本文選題：量子度量學(xué) + 量子費(fèi)舍信息　； 參考：《浙江大學(xué)》2016年博士論文

【摘要】：自從費(fèi)曼等科學(xué)家提出量子計(jì)算機(jī),以及將愛因斯坦的EPR佯謬視為量子通信的鼻祖開始,量子計(jì)算機(jī)和量子通信飛速發(fā)展并最終組合為一門統(tǒng)一的量子信息學(xué)科,而其下又包括量子測(cè)量等分支。隨著摩爾定律極限的逼近和對(duì)保密通信的需求,現(xiàn)實(shí)生活中量子計(jì)算機(jī)和量子通訊也正在變?yōu)楝F(xiàn)實(shí)。本文主要研究的是量子信息下的量子度量學(xué)內(nèi)容。本文的主要內(nèi)容為：(1)第二章中我們回顧了基于對(duì)稱對(duì)數(shù)算符的量子費(fèi)舍信息的概念及導(dǎo)出過程,對(duì)量子費(fèi)舍信息倒數(shù)為無偏估計(jì)參數(shù)測(cè)量精度的下限給出了嚴(yán)格的推導(dǎo)。對(duì)于多參數(shù)估計(jì)問題,我們導(dǎo)出了量子費(fèi)舍信息矩陣并指出了它和度規(guī)間的關(guān)系。(2)第三章中我們回顧了經(jīng)典微分幾何的一些概念,并基于此推導(dǎo)出了幾何相和貝里曲率。對(duì)于幾何相的主要推廣形式,我們給出了數(shù)學(xué)推導(dǎo)并輔以幾何和物理解釋。同時(shí)我們還引入了量子幾何張量并指出了它和貝里曲率以及量子保真度間的關(guān)系。(3)第四章中我們回顧了退相干的經(jīng)典模型：光場(chǎng)和二能級(jí)原子偶極相互作用的Jaynes-Cummings模型及拉比模型,其中我們對(duì)旋波近似進(jìn)行了著重討論。我們?cè)诹銣芈鍌惼澴V下對(duì)一個(gè)假設(shè)旋波近似的量子比特和光場(chǎng)相互作用的退相干模型下對(duì)幾何相進(jìn)行了計(jì)算,并發(fā)現(xiàn)在非馬爾科夫動(dòng)力學(xué)和強(qiáng)耦合的情況下,幾何相存在節(jié)點(diǎn)；利用級(jí)聯(lián)方程這一精確數(shù)值方法,我們對(duì)不含旋轉(zhuǎn)波近似的模型進(jìn)行了精確數(shù)值解并發(fā)現(xiàn)幾何相的節(jié)點(diǎn)消失了。即對(duì)于這個(gè)模型中存在的幾何相節(jié)點(diǎn)是旋波近似的結(jié)果。(4)第五章中我們回顧了幺正演化下參數(shù)生成元的導(dǎo)出,并利用參數(shù)生成元將量子費(fèi)舍信息和貝里曲率簡(jiǎn)潔的表達(dá)了出來。基于純態(tài)在幺正演化下的參數(shù)估計(jì)問題,我們導(dǎo)出了不同參數(shù)的量子費(fèi)舍信息乘積和貝里曲率間的一個(gè)不等式,并提出了量子費(fèi)舍信息壓縮這一概念；基于Robertson-Schrodinger不等式我們導(dǎo)出了另一個(gè)包括費(fèi)舍信息矩陣非對(duì)角元的不等式。最后我們以自旋相干態(tài)為例對(duì)不等式進(jìn)行了計(jì)算,并發(fā)現(xiàn)不等式的效果還是相當(dāng)令人滿意的。(5)附錄中為和正文關(guān)系較大但不便置于正文中的較大段的推導(dǎo),包括量子費(fèi)舍信息矩陣不等式的導(dǎo)出、絕熱定理及自旋相干態(tài)的基本性質(zhì)。文章的最后是結(jié)論和展望。
[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.
【學(xué)位授予單位】：浙江大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O431.2

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