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Quantum States and Their Marginals : from Multipartite Entanglement to Quantum Error-Correcting Codes

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Quantum States and Their Marginals : from Multipartite Entanglement to Quantum Error-Correcting Codes Quantum States and their Marginals: From Multipartite Entanglement to Quantum Error-Correcting Codes DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt von: Felix Michael Huber eingereicht bei der Naturwissenschaftlich-Technischen Fakultat¨ der Universitat¨ Siegen, Siegen 2017 Gutachter: Prof. Otfried Guhne¨ Prof. Jens Siewert Prufer:¨ Prof. Otfried Guhne¨ Prof. Jens Siewert Prof. Mohamed Barakat Prof. Christof Wunderlich Datum der mundlichen¨ Prufung:¨ 15.12.2017 Abstract At the heart of the curious phenomenon of quantum entanglement lies the relation between the whole and its parts. In my thesis, I explore different aspects of this theme in the multipartite setting by drawing connections to concepts from statistics, graph theory, and quantum error-correcting codes: first, I address the case when joint quantum states are determined by their few-body parts and by Jaynes’ maximum entropy principle. This can be seen as an extension of the notion of entanglement, with less complex states already being determined by their few-body marginals. Second, I address the conditions for certain highly entangled multipartite states to exist. In particular, I present the solution of a long-standing open problem concerning the existence of an absolutely maximally entangled state on seven qubits. This sheds light on the algebraic properties of pure quantum states, and on the conditions that constrain the sharing of entanglement amongst multiple particles. Third, I investigate Ulam’s graph reconstruction problems in the quantum setting, and obtain legitimacy conditions of a set of states to be the reductions of a joint graph state. Lastly, I apply and extend the weight enumerator machinery from quantum error correction to investigate the existence of codes and highly entangled states in higher dimensions. This clarifies the physical interpretation of the weight enumerators and of the quantum MacWilliams identity, leading to novel applications in multipartite entanglement. Zusammenfassung Fur¨ das Phanomen¨ der Quantenverschrankung¨ sind die Beziehungen zwischen dem Ganzen und dessen Teilen zentral. In dieser Dissertation untersuche ich ver- schiede Aspekte dieser Thematik im Hinblick auf Mehrteilchen-Systeme, und ziehe Verbindungen zu Konzepten der Statistik, der Graphentheorie, und der Quanten- fehlerkorrektur. Als Erstes untersuche ich die Bedingungen, unter denen Quan- tenzustande¨ durch ihre Reduktionen auf kleine Subsysteme zusammen mit der Anwendung von Jaynes’ Maximum-Entropie-Methode bestimmt sind. Dies kann als eine Ausweitung des Verschrankungsbegriffes¨ gesehen werden, wobei ein weniger komplexer Zustand bereits durch seine Subsysteme kleiner Große¨ festgelegt ist. Zweitens erortere¨ ich notwendige Bedingungen an die Existenz von gewissen hoch verschrankten¨ Zustanden.¨ Insbesondere prasentiere¨ ich die Losung¨ einer langjahri-¨ gen bisher noch ungelosten¨ Frage zur Existenz eines absolut maximal verschrankten¨ Zustandes, welcher aus sieben Quantenbits besteht. Dies wirft neues Licht auf die al- gebraischen Eigenschaften von reinen Quantenzustanden¨ und auf die Restriktionen, welche das Teilen von Verschrankung¨ unter mehreren Parteien limitieren. Drittens untersuche ich gewisse Fragestellungen fur¨ Quantenzustande,¨ welche dem Rekon- struktionproblem von Ulam in der Graphentheorie ahneln.¨ Dies fuhrt¨ zu neuen Bedingungen, damit eine Sammlung von Marginalien einem gemeinsamen Zustand entstammt. Zuletzt wende ich die Theorie der Gewichtszahler¨ aus der Quanten- fehlerkorrektur auf Fragen der Existenz von hoch verschrankten¨ hoher-dimensionale¨ Zustanden¨ und von fehlerkorrigierenden Codes an. Dies klart¨ die physikalische Interpretation der Gewichtszahler¨ und der quantum MacWilliams identitat,¨ und fuhrt¨ zu neuartigen Anwendungen in der Theorie der Mehrteilchen-Verschrankung.¨ Contents List of Illustrations v Figures . v Tables . v Preface vii 1 Basic concepts 1 1.1 Fundamentals . 1 1.1.1 Quantum states . 1 1.1.2 Multiple particles . 2 1.1.3 Operators and maps . 3 1.1.4 Entanglement . 5 1.1.5 Entropy and distance measures . 9 1.1.6 Entanglement detection and measures . 12 1.1.7 Bloch representation . 17 1.2 Further notions . 20 1.2.1 Graph states . 20 1.2.2 Quantum exponential families . 22 1.2.3 Absolutely maximally entangled (AME) states . 27 1.2.4 Quantum error-correcting codes . 29 1.2.5 Semidefinite programming . 39 1.2.6 The quantum marginal problem . 41 2 Ground and thermal states of local Hamiltonians 45 2.1 Introduction . 45 2.2 The setting . 46 2.3 Characterization of conv( ) via semidefinite programming . 47 Qk 2.4 Characterization via the graph state formalism . 52 2.5 Quantum simulation as an application . 56 2.6 Further results . 57 2.6.1 Further results on the information projection . 57 2.6.2 Ground and excited states of local Hamiltonians . 59 2.6.3 States of four parties . 59 2.6.4 Even- and odd-body correlations of qubit states . 62 2.7 Conclusion . 68 iii iv CONTENTS 3 AME state of seven qubits 69 3.1 The Bloch representation . 70 3.2 Properties of AME state reductions . 71 3.3 Scott bound . 72 3.4 Nonexistence of the seven qubit AME state . 73 3.5 Upper bound for the number of maximally mixed reductions . 75 3.6 AME states of n qubits . 76 3.7 Further results . 78 3.7.1 An iterative semidefinite program to find AME states . 78 3.8 Conclusion . 79 4 Ulam’s problems for quantum states 81 4.1 Motivation . 81 4.2 Realizability and uniqueness in graphs . 82 4.3 Graph states . 83 4.4 Weight distribution . 86 4.5 Constraints on the weight distribution . 88 4.6 Detecting illegitimate decks . 90 4.7 When is a weight distribution graphical? . 92 4.8 Conclusion . 93 5 Constraints on QECC and AME states 95 5.1 Introduction . 95 5.2 Motivation . 96 5.3 The Bloch representation . 97 5.4 Quantum error-correcting codes . 98 5.5 The shadow enumerator . 99 5.6 Shor-Laflamme enumerators . 100 5.7 The quantum MacWilliams identity . 103 5.8 The shadow enumerator in terms of the Shor-Laflamme enumerator 106 5.9 New bounds on absolutely maximally entangled states . 108 5.10 Discussion . 110 5.11 Further results . 111 5.11.1 A generalization of the universal state inversion from the shadow inequalities . 111 5.11.2 An application to the quantum marginal problem . 116 5.11.3 A strong subadditivity - like expression for the linear entropy 117 5.11.4 Further non-existence results of qubit-codes . 118 5.11.5 Weight distributions of hypothetical codes . 118 5.12 Conclusion . 118 Summary and outlook 121 Acknowledgments 123 Appendix: Krawtchouk polynomials 125 List of Illustrations Figures 1.1 A graph state of seven qutrits. 21 1.2 The information projection. 27 1.3 Examples of AME graph states. 29 1.4 Parameters of quantum codes. 31 2.1 Exponential families their convex hulls in the state space. 46 2.2 Illustration of the perturbation proof. 49 2.3 Witnesses for conv( ). .......................... 53 Qk 2.4 A witness for interaction certification. 56 2.5 Reconstructing four-party states from marginals. 60 2.6 Other sets of two-body marginals. 61 2.7 Visualization of even and odd-body correlations. 64 3.1 The Fano graph state as an approximation to a seven qubit AME state. 76 4.1 A highly entangled hypergraph state that is LU-inequivalent to graph states. 88 4.2 Example of a quantum deck. 91 4.3 LU-inequivalence and weight distributions. 93 5.1 The combinatorics of the shadow enumerator illustrated. 107 5.2 Table of higher-dimensional AME states. 109 Tables 1.1 Correspondence between qubit stabilizer codes and additive codes over GF (4). ................................. 34 2.1 The fraction of pure five and six qubit states found outside of conv( ). 51 Q2 2.2 Relations between even- and odd-body correlations of qubits. 67 5.1 An overview of weight enumerators and their coefficients. 100 v vi CONTENTS Preface Quantum mechanical states exhibit the two peculiar and closely related phenomena of superposition and entanglement. Exploring the ramifications of these phenomena has been a major scientific challenge during the last century. In particular, the discovered applications of quantum mechanics to information theory have spawned a whole new field of physics that is now known as quantum information theory. It includes all aspects of information processing and computation by means of quantum states, and led to algorithms in the quantum domain which are widely thought to outperform their classical counterparts. Entanglement was recognized early to be one of the key resources for quantum information processing and computation tasks. Loosely speaking, entanglement is present when the ’best possible knowledge of a whole does not necessarily include the best possible knowledge of all of its parts, even though they may be entirely separated and therefore virtually capable of being “best possibly known”’[1]. Thus the individual parts of a quantum system cannot be described independently from each other, making a joint description of all involved particles necessary. However, the characterization of entanglement and of its related notions such as steering and non-locality has turned out to be a far more difficult endeavor than one could have initially anticipated. In particular, there cannot exist any simple means to determine whether any given state is entangled or not [2, 3]. While the understanding of entanglement occurring between two particles has seen strong ad- vances, the deeper investigation of entanglement between multiple parties is a rather recent undertaking. In particular, it has been recognized that entanglement cannot be shared arbitrarily amongst all particles, but that its distribution is constrained in various ways. This thesis explores aspects of multipartite entanglement by drawing connections to concepts from statistics, graph theory, and the theory of quantum error-correcting codes. First, I address the case when quantum states are already determined by their few-body marginals and by Jaynes’ maximum entropy principle [A]. This can be seen as an extension of the notion of entanglement, with less complex states being already determined by their few-body marginals.
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