測試訊息因果論在量子通訊中的正確性

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(1)Department of Physics National Taiwan Normal University PhD Thesis. Information causality and its tests for quantum communications. I-Ching Yu. Advisor: Feng-Li Lin, Ph.D. February. 25. 2013.

(2) c Copyright by I-Ching Yu 2013. ii.

(3) to my family and teachers. iii.

(4) Acknowledgements In the past six years, there were many times that I wanted to give up. Because of the help and the encouragement from teachers, my family and friends, I could hold on straight to the last second and finish this thesis. First, I would like to thank Prof. Feng-Li Lin, my supervisor. Prof. Lin not only taught me how to do the research but also reminded me the correct attitude for doing researches. It is practical and down-to-earth. When I made mistakes in writings or I was not on the right track, Prof. Lin used his patience to modify them again and again. Without his insistence, this thesis could not have the present form. I also want to thank Prof. Hong-Yi Chen and Prof. Li-Yi Hsu. Prof. Chen gave me a lot of help for writing the computer program and building the machine for Message Passing Interface (MPI). Due to Prof. Hsu’s guidance, I have deeper and broader understanding of quantum information. Furthermore, I want to appreciate the committee members of my oral defence, Prof. Chi-Yee Cheung, Prof. Zheng-Yao Su, Prof. Hsi-Sheng Goan and Prof. Min-Hsiu Hsieh. They took time to read my thesis and provided many valuable comments. It benefited me a lot from their comments. My gratitude would go to my friend Ching-Yu. We were classmates over ten years. I can discuss all the worry and problems with her and always get solutions. I will never forget the moment which we went through. I also want to appreciate the assistant, ChunPing and the members in room A201, Chao-Chun, Hsuan-Hao and Pei-Hua. With their suggestions and help, I can have the different point of view to solve my problems and my oral defence can be held without a hitch. In addition, I would like to express my gratitude to my husband, Hsin-Ping for all that he has given to our family. With his support, I did not have the financial stress such that I can concentrate on my study. Finally, I want to thank my parents, Mr. Kuo-Wen Yu and Mrs. Chiu-Feng Huang. Thank them for giving my life and all the supports without conditions. My gratitude can not be written in words. I hope that I can pay them back and let them be proud of me on some day.. iv.

(5) Abstract 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 twolevel 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. Keywords: Information causality, quantum communication, quantum computation, quantum non-locality. v.

(6) 中文摘要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. 關鍵字：訊息因果論、量子通訊、量子計算、量子非定域性.

(7) Table of Contents Page 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. 1.1.3. No-signaling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.2.1. The measurement scenario and the box version . . . . . . 11. 1.1.2.2. No-signaling correlations . . . . . . . . . . . . . . . . . . . 12. 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 nonlocal correlations . . . . . . . . . . . . . . . . . . . . . . . 16. 1.2. Information Causality 1.2.1. Information Causality single out quantum correlation? . . . . . . . 18. 1.2.2. Information Causality and the Tsirelson bound . . . . . . . . . . . 20. 1.2.3 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. 1.2.2.1. The extremal non-locality violates Information Causality . 20. 1.2.2.2. Information Causality derives the Tsirelson bound . . . . 21. Information Causality and the boundary of quantum correlations . 24. Signal propagating and noisy computation . . . . . . . . . . . . . . . . . . 28 1.3.1. The efficient propagation through a noisy channel . . . . . . . . . . 28. 1.3.2. The noisy computation . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.2.1. The model of noisy computation . . . . . . . . . . . . . . 30. vi.

(8) 1.3.2.2. The tolerable error rate and the depth for a reliable computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 1.4. Semidefinite 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 hierarchal semidefinite programming . . . . . . . . . . . . . . . . . . 42. 1.4.2.3. The maximally quantum violation for the general Belltype 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 semidefinite programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.1. Projection operators with quantum behaviors . . . . . . . . . . . . 66. 3.4.2. Hierarchy of the semidefinite programming . . . . . . . . . . . . . . 70. vii.

(9) 3.4.3. The bound of Bell-type inequality and the corresponding mutual information in the hierarchical semidefinite 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. 3.6. 3.5.2. Channels with non-uniform input marginal probabilities. . . . . . . 81. 3.5.3. Information causality for the most general channels . . . . . . . . . 85. 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 Semidefinite programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 D The Tsirelson-type inequality derived from the information causality . . . . . . 99 D.1 Checking the Tsirelson-type bound by semidefinite 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 certificates 105. viii.

(10) List of Figures 1.1. The communication protocol for k = 2 task . . . . . . . . . . . . . . . . . . 20. 1.2. The RAC protocol for k = 4 task . . . . . . . . . . . . . . . . . . . . . . . 23. 1.3 1.4. The model of noisy computation . . . . . . . . . . . . . . . . . . . . . . . . 31 P The example for the failure of inequality I(xm ; Y1 , Y2 , ..., Yk ) ≤ i I(xm ; Yi ) 34. 2.1. RAC protocol for a (n, k, l)-circuit. Each vertex of the circuit corresponds to a NS-box, with its details shown in the big ellipses. . . . . . . . . . . . . 49. 3.1. The NS-box and the channel . . . . . . . . . . . . . . . . . . . . . . . . . . 53. 3.2. Scheme for maximizing the mutual information I over quantum channels. . 56. 3.3. The geometric interpretation of collection Qn. 3.4. Mutual information v.s. (quantum) communication complexity for d = 2,. . . . . . . . . . . . . . . . . 71. k = 2 RAC protocol with i.i.d. and uniform input marginal probabilities. Here, the (quantum) communication complexity is characterized by the CHSH function. The red part can be achieved also by sharing the local correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5. Some points near the top region in Fig. 3.4. . . . . . . . . . . . . . . . . . 80. 3.6. Mutual information vs (quantum) communication complexity for isotropic channels with i.i.d. and uniform input marginal probabilities. . . . . . . . . 81. 3.7. I = I0 vs Pr(a0,1 = 0) for case (i). . . . . . . . . . . . . . . . . . . . . . . . 82. 3.8. Density plot of the Left figure. . . . . . . . . . . . . . . . . . . . . . . . . . 82. 3.9. I0 vs Pr(a0,1 = 0) for case (ii). . . . . . . . . . . . . . . . . . . . . . . . . . 83. 3.10 I1 vs Pr(a0,1 = 0) for case (ii). . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.11 I vs Pr(a0,1 = 0) for case (ii). . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.12 Density plot of the Left figure. . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.13 I0 vs Pr(a0,1 = 0) for case (iii). . . . . . . . . . . . . . . . . . . . . . . . . 84 3.14 I1 vs Pr(a0,1 = 0) for case (iii). . . . . . . . . . . . . . . . . . . . . . . . . 84 3.15 I vs Pr(a0,1 = 0) for case (iii). . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.16 Density plot of the Left figure. . . . . . . . . . . . . . . . . . . . . . . . . . 84. ix.

(11) Chapter 1. Introduction 1.1. No-signaling theory and quantum non-locality. Quantum information science has been developed for nearly a half century. The tasks in the quantum information science such as commutation, computation and cryptography are operated with quantum systems. Since the famous Shor’s factoring algorithm [1] was proposed, people were aware that the quantum computation may be efficient. Therefore, quantum information science became a very important topic in the 1980s. Of course, we cannot ignore the beautiful idea of the BB84 key distribution protocol [2]. Using the uncertainty principle of quantum mechanics, the BB84 protocol keeps the security of the communication. From the above examples, we could know the importance of quantum mechanics to the information science. Nowadays, we know that the quantum mechanics is inherently non-local and it cannot violate the no-signaling theory which states that superluminal signaling is impossible. Due to these properties, here comes a problem: could the quantum correlations be singled out by considering both the non-locality and the no-signaling theory? In this section, we will review the theory of quantum non-locality and the nosignaling theory, and then try to answer this question. 1.1.1. Quantum non-locality. In 1935, Albert Einstein and his collaborators Boris Podolsky and Nathan Rosen (known collectively as EPR) proposed a thought experiment to show the conflict between locality and physical reality in quantum mechanics. Their arguments were latter called the EPR paradox. The paradox implies that quantum mechanics is an incomplete physicals theory [3]. They also gave the definition for a complete physical theory. That is, if every element of the physical reality have a counterpart in the physical theory, we then could judge this physical theory is complete. Furthermore, what is the element of physical reality? The authors said if one could predict the quantity of a measurement with certainly for an 1.

(12) undisturbed system, then there must exist an element of physical reality determining this quantity. However, there are two ways to solve the EPR paradox, the first one is the local hidden variable theory and the other one is the quantum non-locality. Almost thirty years after the proposition of EPR paradox, in 1964, J. S. Bell proposed the famous Bell inequality [5]. The construction of the Bell inequality is based on the local hidden variable theory. Bell then asked, no matter how Alice and Bob measure their particles, could the probability distribution of quantum measurement’s outcomes always be reproduced by the local hidden variable theory? In the other words, could the Bell inequality always be satisfied? The answer is no. Bell found an example to show this. Due to the negative answer, one may think that the local hidden variable theory may be wrong. Therefore, the failure of the local hidden variable theory will imply the validity of the other solution for EPR paradox, quantum non-locality. In this subsection, we will explain what is the EPR paradox and therefore how to use the local hidden variable theory and quantum non-locality to solve the paradox. Moreover, we will introduce the more general Bell inequalities and show the quantum violation, so that we have more confidence to judge that quantum mechanics is a non-local theory. 1.1.1.1. The EPR paradox and the local hidden variable theory. Now, we will study the EPR paradox. We know quantum mechanics can describe physical phenomena at microscopic scales very well, but it still has some limitations. According to the uncertainty principle in quantum mechanics, we cannot precisely obtain two quantities simultaneously if they correspond to non-commute operators such as spin angular momentum operators for x- and z-axis. Therefore, by the definition of the element of physical reality, the elements of physical reality related to the spin angular momentum for x- and z-axis could not exist simultaneously. Bohm’s thought experiment [4] may challenge the uncertainty principle. This experiment is as follows. Suppose Alice and the distant party Bob hold an electron from the emitted pair of electrons separately. The quantum state of the electrons is the singlet state, i.e.,. 1 |ΨiEP R = √ (| + z, −ziAB − | − z, +ziAB ), 2. (1.1). where | + zi and | − zi are used to denote the eigenstates of spin operator along z-axis 2.

(13) (up or down). Note that, | + zi corresponds to the state with the z-component of spin angular momentum being + ~2 , and similarly | − zi to − ~2 . If Alice measures the spin angular momentum along z-axis, once she obtains | + zi, the quantum state then will collapse to | + z, −ziAB . Alice can predict that Bob will get | − zi with certainty if Bob then makes a measurement along the z-axis. Similarly, if Alice obtains | − zi, Bob then gets | + zi. Of course, they can choose x-axis to make the measurement. Since the total spin angular momentum for the singlet state is conserved to be zero, Bob will get the opposite result to Alice’s outcome. These cases show us that Alice can predict Bob’s measurement results along z- and x-axis with certainty before Bob measure his electron. Furthermore, the corresponding elements of physical reality to these quantities must exist simultaneously. Recall the constraint of the uncertainty principle, we cannot predict the spin angular momentum along x- and z-axis with certainty simultaneously. Obviously, this implies that quantum mechanics is not a self-consistent theory. This dilemma is the so-called the EPR paradox. Due to the EPR paradox, the authors of [3] gave a conclusion that quantum mechanics is an incomplete physical theory while the wave function does not provide the complete description to reveal the elements of physical reality. There are two ways to resolve the EPR paradox. The first one is the local hidden variable theory. The other one is the quantum non-locality. The first one was proposed by EPR [3]. The local hidden variable theory satisfies the following conditions. • In a complete physical theory, the measurement results are determined before measurements. • Bob’s measurement will not be disturbed by Alice’s measurement if Alice is far from Bob, i.e., the action in a distance is impossible. Therefore, the non-locality between distant partite is impossible. In the local hidden variable theory, the hidden variable is like a secret code, it can determine the result of measurement. Therefore, if one can know the hidden variable and the wave function of a system, one then can predict the measurement result. Now, we can discuss how to use the local hidden variable to resolve the EPR paradox. Since the hidden variable theory is a local theory, the hidden variable for Bob’s particle will not be modified while Alice measures her own particle. Thus, Alice can obtain the information about the value of the hidden variable when she measures her own particle. 3.

(14) She can then predict the outcome of Bob’s measurement. Let’s take Bohm’s thought experiment for example. Before measuring the spin angular momentum along x- and z-axis, the outcomes of Alice’s and Bob’s measurements are already determined by the hidden variables. Due to the hidden variables, Bob always gets the opposite to Alice’s outcome while Alice and Bob choose the same axes. Thus, before the measurements, the two emitted electrons may be one of the following four types: Alice’s particle. Bob’s particle. (| + zi, | + xi, λ1 ). (| − zi, | − xi, λ1 ). (| + zi, | − xi, λ2 ). (| − zi, | + xi, λ2 ). (| − zi, | + xi, λ3 ). (| + zi, | − xi, λ3 ). (| − zi, | − xi, λ4 ). (| + zi, | + xi, λ4 ). Here λi are used to denote the hidden variable with different values. If Alice and Bob measure their own electron along the different axes, Bob will obtain + ~2 or − ~2 with equal probability no matter what outcome Alice obtains. However, one can find Bob’s measurement result is determined no matter what axis Alice chooses to measure. It is consistent with the property of the locality, i.e., the impossibility of the action at a distance. Thus, the local hidden variable theory not only satisfies the locality but also obtains the same prediction of measurement outcomes as predicted by quantum mechanics. It seems that we have already obtained a complete physical theory. 1.1.1.2. The Bell’s inequality and the CHSH inequality. Bell’s experiment is similar as Bohm’s thought experiment, but Alice and Bob have three choices (axes a, b and c) to measure the spin angular momentum. According to the local hidden variable theory, the determined measurement results for a singlet state (1.1) can be divided into eight types as Tab.1.1. Alice’s and Bob’s measurement results are opposite to each other for each types in order to preserve the angular momentum. We use Ni to denote the number of times for type i while we repeat the same experiment for P8 N = i=1 Ni times. Suppose Alice obtains + ~2 for the measurement along a-axis and Bob also obtains + ~2 when measuring the spin along b-axis. Let Pr(+, +|a, b) be the joint probability of this situation. Obviously, the determined result of particles could be type 3 or 4. Therefore, Pr(+, +|a, b) = 4. N3 + N4 . N. (1.2).

(15) Table 1.1: The determined measurement results Number Alice’s particle N1 (| + ai, | + bi, | + ci) N2 (| + ai, | + bi, | − ci) N3 (| + ai, | − bi, | + ci) N4 (| + ai, | − bi, | − ci) N5 (| − ai, | + bi, | + ci) N6 (| − ai, | + bi, | − ci) N7 (| − ai, | − bi, | + ci) N8 (| − ai, | − bi, | − ci). in the local hidden variable theory Bob’s particle (| − ai, | − bi, | − ci) (| − ai, | − bi, | + ci) (| − ai, | + bi, | − ci) (| − ai, | + bi, | + ci) (| + ai, | − bi, | − ci) (| + ai, | − bi, | + ci) (| + ai, | + bi, | − ci) (| + ai, | + bi, | + ci). Similarly, N2 + N4 N N3 + N7 . Pr(+, +|c, b) = N Pr(+, +|a, c) =. (1.3). Since we know each number Ni should be non-negative, the following inequality will hold, i.e., N3 + N4 ≤ N2 + N4 + N3 + N7 .. (1.4). Therefore, when both sides of the above inequality are divided by N , we can obtain Pr(+, +|a, b) ≤ Pr(+, +|a, c) + Pr(+, +|c, b).. (1.5). This is the so-called Bell inequality. Now, by checking the Bell inequality (1.5), we can check if the prediction of quantum mechanics is always consistent with the local hidden variable theory. Suppose Alice and Bob share a singlet state (1.1) and Alice measures the spin angular momentum along a-axis. In quantum mechanics, Alice has half a chance to obtain + ~2 along a-axis and therefore the singlet state will then collapse to | + a, −ai. Suppose Bob then measures his particle along b-axis, Bob will obtain + ~2 with probability |h+b| − ai|2 = sin2 ( θ2ab ). Here θab is the angle between a- and b-axis. Put this value and the other similar twos into the Bell inequality (1.5), one can rewrite the (1.5) as. sin2 (. θab θac θbc ) ≤ sin2 ( ) + sin2 ( ). 2 2 2. Assume these three axes (a, b and c) are in the x-z plane, i.e., θab =. (1.6) π 2. and θac = θbc = π4 .. One can find the left hand side of (1.6) is 0.5 and the right hand side is 0.2929. Obviously, 5.

(16) the Bell inequality is violated. Thus, the local hidden variable theory and quantum mechanics are not always consistent with each other. Moreover, the non-locality property is revealed by quantum mechanics. In 1969, another form of the Bell inequality named the CHSH inequality was proposed [6]. The CHSH inequality is more convenient for experiments to test the non-locality. As the Bell inequality, the construction of the CHSH inequality is based on the local hidden variable theory, and therefore one could find the quantum violation. Suppose both Alice and Bob has two observables and each observable has two outcomes +1 or −1. We denote Alice’s outcomes for her two observables as A0 and A1 , respectively. Similarly, B0 and B1 are Bob’s outcomes for his observables, respectively. These outcomes would satisfy one of the following conditions. 1. If A0 + A1 = 0, then A0 − A1 = ±2. 2. If A0 + A1 = ±2, then A0 − A1 = 0. One could then find the equality C = (A0 + A1 )B0 + (A0 − A1 )B1 = ±2.. (1.7). |hCi| ≤ h|C|i = 2,. (1.8). Since we know. where h.i is used to denote the expectation value, so that −2 ≤ hA0 B0 i + hA0 B1 i + hA1 B0 i − hA1 B1 i = C0,0 + C0,1 + C1,0 − C1,1 ≤ 2.. (1.9). Note that Ci,j is the expectation value for outcomes Ai Bj . This inequality (1.9) is the well-known CHSH inequality. Now, we want to translate the above into the language to quantum mechanics. In quantum mechanics, the observable is a measurement operator with ±1 eigenvalues. Therefore, the expected value of the outcomes is relied on the type of the measurement operators and the quantum state. Suppose Alice and Bob share a singlet state and Alice’s observables are x and x0 with the outcomes A0 and A1 respectively. Similarly, Bob’s observables are y and y 0 with the outcomes B0 and B1 respectively. Here, the observables r could be expressed as ~r · ~σ , where ~r is a real three-dimensional unit vector and each element of 6.

(17) vector ~σ corresponds to different Pauli matrices, i.e., ~σ = (σx , σy , σz ). Therefore, the expectation value for outcomes A0 and B0 is hA0 B0 i =EP R hΨ|(~x · ~σ )(~y · ~σ )|ΨiEP R = −~x · ~y = − cos θ,. (1.10). where θ is the angle between ~x and ~y . We know that the expectation value is related to the direction of the measurements. Consider the case where Alice and Bob perform measurements with the following obserables: x = σz , x0 = σx ; σz + σx 0 σz − σx y=− √ ,y = √ . 2 2. (1.11). Thus, √ |C0,0 + C0,1 + C1,0 − C1,1 | = 2 2.. (1.12). Obviously, the prediction of quantum mechanics exceeds 2, i.e., the bound from the local hidden variable theory. Therefore, the CHSH inequality is violated by quantum mechanics. Although we have known that the quantum mechanics violates the CHSH inequality, what is the maximal violation? B. S. Cirel’son used the quantum properties of the ob√ servables to obtain the answer, 2 2 [7]. Thus, the maximally quantum violation of the CHSH inequality is called the Tsirelson bound. One can obtain the Tsirelson bound with the following properties [59]: • Since these observables for quantum measurements should be the hermitian operators with eigenvalue ±1, thus the square of each observable is equal to the identity operator, i.e., x2 = x02 = y 2 = y 02 = I.. (1.13). • Alice’s observables commute with Bob’s since they only measure their own particles. Thus, [x, y] = [x, y 0 ] = [x0 , y] = [x0 , y 0 ] = 0.. (1.14). Using the above conditions (1.13) and (1.14), one can obtain (xy + x0 y + xy 0 − x0 y 0 )2 = 4I + [x, x0 ][y, y 0 ]. 7. (1.15).

(18) Using the property of sup norm of a matrix M , i.e., sup|ψi (. kM |ψik k|ψik ),. we can obtain. k [x, x0 ] k≤ 2 k x kk x0 k= 2.. (1.16). √ k xy + x0 y + xy 0 − x0 y 0 k≤ 2 2.. (1.17). Therefore,. This means that the maximal expectation value of the operator xy + x0 y + xy 0 − x0 y 0 √ cannot exceed 2 2. 1.1.1.3. More general Bell-type inequalities. After the publication of the CHSH inequality, more general Bell-type inequalities was proposed such as the Bell-type inequality for higher dimensional system [8] and the case of multi-setting per site [9]. We can have more understanding about quantum non-locality through these more general Bell-type inequalities. As the construction of the Bell inequality, the constructions of these Bell-type inequalities are based on the local hidden variable theory. Therefore, one may find the quantum violation of these Bell-type inequalities since the quantum correlation is non-local and cannot be described by the local hidden variable theory. Here, we would like to review the multi-level Bell-type inequality which is called the CGLMP inequality [8]. In this case, both Alice and Bob have two observables. Alice’s outcomes are Ax , and similarly Bob’s outcomes are By (x, y ∈ {0, 1}). Therefore, in the multi-level system Ax and By ∈ {0, 1, ..., d − 1}. Recall the local hidden variable theory, assuming Alice and Bob’s measurement outcomes are determined by the local hidden variables j, k, l and m, so that the value of observable A0 is j and the one of A1 is k. Similarly, B0 gives l and B1 gives m. The probability for the local hidden variables j, k, l, m is denoted by cjklm , and therefore the P summation of the probabilities should be one, i.e., jklm cjklm = 1. Consider the special case r0 = B0 − A0 = l − j s0 = A1 − B0 = k − l t0 = B1 − A1 = m − k u0 = A0 − B1 = j − m, 8. (1.18).

(19) which automatically yield r0 + s0 + t0 + u0 = 0.. (1.19). This means the relations among the local hidden variables cannot be arbitrary. Although we could determine the relations r0 , s0 and t0 by choosing variables j, k, l and m, but the last term u0 should be constrained by (1.19). This plays the central role in deriving the CGLMP inequality. Consider a function for a d = 3 system as follows. I3 =. [Pr(A0 = B0 ) + Pr(B0 = A1 + 1) + Pr(A1 = B1 ) + Pr(B1 = A0 )] −[Pr(A0 = B0 − 1) + Pr(B0 = A1 ) + Pr(A1 = B1 − 1) + Pr(B1 = A0 − 1)]. (1.20). According to the local hidden variable theory, the maximal value of (1.20) is 2, i.e., I3 (local) ≤ 2. This is because when the three probabilities with the “ + ”-sign are satisfied, the term with the “ − ”-sign will also be satisfied. On the other hand, once all the four terms with “ + ”-sign could be satisfied by non-local correlations, the maximal value of the function (1.20) would be 4. One can then generalize the function from d = 3 to arbitrary d. The generalized function for the multi-level Bell-type inequality is [ d2 ]−1. Id =. X. 2k (1− d−1 )[Pr(A0 =B0 +k)+Pr(B0 =A1 +k+1)+Pr(A1 =B1 +k)+Pr(B1 =A0 +k)]. k=0 −[Pr(A0 =B0 −k−1)+Pr(B0 =A1 −k)+Pr(A1 =B1 −k−1)+Pr(B1 =A0 −k−1)].. (1.21). As for the d = 3 case, the maximal value of Id from the local hidden variable theory is 2, i.e., Id (local) ≤ 2. Therefore, the maximal value achieved by the non-local correlations is still 4. For more detailed proof, please see [8]. Now, we can study the quantum violations respected to the maximum of Id (local). Assuming Alice and Bob share a maximally entangled state d−1. 1 X |jiA ⊗ |jiB , |ψi = √ d j=0. (1.22). and Alice’s observables are x and y 0 , similarly y and y 0 for Bob. One can obtain optimal. 9.

(20) value of (1.21) when these observables correspond to the following eigenvectors [8]: d−1. |kiA,x. 1 X 2π =√ exp(i j(k + αx ))|jiA d d j=0 d−1. 1 X 2π |liB,y = √ exp(i j(l − βy ))|jiB , d d j=0. (1.23). where α0 = 0, α1 = 1/2, β0 = 1/4 and β0 = −1/4. Thus, one can obtain the joint probability for the outcomes Pr(Ax = k, By = l) = =. |hψ|(|kiA,xA,xhk| ⊗ |liB,yB,yhl|)|ψi| 1 . 2 3 2d sin [π(k − l + αx + βy )/d]. (1.24). With this joint probability, one can obtain the bound achieved by the maximally entangled state [ d2 ]−1. Id (QM ) = 4d. X. (1 −. k=0. 2k )(Pr(A1 = B1 + k) − Pr(A1 = B1 − k − 1)) > 2 ≥ Id (local). d−1 (1.25). According to this quantum violation, we have another example such that the local hidden variable theory is not consistent with quantum mechanics. Thus, quantum mechanics is intrinsically non-local. 1.1.2. No-signaling theory. The quantum mechanics is constrained by some natural principle such as the relativistic causality. The relativistic causality gives many important results, one of them is the speed of propagation cannot be faster than the light speed in any reference frame. Due to the fact, here comes the no-signaling theory which states that the speed of the propagating information cannot be faster than the light speed, i.e., the superluminal signaling is impossible. Consider the constraint of no-signaing theory, many related works have been studied. It is interesting that the constraints from the no-signaling theory and quantum mechanics lead to the same behaviors of the correlations such as no-cloning [11, 12], no-broadcasting [13] and monogamy of entanglement [11]. As shown [14, 15], both the no-signaling theory and quantum mechanics preserve the security of the key distribution. 10.

(21) We will study the specific constraint of the no-signaling theory in this subsection. First, we define the measurement scenario for producing the bi-partite correlations. Second, we introduce the correlations satisfying the no-signaling theory. Furthermore, these nosignaling correlation could be divided into two subsets: the local correlations and the non-local ones. We will also discuss the properties of these correlations. 1.1.2.1. The measurement scenario and the box version. First, let us define the measurement scenario for the bi-partite correlations. Both Alice and Bob hold a physical system. They can choose to measure arbitrary observables to probe their systems. We denote the observable for Alice as x and denote the outcome of the observable x as Ax . Similarly, Bob’s observable is denoted as y and the corresponding outcome is By . Therefore, the bi-partie correlation is defined by the conditional joint probability, i.e., Pr(Ax , By |x, y).. (1.26). Note that, the conditional joint probability should be nonnegative and satisfy the normalization conditions, thus the summation of the conditional joint probabilities with fixed observables should be one, i.e.,. X. Pr(Ax , By |x, y) = 1.. (1.27). Ax ,By. One can reformulate the scenario in a more abstract way. Instead of giving a specific physical system, one assumes a bi-partite box shared by Alice and Bob. Both of them put the inputs into the box, he (she) then will obtain the corresponding outputs. The shared box is characterized by the joint probability (1.26). Note that, the number of Alice and Bob’s inputs may not be the same, similarly for the number of their outputs. According to the property of the bi-partite box, one can divide them into many types such as the no-signaling box and the signaling one. We will discuss the property of the bi-partite box later. Hereafter, we will use the terms ”box” and ”correlation” interchangeably when we discuss the property of the joint probability (1.26). For the necessary, one can generalize the above bi-partite box to the multi-partite case.. 11.

(22) 1.1.2.2. No-signaling correlations. Now, we can discuss the no-signaling condition for the bi-partie correlations. Note that, If the shared box can simulate no-signaling correlations, we call the box as a no-signaling box [20]. Otherwise, the shared box is a signaling one. Despite of any non-local correlations previously shared between them, Alice cannot signal to the distant Bob by her choice of inputs due to the no-signaling theory. This indicates that Bob’s marginal probability Pr(By |y) is independent with Alice’s input x. Similarly, Alice’s marginal probability is independent with Bob’s input y. Therefore, the condition for the correlations (box) to be no-signaling is X X Pr(By |y) = Pr(Ax , By |x, y) = Pr(Ax0 , By |x0 , y), Ax. Pr(Ax |x) =. X. ∀By , y, x, x0 ,. Ax0. Pr(Ax , By |x, y) =. By. X. Pr(Ax , By0 |x, y 0 ),. ∀Ax , x, y, y 0 .. (1.28). By0. The no-signaling box can be divided into two types, the local and non-local ones. The local box can simulate the local correlations which can be described by the local hidden variable theory. From the information theoretic point of view, the local correlations can be simulated by the non-communicating observers with the pre-shared classical random data. Therefore, the local correlations should satisfy X Pr(Ax , By |x, y) = Pr(λ) Pr(Ax |x, λ) Pr(By |y, λ),. (1.29). λ. where λ is the pre-shared classical random data and the probability for its occurrence is Pr(λ). The probability of occurrence for Alice’s output Ax with the given random data λ and the input x is denoted as Pr(Ax |x, λ). The marginal probability Pr(By |y, λ) has the similar definition. If the box cannot simulate the local correlation, we will call it the non-local box. Besides using the the local and the non-local as the categories for the correlations, we want to discuss a another way of categorizing, the quantum correlartions. Quantum correlations can be obtained by sharing the quantum resource such as the quantum state. These correlations can be written as Pr(Ax , By |x, y) = Tr(EAx ⊗ EBy ρ),. (1.30). where ρ is the density matrix of the shared quantum state, and EAx and EBy are the projection operators corresponding to Alice and Bob’s measurements, respectively. Note 12.

(23) that, these operators are the elements of a positive-operator valued measure (POVM) [60] and therefore the operators should satisfy ΣAx EAx = ΣBy EBy = I, 1.1.3. ∀x, y.. (1.31). Could the no-signaling theory single out quantum correlations?. Let us recall how to obtain the bound of quantum non-locality. It comes from the mathematic constraint. Therefore, we are curious if there is a natural principle which could imply quantum mechanics. After reviewing the constraint of the no-signaling theory, we may ask a question: could the no-signaling theory be the physical principle which we are looking forward to? More precisely, does the non-locality and the no-signaling theory together imply quantum theory? This question was firstly asked and answered by S. Popescu and D. Rohrlich [21]. Their answer is negative, since quantum mechanics is not the only theory which satisfies the no-signaling constraints and also violates the CHSH inequality. Furthermore, under the constraints of the no-signaling theory, the maximal violation of the CHSH inequality is bigger than the maximally quantum violation. These extremal non-local correlations are called super-quantum correlations in [21]. We will discuss the properties of the super-quantum correlations. We then review the geometric picture of all the no-signaling correlations in [20]. It is interesting that the Bell-type inequality and the facet of the local polytope are related. Finally, under the constraints of the no-signaling theory, one can demonstrate that the extremal non-local correlations lead to the trivial communication complexity from information theoretic point of view. 1.1.3.1. Beyond the quantum correlations. Does the non-locality and the no-signaling theory together imply quantum mechanics? In [21], the authors used the following process to show the negative answer. First, one could take the non-locality and the no-signaling theory as two axioms. Second, one could try to find the maximal violation of the CHSH inequality and notice that the maximal violation cannot be achieved by quantum mechanics. Therefore, the axiom of non-locality implies the quantum correlation is not the most non-local one. One may then guess that the other axiom, the no-signaling theory, might give the constraint of quantum non-locality. However, there is a set of joint probabilities achieves the maximal violation of the CHSH 13.

(24) inequality and satisfies the no-signaling theory (1.28). This means, the non-signaling constraints cannot single out quantum correlations. To be specific, we want to find the maximal non-locality. In [21], they used the CHSH function to characterize the non-locality, where the CHSH function is |C0,0 + C0,1 + C1,0 − C1,1 |.. (1.32). Note that, each term Cx,y in (1.32) is the correlation function for the specific inputs x and y. The definition of Cx,y is Cx,y =. X. (−1)Ax +By Pr(Ax , By |x, y).. (1.33). Ax ,By. Here x, y, Ax and By ∈ {0, 1}. With the definition of Cx,y , we know that each Cx,y lies in [−1, 1]. Therefore, the maximal value of the CHSH function (1.32) should be 4. Since we have known that the local bound and the quantum bound (Tsirelson bound) of CHSH √ function are 2 and 2 2, respectively. Obviously, the maximal value of CHSH function is bigger than the Tsirelson bound and cannot be achieved by the quantum correlations. A question then arises: why quantum mechanics could not be more non-local? Is this because of the constraint from the no-signaling theory? Therefore, one could consider the no-signaling correlations (1.28) and then calculate the corresponding value of CHSH function (1.32). With the no-signaling theory, Alice cannot signal to Bob her choice of inputs, and vise versa. In [21], they found the following joint probabilities such that the no-signaling theory is satisfied and the maximum of the CHSH function (4) is achieved:   1 :A + B (mod 2)=xy; x y 2 Pr(Ax , By |x, y) = (1.34)  0 : otherwise. The no-signaling theory is satisfied, this is because Pr(Ax |x) and Pr(By |y) are equal to 1 2.. Obviously, this set of joint probabilities is not consistent with quantum correlations.. Therefore, the no-signaling theory and the non-locality together cannot imply quantum mechanics. Besides finding the above fact, one could also find the extremal non-local correlations over the non-signaling correlations (1.34). It is called super-quantum correlations in [21]. Once the shared box can simulate the super-quantum correlations, we would say Alice and Bob share a PR-box. 14.

(25) 1.1.3.2. The no-signaling polytope. In [20], the authors gave a geometric picture of the bi-partie no-signaling correlations. The dimensions of the no-signaling polytope could be obtained by the following calculation. Suppose Alice and Bob have kx and ky kinds of inputs, respectively. Each input could have d kinds of outputs. Therefore, we could have kx ky d2 kinds of joint probabilities Pr(Ax , By |x, y)’s. Since we consider the no-signaling box, the constraints of the no-signaling theory (1.28) and the normalization conditions should be imposed on these joint probabilities. This leads to 1. The normalization conditions give kx ky equalities. 2. The no-signaling constraints give. P. xd+. P. y. d = (kx + ky )d equalities.. However, these two types of conditions are not independent. For a fixed x, consider the normalization condition, one of the marginal probabilities Pr(Ax |x)’s could be expressed by the others. Therefore, for a fixed x, one no-signaling condition can be deduced by the normalization condition and the other (d − 1) no-signaling conditions. So that, there are P P kx ky + x (d − 1) + y (d − 1) linear independent equalities. Thus, the number of joint probabilities will be reduced to dim = kx ky d2 − (kx ky +. X x. (d − 1) +. X. (d − 1)).. (1.35). y. This means that the no-signaling boxes form a dim-dimensional polytope [20]. For the two-inputs/two-outputs case, both Alice and Bob have two kinds of inputs, and each input could have two outputs. Therefore, the bi-partite no-signaling boxes yield a 8-dimensional polytope. The no-signaling polytope has 24 vertices, 16 of them correspond to local deterministic boxes and other 8 vertices correspond to the extremal non-local boxes. The local deterministic box implies that the value of the marginal probabilities Pr(Ax |x) and Pr(By |y) is either 0 or 1. The joint probabilities of the local deterministic box should satisfy.   1 :A = αx + β (mod 2) and B = γy + δ (mod 2); x y Pr(Ax , By |x, y) = ,  0 : otherwise.. (1.36). where α, β, γ and δ ∈ {0, 1}. Actually, these local deterministic boxes can yield another convex polytope of locality by themselves. The facets of this polytope can be divided 15.

(26) into two types. The first type of facets restricting the joint probabilities in the polytope should be non-negative. The second type of facets correspond to the Bell-type inequality. Therefore, whenever the Bell-type inequality is violated by a set of joint probabilities, these joint probabilities will lie outside this polytope of locality. In the two-inputs/twooutputs case, the Bell-type inequalities correspond to the CHSH inequalities [22], i.e., (1.9) and its three symmetric partners by shifting the minus sign. There are 8 kinds of the CHSH inequalities. The remainder vertices of the no-signaling polytope correspond to the extremal nonlocal boxes. The joint probabilities should satisfy   1 :A + B = xy + αx + βy + γ (mod 2); x y 2 Pr(Ax , By |x, y) = ,  0 : otherwise.. (1.37). where α, β and γ ∈ {0, 1}. When α = β = γ = 0, the extremal non-local box is the PR-box (1.34). Note that, each extremal non-local box will violate one of the CHSH inequalities. Furthermore, they can achieve the maximal violation, i.e., 4 or −4. It is interesting that one non-local vertex of the no-signaling polytope corresponds to one facet of polytope of locality. There is another example for this connection. Suppose Alice and Bob have two inputs and each input has d outputs, one of the non-local vertex of the no-signaling polytope can achieve the maximal violation of the CGLMP inequality [8], i.e., the value of (1.21) is 4. The corresponding joint probabilities are as follows.   1 :B − A (mod 2)=xy ; y x d , Pr(Ax , By |x, y) =  0 : otherwise.. (1.38). Moreover, one could use this relations to find the complete set of the Bell-type inequalities by other extremal non-local boxes. 1.1.3.3. The communication complexity and the extremal non-local correlations. The communication complexity [23] is used to discuss how much of the communication is needed to solve a distributed decision problem. More specifically, if Alice has a n-bit string ~x = (x1 , x2 , ..., xn ) and Bob also has a n-bit string ~y = (y1 , y2 , ..., yn ). Their goal is to obtain the value of the function f (~x, ~y ). Note that the function f is a Boolean function: f : {0, 1}n × {0, 1}n → {0, 1}. The definition of the communication complexity is the 16.

(27) minimum number of bits exchanged between Alice and Bob in the worst case in order to achieve such a task. If the value of the function f (~x, ~y ) can be obtained with only one bit of communication, one could say that the distributed decision problem has the trivial communication complexity. In practical, Alice and Bob could share some non-local resources to reduce the communication complexity such as an entangle state [26]. One show the power of the extremal non-local correlations in terms of communication complexity. When Alice and Bob share PR-boxes, for any distributed decision problem, the communication complexity is trivial [24]. Let us take an example to show the fact. If Alice and Bob want to determine the inner-product function (IPn : {0, 1}n × {0, 1}n → {0, 1}), where IPn (~x, ~y ) =. n X. xi y i .. (1.39). i=1. Recall the PR-box (1.34), it keeps the inputs and the outputs perfectly correlated, i.e., Pr(Ax + By = xy|x, y) = 1, ∀x, y. If Alice and Bob share n PR-boxes, they could take the i-th bit of the string xi and yi as the input of the i-th PR-box, and obtain the corresponding outputs Axi and Byi , respectively. Thus, n X i=1. xi yi =. n X. Axi + Byi =. i=1. n X. Axi +. i=1. This means that, Bob only needs to send the bit b =. n X. Byi .. (1.40). i=1. Pn. i=1 Byi. to Alice, Alice can then. determine the value of the Inner Product function IPn (~x, ~y ) by her outputs and the bit b. Thus, the communication complexity is trivial in determining the inner-product function. In [24], van Dam showed if one can determine the inner-product function f (~x, ~y ) = ~x ·~y of 2n inputs by one bit communication, then one can also determine any Boolean function f (~x, ~y ) of n inputs with trivial communication complexity. According to the previous example, the inner-product function could have the trivial communication complexity for any number of inputs, we then know that for any distributed decision problem the communication complexity is trivial by sharing the PR-boxes. Moreover, some works [27, 28] show that the trivial communication complexity can indeed be achieved by sharing other non-local boxes. These non-local boxes are not the extremal non-local ones. However, the correlations of these boxes are still more no-local than the quantum correlations, i.e., √ |C0,0 + C0,1 + C1,0 − C1,1 | ≥ 2 2. Therefore, we could not use the trivial communication complexity to single out the quantum correlations. 17.

(28) 1.2 1.2.1. Information Causality Information Causality single out quantum correlation?. In the previous section, we have shown that the no-signaling criterion can not be used to single out the quantum correlations. What principle could be our next candidate? The answer is Information Causality (IC) [29]. IC is presented through the following task which is equivalent to the random access code (RAC) [30]. Alice has a database with k elements, denoted by the vector ~a = (a0 , a1 , ..., k − 1). Each element ai is a binary random variable and is only known to Alice. A second distant party, Bob is given a random variable b ∈ {0, 1, 2, ..., k − 1}. The task of Bob is to optimally guess Alice’s database ab after receiving m-bit string α ~ sent from Alice via the pre-shared correlation between Alice and Bob. Before Alice sends the α ~ , Bob cannot obtain the information about Alice’s database, this is the no-signaling constraint. However, we are concerned about the information gain after Bob receives Alice’s m-bit string α ~ . IC states that the maximal mutual information shared between Alice and Bob cannot exceed the amount of classical communication, even they have pre-shared physical correlations, this condition can be expressed as follows: I=. k−1 X. I(ai ; β|b = i) ≤ m ,. (1.41). i=0. where I(ai ; β|b = i) is Shannon’s mutual information between ai and Bob’s guess d-bit β under the condition b = i. The condition of IC (1.41) is not theory-independent [29]. It holds for quantum information theory. Let us define the pre-shared quantum state for Bob’s part as ρB . In the task, after Alice sends the m-bit string α ~ to Bob, the information possessed by Bob about Alice’s database includes α ~ and ρB . Note that, the mutual information between Alice’s database and Bob’s own information can not be more than m bit, i.e., I(~a; α ~ , ρB ) ≤ m .. (1.42). (1.42) can be proved by some elementary properties and the no-signaling condition, i.e., I(~a; ρB ) = 0. On the other hand, one can prove I ≤ I(~a; α ~ , ρB ) by the elementary properties. Therefore, one can obtain that I ≤ I(~a; α ~ , ρB ) ≤ m. Thus, (1.41) holds for the pre-shared quantum correlations. 18.

(29) IC is not only satisfied by the pre-shared quantum correlations, but also violated by the pre-shared PR-box which can simulate the extremal non-local correlations under the no-signaling constraint [29]. This shows IC is more suitable to single out quantum correlations. Moreover, the Tsirelson bound could be derived from IC. For the multisetting case, we show that more general Tsirelson-type inequalities can be derived by the signal decay theory [50, 51] and IC. One can see Chapter 2 for more detailed proof. According to these results, IC may be the natural principle which we are looking for. Actually, one should prove that IC can exclude all non-quantum correlations so that IC could be used to single out quantum correlations. Therefore, we still need more checks. Since there are some non-quantum correlations below the Tsirelson bound, and therefore we cannot express the entire boundary of quantum correlations by the no-signaling boxes which achieving the Tsirelson bound. Thus, we have to check if the entire quantum boundary can be recovered by IC. In [35], the relation between IC and the quantum boundary for two-inputs/two-outputs case has been studied. In this case, IC is not violated by the quantum correlations, therefore IC is supported. Besides, the relation between bi-partite IC and the more general no-signaling boxes was studied. In the multi-level no-signaling box [36], the relation between bi-partite IC and quantum correlations is unclear. In the multi-partite no-signaling box [37, 38], bi-partite IC is violated by the most non-quantum correlations, but not all the non-quantum correlations violate bi-partite IC. The reason may be, the set of quantum correlations for an arbitrary number of parties can not be singled out by using the principle with bi-partite information concepts [39]. Therefore, although bi-partite IC is satisfied by some non-quantum correlations, it does not mean IC cannot single out the quantum correlations. It just means that we need to generalize the bi-partite IC to the multi-partite one. However, even in this simplest case (the two-inputs/two-outputs no-signaling box) [35], it is still unclear that if IC could exclude all the non-quantum correlations or not. Why it is so difficult to prove the criterion? The reason is that there are too many strategies for using the no-signaling boxs. Even sharing the same no-signaling box, different strategies for using the box will yield different probabilities to win the task. Therefor, the mutual information between Alice’s database ab and Bob’s guessing bit β will be different. So, it is hard to ascertain if the non-quantum correlations violate IC or not. However, we can have more understanding about the relation between IC and quantum 19.

(30) correlations by testing IC for different quantum communication protocols or more general settings. We will study these more general cases in chapter 3. In the following subsections, we will review some important relations between IC and quantum correlations which we mentioned in this subsection. They include how to derive the Tsirelson bound from IC and how to compare quantum boundary and IC for the two-inputs/two-outputs case. 1.2.2. Information Causality and the Tsirelson bound. 1.2.2.1. The extremal non-locality violates Information Causality. As shown in [29], IC will be violated by the PR-box which is the extremal non-local box under the constraint of no-signaling theory. One can use the specific RAC protocol to show the fact.. Figure 1.1: The communication protocol for k = 2 task. Let us consider the simplest case, it is described in Fig1.1. The protocol is used to solve k = 2 task. To be specific, first, Alice encodes her database as the input x of the no-signaling box, where x = a0 + a1 . On the other hand, the distant Bob takes his given bit b as the input y of the no-signaling box. They then obtain the outputs Ax and By , respectively. Note that, in this case x, y, Ax and By ∈ {0, 1}. Second, after obtaining their outputs, Alice then sends a bit α = Ax + a0 to Bob. Finally, Bob can decode Alice’s database ab by calculating the bit β = α + By . Since β = α + By = Ax + By + a0 , if 20.

(31) Alice’s and Bob’s inputs and outputs are perfectly correlated, i.e., Ax + By = xy, ∀x, y, Bob can then obtain β = xy + a0 = (a0 + a1 )y + a0 (mod 2). Obviously, either if y = 0, β = a0 or if y = 1, β = a1 . Thus, Bob can guess Alice’s database perfectly. If not the PR-box, Bob’s successful probability depends on the value of the joint probability Pr(Ax + By = xy|x, y). In this case, the successful probability for the task is related to the CHSH function (1.32). The successful probability to guess a0 right under the condition b = 0 is denoted as P0 , similarly P1 for a1 . If Alice’s input is unbiased, i.e., the marginal probabilities Pr(x = i) =. 1 2. i ∈ {0, 1}, the successful probabilities can be rewritten as. 1 P0 = (Pr(Ax + By = 0|x = 0, y = 0) + Pr(Ax + By = 0|x = 1, y = 0)) 2 1 P1 = (Pr(Ax + By = 0|x = 0, y = 1) + Pr(Ax + By = 1|x = 1, y = 1)). 2 Note that, one can obtain Pr(Ax + By = 0|x = 0, y = 0) =. 1+C0,0 2. (1.43). by the definition (1.33). of the correlation function C0,0 and the normalization condition of the joint probability. 1+C1,0 1+C0,1 2 , Pr(Ax + By = 0|x = 0, y = 1) = 2 1−C1,1 2 . Consider these relations, if the marginal. Similarly, Pr(Ax + By = 0|x = 1, y = 0) = and Pr(Ax + By = 1|x = 1, y = 1) =. probability Pr(b) is uniform, one can then find that the successful probability is equivalent to the CHSH function, i.e., 1 1 P = (P0 + P1 ) = (4 + C0,0 + C0,1 + C1,0 − C1,1 ). 2 8. (1.44). Since we have already know the locality bound and the Tsirelson bound for the CHSH function, we can estimate the local bound and the quantum bound for the function P . The local bound for P is. 3 4. and the quantum one is. √ 2+ 2 4 .. Under the no-signaling condition,. we can also know that the bound for the PR-box is 1. In this case, Bob can always guess Alice’s database with certainty. Therefore, the mutual information between ai and β under the condition b = i is 1, i.e., I(a0 ; β|b = 0) = I(a1 ; β|b = 1) = 1. That is, the mutual information I is 2. Thus, IC is violated by sharing a PR-box. 1.2.2.2. Information Causality derives the Tsirelson bound. The Tsirelson bound emerges from IC, it is one of the important results in [29]. In order to show the fact, one has to connect the mutual information I and the CHSH function. Recall the CHSH function is equivalent to the successful probability of the task. Therefore, 21.

(32) one should rewrite the mutual information I in terms of the successful probability of the P task. First, one can rewrite the mutual information I as i H(ai |b = i) − H(ai |β, b = i), where H is the binary entropy function. Assuming that the marginal probability Pr(ai ) is uniform, thus the entropy function H(ai |b = i) = 1 for each ai . On the other hand, one can use the chain rule to obtain H(ai |β, b = i) ≤ H(ai +β(mod 2)|b = i) = H(Pi ), where Pi is the successful probability for Bob to guess ai . Therefore, the mutual information becomes I≥. k−1 X. [1 − H(Pi )].. (1.45). i=0. Here, in order to reveal the Tsirelson bound from IC, one needs a specific strategy to use the no-signaling box. One way is to nest many two-inputs/two-outputs no-signaling boxes. In this way, Alice’s database ~a = (a0 , a1 , ..., ak−1 ) has k = 2n elements with n an integer. Guessing ab perfectly is Bob’s task. Thus, Bob is given n-bit string ~y = (y0 , y1 , ..., yn−1 ) i (i) used to encode b = Σn−1 i=0 yi 2 . Alice’s output of the i-th box is denoted as A , similarly. B (i) for Bob. One may need to nest k − 1 two-inputs/two-outputs no-signaling boxes to solve the task. Let us take k = 4 and m = 1 task for example. The RAC protocol is described in Fig. 1.2. Since there are 4 elements in Alice’s database, she can divide them into two subset and therefore each subset has two elements. As for the k = 2 simplest RAC protocol, she can then encode each subset as the input of the no-signaling box. In this case, Alice can input a0 + a1 in the first box and then send α(1) = A(1) + a0 to make Bob able to guess a0 and a1 . Similarly, she inputs a2 + a3 in the second box and sends α(2) = A(2) + a2 for the same purpose. Note that, Alice only allows to send 1 bit, thus she needs the third box to make Bob able to guess α(1) and α(2) . Similarly, Alice inputs α(1) + α(2) and sends the bit α = A(3) + α(1) to Bob. On the other hand, suppose Bob’s task is to guess a0 , the bit b = 0 is encoded as ~y = (y0 = 0, y1 = 0). Therefore, Bob inputs y0 and y1 in the first and the third box, respectively. Bob can get α(1) by calculating α + B (3) . Similarly, he can then use the output B (1) of the first box to get a0 by calculating α(1) + B (1) . Therefore, Bob’s optimal guessing bit β = α + B (3) + B (1) . This communication protocol can be generalized to any k = 2n case. In this RAC protocol, the successful probability for Bob to guess ab right is given by 1 Pbn = (1 + E1n−s E2s ), 2 22. (1.46).

(33) , ,. , + A. =. +. +. =. +. = β =. +B. Figure 1.2: The RAC protocol for k = 4 task. where Ei is related to the successful probability Pi (1.43) in the k = 2 RAC protocol, the relation is Ei = 2Pi − 1. The value of s depends on the encoded vector ~y , the definition Pn−1 of s is s = i=0 yi . Now, we are ready to calculate the bound from IC. Put (1.46) into I (1.45) and impose. 23.

(34) the condition 1 − H( 1+Z 2 )≥ I. ≥. Z2 2ln2 , k−1 X. we can then obtain. [1 − H(Pbn )],. b=0 n X. . n. .  [1 − H( 1 (1 + E1n−s E2s ))], 2 s s=0   n X n 1   (E12 )n−s (E22 )s , ≥ 2 ln 2 s s=0 =. =. . 1 (E 2 + E22 )n . 2 ln 2 1. (1.47). One can find that if E12 + E22 > 1,. (1.48). then the mutual information I will be bigger than 1 for some n, i.e., IC is violated. Recall the definition (1.43) of the successful probability Pi , one can rewrite the constraint as (C0,0 + C1,0 )2 + (C0,1 − C1,1 )2 > 4.. (1.49). In fact, this is nothing but the violation of Uffink’s quadratic inequality [41]. Consider the special case, let P1 = P2 = P and therefore E1 = E2 = E. In this case, IC will be violated while 2E 2 > 1. Recall the relation between the successful probability and the CHSH function (1.44), one can find that the Tsirelson bound is violated while E >. √1 . 2. This case shows that the Tsirelson bound emerges from IC. 1.2.3. Information Causality and the boundary of quantum correlations. The authors in [35] tried to answer that if IC can recover the entire quantum boundary or not. In their proof, they considered the simplest case, n = 1, k = 2 RAC protocol and the quantum correlations for two-inputs/two-outputs no-signaling box. They studied the no-signaling correlations by several two dimensional slices of the non-signalling polytope, that is equivalent to considering the following noisy PR-box. P Rλ,µ = λP R + µB + (1 − λ − µ)I,. (1.50). where P R is the PR-box and B could be another extremal non-local box (1.37) or local deterministic box (1.36). Thus, the noisy PR-boxes can be grouped into two families. In 24.

(35) [35], the Uffink’s inequality (1.49) is taken as the condition for the violation of IC. On the other hand, they imposed two quantum constrains to obtain the quantum boundary. These constraints correspond to different families of noisy PR boxes. Finally, one may compare the condition for the violation of IC and the quantum boundary. Two results were arrived in [35]. First, the bi-partite quantum correlations satisfy IC. Second, only parts of the quantum boundary emerges from IC. For the first family, B is one of the extremal non-local box (1.37) except the PR one. In this family, the marginal probabilities Pr(Ax |x) and Pr(By |y) are uniform. The correlation functions Cx,y can be rewritten as C0,0 = λ + (−1)γ µ, C0,1 = λ + (−1)β+γ µ, C1,0 = λ + (−1)α+γ µ, C1,1 = λ + (−1)α+β+γ µ, (1.51) where α, β and γ are used to label the types of extremal non-local boxes (1.37). For fixed α, β and γ (or equivalently, a chosen extremal non-local box B), one could plug (1.51) into the condition (1.49) for the violation of IC so that IC and the noisy PR-box are related. For example, if (α, β, γ) = (0, 1, 0), one may find that IC is violated when 1 λ2 + µ2 > . 2. (1.52). On the other hand, if a set of correlation functions Cx,y can be reproduced by the quantum system (quantum operators and quantum state), this set of correlation functions need to satisfy the following quantum condition, Xq 2 )(1 − C 2 ). |C0,0 C1,0 − C0,1 C1,1 | ≤ (1 − C0,j 1,j. (1.53). j=0,1. The above condition was proposed by Landau [42], and an equivalent set of conditions were proposed by Masanes [43] expressed in different form. Note that, if (1.53) is satisfied by a set of correlation functions Cx,y , the corresponding marginal probabilities Pr(Ax |x) and Pr(By |y) must be uniform. As for the condition for the violation of IC, for fixed α, β and γ, one can plug (1.51) into the quantum condition (1.53) and find the associated quantum boundary. For example, when (α, β, γ) = (0, 1, 0), the boundary of quantum correlations and the condition for the violation of IC are the same. In this case, the quantum boundary emerges from IC. However, this is not always true, for example for (α, β, γ) = (1, 1, 1). There are some non-quantum correlations satisfying IC. This does 25.

(36) not mean IC cannot exclude these non-quantum correlations, because if one use another strategy for the protocol such as nesting some non-quantum boxes mentioned in the previous subsection, these non-quantum boxes may violate IC. In the second family of the noisy PR-box, B is the local deterministic box considered in (1.36). Among these local boxes, one could consider the special ones which lie on the CHSH facets of the polytope of locality discussed in section 1.3.2. Therefore, the parameters of these special boxes given by (1.36) should satisfy αγ + β + δ = 0 mod 2. As for the first family, one can use the parameters of the noisy PR-box given in (1.50) to rewrite the correlation function Cx,y . For example, if (α, β, γ, δ) = (0, 0, 0, 0), one then obtains C0,0 = C0,1 = C1,0 = λ + µ, C1,1 = µ − λ.. (1.54). Note that, the marginal probabilities Pr(Ax |x) and Pr(By |y) are no longer uniform. Therefore, the marginal correlations Cx and Cy are not zero. The marginal correlations are the linear combinations of the marginal probabilities. The definition for marginal correlations is Cx = Pr(Ax = 0|x) − Pr(Ax = 1|x) and Cy = Pr(By = 0|y) − Pr(By = 1|y). Here, the marginal correlations Cx=0 = Cx=1 = Cy=0 = Cy=1 = µ. In this case, the quantum constraint (1.53) is not suitable for picking up the quantum correlations. Instead, one may use the following condition and its three symmetric partners by shifting the minus sign. These conditions include the marginal correlations, i.e., | arcsin(D0,0 ) + arcsin(D0,1 ) + arcsin(D1,0 ) − arcsin(D1,1 )| ≤ π, x Cy where Dx,y = √ Cx,y −C 2. (1−Cx )(1−Cy2 ). (1.55). . This condition was proposed in [44, 45]. Note that, (1.55) is. the quantum constraint in the first step of the hierarchal semidefinite programming [45]. This means, if a set of correlation functions Cx,y , Cx and Cy satisfies (1.55), we could not make sure if it could be reproduced by a quantum system, unless it can satisfy all the quantum constraint in each step of the hierarchical semidefinite programming. Plug all the correlation functions (1.54) into the quantum constraint (1.55), one can obtain the associated quantum boundary. Similarly, plug (1.54) into the condition (1.49) for the violation of IC, one may find that IC is violated when (λ + µ)2 + λ2 > 21 . In this case, the quantum boundary from the quantum constraint (1.55) and the condition of IC are not the same. There are some non-quantum correlations satisfying IC. As for the case in 26.

(37) the first family, one may use another strategy such that these non-quantum correlations will still violate IC. However, with the above examples, one may only know that quantum correlations satisfy IC and could not make sure if IC can exclude all the non-quantum correlations. Moreover, one may find there are some assumptions and approximations in arriving the condition (1.48) for the violation of IC. Therefore, instead of using this condition, we directly calculate the mutual information I of the quantum correlations and then compare with IC. See Chapter 3 for more details.. 27.

(38) 1.3. Signal propagating and noisy computation. Signal propagating suffers signal decay. It is a very important issue when considering communication systems. On the other hand, one could image the noisy computation as a sequence of steps [49]. Each step has an input and an output. The output has some information about the associated input, and the output will be the input of the next step. Once some step produces a random noise, one can interleave some parallel step to control the error rate of the entire computation. In this way, each step of the noisy computation can be seen as a noisy communication channel. Thus, one may expect that the issue of signal decay will also appear for the noisy computation. von Neumann was the first to be aware of this fact [47]. He suggested that the error of the computation should be treated by thermodynamical method as the treatment for the communication in Shannon’s work [46]. This means that, for the noisy computation, one should use some information-theoretic methods related to the noisy communication. After thirty years from the publication of von Neumann’s work, Pippenger took some information theoretic arguments to study the reliable noisy computation [48]. After that, Evans and Schulman proposed a new result for the efficient signal propagating and then use it to study the threshold of noisy computation [50, 51]. In this section, we want to review Evans and Schulman’s work and also compare it with the previous results obtained by von Neumann and Pippenger. 1.3.1. The efficient propagation through a noisy channel. In the process of signal propagation, we will face the problem of signal decay. To be specific, it is manifested through the data processing inequality. Consider two communication channels and three random variables. Let variable X be the input of the first channel, the variable Y is the output of the first channel and also is the input of the second one, and the variable Z is the output of the second channel, i.e., the cascade of two communication channels: X → Y → Z. No matter what are the properties of the communication channels, the data processing inequality states that I(X; Z) ≤ I(X; Y ). The mutual information I(X; Y ) = H(Y ) −. P. i Pr(X. (1.56) = i)H(Y |X = i), where H(Y ). and H(Y |X) are the Shannon entropies for the probability Pr(Y ) and the conditional 28.

(39) probability Pr(Y |X), respectively. Here are some properties of the communication channel which we will use later. If a communication channel has m inputs and n outputs, this communication channel could be expressed by a m × n matrix. The (i, j)-th element of this matrix is the conditional probability of obtaining the output j under the condition of the input i. Let us take the binary channel for example. A general binary channel is specified by   sin2 α cos2 α . A= 2 2 cos β sin β. (1.57). Let the random variables G and F be the input and output of the binary channel A, P1 respectively. Since the output probability can be expressed as Pr(F = j) = i=0 Pr(F = j|G = i) Pr(G = i), we can plug in the elements of the channel to obtain Pr(F = 0) = sin2 α Pr(G = 0) + cos2 β Pr(G = 1) and Pr(F = 1) = cos2 α Pr(G = 0) + sin2 β Pr(G = 1). Note that, there is a special channel called the symmetric channel. The rows of the symmetric channel are cyclic under permutations, so are the columns. The binary symmetric channel has the following form:  A=. 1−ξ 2 1+ξ 2. 1+ξ 2 1−ξ 2.  ,. (1.58). where ξ is the noisy parameter and 0 ≤ ξ ≤ 1. Now, we can study the new result proposed by Evans and Schulman. Naively, by the data processing inequality one may expect the upper bound of the ratio 1. In [48], Pippenger proposed that the upper bound of the ratio. I(X;Z) I(X;Y ). I(X;Z) I(X;Y ). to be. for the binary. symmetric channel should be tighter than 1. In [50, 51], Evans and Schulman further gave the upper bound of the ratio. I(X;Z) I(X;Y ). for the general binary channels. It is amazing. that this bound is a function of the second communication channel in the the process of signal propagation. If the second channel is specified by (1.57), one can obtain the upper bound of the ratio the maxima of the. I(X;Z) 2 I(X;Y ) to be sin (α − β). For the binary I(X;Z) 2 ratio I(X;Y ) is ξ . Note that, this bound is. symmetric channel (1.58), tighter than ξ proposed by. Pippenger [48]. Evans and Schulman used the following fact to reduce the complexity of the proof. They found that the maxima of the ratio. I(X;Z) I(X;Y ). is achieved when the conditional probabilities. Pr(Y |X = 0) and Pr(Y |X = 1) are almost indistinguishable. This condition implies that 29.