Separability Of The Quantum State, The Fidelity Of Quantum Teleportation, And Quantum Entanglement Metrics

Entanglement is the characteristic trait of quantum mechanics, and it re?ects theproperty that a quantum system can simultaneously appear in two or more di?erentstates [1]. Although the nonclassical nature of entanglement has been recognizedfor many years, considerable e?orts have been taken to understand and characterizeits properties recently. However the physical character and mathematical structureof entangled states have not been well understood. Basically, how can we knowa given quantum state is entangled or not and how to qualify the entanglementof an arbitrary quantum state? Furthermore, how can we accomplish quantuminformation tasks perfectly? Many questions need further investigation. The resultsin this thesis mainly concern three aspects as described below.We give an introduction to quantum information and quantum entanglement inthe first section. Then we introduce some basic definitions and concepts in quantummechanics and the theory of quantum entanglement. The third section is devotedto the study of separability of quantum states. We first introduce several importantseparability criteria derived in recent years. Then we give new criteria of separabilitybased on the Bloch representation and the covariance matrix of quantum states.We further derive the normal form of multipartite quantum states, from which newseparability criterion is also derived. We show that the ability of these criterions torecognize entanglement can be greatly improved by first transforming the quantumstates into their normal forms.Concurrence, a well defined quantum entanglement measure, is studied in sec-tion four. We derive a lower bound of concurrence for bipartite arbitrary dimensionalquantum states by using the covariance matrix criterion for separability. The lowerbound is independent of other bounds and can be used to make a better estimationof concurrence. We further prove that although we can not distill a singlet from many pairs of bound entangled states, the concurrence of two entangled quantumstates is always strictly larger than that of one, even both the two entangled quan-tum states are bound entangled. The subadditivity of concurrence is proved at theend of the section.The fifth section mainly concerns the Fully Entangled Fraction (FEF), whichis tightly related to the fidelity of optimal teleportation and many other quantuminformation processing. Unfortunately there is no general formula of FEF yet foran arbitrary bipartite quantum state. We first derive a tight upper bound of FEFfor bipartite quantum state with arbitrary dimensions. The upper bound is shownto be exact for two qubits system. For bipartite quantum systems with higherdimensions the upper bound can be used not only to give tight estimation of FEFbut also to improve the distillation protocol. Other upper bounds of FEF are alsoderived. These make complements on estimation of the value of FEF. These upperbounds make complements on the estimation of the value of FEF. For weakly mixedquantum states, an upper bound is shown to be very tight to the exact value ofFEF.At last we investigate the relation between the FEF and concurrence for bothbipartite high dimensional systems and three-qubit systems.