially the Lee-Yang zeros theory.This thesis mainly explores the relationship between the Lee-Yang zero point and quantum parameter estimation.The development of quantum thermodynamics,and the research content and progress of quantum parameter estimation are briefly introduced.The Lee-Yang theorem,given by Yang and Lee in 1952 when they studied the equation of state and phase transition,is introduced,and the Lee-Yang zeros of the coordination function of a general quantum system in the complex plane is given.Further,the basic concepts of quantum parameter estimation are introduced.In single-parameter estimation,the lower bound of the parameter accuracy is the inverse of the quantum Fisher information,while in multiparameter estimation,this lower bound is the inverse matrix of the quantum Fisher information matrix,which is described by the quantum Cramér-Rao inequality,i.e.,the lower bound of the covariance matrix of the multi-parameter is the inverse of the quantum Fisher information matrix.The lower bound inscribed by the quantum Fisher information matrix in the quantum Cramér-Rao theory is not always tight,and conditions are given for the parameter estimates to be accessible to the quantum CramérRao inequality,i.e.,the classical Fisher information matrix is equal to the quantum Fisher information matrix when the measurement is optimal.And the expressions of the quantum Fisher information matrix for different scenarios are summarized.The Lee-Yang zeros fully characterize various properties of thermodynamic systems,especially the phase transition theory.Exploring quantum phase transitions using quantum Fisher information has been widely studied,and the relationship between the Lee-Yang zeros and quantum Fisher information is a problem to be solved.firstly,the distribution of the Lee-Yang zeros in the quantum Ising model is reviewed,and the system’s partition function can be written in the form of an Nth order polynomial with the Lee-Yang zeros distributed on the unit circle.Further,the detection of the Lee-Yang zeros based on quantum parameter estimation theory is investigated.A nonlinear quantum spin model is proposed,extending the roots of the model’s partition function to the complex space by analytic extension,and studying the distribution of the Lee-Yang zeros on the complex space under different scenarios,giving four theorems,which theorems illustrate the distribution of the Lee-Yang zeros on the complex space.Theorem 4.1 describes that not all Lee-Yang zeros can be distributed on the unit circle simultaneously when the nonlinear coefficients are odd.Theorem 4.2 shows that the point(-1,0)is always one of the Lee-Yang zeros when the nonlinear coefficients are even and the number of spins is odd.Theorem 4.5,when the nonlinear coefficients are even,the modulus of the Lee-Yang zeros is always 1.Theorem 4.6,when the parameter βγis sufficiently small,all Lee-Yang zeros are always distributed on the unit circle.In addition,the detection of the Lee-Yang zeros by probe qubit is discussed in depth.Using the coupling between the probe qubit and the nonlinear system,all Lee-Yang zeros can be detected in the dynamics of the probe qubit by adjusting the coupling strength and nonlinear coefficients of the nonlinear system.In addition,the expression of the quantum Fisher information matrix at the Lee-Yang zeros is analyzed,and it is found that both the coupling strength and temperature can reach the accuracy limit at the Lee-Yang zeros at the same time,but the information of temperature cannot be obtained at the Lee-Yang zeros using the probe qubit if the Lee-Yang zeros are located on the unit circle.Specific examples are given to explore the relationship between the Lee-Yang zeros and the quantum Fisher information matrix.Furthermore,the anisotropic XY model is further investigated and the Lee-Yang zeros distribution of the model in the complex magnetic field plane is explored.The anisotropic XY model is investigated using an auxiliary line.The relationship between the quantum coherence of the isotropic XY model and the Lee-Yang zeros under complex space is given.The evolution density matrix under the action of auxiliary qubits is talked about,and the relationship between the quantum Fisher information and the Lee-Yang zeros under the isotropic XY model is explored.Finally,under the anisotropic XY model,the relations between the ground state energy,the free energy,and the quantum fluctuation of the energy based on the analytic expressions of the partition function are also explored.Finally,the main contents of this thesis are summarized and an outlook for further research directions is presented.