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研究生： 余怡青
I-Ching Yu
論文名稱： 測試訊息因果論在量子通訊中的正確性
Information causality and its tests for quantum communications
指導教授： 林豐利
Lin, Feng-Li
學位類別： 博士
Doctor
系所名稱： 物理學系
Department of Physics
論文出版年： 2013
畢業學年度： 101
語文別： 英文
論文頁數： 122
中文關鍵詞： 訊息因果論 、量子通訊 、量子計算 、量子非定域性
英文關鍵詞： Information Causality, quantum communication, quantum computation, quantum non-locality
論文種類： 學術論文
相關次數： 點閱：162  下載：15 
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    • 在基於物理理論的通訊協定中, 例如:量子力學, 訊息因果論限制傳送者與
    接受者之間的最大共有訊息量.
    我們通過更廣泛的框架, 即使用討論訊號傳送及錯誤計算的結果, 來重新
    了解訊息因果論與量子力學的關係. 在我們的框架中, 訊息因果論將導致一組
    在二態量子系統下的Tsirelson 不等式 (量子系統的極限值). 基於這樣的結
    果, 訊息因果論對使用物理系統的實驗產生限制. 此外, 在我們的框架中, 可
    信賴的非定域性的計算是不可行的. 訊息因果論的限制將使得物理系統的計算
    線路無法進行可信賴的計算.
    另外, 我們直接計算共有訊息量, 藉以測試訊息因果論在更普遍的通訊協
    定中的正確性, 這些普遍的通訊協定包含多態的系統及非對稱的通訊管道. 我
    們的結果支持訊息因果論, 意思是在這些普遍的通訊協定中, 共有訊息量不會
    超過訊息因果論給的限制. 此外, 如果通訊管道包含兩個輸出及兩個輸入, 我
    們發現共有的量子系統擁有最大非定域性時 (滿足Tsirelson 不等式的限制),
    共有訊息量的值不是最大的. 最大的共有訊息量出現在共有的系統恰好滿足定
    域性理論給出的極大值時 (Bell 不等式給的限制), 且此時共有的訊息量和訊
    息因果論給的限制相同. 這個結果指出共享一個量子非定域系統, 並不一定產
    生較多的共有訊息量.

    Information causality has been proposed to constrain the maximal mutual information shared between sender and receiver in a communication protocol based on physical theories such as quantum mechanics.

    We reformulate the information causality in a more general framework by adopting the results of signal propagation and computation in a noisy circuit. In our framework, the information causality leads to a broad class of Tsirelson inequalities for the two-level quantum systems. This fact allows us to subject the information causality to
    the experimental scrutiny. A no-go theorem for reliable nonlocal computation is also derived. Information causality prevents any physical circuit from performing reliable computations.

    Moreover, we test the information causality for the more general quantum communication protocols with multi-level and (non-)symmetric channels by directly evaluating the mutual information. Our results support the information causality which is never violated for the more general settings discussed in this work. For the two-inputs/two-outputs cases, we also find that the information causality is saturated not for the channels with the maximal quantum
    non-locality associated with the Tsirelson inequality but for the marginal cases saturating the Bell's inequality. This indicates that the more quantum non-locality may not always yield the more mutual information.

    Table of Contents . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 No-signaling theory and quantum non-locality . . . . . . . . . . . . . . . . 1 1.1.1 Quantum non-locality . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1.1 The EPR paradox and the local hidden variable theory . . 2 1.1.1.2 The Bell's inequality and the CHSH inequality . . . . . . 4 1.1.1.3 More general Bell-type inequalities . . . . . . . . . . . . . 8 1.1.2 No-signaling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.2.1 The measurement scenario and the box version . . . . . . 11 1.1.2.2 No-signaling correlations . . . . . . . . . . . . . . . . . . . 12 1.1.3 Could the no-signaling theory single out quantum correlations? . . . 13 1.1.3.1 Beyond the quantum correlations . . . . . . . . . . . . . . 13 1.1.3.2 The no-signaling polytope . . . . . . . . . . . . . . . . . . 15 1.1.3.3 The communication complexity and the extremal non-local correlations . . . . . . . . . . . . . . . . . . . . . . . 16 1.2 Information Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.1 Information Causality single out quantum correlation? . . . . . . . 18 1.2.2 Information Causality and the Tsirelson bound . . . . . . . . . . . 20 1.2.2.1 The extremal non-locality violates Information Causality . 20 1.2.2.2 Information Causality derives the Tsirelson bound . . . . 21 1.2.3 Information Causality and the boundary of quantum correlations . 24 1.3 Signal propagating and noisy computation . . . . . . . . . . . . . . . . . . 28 1.3.1 The e efficient propagation through a noisy channel . . . . . . . . . . 28 1.3.2 The noisy computation . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.2.1 The model of noisy computation . . . . . . . . . . . . . . 30 1.3.2.2 The tolerable error rate and the depth for a reliable computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4 Semidefi nite programming and the quantum correlations for the bi-partite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.4.1 The quantum correlations for two-level quantum systems . . . . . . 37 1.4.1.1 Characterizing quantum correlations by Tsirelson's theorem 37 1.4.1.2 The maximally quantum violation for the CHSH-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.4.2 The quantum correlations for more general quantum systems . . . . 39 1.4.2.1 The constraints for bi-partite quantum probabilities . . . . 40 1.4.2.2 Bounding the quantum correlations with the hierarchical semidefi nite programming . . . . . . . . . . . . . . . . . . 42 1.4.2.3 The maximally quantum violation for the general Bell-type inequality . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Information Causality and Noisy Computations . . . . . . . . . . . . . . . . . . 44 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 Tsirelson-type inequalities from the information causality . . . . . . . . . . 46 2.3 Noisy nonlocal computation . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Testing Information Causality for General Quantum Communication Protocols . 52 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Multi-level Bell-type inequality from signal decay theorem . . . . . . . . . 58 3.3 Convexity and mutual information . . . . . . . . . . . . . . . . . . . . . . 61 3.3.1 Feasibility for maximizing mutual information by convex optimization? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.2 Convex optimization for symmetric and isotropic channels with i.i.d. and uniform input marginal probabilities . . . . . . . . . . . . . . . 64 3.4 Finding the bound of Bell-type inequality from the hierarchical semidefi nite programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.1 Projection operators with quantum behaviors . . . . . . . . . . . . 66 3.4.2 Hierarchy of the semidefi nite programming . . . . . . . . . . . . . . 70 3.4.3 The bound of Bell-type inequality and the corresponding mutual information in the hierarchical semidefi nite programming . . . . . . 72 3.5 Maximizing mutual information for general quantum communication channels 76 3.5.1 Symmetric channels with i.i.d. and uniform input marginal probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.2 Channels with non-uniform input marginal probabilities . . . . . . 81 3.5.3 Information causality for the most general channels . . . . . . . . . 85 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Appendix A Signal decay and data processing inequality for multi-nary channels . . . . . . . 89 A.1 Sketch of the proof in [50] . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.2 Generalizing to the multi-nary channels . . . . . . . . . . . . . . . . . . . . 91 B The concavity of mutual information . . . . . . . . . . . . . . . . . . . . . . . . 94 C Semidefi nite programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 D The Tsirelson-type inequality derived from the information causality . . . . . . 99 D.1 Checking the Tsirelson-type bound by semidefi nite programming . . . . . . 100 E The quantum constraints for n = 1 and n = 1 + AB certificate . . . . . . . . . . 104 E.1 The quantum constraints for n = 1 and n = 1 + AB certificate . . . . . . . 104 E.2 Estimating the number of constrains for n = 1 and n = 1 + AB certificate 105

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