- 學位論文
量子資訊理論中的錯誤率分析
Error Exponent Analysis in Quantum Information Theory
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摘要
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。
並列摘要
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.
參考文獻
[2] B. C. H, . quantum cryptography using any two nonorthogonal states, Physical Review Letters, vol. 68, p. 3121 3124, 1992.
[3] B. C. H, B. G, C. C, Peres, and Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Physical Review Letters, vol. 70, p. 1895, 1993.
[4] C. I. L. Gottesman D, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature, vol. 402, p. 390, 1999.
[6] E. Gibney, Chinese satellite is one giant step for the quantum internet, Nature, vol. 535, no. 7613, pp. 478 479, jul 2016.
[7] M. Tomamichel, M. Berta, and J. M. Renes, Quantum coding with nite resources, Nature Communications, vol. 7, p. 11419, may 2016.