The Studies Of Quantum Correlations And Quantum Uncertainty Relation

The existence of quantum correlations between different quantum systems is one of the most notable features in quantum physics.Quantification of quantum correlation is one of the main subject related to the understanding and efficient utilization of the state for various quantum information processing schemes.Quantum entanglement is one of the vital quantum correlation.However,It turns out that entanglement is not the only way to quantify quantum correlations.In this thesis,we are mainly interested in quantum discord,which proposed by Ollivier and Zurek in 2001,as a measure of quantum correlations by quantizing concepts from classical information theory.The uncertainty principle is the fundamental of quantum mechanics,which is another important subject of quantum information theory.Quantum uncertainty relation is widely used in the detection of quantum entanglement and other related tasks.Following the two directions,this dissertation is devoted to the study of quantum correlation and quantum uncertainty relation.The main research results are listed as f'ollows:(i)We investigate two problems of quantum discord in quantum correlation:(a)Quantum discord of X-states as optimization of a multi-variable function with five parameters.First,we propose a new method to reduce the associated optimization problem into that of a one-variable entropy-like function F(z)on the closed interval[0,1],which in principle solves the problem of the quantum discord.Second,we give exact and analytical solutions for the general X-type state for several nontrivial regions of the parameters and prove rigorously that the answer is mostly given by the end-points of[0,1].Third,for the exceptional cases not covered by the second step and when the maximum is at an interior point of(0,1),we have formulated an effective algorithm to pin down the exotic solutions using Newton's formula.We remark that the third step covers the situation when all previous methods cannot solve the quantum discord.Combined with the end-points,the iterative formula has completely resolved the problem of the quantum discord for the general X-type state.In the last,We discuss the relationship between quantum discord and quantum concurrence for the rank two mixed X states.(b)Super quantum discord(SQD)for general two qubit X states.First,We propose the same idea to compute the super quantum discord by reducing the optimization to that of one-variable function.We also give analytical formulas of the SQD for several nontrivial regions of the parameters.In this section,iterative formula completely solves the problem in principle.Second,we analyze the dynamic behavior of super quantum discord through phase damping channel.(ii)We study the distribution of spin correlation strengths in multipartite systems.First,we generalize the concept of isotropic strength in the recent interesting work of Cheng to the general qudit system,and the strength distributions for tripartite and quadripartite qudit systems are thoroughly investigated.We show that the sum of relative isotropic strengths of any three qudit state over d-dimensional Hilbert space cannot exceed d-1,which generalizes of the case d=2.The trade-off relations and monogamy-like relations of the sum of spin correlation strengths for pure three-and four-partite systems are derived.Moreover,two applications are given on how the distribution of spin correlation strengths among different subsystems of a multipartite state are used in analyzing quantum correlation.In the first application we obtain a trade-off relation of maximal violation of Bell inequality for any four-qubit state;In another example,we present the necessary conditions of the pure biseparable four-qudit state,and give a criterion to detect genuine multipartite entanglement for any four-qudit state,which generalizes Vicente-Huber's result for the four-qubit state.(iii)We investigate variance-based uncertainty relations in the product form for uni-tary operators.In this chapter we first provide a new set of inequalities with the help of the geometric-arithmetic mean inequality.As these inequalities are "fine-grained" com-pared with the well-known Cauchy-Schwarz inequality,our framework naturally improves the results based on the latter.Then We present uncertainty principles for unitary oper-ators using this sequence of inequalities.As such,the unitary uncertainty relations based on our method outperform the best known bound introduced in[Phys.Rev.Lett.120,230402(2018)]to some extent.