The Bloch Representations Of Quantum States And Its Application In The Theory Of Entanglement

Quantum entanglement is a kind of important resource in the research of quantum information theory.Bloch representation is a common tool for research of quantum information,and concurrence is one of the most wildly used entanglement metric.This paper mainly studies concurrence and some other entangled measures by Bloch representation of quantum states' density matrices and give a valid lower bound estimation of them,and study the relation of Bloch vectors,correlaton matrices and correlaton tensors in Bloch representation,and give an application in Bell inequalities.The first part of the thesis is an introduction,which briefly introduces the development of quantum entanglement theory,the relationship between quantum entanglement and quantum information and recommends the domestic and international development background and current situation.The second part of the paper introduces some fundamental framework and theoretical knowledge related to quantum information,such as matrix decomposition,the density matrix and reduced density matrix of quantum states,quantum separability and entanglement,some classical quantum entanglement criteria,entangled measures and the definition of concurrence,etc.The third part of the thesis uses Bloch representation of density matrices to study the properties of the trace norm of its correlated matrix,and consider entanglement metrics by combining some classical quantum entanglement criteria.Finally we give new lower bound estimation of concurrence and some other entanglement metrics.In the fourth part of the thesis,we get the upper bound of the sum of Bloch vectors' norms for all three-partite reduced quantum states in four-partite quantum systems through the Bloch representation of quantum states,and we derive a new kind of trade-off relation in multipartite qubit quantum systems by using this outcome.The fifth part of the thesis is the part of examples.In the formal three examples,we compare our results with some existed lower bound of entangled measures.Then through some matlab experiments,we show that our lower bound is bigger than other existing correlated results,that is,our lower bound of concurrence are valid.The last example verify the trade-off relation and show that the reduced triple qubits cannot reach the maximal violation value of Svetlichny inequality simultaneously in four-qubit systems.