摘要
量子纠缠态是量子信息与量子计算理论中最主要的物理资源.利用两体或多体量子态的纠缠性质可以实现很多经典信息理论中无法完成的任务,比如量子隐形传态,量子密钥分配,量子并行计算等等.在过去的几十年里,人们不断地对量子纠缠现象进行研究,并且取得了很多重要的进展.然而我们仍然没有完全理解它的本质.最基本的,给定一个量子态,如何判断它是否纠缠?进一步地,如何来度量一个量子态的纠缠程度?再比如,如何利用量子纠缠态更好地完成量子信息传递的任务?很多问题需要更加深入的研究.本文将从三个主要方面对量子纠缠态进行研究,取得了如下成果.
在论文的第三部分,研究了量子态的可分性问题.本章节首先简单介绍了几个已有的可分性判据.在此基础上,分别利用量子态的Bloch表示,量子态的协方差矩阵等方法给出了量子态的可分性判据.我们还给出了多体量子态的标准型,进而把量子态的可分性与量子态的标准型联系在一起,即通过将量子态的密度矩阵转化成其标准型,提升已有可分性判据识别纠缠态的能力.
论文的第四部分研究了纠缠度量concurrence.首先将基于量子态的协方差矩阵给出的可分性判据与concurrence联系在一起,从而给出了高维量子纠缠态的concurrence的一个下界.这个下界独立于已有的结论,从而改进了两体纠缠度量concurrence的估计值.本文还研究了两体和多体纠缠度量concurrence的关系,并且给出了多体纠缠度量concurrence的更好的估计值.本文还证明了虽然无法从许多束缚纠缠态当中提纯出最大纠缠的纯态,但是两个纠缠态的concurrence总是大于其中的任意一个,即使这两个纠缠态都是束缚纠缠态.在这一部分的最后给出了concurrence次可加性的简单证明.
论文的第五部分对量子隐形传态的最优保真度进行了研究.Horodecki等指出量子隐形传态的最优保真度与量子态的Fully Entangled Fraction(FEF)有直接的关系.然而对于FEF却没有一般的计算公式.我们首先给出了两体任意维量子态的FEF的一个上界值,进一步地指出这个上界对于两个量子比特的情形就是FEF的精确值.对于高维情形,我们指出这个上界不仅可以用来很好地估计量子隐形传态的最优保真度,还可以用来改进量子纠缠提纯方案.随后利用拉格朗日乘子法给出了FEF的另外的一个上界,并指出它不同于前者.对于弱混合的量子态,我们得到了一个专门的上界,它可以非常精确地估计FEF.在本章节的最后,我们给出了两体任意维量子态的FEF与它的concurrence之间的关系并且研究了三个量子比特的concurrence和其中两个量子比特的FEF之间的控制关系.
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.
引文
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