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Signature Redacted Certified By Quantum Correlations, Quantum Resource Theories and Exclusion Game ARCHNES MASSACHUSETTS INSTITUTE by OF TECHNOLOLGY Zi-Wen Liu JUL 3 0 2015 Submitted to the Department of Mechanical Engineering LIBRARIES in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015 @ Massachusetts Institute of Technology 2015. All rights reserved. Author ..... Signature ...................... Department of Mechanical Engineering May 8, 2015 Z- / 7/ Signature redacted Certified by.... .........0......................... ... Seth Lloyd Professor of Mechanical Engineering Thesis Supervisor Accepted by. Signature redacted ............... David E. Hardt Ralph E. and Eloise F. Cross Professor of Mechanical Engineering Graduate Officer, Department of Mechanical Engineering 2 Quantum Correlations, Quantum Resource Theories and Exclusion Game by Zi-Wen Liu Submitted to the Department of Mechanical Engineering on May 8, 2015, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract This thesis addresses two topics in quantum information theory. The first topic is quantum correlations and quantum resource theory. The second is quantum commu- nication theory. The first part summarizes an ongoing work about quantum correlations beyond entanglement and quantum resource theories. We systematically explain the concept quantum correlations beyond entanglement, and introduce a unified framework of measuring such correlations with entropic quantities. In particular, a new measure called Diagonal Discord (DD), which is simpler to compute than discord but still possesses several nice properties, is proposed. As an application to real physical scenarios, we study the scaling behaviors of quantum correlations in spin lattices with these measures. On its own, however, the theory of quantum correlations is not yet a satisfactory quantum resource theory. Some partial results towards this goal are introduced. Furthermore, a unified abstract structure of general quantum resource theories and its duality is formalized. The second part shows that there exist (one-way) communication tasks with an infinite gap between quantum communication complexity and quantum information complexity. We consider the exclusion game, recently introduced by Perry, Jain and Oppenheim [80], which exhibits the property that for appropriately chosen parame- ters of the game, there exists an winning quantum strategy that reveals vanishingly small amount of information as the size of the problem n increases, i.e., the quantum (internal) information cost vanishes in the large n limit. For those parameters, we prove the quantum communication cost (the size of quantum communication to suc- ceed) is lower bounded by Q (log n), thereby proving an infinite gap between quantum information and communication costs. This infinite gap is further shown to be robust against sufficiently small error. Some other interesting features of the exclusion game are also discovered as byproducts. Thesis Supervisor: Seth Lloyd 3 Title: Professor of Mechanical Engineering 4 Acknowledgments First of all, I would like to thank my academic and research advisor, Professor Seth Lloyd, for leading me into the magnificent world of quantum information that I knew little of prior to coming to MIT, and for all his generous support and insightful guidance over the last two years. I can never know how blessed I am to have such an opportunity. I would like to thank Professors Scott Aaronson, Harry Asada, Gang Chen, Isaac Chuang, Eddie Farhi, Aram Harrow, Mehran Kardar, Seth Lloyd, Hong Liu, Peter Shor, Wati Taylor, Salil Vadhan, Evelyn Wang, Xiao-Gang Wen and many more, from whom I learned priceless knowledge through courses and/or discussions. I would like to thank my fellow graduate students and close collaborators, es- pecially Can Gokler, Dax Koh, Kevin Thompson, Elton Zhu and Quntao Zhuang, for all the happy time we spent together talking about everything from quarks to universe(s). I would also like to thank all my friends, for sharing my happiness and sorrow, and for allowing me to do the same for them. Thanks to all the people above, for making me believe I am doing the right thing, at the right place. And at last, I would like to thank my parents and my girlfriend Xiaoyu for en- couraging and trusting me, unswervingly. Miraculously, I feel that you are all right here with me, when I write down these words. 5 6 Contents I Quantum correlations and resource theories 14 1 Introduction 15 2 Quantum correlations beyond entanglement 19 2.1 Purely classical correlations ........ ......... ...... 19 2.1.1 Classically correlated states ...... ....... ...... 20 2.1.2 Creating nonclassical correlations ... ....... ...... 22 2.2 Entropic measures .... ...... ...... ....... ..... 25 2.2.1 Candidates ......... ........... ........ 26 2.2.2 Hierarchy ............................. 34 2.2.3 Multipartite generalization ........... ......... 39 2.3 Scaling behaviors in spin lattices ..... ............. ... 42 2.3.1 Quantum correlation between two spins ............. 43 2.3.2 Example: 1D Heisenberg XXZ chain . ............. 46 2.3.3 Area laws ...... ....................... 48 2.3.4 O utlook ....... ....................... 52 2.4 Diagonal Discord (DD) ... ............ ........... 52 2.4.1 M otivation ............................. 52 2.4.2 Properties ................ ............. 54 2.4.3 Physical interpretation ...................... 58 3 Quantum Resource Theories (QRTs) 61 3.1 Unified framework ............................ 62 7 3.1.1 Elements ... ....................... .... 62 3.1.2 Perfect QRT .......... .................. 68 3.1.3 Hierarchical structure ......... ........... ... 72 3.1.4 Combining QRTs ......................... 73 3.2 Quantum correlations as a resource ................... 76 3.2.1 Free states ................ ............. 78 3.2.2 So .................... ............. 79 3.2.3 Promotion to S ....... ........... ........ 83 3.2.4 Map zoo ........ ............ .......... 85 3.2.5 Remarks ...... ............ ........... 87 3.3 Dual QRTs . ............. ............ ...... 87 3.3.1 General structure ...... ........... ........ 88 3.3.2 Examples .. ....................... .... 91 4 Summary and outlook 95 II Exclusion game 97 5 Introduction 99 6 Preliminaries 103 6.1 General formulation of communication tasks .............. 103 6.1.1 Mathematical structure ............ .......... 103 6.1.2 Exclusion game ................. ......... 104 6.2 Information and communication ..... ............. ... 105 6.3 Classical communication complexity . ......... ........ 106 6.4 PJO strategy .......... ............. ........ 106 6.4.1 Protocol ........ ............ 107 6.4.2 Quantum information cost ........ ........... 108 8 7 Quantum communication complexity 109 7.1 Zero error .... ..... ....... ... .... ... .... ... 109 7.1.1 Classical encodings of quantum states ........... ... 110 7.1.2 Lower bound of Qcc . ........... .. .... .... 111 7.1.3 G aps ..... ....................... .... 113 7.2 Robustness against error .......... ............... 114 7.2.1 Lem m as ........... ........... ........ 114 7.2.2 Quantum-classical separation of communication ....... 118 7.2.3 Maximum error ........ .................. 119 8 Concluding remarks 121 Appendix 123 A QD and DD of real X states 123 A.1 Preparations ....... ......................... 123 A.2 Pairwise quantum correlation ......... ............ 125 A.2.1 Optimal basis .............. ............. 125 A.2.2 Explicit calculation of pairwise QD ....... 126 A.2.3 Pairwise DD ...... ............ .......... 130 9 10 List of Figures 2-1 (Adapted from [104]) Area law. Intuitively speaking, under locally interacting Hamiltonians, the correlation length in noncritical phases is finite so that sites in A and B that are separated by a distance further than the correlation length (the shaded stripe) should not contribute to the mutual information or correlation measures between A and B, hence bounded by the number of sites at the boundary and therefore scales as the boundary area. ........... ........... 49 3-1 Intuitive illustration of the basic content and structure of a QRT. .. 63 3-2 A hierarchical structure of maps. Columns represent correspondences, and rows represent strict hierarchies. ..... ............. 73 3-3 A sketch of a strategy for determining a qubit state that queries the QRTs of coherence and purity. The dashed circle represents the states that has the same entropy (connected by unitary transformations), and the solid line represents the states that are diagonal in the appointed basis (incoherent states). For an arbitrary state p, the unitary U brings it to one of the incoherent states, while preserving purity/entropy. .. 76 3-4 Geometrical (Bloch sphere) demonstration of the effect of a unital map on a qubit. The unital channel keeps the two basis vectors symmetric with respect to the center of the Bloch sphere, and the output state can be diagonalized in the orthonormal basis corresponding to the in- tersections of the connecting line and the surface of the Bloch sphere. 82 11 3-5 (Adapted from [91]) The detailed hierarchy in between mixture of uni- taries and unital maps. AQBP denotes "Asymptotic Quantum Birkhoff P roperty". ..... ...... ...... ...... ..... ..... 86 3-6 Illustration of the dual QRT. The parts with grey fill means "free". Dashed lines represent the pair of elements in each theory, and arrow represent partial orders,
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