| Quantum entanglement,quantum discord and quantum coherence not only reflect the basic properties of quantum mechanics,but also play an important role in quantum information tasks,which are very important physical resources.For example,quantum entanglement has a deci-sive place in quantum teleportation,quantum dense coding and quantum key distribution;quan-tum coherence performance enables quantum computers to realize quantum parallel computing;quantum discord is shown to be proportional to quantum efficiency in the Knill-Laflamme al-gorithm and so on.How to judge whether such physical resources exist in a quantum system and how to quantify these resources have become the core problems in quantum resource the-ory.In this paper,we mainly study measures of quantum properties such as quantum discord and quantum coherence.By various possible methods,we can establish analytically calculated resource measures or give analytical solutions to related problems.This thesis mainly includes six chapters,and the main work of this thesis is in chapters three to five.In chapter 1,we introduce the development of quantum information,elucidate the impor-tance of quantum resource theory,and mainly review quantum resources theory of quantum entanglement,quantum discord,and quantum coherence.In chapter 2,we introduce the fundamental concepts of quantum information,such as quan-tum measurement,quantum state distance,etc.,and expound on quantum resource theory about free states and free operations.We also present quantitative methods from two perspectives:the general theoretical framework and the specific application of some quantum properties.In chapter 3,we use skew information and rank-one operator to redefine Measurement-Induced Nonlocality.It can be analytically calculated for any pure state,（2（?）d）-dimensional quantum state and some special higher-dimensional quantum states.For general high-dimensional quantum state,the approximate analytical solution can be obtained by using a simple and effi-cient inverse approximate joint diagonalization algorithm.Our analytical results are in perfect agreement with the numerical results,which further verify our conclusions.The new mea-sure not only inherits the advantages of the original definition such as non-negativity and local unitary invariance but also has obvious operational significance in quantum metrology.More importantly,the new measure based on the skew information satisfies the contractivity and has good computability.In chapter 4,we also use the method of skew information and rank-one operator in sym-metric quantum correlation,and give new measures of symmetric measurement-induced non-locality（SMIN）and symmetric quantum discord（SQD）.Based on this method,we give the analytical expression of SMIN for any 2-qubit quantum state,and the analytical expression of the SQD for”X”-type quantum states and block-diagonal quantum states respectively.All the analytical solutions have been tested numerically.Both of the measures not only keep the quan-tum correlation value unchanged after attaching an auxiliary quantum state but also have good analytical computability.In chapter 5,we study the optimal approximation problem of target quantum states through convex combinations of N given quantum states.For 2-dimensional target quantum state,we use fidelity as the similarity of quantum states,and obtain an analytical expression of the op-timization problem for N=2,3 quantum states in a given set.We also use trace norm to quantify the states distance,and give the analytical expression of the optimization problem for N=2,3,4 quantum states in a given set.For the general d-dimensional case,we prove that for the optimal approximation problem of N≥d2given quantum states,the optimal distance can be achieved by a convex combination of no more than d2given quantum states.Thus,we consider l2norm as state distance measure,and give the closed solution to the problem of N≤d2convex combination of given quantum states approaching the target quantum state,then we solve the optimal approximation problem of d-dimensional target quantum states in general.Finally,numerical examples are given to verify our results.In chapter 6,we give a summary and prospects. |
