【作者】 郭玮；

【导师】 王晓光；

【作者基本信息】 浙江大学 ， 理论物理， 2016， 博士

【摘要】 自从费曼等科学家提出量子计算机,以及将爱因斯坦的EPR佯谬视为量子通信的鼻祖开始,量子计算机和量子通信飞速发展并最终组合为一门统一的量子信息学科,而其下又包括量子测量等分支。随着摩尔定律极限的逼近和对保密通信的需求,现实生活中量子计算机和量子通讯也正在变为现实。本文主要研究的是量子信息下的量子度量学内容。本文的主要内容为：(1)第二章中我们回顾了基于对称对数算符的量子费舍信息的概念及导出过程,对量子费舍信息倒数为无偏估计参数测量精度的下限给出了严格的推导。对于多参数估计问题,我们导出了量子费舍信息矩阵并指出了它和度规间的关系。(2)第三章中我们回顾了经典微分几何的一些概念,并基于此推导出了几何相和贝里曲率。对于几何相的主要推广形式,我们给出了数学推导并辅以几何和物理解释。同时我们还引入了量子几何张量并指出了它和贝里曲率以及量子保真度间的关系。(3)第四章中我们回顾了退相干的经典模型：光场和二能级原子偶极相互作用的Jaynes-Cummings模型及拉比模型,其中我们对旋波近似进行了着重讨论。我们在零温洛伦兹谱下对一个假设旋波近似的量子比特和光场相互作用的退相干模型下对几何相进行了计算,并发现在非马尔科夫动力学和强耦合的情况下,几何相存在节点；利用级联方程这一精确数值方法,我们对不含旋转波近似的模型进行了精确数值解并发现几何相的节点消失了。即对于这个模型中存在的几何相节点是旋波近似的结果。(4)第五章中我们回顾了幺正演化下参数生成元的导出,并利用参数生成元将量子费舍信息和贝里曲率简洁的表达了出来。基于纯态在幺正演化下的参数估计问题,我们导出了不同参数的量子费舍信息乘积和贝里曲率间的一个不等式,并提出了量子费舍信息压缩这一概念；基于Robertson-Schrodinger不等式我们导出了另一个包括费舍信息矩阵非对角元的不等式。最后我们以自旋相干态为例对不等式进行了计算,并发现不等式的效果还是相当令人满意的。(5)附录中为和正文关系较大但不便置于正文中的较大段的推导,包括量子费舍信息矩阵不等式的导出、绝热定理及自旋相干态的基本性质。文章的最后是结论和展望。更多还原

【Abstract】 Since Feynman et al introduced the concept of quantum computation and the EPR paradox raised by Einstein et al showed the possibility to use quantum entanglement to transmit informa-tion, the field of quantum computer and quantum communication has been developing rapidly and unified into the subject of Quantum Information. As Moore’s law reaches its limit and the need for secure communication, quantum computer and quantum communication are now becoming reality. This thesis is devoted to study a sub-field of quantum information, which is quantum metrology.The content of thesis is as follows:(1) In Chapter 2, we review the definition of quantum fisher information based on symmetric logarithmic derivative operators, and showed that the inverse of quantum fisher information gives the upper bound of the accuracy of estimation with rigorous proof. As for multiparameter estimation problem, we derived the quantum fisher information matrix and pointed out its relationship with metric in Hilbert space.(2) In Chapter 3, we reviewed the notions in classical differential geometry, especially the parallel transport law and the holonomy, and we introduced the geometric phase together with Berry curvature. For the major forms of generalization of geometric phase, we give the derivation based on physical fact and from the math based on Fiber bundle. Meanwhile, we introduced the quantum geometric tensor and its relation with fidelity and Berry curvature.(3) In Chapter 4, we review the classical decoherence model based on dipole interaction be-tween atom and light field, i.e., Jaynes-Cummings model and Rabi model, with emphasis on the rotating-wave approximation. Under the Lorentzian spectrum at zero temperature, we calculated the geometric phase of a qubit under decoherence with rotating-wave approximation analytically, and we found that under non-Markovian dynamics and strong coupling, the geometric phase has nodal structure. Utilizing Hierarchy equation of motion, we investigated the evolution of qubit be-yond rotating-wave approximation and we found the nodal structure in geometric phase is gone. Thus we concluded that the nodal structure of geometric phase is a result of rotating-wave approx- imation.(4) In Chapter 5, we review the derivation of parameter generators and use them to express the quantum fisher information and Berry curvature. Within the framework of estimating param-eters under unitary evolution of pure state, we derived an inequality with the product of different parameters’ quantum Fisher information on onside and the four times of square of Berry curva-ture on the other side, and upon this we introduced the concept of quantum Fisher information squeezing. Based on Robertson-Schrodinger inequality we derived another inequality that involves off-diagonal elements of quantum Fisher information matrix. We test the inequality to a specific parameter estimation problem with spin-coherent state and the inequality works satisfactorily.(5) Within the Appendix are the derivations that is too long to plug into the main body, some particularly related subject involves the matrix inequality, adiabatic theorem and the introduction to spin-coherent state.The last Chapter contains the summary and possible directions to push forward.更多还原