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研究生： 李易展
Lee, Yi-Chan
論文名稱： 量子資訊科學衍生的限制與可能性
The constraint and possibility derived from quantum information science
指導教授： 李瑞光
Lee, Ray Kuang
口試委員: 蘇正耀
吳欣澤
徐立義
陳柏中
學位類別： 博士
Doctor
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 94
中文關鍵詞： 量子資訊 、非侷域性 、宇稱-時間對稱 、錯誤更正碼
外文關鍵詞： quantum information, nonlocality, parity-time symmetry, error correction code
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    • 量子資訊科學是一個跨領域的學科，其在物理、應用數學、電腦科學等相關領域發展了許多有趣的連結。量子資訊科學的概念亦在這些領域中產生了許多不同的影響。在量子資訊科學中，即使最大量子糾纏態具備了古典物理系統沒有的非侷域性，其應用仍舊被要求遵守狹義相對論的光速為資訊傳遞最高速原則。在這篇論文中，我們利用了此量子科學資訊中的基本原則來檢視宇稱-時間對稱量子理論($\mathcal{PT}$對稱量子理論)。我們的結果顯示對於$\mathcal{PT}$對稱量子理論的不適當解讀會造成資訊傳遞超越光速的情況，同時定下了發展這個新穎的量子理論必須遵守的限制。在這篇論文的第二部分，我們研究了在Kitaev蜂巢模型上實現量子錯誤更正碼的可能。我們首先證明了該模型基態的準簡併性(quasi-degeneracy)以及其近似侷域不可分辨性(approximate local-indistinguishability)。我們更進一步的提出了在古典電腦上模擬該錯誤更正碼在熱平衡過程中的表現的可能性。藉著結合量子資訊科學以及物理兩個領域的方法，我們的結果顯示出兩個領域的結合有更多的可能性。

    Quantum information science, a interdisciplinary field, develops many interesting connections between the fields of physics, mathematics, computer science and other related subjects. The intuitive ideas in quantum information also has different influence in these areas. In this thesis, we use no-signalling condition in special relativity theory which is required to be satisfied in quantum information science, even under the condition that a maximally entangled state is given, to test parity-time symmetry quantum theory ($\mathcal{PT}$-symmetric theory). Our result shows that improper interpretation of $\mathcal{PT}$-symmetry quantum mechanics will cause the violation of no-signalling principle and set the constraints which should be obeyed for developing this novel theory. In the second part of thesis, we study the error correction in quantum information science on Kitaev honeycomb model, and establish the connection with the researches done by physicists. Our result formally proves the quasi-degeneracy and the approximate local-indistinguishability of the ground states of Kitaev honeycomb model. We further point out that the simulation of the information lifetime under the thermalization process is possible. By combining the methods from physics society and quantum information society, we shows that there are more possibility in the connection between these two fields.

    I Impossible and possible alternatives of parity-time symmetric Hamiltonian 1 1 Introduction 2 2 Background knowledge of parity-time symmetric Hamiltonian 5 2.1 The first PT -symmetric Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Simplest two-level PT symmetric Hamiltonian . . . . . . . . . . . . . . . . . . . 8 2.3 Proposed applications of finite dimensional PT-symmetric Hamiltonian . . . . . 9 2.3.1 Ultra-fast spin flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.2 Distinguish non-orthogonal state with single-shot measurement . . . . . . 10 2.4 Equivalence of PT symmetric Hamiltonian and Hermitian Hamiltonian in finite dimensional system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Violation of no-signalling principle 14 3.1 No-signalling principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Transmission of one bit classical information through PT box . . . . . . . . . . . 15 3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 General parity-time symmetric two-level system 19 4.1 Naimark dilation of PT -symmetric Hamiltonian . . . . . . . . . . . . . . . . . . 19 4.2 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Future work and conclusion of PT -symmetric Hamiltonian 26 II Quantum error correction on Kitaev honeycomb model 28 6 Introduction 29 7 Background knowledge of QECC and KHM 32 7.1 Quantum error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.1.1 Bit-flip code and Phase-flip code . . . . . . . . . . . . . . . . . . . . . . . 34 7.1.2 Shor code and discretization of errors . . . . . . . . . . . . . . . . . . . . 36 7.1.3 Stabilizer formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.1.4 Toric code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.1.5 Quantum error-correction condition . . . . . . . . . . . . . . . . . . . . . 46 7.2 Kitaev honeycomb model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8 Quasi-degeneracy and approximate local indistinguishability 56 8.1 Quasi-degeneracy of ground states . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.2 Local indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3 Approximate local-indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3.1 Proof of approximate local-indistinguishability . . . . . . . . . . . . . . . 60 8.3.2 Argument for arbitrary local operators . . . . . . . . . . . . . . . . . . . . 65 9 Numerical simulation results 68 9.1 Methods of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 9.2 Results of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9.2.1 Logical error rate in medium- and low-temperature regions . . . . . . . . 73 9.2.2 Logical error rate in high-temperature region . . . . . . . . . . . . . . . . 77 10 Conclusion and future work of QECC on KHM 84 11 Conclusion of thesis 86 Bibliography 87

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