Theoretical Studies On Quantum Correlations And Quantum Uncertainty Relation

Quantum information theory is a burgeoning interdiscipline which applied the basic properties of quantum mechanics with the information science. It mainly consists of quantum communication, quantum computation and so on. Quantum entanglement is an important physical resource and plays a vital role in quantum information processing tasks. The latest research results show that quantum entanglement is just part of quantum correlation. Sepa-rable quantum states may have non-classical correlations. It also plays a particular important role in quantum information processing. Hence the quantification of quantum correlation be-comes a vital subject of quantum information theory. Generally speaking, quantification of quantum correlation based on different methods will provide different view point of under-standing of quantum correlation. There is a relation between geometric discord and quantum nonlocality, a relation between quantum discord based on information theory and quantum uncertainty relation. In addition, uncertainty relation has a wide range of applications such as entanglement detection, security analysis in quantum cryptography and so on. The tight bounds of the quantum uncertainty relation will directly affect many quantum tasks. As a consequence studies on measure of quantum correlation and the tighter bounds of the quan-tum uncertainty relation are the main issues of this thesis. This thesis includes eight chapters and our main research work are given in chapters 3 to 7. The summary and prospection are presented in chapter 8.In Chapter 1, we introduce the background of this thesis and the developments of quan-tum correlation and quantum uncertainty relation. In Chapter 2, we introduce the prelimi-nary concepts and knowledge of quantum information theory that will be used in this thesis.In Chapter 3, the quantification of quantum correlation based on the distance measures are studied. First, we study the nonclassical correlations in quantum state by the perturbing local unitary operation method. During the optimization of unitary process, one can divide the set of the local unitary operation into different subsets based on the different constraint condition. We define a new quantity called generalized measurement-induced nonlocality which is the dual definition of geometric discord. We presents a unified understanding between geometric discord, measurement-induced nonlocality and generalized measurement-induced nonlocality subject to the Hilbert-Schmidt distance. In addition, we investigate the quantum correlation in weak measurement processing based on trace distance and redefine the quantum correlation cost.In Chapter 4, based on the definition of the symmetric quantum discord, we give the analytic expression for the information-theoretical for the information-theoretical symmetric discord for one type of two-qubit X states.In Chapter 5, we study the roles of quantum correlations in quantum information pro-cessing. The results show that the overlap measurement scheme (OMS) could require the presence of quantum entanglement, classical correlations, and even no correlation based on different types of input states. And in quantum cloning, the results show that quantum corre-lations (quantum entanglement, quantum discord, tripartite entanglement)can not uniquely determine the characteristic parameters of the cloning such as the fidelity and success proba-bility. These parameters also depend on the quantum cloning scheme and the cloned states. In particular, we even find that the successful quantum cloning does not need any quantum correlation at all.In Chapter 6, we investigate; the entropie uncertainty relation. By employing the Renyi entropy, we give the explicit uncertainty relations for a pair of observables which are operated the qubit state successively. The results cover a large family of the entropy (including the Shannon entropy) uncertainty relation with a lower optimal bound. Next, we present tighter bounds on both entropie uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory.In Chapter 7, we skillfully utilize quantum superposition principle to establish the up-per and lower bounds on the stronger uncertainty relation for two observables and multiple observables. The advantage is that our bounds include some free parameters which not only guarantee the nontrivial bounds but also can effectively control the bounds as tightly as one expects. The summary and prospection are given in Chapter 8.