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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 葉丙成 | |
dc.contributor.author | Hao-Chung Cheng | en |
dc.contributor.author | 鄭皓中 | zh_TW |
dc.date.accessioned | 2021-05-12T09:35:03Z | - |
dc.date.available | 2018-05-31 | |
dc.date.available | 2021-05-12T09:35:03Z | - |
dc.date.copyright | 2018-03-01 | |
dc.date.issued | 2018 | |
dc.date.submitted | 2018-02-22 | |
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dc.identifier.uri | http://tdr.lib.ntu.edu.tw/handle/123456789/1256 | - |
dc.description.abstract | 資訊理論中最基本的問題之一是刻劃三個重要參數的取捨—資訊處理的品質優劣、錯誤更正碼的區塊長度、以及傳輸率。錯誤率指數分析即為一個強大且有效的方法來研究當傳輸率固定時錯誤概率如何隨著編碼區塊增大進而指數遞減。在本論文中,我們討論兩個重要的量子資訊處理協定—經由量子資訊的協助來壓縮經典數據、以及經典數據經由量子信道傳輸—之錯誤率指數分析。
我們首先證明錯誤指數函數的諸多重要性質,使我們得以更深刻理解量子資訊協定的錯誤率行為模式。第二、在有限的區塊編碼長度下我們對研究的兩種量子資訊協定求得精確的錯誤率分析,為次世代量子資訊科技的設計提供了更佳的品質估計準則。最後,我們研究當傳輸率趨近重要的閾值時的錯誤概率行為—當壓縮率緩慢逼近條件熵值時,被壓縮的經典數據得以完美恢復、以及當傳輸率緩慢逼近信道容量時數據得以無暇傳輸。 此論文呈現方式力求以經典資訊理論的架構來撰寫,因此讀者不限於具有量子資訊理論背景之學者。工程師、科技系統設計者以及任何對量子資訊理論有興趣者皆能藉由閱讀此論文來探索此豐富且深邃的研究課題。 | zh_TW |
dc.description.abstract | One of the fundamental problems in information theory is to clarify the trade-offs between the performance of an information task, the size of the coding scheme, and the coding rate that determines the efficiency of the task.
Error exponent analysis was proposed as a powerful methodology to study how rapidly the error probability exponentially decays with an increase of coding blocklength when the rate is fixed.In this thesis, we give an exposition of error exponent analysis to two important quantum information processing protocols - classical data compression with quantum side information, and classical communications over quantum channels. We first prove substantial properties of various exponent functions, which allow us to better characterize the error behaviors of the tasks. Second, we establish accurate achievability and optimality finite blocklength bounds for the optimal error probability, providing useful and measurable benchmarks for future quantum information technology design. Finally, we study the error probability under the scenario that the coding rate converges to certain limits, a research topic known as moderate deviation analysis. In other words, we show that the data recovery can be perfect when the compression rate approaches the conditional entropy slowly, and the reliable communication over a classical-quantum channel is possible as the transmission rate approaches channel capacity slowly. The audience of this thesis are not restricted to researchers with backgrounds in quantum information theory. Engineers, technology providers, and people who interest in information processing are welcome to explore the frontiers along this line of research. | en |
dc.description.provenance | Made available in DSpace on 2021-05-12T09:35:03Z (GMT). No. of bitstreams: 1 ntu-107-F99942118-1.pdf: 3485401 bytes, checksum: 44037960afe6a81eb9b42e5cd981fb90 (MD5) Previous issue date: 2018 | en |
dc.description.tableofcontents | 1 Introduction 1
1.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quantum Information Processing Protocols . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Information Storage with a Quantum Helper (Source Coding) . . . . . . . . . . 5 1.2.2 Information Transmission over a Quantum Channel (Channel Coding) . . . . . 7 1.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 I Fundamentals 12 2 Mathematical Tools 13 2.1 Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Large Deviation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Quantum Entropic Quantities and Notation 26 3.1 Quantum Rényi divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Conditional Rényi Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Rényi Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Quantum Hypothesis Testing 39 4.1 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.1 Nussbaum-Szkoªa Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.2 Proofs of Theorem 4.4 and Corollary 4.1 . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Moderate Deviation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3.1 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.2 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 II Information Storage with a Quantum Helper 52 5 Error Exponent Functions (Source Coding) 53 5.1 Variational Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Properties of Auxiliary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.1 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.2 Proof of Proposition 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3 Properties of Error Exponent Functions and Saddle-Point . . . . . . . . . . . . . . . . 60 6 Achievability (Source Coding) 64 7 Optimality (Source Coding) 67 7.1 One-Shot Converse Bound (Hypothesis Testing Reduction) . . . . . . . . . . . . . . . 67 7.2 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8 Moderate Deviation Analysis (Source Coding) 72 8.1 Asymptotic Expansion of Error Exponent around Slepian-Wolf Limit . . . . . . . . . . 75 III Information Transmission over a Quantum Channel 78 9 Error Exponent Functions (Channel Coding) 79 9.1 Variational Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9.2 Properties of Auxiliary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 9.2.1 Proof of Proposition 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 9.2.2 Proof of Proposition 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 9.2.3 Proof of Proposition 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.2.4 Proof of Proposition 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.3 Properties of Error Exponent Functions and Saddle-Point . . . . . . . . . . . . . . . . 95 9.3.1 Proof of Proposition 9.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.3.2 Proof of Proposition 9.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10 Achievability (Channel Coding) 103 11 Optimality (Channel Coding) 107 11.1 Literature Review of Classical Sphere-Packing Bound . . . . . . . . . . . . . . . . . . 108 11.2 A Weak Sphere-Packing Bound via Wolfowitz Strong Converse . . . . . . . . . . . . . 110 11.2.1 Proof of Wolfowitz's Strong Converse, Proposition 11.1 . . . . . . . . . . . . . . 112 11.3 A Strong Sphere-Packing Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 11.3.1 One-Shot Converse Bound (Hypothesis Testing Reduction) . . . . . . . . . . . 114 11.3.2 Chebyshev's Type Converse Bound . . . . . . . . . . . . . . . . . . . . . . . . . 114 11.3.3 A Sharp Converse Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 11.3.4 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 11.3.5 Proofs of Theorem 11.1 and Corollary 11.1 . . . . . . . . . . . . . . . . . . . . 124 11.4 Symmetric Classical-Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 127 12 Moderate Deviation Analysis (Channel Coding) 130 12.1 Proof of Achievability, Theorem 12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 12.2 Proof of Converse, Theorem 12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 12.3 Asymptotic Expansions of Error-Exponent around Capacity . . . . . . . . . . . . . . . 136 13 Conclusions and Open problems 140 13.1 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 13.1.1 Properties of Error Exponent Functions and Auxiliary Functions . . . . . . . . 141 13.1.2 Achievaibility: Random Coding Bound . . . . . . . . . . . . . . . . . . . . . . . 142 13.1.3 Optimality: Sphere-Packing Bound . . . . . . . . . . . . . . . . . . . . . . . . . 143 13.1.4 Moderate Deviation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Bibliography 145 | |
dc.language.iso | en | |
dc.title | 量子資訊理論中的錯誤率分析 | zh_TW |
dc.title | Error Exponent Analysis in Quantum Information Theory | en |
dc.type | Thesis | |
dc.date.schoolyear | 106-1 | |
dc.description.degree | 博士 | |
dc.contributor.oralexamcommittee | 鐘楷閔,王奕翔,管希聖,賴青沂,謝明修 | |
dc.subject.keyword | 錯誤指數分析,中偏差分析,大偏差分析,量子資訊理論,經典量子信道,量子輔助資訊,可靠度函數,矩陣分析, | zh_TW |
dc.subject.keyword | error exponent analysis,moderate deviation analysis,quantum information theory,classical-quantum channel,Slepian-Wolf coding,quantum side information,reliability function,large deviation theory,matrix analysis, | en |
dc.relation.page | 154 | |
dc.identifier.doi | 10.6342/NTU201800597 | |
dc.rights.note | 同意授權(全球公開) | |
dc.date.accepted | 2018-02-22 | |
dc.contributor.author-college | 電機資訊學院 | zh_TW |
dc.contributor.author-dept | 電信工程學研究所 | zh_TW |
顯示於系所單位: | 電信工程學研究所 |
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