Multipartite entangled quantum states: Transformation, Entanglement monotones and Application by Wei Cui A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto c Copyright 2013 by Wei Cui Abstract Multipartite entangled quantum states: Transformation, Entanglement monotones and Application Wei Cui Doctor of Philosophy Graduate Department of Physics University of Toronto 2013 Entanglement is one of the fundamental features of quantum information science. Though bipartite entanglement has been analyzed thoroughly in theory and shown to be an important resource in quantum computation and communication protocols, the theory of entanglement shared between more than two parties, which is called multi- partite entanglement, is still not complete. Specifically, the classification of multipartite entanglement and the transformation property between different multipartite states by local operators and classical communications (LOCC) are two fundamental questions in the theory of multipartite entanglement. In this thesis, we present results related to the LOCC transformation between multi- partite entangled states. Firstly, we investigate the bounds on the LOCC transformation probability between multipartite states, especially the GHZ class states. By analyzing the involvement of 3-tangle and other entanglement measures under weak two-outcome measurement, we derive explicit upper and lower bound on the transformation probabil- ity between GHZ class states. After that, we also analyze the transformation between N-party W type states, which is a special class of multipartite entangled states that has an explicit unique expression and a set of analytical entanglement monotones. We present a necessary and sufficient condition for a known upper bound of transformation probability between two N-party W type states to be achieved. We also further investigate a novel entanglement transformation protocol, the ran- dom distillation, which transforms multipartite entanglement into bipartite entanglement ii shared by a non-deterministic pair of parties. We find upper bounds for the random dis- tillation protocol for general N-party W type states and find the condition for the upper bounds to be achieved. What is surprising is that the upper bounds correspond to en- tanglement monotones that can be increased by Separable Operators (SEP), which gives the first set of analytical entanglement monotones that can be increased by SEP. Finally, we investigate the idea of a new class of multipartite entangled states, the Absolutely Maximal Entangled (AME) states, which is characterized by the fact that any bipartition of the states would give a maximal entangled state between the two sets. The relationship between AME states and Quantum secret sharing (QSS) protocols is exhibited and the application of AME states in novel quantum communication protocols is also explored. iii Acknowledgements Firstly, I want to thank my supervisor, Prof Hoi-Kwong Lo. During these five years, his endless help and inspired discussion guided me to explore the fantastic world of quantum information theory, which was a really exciting and enjoyable journey because of him. He taught me not only in science but also in other aspects of life. He showed my how to do presentation, how to improve my English, and more importantly, how to treat and work with other people. All the above and his kind help on my nonacademic life will be valuable and remembered for a lifelong time. Secondly, I really appreciate the advices and suggestions from my committee members, Daniel James and Aephraim Steinberg. It has been my great pleasure to work with a group of pleasant and brilliant col- leagues. I want to show my acknowledgement to Eric Chitambar, Wolfram Helwig, Bing Qi, Christian Weedbrook, Xiongfeng Ma, Benjamin Fortescue, Yi Zhao, Yuemeng Chi, Viacheslav Burenkov, Feihu Xu, Kero Lau, Zhiyuan Tang, Felix Liao, and He Xu. Special thanks to Eric Chitambar for his brilliant discussions and endless passion on the subject. And to Bing Qi for his support on both of my academic and nonacademic life. I have benefited a great deal from the discussion with many excellent scientists. Specif- ically, I wish to thank Lin Chen, Daniel Gottesman, Fred Fung, Debbie Leung, Jonathan Oppenheim, and David Gosset. I would like to thank Viacheslav Burenkov for his suggestions and proof reading. Responsibility for any remaining mistakes rests entirely with the author. Also, I wish to thank Krystyna Biel and Diane Silva for their great job in adminis- trative help. The help from the Center of International Experience, family care office and family housing of the University of Toronto is also acknowledged. With their help, I had a really harmonious life with my family while studying in the University of Toronto as an international student. Finally, the love and support from my family is greatly appreciated. This thesis is dedicated to my parents, my wife Bilian, and my lovely son Stephen. iv Contents 1 Introduction 1 1.1 Our results . 2 1.1.1 List of papers and presentations . 4 2 Background Information 6 2.1 Entanglement . 6 2.1.1 Entanglement in quantum physics . 6 2.1.2 Entanglement in Hilbert space . 7 2.1.3 Entanglement as a resource . 8 2.2 Quantum operations and entanglement measures . 10 2.2.1 Quantum Operators . 10 2.2.2 Local Operators and Classical Communications . 11 2.2.3 Separable operators . 13 2.2.4 Entanglement measures for pure bipartite states . 14 2.2.5 Entanglement measures for mixed states . 17 2.3 Multipartite entangled pure states . 19 2.3.1 Tripartite entangled states . 19 2.3.2 W type entangled states . 21 3 LOCC transformation bounds between multipartite pure states 23 3.1 Introduction . 24 3.2 Upper Bound for the Conversion from GHZ state to a GHZ class state . 25 3.3 Failure Branch . 30 3.3.1 Conservation of interference term . 31 3.3.2 Conservation of normalization . 32 3.4 Upper Bound for a general case . 35 3.4.1 interference term and the maximal value of the 3-tangle of a GHZ- class state . 35 v 3.4.2 "stop and reconstruct" procedure . 36 3.4.3 Example: GHZ φ = γ( 000 + aaa ) . 38 j i ! j i j i j i 3.4.4 general case . 45 3.5 Lower Bound for the Transformation . 48 3.6 Summary and Concluding Remarks . 56 4 Optimal entanglement transformations among N-qubit W-type states 57 4.1 Introduction . 57 4.2 Upper bounds . 59 4.3 Lower bounds . 63 4.4 General Features of symmetric transformations . 66 4.5 Conclusion . 68 5 Random distillation for W type states 69 5.1 Introduction . 69 5.2 Previous results and notation . 74 5.2.1 The generalized Fortescue-Lo protocol . 74 5.2.2 Additional notation and the Kintas-Turgut monotones . 75 5.3 The least party out protocol . 76 5.3.1 Phase I: Remove x0 component . 76 5.3.2 Phase II: Equal or vanish (e/v) subroutine . 77 5.3.3 Phase III: Obtaining EPR pairs . 77 5.4 Main results: The LPO protocol on multipartite W type states . 81 5.4.1 Summary of results . 81 5.4.2 Three qubits . 81 5.4.3 Four qubits . 83 5.4.4 n qubits and the entanglement monotones . 89 5.4.5 Interpretation of monotones . 92 5.5 SEP VS LOCC . 93 5.5.1 Random distillation by Separable transformations . 93 5.5.2 Comparison between SEP and LOCC . 96 5.6 Applicaiton to the transformation φ WN . 98 j i1;··· ;N ! j i 5.7 Conclusion . 99 5.7.1 Open questions and concluding remarks . 99 6 Absolutely maximal entangled state and quantum secret sharing 102 6.1 Introduction . 102 vi 6.2 Definition of AME states . 104 6.3 Parallel Teleportation . 105 6.4 Quantum Secret Sharing. 107 6.5 Conclusion . 110 7 Conclusion 112 7.1 Future work . 113 7.2 Concluding words . 114 8 Appendix 115 8.1 Appendix: Proof of Theorem 5 . 115 8.2 Appendix: proof of Theorem 14 . 117 8.3 Dual solution to WN distillation by SEP . 119 j i Bibliography 122 vii List of Figures 2.1 Structure of states that can be obtained from W3 state by SLOCC. The first level is the true W3 type state which is also the genuine W class state. The second level are bipartite entangled states, such as (AB)-C ( φ ), (AC)-B ( φ ) and (BC)-A ( φ ), and the third j iAB j iC j iAC j iB j iBC j iA level is the product state φ1 φ2 φ3 . 22 j iA j iB j iC 3.1 mapping type 1. c 2010 American Physical Society . 27 3.2 mapping type 2. c 2010 American Physical Society . 27 y 1 U 3 3.3 The value of p as a function of a. In this figure, a = ( y−1 ) . So when a goes from 0 to 1, y goes from 0 to . Note that as y goes to infinity, a 1 goes to 1. We express the value as a function of a because it will be easier for us to combine different graphs into one graph later. c 2010 American Physical Society. 34 3.4 "stop and reconstruct" for a two-outcome measurement. c 2010 American Physical Society. 36 3.5 The original protocol written in the many two-outcome measurements form. c 2010 American Physical Society. 37 3.6 "stop and reconstruct" for general protocol, I stands for the interference term. c 2010 American Physical Society. 38 3.7 The new protocol, which can reconstruct the original one. c 2010 American Physical Society. 39 3.8 the relation between p¯s and p¯τ . c 2010 American Physical Society. 43 ABC 3.9 The upper bound for the transformation .