Research Of Majorana’s Stellar Representation Of Single-particle Reduced Density Matrix For Completely Symmetric States

Completely symmetric(CS)states have theoretical and practical value in various quantum information processing(QIP)research,especially in the field of quantum entanglement.Over the past few decades,the model has gotten lots of attention.However,these experiments all focus on few-qubit systems.When N increases,the composite system is not only hard to build in the experiment but also hard to process in numerical simulation due to the exponential growth of the dimensional of the Hilbert space.In this case,it is necessary to find further analytical solutions to this model.In this thesis,we calculate the reduced density matrix(RDM)for a single particle of the completely symmetric system coupled by N spin-1/2 particles,because it helps to investigate the evolution of expectation value for the observable and to calculate the entanglement between the subsystems.Furthermore,we use Majorana’s stellar representation(MSR)to represent the results because it provides an intuitive geometric perspective to comprehend the quantum states in the highdimensional Hilbert space with trajectories of the Majorana stars on a Bloch sphere.The results exhibit the relations between the composite systems and the subsystems.Moreover,our result of reduced density matrix in Majorana’s stellar representation transforms those high-order matrices of high-dimensional Hilbert space into ordinary algebraic operations,which can be simplified using relations in combinatorics,thus providing a new idea for the numerical solution of multi-qubit systems.In Chapter 1 of this thesis,we introduce some basic concepts and background related to our research,including a completely symmetric many-body system,reduced density matrix,and quantum entanglement.In Chapter 2,we introduce the representation method of the problem we studied:Majorana’s stellar representation and some related basic knowledge,including spin,Bloch vector,Majorana’s stellar representation,and Schwinger representation.In Chapter 3,we first take some few-qubit states as the examples and then calculate the general formula of the reduced density matrix in Majorana’s stellar representation for the composite system of N-qubit with the properties of the generalized many-body anticommutators.In the process,we also reproduce the normalization constant of these states.In Chapter 4,due to the symmetry of the system,our results can be simplified in lots of cases.As the application and verification,we calculate the single-qubit reduced density matrix for the Dicke states with the results and find the coupling coefficient in the reduced density matrix.In Chapter 5,we use these results to further solve the reduced density matrix of the spin-N/2 state in a uniform magnetic field and also find the coupling coefficient in the reduced density matrix.Using these results,we also study the systems with symmetric structures on the Bloch sphere and further obtain the reduced density matrices of some symmetric structures on the Bloch sphere.In Chapter 6,we provide a summary and outlook.