DOCSLIB.ORG

Multipartite Entangled Quantum States: Transformation, Entanglement Monotones and Application

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multipartite Entangled Quantum States: Transformation, Entanglement Monotones and Application 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 .
Load more

Recommended publications
  • Completeness of the ZX-Calculus
    Completeness of the ZX-calculus Quanlong Wang Wolfson College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 2018 Acknowledgements Firstly, I would like to express my sincere gratitude to my supervisor Bob Coecke for all his huge help, encouragement, discussions and comments. I can not imagine what my life would have been like without his great assistance. Great thanks to my colleague, co-auhor and friend Kang Feng Ng, for the valuable cooperation in research and his helpful suggestions in my daily life. My sincere thanks also goes to Amar Hadzihasanovic, who has kindly shared his idea and agreed to cooperate on writing a paper based one his results. Many thanks to Simon Perdrix, from whom I have learned a lot and received much help when I worked with him in Nancy, while still benefitting from this experience in Oxford. I would also like to thank Miriam Backens for loads of useful discussions, advertising for my talk in QPL and helping me on latex problems. Special thanks to Dan Marsden for his patience and generousness in answering my questions and giving suggestions. I would like to thank Xiaoning Bian for always being ready to help me solve problems in using latex and other softwares. I also wish to thank all the people who attended the weekly ZX meeting for many interesting discussions. I am also grateful to my college advisor Jonathan Barrett and department ad- visor Jamie Vicary, thank you for chatting with me about my research and my life. I particularly want to thank my examiners, Ross Duncan and Sam Staton, for their very detailed and helpful comments and corrections by which this thesis has been significantly improved.
    [Show full text]
  • Unifying Cps Nqs
    Citation for published version: Clark, SR 2018, 'Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks', Journal of Physics A: Mathematical and General, vol. 51, no. 13, 135301. https://doi.org/10.1088/1751- 8121/aaaaf2 DOI: 10.1088/1751-8121/aaaaf2 Publication date: 2018 Document Version Peer reviewed version Link to publication This is an author-created, in-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1751-8121/aaaaf2/meta University of Bath Alternative formats If you require this document in an alternative format, please contact: [email protected] General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 04. Oct. 2021 Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks Stephen R Clarkyz yDepartment of Physics, University of Bath, Claverton Down, Bath BA2 7AY, U.K. zMax Planck Institute for the Structure and Dynamics of Matter, University of Hamburg CFEL, Hamburg, Germany E-mail: [email protected] Abstract.
    [Show full text]
  • Remote Preparation of $ W $ States from Imperfect Bipartite Sources
    Remote preparation of W states from imperfect bipartite sources M. G. M. Moreno, M´arcio M. Cunha, and Fernando Parisio∗ Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901,Recife, Pernambuco, Brazil Several proposals to produce tripartite W -type entanglement are probabilistic even if no imperfec- tions are considered in the processes. We provide a deterministic way to remotely create W states out of an EPR source. The proposal is made viable through measurements (which can be demoli- tive) in an appropriate three-qubit basis. The protocol becomes probabilistic only when source flaws are considered. It turns out that, even in this situation, it is robust against imperfections in two senses: (i) It is possible, after postselection, to create a pure ensemble of W states out of an EPR source containing a systematic error; (ii) If no postselection is done, the resulting mixed state has a fidelity, with respect to a pure |W i, which is higher than that of the imperfect source in comparison to an ideal EPR source. This simultaneously amounts to entanglement concentration and lifting. PACS numbers: 03.67.Bg, 03.67.-a arXiv:1509.08438v2 [quant-ph] 25 Aug 2016 ∗ [email protected] 2 I. INTRODUCTION The inventory of potential achievements that quantum entanglement may bring about has steadily grown for decades. It is the concept behind most of the non-trivial, classically prohibitive tasks in information science [1]. However, for entanglement to become a useful resource in general, much work is yet to be done. The efficient creation of entanglement, often involving many degrees of freedom, is one of the first challenges to be coped with, whose simplest instance are the sources of correlated pairs of two-level systems.
    [Show full text]
  • ZX-Calculus for the Working Quantum Computer Scientist
    ZX-calculus for the working quantum computer scientist John van de Wetering1,2 1Radboud Universiteit Nijmegen 2Oxford University December 29, 2020 The ZX-calculus is a graphical language for reasoning about quantum com- putation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurement- based quantum computation, and quantum foundations. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those fa- miliar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus that they could use it in their daily work for small computations on quantum circuits and states. The lat- ter sections give a condensed overview of the literature on the ZX-calculus. We discuss Clifford computation and graphically prove the Gottesman-Knill theorem, we discuss a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and we discuss the recent completeness theorems for the ZX-calculus that show that, in principle, all reasoning about quantum computation can be done using ZX-diagrams. Ad- ditionally, we discuss the categorical and algebraic origins of the ZX-calculus and we discuss several extensions of the language which can represent mixed states, measurement, classical control and higher-dimensional qudits. Contents 1 Introduction3 1.1 Overview of the paper . 4 1.2 Tools and resources . 5 2 Quantum circuits vs ZX-diagrams6 2.1 From quantum circuits to ZX-diagrams . 6 2.2 States in the ZX-calculus .
    [Show full text]
  • Categorical Tensor Network States
    Categorical Tensor Network States Jacob D. Biamonte,1,2,3,a Stephen R. Clark2,4 and Dieter Jaksch5,4,2 1Oxford University Computing Laboratory, Parks Road Oxford, OX1 3QD, United Kingdom 2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 3ISI Foundation, Torino Italy 4Keble College, Parks Road, University of Oxford, Oxford OX1 3PG, United Kingdom 5Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unifi- cation of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition prob- lem which has several appealing features. Specifically, given an n-body quantum state |ψi, we present a new and general method to factor |ψi into a tensor network of clearly defined building blocks. We use the solution to expose a previously un- known and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states. PACS numbers: 03.65.Fd, 03.65.Ca, 03.65.Aa arXiv:1012.0531v2 [quant-ph] 17 Dec 2011 a [email protected] 2 I. INTRODUCTION Tensor network states have recently emerged from Quantum Information Science as a gen- eral method to simulate quantum systems using classical computers.
    [Show full text]
  • Quantum Protocols Involving Multiparticle Entanglement and Their Representations in the Zx-Calculus
    Quantum Protocols involving Multiparticle Entanglement and their Representations in the zx-calculus. Anne Hillebrand University College University of Oxford A thesis submitted for the degree of MSc Mathematics and the Foundations of Computer Science 1 September 2011 Acknowledgements First I want to express my gratitude to my parents for making it possible for me to study at Oxford and for always supporting me and believing in me. Furthermore, I would like to thank Alex Merry for helping me out whenever I had trouble with quantomatic. Moreover, I want to say thank you to my friends, who picked me up whenever I was down and took my mind off things when I needed it. Lastly I want to thank my supervisor Bob Coecke for introducing me to the subject of quantum picturalism and for always making time for me when I asked for it. Abstract Quantum entanglement, described by Einstein as \spooky action at a distance", is a key resource in many quantum protocols, like quantum teleportation and quantum cryptography. Yet entanglement makes protocols presented in Dirac notation difficult to follow and check. This is why Coecke nad Duncan have in- troduced a diagrammatic language for multi-qubit systems, called the red/green calculus or the zx-calculus [23]. This diagrammatic notation is both intuitive and formally rigorous. It is a simple, graphical, high level language that empha- sises the composition of systems and naturally captures the essentials of quantum mechanics. One crucial feature that will be exploited here is the encoding of complementary observables and corresponding phase shifts. Reasoning is done by rewriting diagrams, i.e.
    [Show full text]
  • Generative Machine Learning with Tensor Networks: Benchmarks on Near-Term Quantum Computers
    PHYSICAL REVIEW RESEARCH 3, 023010 (2021) Generative machine learning with tensor networks: Benchmarks on near-term quantum computers Michael L. Wall, Matthew R. Abernathy , and Gregory Quiroz The Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland 20723, USA (Received 16 October 2020; revised 3 March 2021; accepted 7 March 2021; published 2 April 2021) Noisy, intermediate-scale quantum (NISQ) computing devices have become an industrial reality in the last few years, and cloud-based interfaces to these devices are enabling the exploration of near-term quantum computing on a range of problems. As NISQ devices are too noisy for many of the algorithms with a known quantum advantage, discovering impactful applications for near-term devices is the subject of intense research interest. We explore a quantum-assisted machine learning (QAML) workflow using NISQ devices through the perspective of tensor networks (TNs), which offer a robust platform for designing resource-efficient and expressive machine learning models to be dispatched on quantum devices. In particular, we lay out a framework for designing and optimizing TN-based QAML models using classical techniques, and then compiling these models to be run on quantum hardware, with demonstrations for generative matrix product state (MPS) models. We put forth a generalized canonical form for MPS models that aids in compilation to quantum devices, and demonstrate greedy heuristics for compiling with a given topology and gate set that outperforms known generic methods in terms of the number of entangling gates, e.g., CNOTs, in some cases by an order of magnitude. We present an exactly solvable benchmark problem for assessing the performance of MPS QAML models and also present an application for the canonical MNIST handwritten digit dataset.
    [Show full text]
  • Single-Photon Three-Qubit Quantum Logic Using Spatial Light Modulators
    ARTICLE DOI: 10.1038/s41467-017-00580-x OPEN Single-photon three-qubit quantum logic using spatial light modulators Kumel H. Kagalwala1, Giovanni Di Giuseppe 1,2, Ayman F. Abouraddy1 & Bahaa E.A. Saleh1 The information-carrying capacity of a single photon can be vastly expanded by exploiting its multiple degrees of freedom: spatial, temporal, and polarization. Although multiple qubits can be encoded per photon, to date only two-qubit single-photon quantum operations have been realized. Here, we report an experimental demonstration of three-qubit single-photon, linear, deterministic quantum gates that exploit photon polarization and the two-dimensional spatial-parity-symmetry of the transverse single-photon field. These gates are implemented using a polarization-sensitive spatial light modulator that provides a robust, non-interferometric, versatile platform for implementing controlled unitary gates. Polarization here represents the control qubit for either separable or entangling unitary operations on the two spatial-parity target qubits. Such gates help generate maximally entangled three-qubit Greenberger–Horne–Zeilinger and W states, which is confirmed by tomographical reconstruction of single-photon density matrices. This strategy provides access to a wide range of three-qubit states and operations for use in few-qubit quantum information processing protocols. 1 CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA. 2 School of Science and Technology, Physics Division, University of Camerino,
    [Show full text]
  • Arxiv:1704.06359V2 [Quant-Ph] 22 Aug 2017 Heat Engines [21] and Quantum Key Distribution [22]
    N identical particles and one particle to entangle them all Bruno Bellomo,1 Rosario Lo Franco,2, 3, ∗ and Giuseppe Compagno3 1Institut UTINAM - UMR 6213, CNRS, Observatoire des Sciences de l’Univers THETA, Université Bourgogne Franche-Comté, F-25000 Besançon, France 2Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy 3Dipartimento di Fisica e Chimica, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy (Dated: September 4, 2018) In quantum information W states are a central class of multipartite entangled states because of their robustness against noise and use in many quantum processes. Their generation however remains a demanding task whose difficulty increases with the number of particles. We report a simple scalable conceptual scheme where a single particle in an ancilla mode works as entanglement catalyst of W state for other N separated identical particles. A crucial novel aspect of the scheme, which exploits basically spatial indistinguishability, is its universality, being applicable without essential changes to both bosons and fermions. Our proposal represents a new paradigm within experimental preparation of many-particle entanglement based on quantum indistinguishability. I. INTRODUCTION states with the largest number of particles observed in the lab- oratory consist of eight two-level trapped ions [51], while for Quantum entanglement represents nonclassical correlations instance this number is lowered to four particles for polarized among constituents of composite systems which make them photons [50]. A key step towards simpler reliable generation intertwined independently of how far they are each other protocols is to find conceptual schemes based on fundamen- [1].
    [Show full text]
  • Constructions and Applications of W-States
    Constructions and Applications of W-States A Major Qualifying Project Submitted to the Faculty of Worcester Polytechnic Institute in partial fulfillment of the requirements for the Degree in Bachelor of Science in Physics and Mathematical Sciences By James McClung Date May 17, 2020 Advisors Professor P. K. Aravind Professor Umberto Mosco This report represents work of WPI undergraduate students submitted to the faculty as evidence of a degree requirement. WPI routinely publishes these reports on its web site without editorial or peer review. For more information about the projects program at WPI, see http://www.wpi.edu/Academics/Projects. Abstract After introducing the fundamentals of quantum circuits for a general audience, this report evaluates two means of constructing W-states: the probabilistic naive construction (PNC) and the optimized splitter construction (OSC). These methods are compared in terms of asymptotic gate cost and other parameters. Furthermore, a proof of Bell's theorem is explored with new results concerning the probability of contradicting elements of reality using W-states. i Acknowledgements Thanks to Connor Anderson for discovering the connection between (96) and Eulerian numbers. Thanks also to Ethan Washock for helping with the same. Thanks to all my friends in the math lounge, especially Kyle Dituro, for making my time there brighter, and to Mr. Greg Aubin for motivating me on late nights spent working. Thanks to my parents and to Emily Staknis for their constant support. Most of all I thank Professor P. K. Aravind for putting me to work on Qudoku over a year ago, and guiding my studies in quantum information theory all the way up to this point.
    [Show full text]
  • Multi-Partite Entanglement
    Multi-partite entanglement Michael Walter∗1, David Grossy2,3, and Jens Eisertz4 1Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA 2Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany 3Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia 4Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany Contents 1 Introduction 1 2 General theory 2 2.1 Classifying pure state entanglement . .2 2.2 Local unitary equivalence . .3 2.3 Equivalence under local operations and classical communication . .5 2.4 Asymptotic manipulation of pure multi-partite quantum states . .9 2.5 Quantifying pure multi-partite entanglement . 11 2.6 Classifying mixed state entanglement . 12 2.7 Detecting mixed state entanglement . 14 3 Important classes of multi-partite states 15 3.1 Matrix-product states and tensor networks . 15 3.2 Stabilizer states . 17 3.3 Bosonic and fermionic Gaussian states . 18 4 Specialized topics 20 4.1 Quantum Shannon theory . 20 4.2 Quantum secret sharing and other multi-party protocols . 20 4.3 Quantum non-locality . 21 4.4 Measurement-based quantum computing . 22 arXiv:1612.02437v2 [quant-ph] 5 Feb 2017 4.5 Metrology . 23 1 Introduction In this chapter, we generalize entanglement theory from two-partite to multi-partite systems. The term multi-partite may refer here to quantum systems composed of a macroscopic number of subsystems, ∗[email protected] [email protected] [email protected] 1 such as the parts of an interacting many-body system as studied in condensed matter physics, or it may mean merely “three”.
    [Show full text]
  • Matrix Product States: Entanglement, Symmetries, and State Transformations
    Matrix Product States: Entanglement, symmetries, and state transformations David Sauerwein∗ Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany and Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21A, 6020 Innsbruck, Austria Andras Molnar† and J. Ignacio Cirac‡ Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany and Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen,Germany Barbara Kraus§ Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21A, 6020 Innsbruck, Austria (Dated: January 24, 2019) We analyze entanglement in the family of translationally-invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by SLOCC transformations, a central question in entanglement theory. We use that criterion to determine SLOCC classes, and explicitly carry out this classification for the simplest, non-trivial MPS. We also characterize all symmetries of MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts. 1. Introduction: Entanglement is a resource for numer- formations can be reduced to families of non-generic, but ous striking applications of quantum information theory physically relevant states. [1, 2]. Furthermore, it is key to comprehend many pecu- In this Letter we present a systematic investigation of liar properties of quantum many-body systems [3, 4] and state transformations for Matrix Product States (MPS) has become increasingly important in areas like quantum that describe translationally invariant systems (with pe- field theory or quantum gravity [5, 6]. Despite its rele- riodic boundary conditions) [19, 20].
    [Show full text]