量子费舍信息和几何相位在量子光学中的应用

发布时间：2018-06-23 20:47

本文选题：量子度量学 + 量子费舍信息　；参考：《浙江大学》2016年博士论文

【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。
[Abstract]:Ever since Feynman and other scientists proposed quantum computers and regarded Einstein's EPR paradox as the ancestor of quantum communication, quantum computers and quantum communications have developed rapidly and finally combined into a unified subject of quantum information. And it also includes the quantum measurement and other branches. With the approaching of the limit of Moore's law and the demand for secure communication, quantum computer and quantum communication are becoming reality in real life. This paper focuses on quantum metrics under quantum information. The main contents of this paper are as follows: (1) in Chapter 2, we review the concept and derivation process of quantum Fisher information based on symmetric logarithmic operator, and give a strict derivation of the lower limit of the measurement accuracy of the inverse of quantum Fisher information for unbiased estimation parameters. For the problem of multi-parameter estimation, we derive the quantum Fisher information matrix and point out the relationship between it and metric. (2) in chapter 3, we review some concepts of classical differential geometry, and derive the geometric phase and Berry-curvature. For the main generalized forms of geometric phase, we give the mathematical derivation, supplemented by geometric and physical explanations. At the same time, we introduce the quantum geometry Zhang Liang and point out the relationship between it and the Bayesian curvature and quantum fidelity. (3) in Chapter 4, we review the classical model of decoherence: the Jaynes-Cummings of the dipole interaction between the light field and the two-level atom. Model and rabbi model, Among them, we focus on the discussion of the spin wave approximation. In this paper, we calculate the geometric phase in a decoherence model of a hypothetical spin wave approximation of quantum bit and light field under the zero-temperature Lorentz spectrum, and find that there are nodes in the geometric phase under the condition of non-Markov dynamics and strong coupling. By using the exact numerical method of cascade equations, we obtain the exact numerical solution of the model without the approximation of rotational wave and find that the nodes of geometric phase disappear. In chapter 5, we review the derivation of the parametric generator in unitary evolution, and express the quantum Fisher information and the Berry-curvature concisely by using the parametric generator. Based on the parameter estimation problem of pure states in unitary evolution, we derive an inequality between the product of quantum Fisher information and the Berri curvature of different parameters, and propose the concept of quantum Fisher information squeezing. Based on Robertson-Schrodinger inequality, another inequality including non-diagonal elements of Fisher information matrix is derived. Finally, we calculate the inequality by taking the spin coherent state as an example, and find that the effect of the inequality is quite satisfactory. (5) the derivation of the larger section in the appendix, which has a large relation with the text but is not convenient to be placed in the text. It includes the derivation of quantum Fisher's information matrix inequality, the adiabatic theorem and the basic properties of spin coherent states. The last part of the article is the conclusion and prospect.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O431.2

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