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The Mathematical Structure of Tensor Networks

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The Mathematical Structure of Tensor Networks The Mathematical Structure of Tensor Networks Gert Vercleyen Promotor: Prof. dr. Frank Verstraete Masters Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Science in Mathematics Academic Year: 2017-2018 I dedicate this thesis to Zempo, Spock, James Tiberius Kirk, Leonard ‘bones’ McCoy, Yanick, Jenny, Chris, Febe, and the other ten fish swimming gracefully in our pond. Acknowledgements One of the great Zen masters had an eager disciple who never lost an opportunity to catch whatever pearls of wisdom might drop from the master's lips, and who followed him about constantly. One day, deferentially opening an iron gate for the old man, the disciple asked, `How may I attain, enlightenment?' The ancient sage, though withered and feeble, couldbe quick, and he deftly caused the heavy gate to shut on the pupil's leg, breaking it. It would seem to the unenlightened as though the master, far from teaching his disciple, had left him more perplexed than ever by his cruel trick. To the enlightened, the anecdote expresses a deep truth. (Carl E. Linderholm, Mathematics Made Difficult) To me the quote above contains a big message when it comes to research. An important part of research consists of doing things yourself, even if this implies bashing your head against a wall until either the wall or your head breaks. Research consists of stepping in the dark. Research consists of spending weeks of time inventing stuff only to discover the next day that the stuff you’ve invented not only already exists, but is also better formulated, deeper in structure and is moreover contained in one of the references you’ve overlooked at the start of the project. Ironically those wasted weeks might turn out to be the most productive weeks, or at least in my case they were. To stay motivated would be a challenge were it not for the fact that I received a lot of support. Therefore I would like to thank everyone I could count on to make this masters thesis the document it is now. In particular I would like to thank Frank Verstraete, without whom none of this would have been possible. I am very grateful for the lectures on tensor networks, for the conferences and references on the subject, for the hospitality in the quantum group, for the freedom in the content of the research, but most of all for the endless confidence given to me. Of course I would also like to thank all members of the quantum group in Ghent for making me feel like part of the research group. Special thanks goes to i Sam, Tommeke, Spoon and Nadia who proofread the pile of errors that I presented as being a thesis. Of course many thanks goes to my family for all the support along the way and the same holds for everyone attending the weekly mindfulness sessions with me. Last but not least I would like to thank Natasha and her little wonder Eden for the truly wonderful time spent together. ii Terms of use The author grants permission to make this master’s thesis available for consultation and to copy parts of the master’s thesis for personal use. Any other use falls under the limitations of copyright, in particular with regard to the obligation to explicitly mention the source when citing results from this master’s thesis. Date: 31/05/2018 iii iv Contents 1 Quantum Mechanics 5 1.1 Introduction .................................. 5 1.2 States and Observables ........................... 6 1.2.1 Observables ............................. 6 1.2.2 States ................................. 7 1.3 Measurement ................................. 8 1.4 Composite Systems and Entanglement .................. 10 1.5 Entanglement Entropy ............................ 11 2 Graphical Language of Category Theory 13 2.1 Categories, Functors and Natural Transformations ............ 13 2.1.1 Categories .............................. 13 2.1.2 Functors between Categories .................... 14 2.1.3 Natural Transformations between Functors ............ 15 2.1.4 Graphical Language ......................... 16 2.2 Monoidal Categories ............................. 18 2.2.1 Defintition ............................... 18 2.2.2 Graphical Language ......................... 19 2.3 Braided Monoidal Categories ........................ 22 2.3.1 Definition ............................... 22 2.3.2 Graphical Language ......................... 23 2.4 Symmetric Monoidal Categories ...................... 24 2.4.1 Definition ............................... 24 2.4.2 Graphical Language ......................... 24 2.5 Compact Closed Categories ........................ 25 2.5.1 Definition ............................... 25 2.5.2 Graphical Language ......................... 26 2.6 Dagger Categories .............................. 28 2.6.1 Definition ............................... 28 2.6.2 Graphical Language ......................... 29 v 2.7 Dagger Compact Categories ........................ 29 2.7.1 Definition ............................... 29 2.7.2 Graphical Language ......................... 30 3 Category Theory Redefined 31 3.1 Categories, Functors and Natural Transformations ............ 31 3.1.1 Categories .............................. 31 3.1.2 Functors between Categories .................... 33 3.1.3 Natural Transformations between Functors ............ 35 3.2 Strict Monoidal Categories ......................... 36 3.3 Symmetric Monoidal Categories ...................... 37 3.4 Compact Closed Categories ........................ 38 3.5 Dagger Compact Categories ........................ 40 4 Categorical Quantum Mechanics 43 4.1 Hilbert Space Formalism .......................... 43 4.2 Density Matrix Formalism .......................... 46 5 Topological Quantum Field Theory 53 5.1 Cobordisms .................................. 53 5.2 Axioms of TQFT ............................... 56 6 Finite Spin Lattices and Quantum Phases 59 6.1 Finite Spin Lattices .............................. 59 6.1.1 Definitions ............................... 59 6.1.2 Locality ................................ 61 6.2 Quantum Phases ............................... 64 7 Matrix Product States 65 7.1 Introductory Example ............................ 65 7.2 Definitions ................................... 68 7.3 Basic Properties ............................... 70 7.3.1 Span of generalized MPS ...................... 70 7.3.2 Entanglement ............................. 71 7.3.3 Expectation Values .......................... 71 7.4 Canonical Form and Injectivity ....................... 72 Appendices 77 A Dutch summary 77 A.1 Spin Roosters ................................. 77 vi A.2 Categorie Theorie .............................. 78 B Rigorous Graphical Calculus 79 B.1 Monoidal Categories and Their Diagrams ................. 79 B.1.1 Monoidal Categories ......................... 79 B.1.2 Graphs ................................ 82 B.1.3 Progressive Plane Diagrams .................... 84 B.1.4 Free Monoidal Categories ...................... 89 B.2 Symmetric Tensor Categories and Progressive Polarised Diagrams .. 96 B.2.1 The Value of a Progressive Polarised Diagram .......... 96 B.2.2 Free Symmetric Tensor Categories ................. 102 1 2 Introduction This thesis has two goals. The first goal is to translate concepts in category theory to diagrams, in the hope to facilitate communication between mathematicians and physicists. Category theory or “abstract nonsense seems” to be taking over mathematics and this trend affects the language used in papers on mathematical-and theoretical physics. Often categorical concepts are used where a traditional set-based approach would suffice. In such cases no insight in category theory is required, although it might seem to be the case merely because of the language used. This paper hopes to resolve this issue by translating some of the common categorical notions to a simple diagrammatic language. It turns out that the language developed, simple as it may be, is both sound and coherent for many important categories. The second goal is to provide an introduction to tensor networks for those people with a mathematical background. Category theory, and the corresponding diagrammatic language, can then be used to rephrase the postulates of quantum mechanics. This would seem overkill were it not that the, otherwise abstract, notion of a topological quantum field theory also fits in the same scheme. The graphical language used to define quantum mechanics is also the same as the one used for constructing tensor networks. This way we can truly regard tensor networks as being built up of maximally entangled pairs. The structure of this thesis is as follows. In chapter 1 a brief introduction to quantum mechanics is given. People familiar with the concept of entanglement might wish to skip this chapter. In chapter 2 various categorical concepts are defined together with their corresponding diagrammatic representations. People traumatised by category theory are invited to skip this chapter, only to return to it when a deeper understand- ing is necessary. In chapter 3 the categorical definitions are given using diagrams only. Various tables that translate the diagrams to conventional categorical nomencla- ture are included. Chapter 4 translates the postulates of finite dimensional quantum mechanics to the graphical language. This is done both for the ‘Hilbert space formal- ism’ as the ‘density matrix formalism’. Chapter 5 introduces the notion of a topological 3 quantum field theory. It serves as an example of the power of categorical (read: di- agrammatic) techniques. Chapter 6 contains a brief summary of some central defini- tions
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