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研究生： 尤瑞揚
Yu, Rui-Yang
論文名稱： 可視化的3-局域關聯集合和3-量子關聯集合
Visualization the set of 3-local and 3-quantum correlation
指導教授： 梁永成
Liang, Yeong-Cherng
共同指導教授: 朱德明
Denis Rosset
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 55
中文關鍵詞： Convexity 、causal structure 、n-locality 、Partially Entangled state 、nonlinear
外文關鍵詞： Convexity, causal structure, n-locality, Partially Entangled state, nonlinear
相關次數： 點閱：118  下載：2 
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    • 根據貝爾定理，量子系統表現出比由LHV（局域隱變量）描述的經典系
    統更強的相關性。在標準Bell 事件中，所有觀察者之間共享同樣的隱變量。然而, 在量子網絡中, 隱變量是根據不同的時空拓撲結構有所分佈限制; 因此特別難以表徵其複雜時空拓譜結構下的經典集與量子集之特性。首先我們考慮到最簡單的循環量子網絡, 三角形, 其中各方只有二元輸出。對於相應的三局域集合，可以使用非線性不等式做外部近似，特別是通過膨脹法獲得的不等式（由Wolfe 等人）。在另一方面，只有幾個顯著的點是已知的三局域相關的點。因此，大部分的三局域集合相關空間的狀態是未知的。我們設計出一種方法, 可以找到更多額外的三局域相關點來填充這個未知區域的，從而提供一個內部近似。我們建立一個可視化的2 維和3 維對稱相關的子空間架
    構來比較所有這些結果。同時我們嘗試用隨機產生的三量子點和特定架構的三量子點來解答另一個未知的問題, 是否存在不是三局域的三量子相關事件點? 最終我們得到幾個有可能的候選點

    According to Bell’s theorem, quantum systems exhibit stronger correlations than classical systems described by LHV (local hidden variables). In standard Bell scenarios,
    the LHV is shared between all observers. In quantum networks however, resources have a distribution restricted according to a specific topology; the resulting local and quantum sets are particularly difficult to characterize. We consider the
    simplest cyclic quantum network, the triangle, where the parties have only binary outputs. For the corresponding 3-local set, outer approximations are available using nonlinear inequalities, in particular the ones obtained by the inflation method (Wolfe et al.). On the other hand, only a few remarkable points are known to be 3-local. Thus, the status of a large part of the correlation space is unknown. We devise a method to find additional 3-local points to fill this unknown area, thus providing an inner approximation. We compare all these results by providing 2D and 3D visualizations of the subspace of symmetric correlations. Another open question is the existence of 3-quantum correlations that are not 3-local. We work in that direction
    by producing random and structured 3-quantum distributions, and identifying the most promising candidates.

    Cover i Oral presentation document ii Chinese version . . . . . ii Abstract (Chinese) iii Abstract (English) iv Acknowledgments v Table of Contents vi List of Tables ix List of Figures x Chapter 1. Introduction 1 1.1 Bell locality . . . 2 1.2 The n-locality assumption . . . 2 1.3 The triangle scenario . . . . . 3 1.3.1 Triangle 3-local . . . . . . . 4 1.3.2 3-quantum correlations in the triangle scenario . . 6 1.3.3 Correlation basis in the triangle scenario . . . . .7 1.4 Prior works . . . . . . . . . . . . . . . . . . . . 7 1.5 Project objectives . . . . . . . . . . . . . . . . . 9 Chapter 2. 3-local behaviors 11 2.1 How to visualize and generate 3-local points ? . . . 11 2.1.1 Natural constraints in the symmetric subspace . . . 12 2.2 The sampling method and its results . . . . . . . . . 13 2.2.1 2D subspaces and approximate slices . . . . . . . . 14 2.2.2 2D subspaces in the correlation basis . . . . . . . 16 2.2.3 Paradox in sampling with increasing d . . . . . . . 16 2.3 Approximate P̸= by fixed response functions . . . . . .18 2.4 Optimizing 3-local models to approximate a given distribution . . . . . . . . . . . . . . . . . . .. . . . 19 2.5 Connecting gaps between 3-local points (white area in W set) . . . . . . . . . . . . . . . . . . . . . . . . . 21 Chapter 3. 3-quantum-behaviors ...........................24 3.1 Sets of symmetric 3-quantum distributions . . . . . . 24 3.2 Random sampling . . . . . . . . . . . . . . . . . . . 25 3.3 Parametrized shared entangled state and joint measurement . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 4. Conclusion . . . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . .32 Appendix A. Relaxation of the 3-locality scenarios using the inflation method . . . . . . . . . . . . . . . . . . . . .33 A.1 The triangle 3-local scenario and its simple inflation 33 A.1.1 Rules of inflation scenarios . . . . . . . . . . . .34 A.2 Relations between the original scenario and the inflation . . . . . . . . . . . . . . . . . . . . . . . . 35 A.2.1 Marginal distribution of Alice and Bob/Bob and Charlie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A.2.2 Marginal distribution of Alice and Charlie . . . . 36 A.3 Solving the system of inequalities . . . . . . . . . .37 A.4 Testing the distribution P= . . . . . . . . . . . . . 38 Appendix B. The response function form of P̸= .............39 B.1 Model with alpha, beta, gamma . . . . . . . . . . . . 39 B.2 Reducing the model to have beta take two values . . . 39 B.2.1 Coefficient for P̸=(a = 0; b = 0; c = 0) and P̸=(a = 1; b =1; c = 1) . . . . . . . . . . . . . . . . . . . . . . 40 B.2.2 Coefficient for P̸=(a = 0; b = 0; c = 1) . . . . . . 42 Appendix C. Fixed response functions to approximate P̸= with finite d . . . . . . . . . . . . . . . . . . . . . . . . .44 C.1 The coefficients PABC(000) and PABC(111) are equal . 44 C.2 The coefficients PABC(001) and PABC(011) are equal . 46 C.3 How to approach P̸= ? . . . . . . . . . . . . . . . . .47 C.4 What are the possible values of mu? . . . . . . . . . 48 C.4.1 Derivation . . . . . . . . . . . . . . . . . . . . .48 Appendix D. Mixed response function . . . . . . . . . . . 50 Appendix E. Relaxation of the 3-local set using 2-party constraints . . . . . . . . . . . . . . . . . . . . . . . 52 E.1 Bell-like scenario . . . . . . . . . . . . . . . . . .52

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