Anyon Theory in Gapped Many-Body Systems from Entanglement

Anyon theory in gapped many-body systems from entanglement

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Bowen Shi, B.S.

Graduate Program in Department of Physics

The Ohio State University

2020

Dissertation Committee:

Professor Yuan-Ming Lu, Advisor Professor Daniel Gauthier Professor Stuart Raby Professor Mohit Randeria Professor David Penneys, Graduate Faculty Representative c Copyright by

Bowen Shi

2020 Abstract

In this thesis, we present a theoretical framework that can derive a general anyon theory for

2D gapped phases from an assumption on the entanglement entropy. We formulate 2D quantum states by assuming two entropic conditions on local regions, (a version of entanglement area law that we advocate). We introduce the information convex set, a set of locally indistinguishable density matrices naturally deﬁned in our framework. We derive an isomorphism theorem and structure theorems of the information convex sets by studying the internal self-consistency. This line of derivation makes extensive usage of information-theoretic tools, e.g., strong subadditivity and the properties of quantum many-body states with conditional independence.

The following properties of the anyon theory are rigorously derived from this framework.

We deﬁne the superselection sectors (i.e., anyon types) and their fusion rules according to the structure of information convex sets. Antiparticles are shown to be well-deﬁned and unique.

The fusion rules are shown to satisfy a set of consistency conditions. The quantum dimension of each anyon type is defined, and we derive the well-known formula of topological entanglement entropy. We further identify unitary string operators that create anyon pairs and study the circuit depth. We define the topological S-matrix and show it satisfies the Verlinde formula.

It follows that the mutual braiding statistics of the sectors are nontrivial (they are anyons); moreover, the underlying anyon theory is modular.

Three additional things, closely related to this framework, are presented: (1) The framework on a discrete lattice; (2) A calculation of information convex set based on solvable Hamiltonians;

(3) A conjecture concerning the generality of our assumptions.

ii To my mother, who brought me images

iii Acknowledgments

It takes unexpected good luck and a fruitful journey for me to meet this wonderful research project and participate in it. None of this work would have been possible without the help of three people: Stuart Raby, Yuan-Ming Lu, and Isaac Kim. Stuart was my research advisor for my ﬁrst three years at OSU. I gained valuable experience in research during that time.

As a person who seeks the best experience for students, Stuart encouraged me to look into possible connections to diﬀerent sub-ﬁelds of physics that I was interested in. I got to know some connection between high-energy theory and quantum many-body physics at that time. I am also very grateful to Stuart’s genuine support during my transition of research direction.

Yuan-Ming is an enthusiastic junior faculty who introduced me to the fast-developing ideas in the research ﬁeld of quantum many-body physics. I feel lucky that Yuan-Ming had moved to

OSU and brought the related research directions to our department. Ever since I knocked on his door, we have done a few exciting works together. We had a reading club on anyons, in which I learned the valuable background of anyon theory. Isaac is my friend and a wonderful collaborator. After I read his compact and thought-provoking original works, my mind ﬁlled up with questions. The attempts to answer some of these questions during the years had gradually brought me to the exciting research project that this thesis describes.

I thank all the physics education I got at OSU. The knowledge and idea matter so does the way to communicate these ideas. Here are some memorable courses. The quantum mechanics course by Mohit Randeria has a clear focus on the physical principle and logical reasoning. The statistical mechanics course by Ciriyam Jayaprakash is always ﬁlled with intriguing physical examples and intuitive explanations. In Stuart’s elementary particle physics courses, he always

iv welcomes all kinds of questions from the students, and he shows an amazing ability to answer these questions; he is very generous in his time for the course and other (reading) courses. Chris

Hirata has an admirable ability to break down a complex phenomenon into much simpler ones.

Samir Mathur is both knowledgeable and humble; he pays great attention to what a student may say.

I thank Joseph McEwen for organizing the “geometry club” during my 1st and 2nd year. It was an informal graduate student meeting on any topic we ﬁnd interesting. Most of the topics are related to geometry and topology. I shared this good experience with friends Alexander

Davis, Noah Charles, Zaq Carson, Nicholas Mazzucca, Waylon Chen, and several others. These meetings sparkled with our curiosity for science, which makes it memorable.

I attempted to work in several research areas. This experience gave me a valuable chance to meet and learn from people working on diﬀerent research areas: cosmology, high energy physics, condensed matter physics, quantum information, and math. I thank Mingzhe Li and Xuejun Guo for some memorable instructions on research dating back to when I was at Nanjing University.

I thank Eric Braaten, Chris Hirata, and Chris Hill to be my committee members during the time I was doing research on high energy physics. I thank Daniel Gauthier, Mohit Randeria, and Stuart Raby to be my committee members after I switched to condensed matter theory.

I thank Junko Shigemitsu, Chris Hirata, Linda Carpenter, and Ulrich Heinz for everything I learned during the time I work with them as a grader. I thank Tin-Lun Ho, Nandini Trivedi,

Ilya Gruzberg, and Brian Skinner for things I learned from them during various discussions.

I am grateful to have learned many things about tensor categories from researchers in the math department at OSU, including Yilong Wang, David Penneys, Corey Jones, and Peter

Huston. Among them, I ﬁrst met Yilong during a conference at Indiana University Bloomington;

I attended a course on quantum algebra taught by Dave; Corey is an energetic postdoc who can easily switch back and forth between math and physics terminology. In an occasional conversation with Peter, he encouraged me to talk about my work in the math department.

v I have been fortunate to have the opportunity to travel to multiple places for short term summer (winter) schools and conferences. Furthermore, I thank Beni Yoshida and Timothy

Hsieh for inviting me to the Perimeter Institute. I got to know many friends during these travels, including: Meng Hua, Shuoguang Liu, Rui-Xing Zhang, Isaac Kim, Xueda Wen, Liujun

Zou, Zhehao Dai, Hassan Shapourian, Byungmin Kang, Shudan Zhong, Aaron Szasz, Lukasz

Fidkowski, Junyi Zhang, Lauren McGough, Huan He, Jie Wang, Yunqin Zheng, Kohtaro Kato,

Xie Chen, Tian Zhang, Kevin Slagle, Sharmistha Sahoo, Xiao Chen, Alex Thomson, Shubhayu

Chatterjee, Zhen Bi, Yingfei Gu, Wenbo Fu, Akhil Sheoran, Zhu-Xi Luo, Hao-Yu Sun, Zi-

Wen Liu, Sam Roberts, Dominic Williamson. Your presence enriched my travel experience.

Especially, I was fortunate to have met Kohtaro Kato in a conference at KITP, during which he taught me how to merge quantum Markov states. Later on, this merging technique turned out to be crucial in the derivations in “fusion rules from entanglement”, a work we collaborated with Isaac.

Finally, I wish to thank my fellow physics graduate students and postdocs for everything I learned from them as well as their accompanying in various weekly events/activities related to physics during diﬀerent stages of my career as a graduate student, including Yunlong Zheng,

Yan Yan, Yiming Pan, Qing Wang, Yifei He, Archana Anandakrishnan, B. Charles Bryant, Zijie

Poh, Xiao Fang, Shaun Hampton, Bin Guo, Liping He, Lipei Du, Hong Zhang, Bei Zhou, James

Rowland, Jiaxin Wu, Cheng Li, Fuyan Lu, Biao Huang, David Ronquillo, Tamaghna Hazra,

Wenjuan Zhang, X. Y. Yin, Saad Khalid, Yanjun He, Xiaozhou Feng, Shuangyuan Lu, Chang-

Yan Wang, Mohammed Karaki, Yonas Getachew, Joseph Szabo, Xin Dai, Alex Rasmussen,

Wayne Zheng and many others too numerous to mention. Being able to discuss with friends makes physics even more fun.

vi Vita

2010 ...... B.S. Physics, Nanjing University

Publications

Research Publications

7. B. Shi, Verlinde formula from entanglement, [arXiv: 1911.01470]

6. B. Shi, K. Kato, and I. H. Kim, Fusion rules from entanglement, [arXiv: 1906.09376]

5. B. Shi, Seeing topological entanglement through the information convex, Phys. Rev. Re- search 1, 033048 (2019) [arXiv: 1810.01986]

4. B. Shi and Y.-M. Lu, Characterizing topological orders by the information convex, Phys. Rev. B 99, 035112 (2019) [arXiv: 1801.01519]

3. B. Shi and Y.-M. Lu, Deciphering the nonlocal entanglement entropy of fracton topological orders, Phys. Rev. B 97, 144106 (2018) [Editors’ suggestion] [arXiv: 1705.09300]

2. F. Lu, B. Shi, Y.-M. Lu, Classiﬁcation and surface anomaly of glide symmetry protected topological phases in three dimensions, New J. Phys. 19, 073002 (2017) [arXiv: 1701.00784]

1. B. Shi and S. Raby, Basis invariant descriptions of chemical equilibrium with implications for a recent axionic leptogenesis model, Phys. Rev. D 92, 085008 (2015) [arXiv: 1507.08392]

Fields of Study

Major Field: Department of Physics

vii Table of Contents

Page

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... vii

List of Tables ...... xi

List of Figures ...... xii

1. Introduction ...... 1

1.1 Organization of the thesis: a road map and how to read ...... 3 1.2 Physics of quantum many-body systems ...... 6 1.2.1 Quantum many-body systems ...... 6 1.2.2 Quantum Phases: its universal properties and classiﬁcations ...... 8 1.2.3 Emergent laws ...... 9 1.3 Anyon theory, 2D gapped phases, and entanglement ...... 9 1.3.1 Algebraic theory of anyons and chiral central charge ...... 11 1.3.2 Anyon theory and quantum entanglement ...... 14

2. Entangled quantum states: background ...... 17

2.1 Quantum entanglement ...... 17 2.2 Quantum state and convex set ...... 20 2.3 Quantifying the distance between quantum states ...... 22 2.4 Properties of the von Neumann entropy ...... 24 2.5 Quantum Markov states: ...... 25

viii 3. Formulating quantum states with an area law ...... 28

3.1 Setup and axioms ...... 29 3.2 Renormalization group ﬁxed point ...... 32 3.3 Anyon data are the “order parameters” of topological orders ...... 36

4. Information convex set and isomorphism theorem ...... 39

4.1 Information convex set ...... 40 4.2 Elementary steps and Isomorphism Theorem ...... 44 4.3 Merging of elements in the information convex sets ...... 54 4.3.1 An alternative formulation of the information convex set ...... 54 4.3.2 Merging in the alternative formulation ...... 55 4.3.3 Equivalence of the deﬁnitions ...... 57

5. Fusion rules from entanglement ...... 59

5.1 Superselection sectors ...... 60 5.2 Fusion rules and fusion spaces ...... 65 5.3 Derivation of the axioms of the fusion rules ...... 68 5.4 Extreme points (details) ...... 78 5.4.1 Implication of the orthogonality ...... 82 5.5 Fusion space (details) ...... 83

6. Topological entanglement entropy ...... 87

6.1 The derivation of TEE ...... 88 6.2 Implications ...... 90

7. String operators and circuit depth ...... 93

7.1 Heuristic discussions ...... 93 7.2 String operators from entanglement ...... 96 7.3 Circuit depth of the string operators ...... 97

8. Verlinde formula from entanglement ...... 101

8.1 Our deﬁnition of the S-matrix ...... 102 8.2 The proof of the Verlinde formula ...... 104

9. Discrete version of the framework ...... 109

9.1 Setup and axioms on the lattice ...... 110 9.2 Reference state properties ...... 111 9.3 The information convex set ...... 112 9.4 The isomorphism theorem ...... 113

ix 9.5 Structure theorems ...... 117 9.6 Merging with a change of topology ...... 117 9.7 Topological entanglement entropy ...... 119

10. Information convex set and Hamiltonians ...... 121

10.1 Information convex set for frustration-free Hamiltonian ...... 122 10.2 Equivalence between Σ(Ω σ) and Σ(Ω H) under conditions ...... 125 10.3 Explicit calculation for quantum| double| models ...... 126

11. On the generality of area law: an RG point of view ...... 132

11.1 A conjecture ...... 133 11.2 Testing the conjecture with known examples ...... 133

Bibliography ...... 136

x List of Tables

Table Page

5.1 Physical data that can be extracted from disks with diﬀerent number of holes. . 60

5.2 A partition of B used in the proof of Theorem 5.5...... 84

xi List of Figures

Figure Page

1.1 A road map to the thesis is illustrated. The boxes labeled by numbers are the chapters. The arrows indicate the relationships between these chapters...... 5

2.1 Examples of convex sets: (a) A simplex; (b) A solid ball...... 21

3.1 The reference state σ of a 2D quantum many-body system. Some of the µ-disks are shown. The zoomed-in depiction of µ-disk b with partition BC (BCD) relevant to Axiom A0 (A1). Three relevant length scales , r and ∆ are illustrated. All of them are larger than the correlation length and they can be much smaller than the system size...... 30

3.2 An illustration of the growth procedure of a disk from AB to ABC. Here A can be large and BCD is contained in a µ-disk in a manner similar to Fig. 3.1. . . . 33

3.3 The extension of the axioms. A disk is divided into either BC or BCD.A µ-disk is on a smaller length scale, i.e., the small dashed circle surrounding the colored region. These ﬁgures represent three ways of enlarging C by a small step. (a) bc B and d C; (b) bc B and d C; (c) bc B, d C and d0 D..... 34 ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ 3.4 Subsystem E is a square of length scale ∆. F is an annulus around it. The thickness of F is at least ...... 38

4.1 This ﬁgure is a schematic depiction of regions involved in the deﬁnition of information convex set Σ(Ω σ). Here Ω is the annulus between the black circles. Annulus Ω0 Ω is a thickening| of Ω. It is the region between the gray circles. ⊇ Any element in Σ(Ω˜ 0 σ) is consistent with the reference state σ on every µ-disk b contained in Ω0. We| choose Ω to be an annulus for illustration purposes. Other topologies are allowed as well...... 41

4.2 A µ-disk can grow until it covers a larger disk ω...... 43

xii 4.3 Smooth deformations of subsystems (ABC ABCD). The subsystem labels are chosen to make the (later) discussion of↔ merging convenient. (a) A smooth deformation of a disk. (b) A smooth deformation of an annulus. The annulus topological is chosen for illustration purposes. Other topologies are allowed as well. 45

4.4 A schematic depiction of the merging process (Lemma 4.6). A pair Markov states a x ax ρABC and λBCD are merged into yet another quantum Markov state τABCD.... 47

4.5 A schematic depiction of subsystem ABCD. The partition B0C0 = BC is chosen such that no µ-disk overlaps with both AB0 and CD. Note that, the subsystems A, B, C, D are allowed to take a variety of topologies...... 49

4.6 The detailed partition relevant to the elementary step of deformation Ω Ω0 ↔ is illustrated, where Ω = ABC and Ω0 = ABCD. BCD is a disk, and CD is contained in a µ-disk. Only a part of A is shown. The topology of A can be arbitrary. B0C0 = BC. A and D are separated by at least 2r + , so that on µ-disk overlaps with both AB0 and CD...... 51

4.7 A partition of the subsystem Ω for deﬁning Σ(Ω);ˆ see the second condition of Deﬁnition 4.3. Let Ω = ABC where BC is a subsystem contained in a µ-disk. The horizontal line is the boundary of Ω. Only part of A is shown for illustration purposes...... 55

4.8 A partition of the subsystem Ω for deﬁning Σ(Ω);ˆ see the third condition of Deﬁnition 4.3. Let Ω ABC where BC is a subsystem contained in a µ-disk. ⊇ The horizontal line is the boundary of Ω. Only part of A is shown for illustration purposes...... 55

5.1 (a) Division of an annulus X into three thinner annuli L, M, R. (b) A path (extensions-extensions-restrictions-restrictions) which generates an isomorphism Σ(L) ∼= Σ(X) ∼= Σ(R). (c) A schematic depiction of the simplex structure of Σ(X). The extreme points are the “corners” of the simplex. If the annulus X is contained in a disk, then one of the extreme points has the vacuum label “1”. . 61

5.2 A pair of annuli X0 and X1 on a torus. They cannot be connected by any smooth path because X0 is contractible and X1 is non-contractible...... 63

5.3 Both C and BC are disks. B is an annulus. X0, X1, X˜ 0 and X˜ 1 are annuli. Note that X0 and X1 are subsets of C. In the proof of Lemma 5.3, we construct an extension Xt X˜ t with B X˜ t...... 64 → ⊆ 0 5.4 A 2-hole disk Y = BY , with B = B1B2B3. B1, B2 and B3 are three annuli surrounding the three boundaries of Y . YD is a disk, where D = D1D2. D1 and D2 are the two disks surrounded by annuli B1 and B2...... 67

xiii 5.5 A hole with the vacuum charge can be merged with a disk. The case shown in this diagram involves an annulus and a disk, but the idea works for any n-hole disk with n 1. The left side shows the topology of the subsystems before they are merged.≥ Also, the number “1” is the vacuum sector. The merged subsystem is shown on the right. The three concentric lines partition the disk into the four subsystems used in the merging process...... 70

5.6 Merging two annuli and obtain a 2-hole disk. On the right side, there are two thin disk-like regions in the middle, which are chosen to be the B and C subsystem in the merging lemma(Lemma 4.6)...... 72

5.7 Merging a pair of annuli to obtain a 2-hole disk. We ﬁrst deform the annulus associated to 1 so that it becomes “longer” vertically. Then, the annulus associated to a is merged into the interior of this deformed annulus. The two thin U-shaped disk-like regions are chosen to be the subsystem B and C in the merging lemma(Lemma 4.6)...... 74

5.8 Merging a pair of 2-hole disks to obtain a 4-hole disk...... 75

¯ 5.9 The fusion of (a, b) and (b, a¯), and matching the fusion probabilities P(a×b→c) and P(¯b×a¯→c¯)...... 75

5.10 Merging a pair of 2-hole disks to obtain a 3-hole disk. Here Z = Y Y = Y Y . l ∪ R L ∪ r Here a, b, c, d, i, j are labels of the superselection sectors...... 77

5.11 Merging three annuli to obtain a 3-hole disk...... 78

5.12 Here Ω = ABC is an arbitrary subsystem with a boundary. B and C are concentric annuli covering the boundary...... 80

5.13 A partition of B used in the proof of Theorem 5.5. This ﬁgure does not represent the actual underlying geometry. Rather, it represents the relative distance be- 0 0 tween the “inner” part of Y (i.e., Y ) and the annuli surrounding Y (i.e., BR,BM , and BL). Auxiliary system E is introduced to purify the extreme points. Here BR 0 is the “innermost” part of B that is directly in contact with Y , BM is a disjoint union of annuli surrounding BR, and BL surrounds BM ...... 84

6.1 (a) The Kitaev-Preskill partition; (b) the Levin-Wen partition...... 88

6.2 Merging a pair of disks to obtain an annulus. Two disks are deformed so that, once merged together, they form an annulus...... 89

xiv 7.1 Disk ω is the union of 2-hole disk Y and its two holes. W is the complement of ω. The topological charges a anda ¯ within the two holes are created by unitary string operator U (a,a¯). The support of U (a,a¯) is the union of the deformable gray area and the two holes shown in red...... 96

7.2 (a) Anyon pairs (a, a¯) and (b, ¯b) are independently created on the reference state. (b) The annulus X = LMR detects the fusion result of a and b. L, M and R are concentric annuli and L and R are separated by distance l...... 98

a 8.1 An annulus X and string operators supported within it. (a) String operator UR which creates a pair of anyons a anda ¯ on the reference state. (b) String operator a a UL is obtained by deforming UR on the reference state...... 103

8.2 Two distinct ways to create four sectors: (a) with UR and VL, (b) with UL and VR. Here UL ψ = UR ψ and VL ψ = VR ψ . Depending on the context of the discussion, an| operatori | mayi either| correspondsi | i to a string carrying a ﬁxed sector or a string bundle...... 104

8.3 (a) A single string. (b) A string bundle. In this particular ﬁgure, the string bundle consists of two strings...... 105

1 8.4 (a) The merging of σABC and σCD, where C = C1C2. ABCD is not a subsystem of the original system, and it has a topology equivalent to a torus with one hole. a (b) The unitary string operator UR is supported on BC. (c) The unitary string a a operator UL is supported on CD. It is obtained from the deformation of UR... 107

10.1 This ﬁgure is a schematic depiction of regions involved in the deﬁnition of information convex set Σ(Ω, Ω0 H). Here Ω is the annulus between the black circles. Annulus Ω0 Ω is a thickening| of Ω. It is the region between the gray circles. ⊇ The small disk-like region is the support of a term hi of the Hamiltonian HΩ0 . We choose Ω to be an annulus for illustration purposes. Other topologies are allowed as well...... 124

10.2 The quantum double model on a square lattice. Notation: v is a vertex, e is an edge and f is a face. When we ﬂip the arrow on an edge, the local Hilbert space has a basis change g g¯ , whereg ¯ is the inverse of g...... 127 | i → | i 10.3 The minimal diagrams for three topology types: a disk, an annulus and a 2-hole disk. We have ﬁxed the group elements for the links on the boundary of the subsystems...... 129

11.1 With a topological defect: A µ-disk is divided into BCD. An isolated topological defect x is in C...... 134

xv Chapter 1: Introduction

In many ways, the law of quantum mechanics is well-established by now. Does this mean that one can always understand how a quantum system work by making derivations from the law of quantum mechanics? The answer to this question is complicated.

People could solve many important physical problems by making derivation from the principle

(or axioms) of the quantum mechanics, either find exact solutions or make reasonable approximations as perturbation theory does. Nevertheless, there are (well-defined) physical problems that are either too hard to be solved in a reasonable amount of time or that an effective approach to solving the problem is unknown. A prominent class of such physical problems comes from the study of interacting quantum many-body systems.

Generally speaking, a quantum many-body system is a physical system that consists of many particles. In the absence of interaction, a lot of many-body systems have been solved. This has led to the explanations of many important physical phenomena, e.g., the conductivity of metals, integer quantum Hall eﬀect, etc. In the presence of interaction, however, the problems are, in general, very diﬃcult to solve analytically. Up to now, a classical computer can solve the energy spectrum of an interacting system only for a system size up to 20-30 qubits (e.g., spins or electrons). The computational cost grows exponentially with the system size.

Despite the diﬃculty, interacting quantum many-body systems attract a huge among of studies (both theoretical and experimental). This is because interacting quantum many-body systems are responsible for a large class of exotic phenomena. Given the lack of ability to solve these problems by applying the axioms of quantum mechanics, people usually have to

1 make nontrivial assumptions and discuss eﬀective theories of such systems. These eﬀective theories are sometimes referred to as emergent physical laws. Examples of emergent laws include quantized values of critical exponents of second-order phase transitions and the algebraic theory of anyon [1].

In some circumstances, exactly solvable models are available to particular phases. Solving these models is useful to collect concrete intuition for nontrivial physical properties, and to show the existence of nontrivial phases. However, exactly solved models known so far only cover a subset of interesting phases. It is not clear if every interesting quantum phase has an exactly solvable model.

A different approach to understanding interacting many-body systems is to find a reasonably general physical assumption, from which nontrivial conclusions can be deduced logically. Such a physical assumption should be a criterion that is insensitive to the detailed constitution of the system and arguably holds for a large class of models. If such a physical assumption is identified, a physical understanding is gained on why a property holds for a variety of models differing in their details.

Physical assumptions that can describe a large class of nontrivial physical phenomena in interacting many-body systems are rare. One example is the entanglement area law in one dimensional (1D) gapped systems [2]. Another example is the conformal symmetry in 1+1D critical systems, see the rigorous formulation in the framework of conformal ﬁeld theory (CFT)

[3]. Enormous constraints of the theory are derived from the assumption of conformal theory and unitarity.1 It can give strong constraints to the values of critical exponents in second-order phase transitions.

The main topic of the thesis is a theoretical framework that attempts to derive a general theory of anyons in 2-dimensional (2D) gapped phases. Anyons [4, 5] are particles with exotic mutual braiding statistics. They are neither fermions nor bosons, and they are not an elementary

1Note that non-unitary CFTs exist and they have physical relevance.

2 excitation of the vacuum of our universe. Nonetheless, anyons can emerge in interacting 2D gapped systems as quasiparticles. Is there a general physical mechanism for anyons to emerge?

The underlying physical assumption we identify are two local conditions on the entanglement entropy (see axiom A0 and A1 in Chapter 3). This axiom set is a local and compact version of the well-known (conjectured) form of 2D entanglement area law [6, 7]. With these assumptions, we derive the fusion rules of anyons, recover the dependence of the value of the constant term in the area law (i.e., identify the value of the topological entanglement entropy), and also begin to uncover the nontrivial braiding properties. The results derived from this framework matches that of the algebraic theory of anyon, summarized in Appendix E of Ref. [1].

1.1 Organization of the thesis: a road map and how to read

In this section, we explain the organization of the thesis. A road map of the thesis is presented in Fig. 1.1.

Chapter 1 and 2 are some background. Chapter 1 provides a review of quantum many- body physics. Attention is paid to 2-dimensional gapped phases that host anyons. It provides relevant background and the motivation for the research presented in this thesis. Chapter 2 provides a review of the relevant tools of quantum information theory. These tools are necessary to understand our derivations. In fact, the thesis is (almost) self-contained. No knowledge beyond the background presented in Chapter 2 is necessary to understand our framework.

The main content of the thesis is a theoretical framework presented in the following chapters:

Chapter 3 introduces the setup and basic assumptions (axiom A0 and A1) of the theo- • retical framework. These axioms correspond to a compact version of entanglement area

law. The results presented in chapter 4, 5, 6, 7 and 8 are built upon these axioms. These

chapters are the main content of the thesis.

3 Chapter 4 introduces the concept of information convex set. We prove that the structure • of the information convex set is invariant (isomorphic) under the smooth deformation of

the subsystem.

Chapter 5 describes the structure theorems. The self-consistency relation of the informa- • tion convex sets is derived by merging diﬀerent subsystems to induce a topology change.

We deﬁne the superselection sectors and derive their fusion rules following this line of logic.

Chapter 6 contains a derivation of topological entanglement entropy γ = ln . • D

Chapter 7 discusses the unitary string operators in our framework. The circuit depth of • the string operator is studied.

Chapter 8 show that a unitary topological S-matrix can be deﬁned in our framework. The • Verlinde formula, which relates the topological S matrix and the fusion multiplicities, is

derived.

Note that the only input of this theoretical framework is a reference state σ satisfying the axioms.

This approach is completely Hamiltonian independent.

The last three chapters (9, 10 and 11) are not part of the theoretical framework. Nevertheless, the content is closely related. Chapter 9 presents a discrete version of the framework put on a lattice. It can be thought of as a concrete alternative formulation of the framework.

Chapter 10 discusses another deﬁnition of the information convex set. This deﬁnition is based on a frustration-free Hamiltonian. Under certain conditions, the Hamiltonian formulation of the information convex set is equivalent to the one based on a quantum state. Chapter 11 present some thought on the generality of the area law assumption.

4 (1) (2)

Quantum Many-body Physics Quantum Information Theory

(3) (9) (11) Axiom A0: A1: Discrete Version: Conjecture: A1 Reference State σ RG Fixed Point ≈ RG Fixed Point is Stable (4) Information Convex Set Ω Σ(Ω σ) | (10) Isomorphism Theorem Hamiltonian-based P H = i hi (5) Σ(Ω, Ω0 H) | Σ(X) = Under conditions: Σ(Ω σ) = Σ(Ω H) | | Structure Theorems Merging and Fusion (6) (7) Unitary String Operator a a¯

TEE: γ = ln Circuit Depth D (8)

Mutual Braiding

Figure 1.1: A road map to the thesis is illustrated. The boxes labeled by numbers are the chapters. The arrows indicate the relationships between these chapters.

5 Comments on the content and writing style: This thesis is based on my published works [8, 9] and part of [10, 11]. A few unpublished contents are included as well. I choose to write the thesis in such a way that a second or third-year graduate student, who is interested in this research direction, may beneﬁt from this thesis. Physical intuition has the priority, and for this reason, some content of the original papers are addressed in an alternative way. For some proof, only a sketch is presented. Some other proofs are omitted.

Some contents related to [10, 11] are addressed in a diﬀerent way, because some statements are now known to hold with higher generality. In other words, the results hold under simpler assumptions. Proposition 7.3 and Theorem 10.1 are added for this propose. The details of the discrete version (Chapter 9) has not been presented elsewhere. However, it is a straightforward translation of the abstract version of the framework onto discrete lattices. The discrete lattice is similar to Ref. [12] and [13]; see also the discussion around Fig. 1 of [8]. Conjecture 11.1 is new.

1.2 Physics of quantum many-body systems

1.2.1 Quantum many-body systems

We review relevant setup in the study of interacting quantum many-body systems. Lattice models are frequently considered in quantum many-body problems. Local degrees of freedom are assigned to positions localized in space. Nearby degrees of freedom can interact.

Most theoretical many-body physics studies start with a speciﬁc form of Hamiltonian. This approach had lead to fruitful development ever since the early age of sold state physics. The key concept, the energy gap, is deﬁned in terms of a Hamiltonian. Quantum many-body wave function, on the other hand, is directly linked to the correlation of the system. Therefore, it is useful to determine whether a quantum phase has a certain order. This approach plays an increasingly important role in recent researches. Quantum many-body wave function will play

6 a central rule in the main topic of the thesis because our theoretical framework is Hamiltonian independent, and the only input is a quantum state.

Hamiltonians: Because the atoms in our 3+1 dimensional universal are either fermions or bosons, a Hamiltonian relevant to the real world (eﬀective Hamiltonian not included) are built out of fermions, bosons or spins.

By fermionic local degrees of freedom, we mean fermionic creation/annihilation operators • with f , f = f †, f † = 0, f , f † = δ and the Hilbert space structure associated { j k} { j k } { j k } j,k with it. Note that the fermionic creation and annihilation operators can be rewritten as

Majorana fermion operators.

By bosonic local degrees of freedom, we mean bosonic creation/annihilation operators with • † † † [bj, bk] = [bj, bk] = 0, [bj, bk] = δj,k and the Hilbert space structure associated with it.

By spin local degrees of freedom, we mean a model with ﬁnite dimensional local Hilbert • space dimension, dim = d; e.g., d = 2 is refereed to as spin- 1 and it has a qubit on Hj 2 each site. For d = 2, the operator algebra on each site is generated by a subset of Pauli

operators, X ,Z . The generation to d 2 is well-understood. { j j} ≥

A few remarks are in order. If the Hamiltonian is well-deﬁned, the physical problem, e.g., the energy spectrum, can, in principle, be solved. However, interacting systems are, in general, hard to solve. The diﬃculty grows exponentially with the system size. An interacting quantum many-body system can have emergent particles. Anyons are not excitations of the vacuum of our 3+1 D universe. However, they can emerge as collective excitations (quasi-particles) of a quantum many-body system.

Quantum many-body wave function: A quantum many-body wave function is an important object to study for multiple reasons. It may reveal the universal properties of a quantum phase. Therefore, it may help with the classiﬁcation of quantum phases. Given a many-body

7 wave function, we can calculate all kinds of correlation functions. In addition, we can consider the entanglement between the subsystems.

The number of parameters to specify either a many-body Hamiltonian or a many-body wave function grows fast with the system size. Moreover, even if we know all these parameters, it is generically diﬃcult to compute a physical quantity from them. To study universal physical properties, people usually have to start with some extra assumptions and make approximations.

(Occasionally, later developments can prove/disprove assumptions used in previous works. These prove/disprove usually built upon more fundamental assumptions or assumptions from a newly developed research ﬁeld.) This situation further motivates several broad viewpoints: quantum phases, universal properties, and emergent physical laws.

1.2.2 Quantum Phases: its universal properties and classiﬁcations

We are interested in the universal properties and the classiﬁcation of quantum many-body systems. Below are some concepts used in modern literature.

Energy gap and the definition of quantum phase: A prominent property of a quantum system in the thermodynamic limit is the presence/absence of an energy gap. The quantum system is gapped if the energy difference between the ground state and the first excited state remains finite in the thermodynamic limit. (A finite number of degenerate ground states are allowed.)

If two gapped Hamiltonians can be tuned to each other without closing an energy gap, they belong to the same phase. It is expected that universal properties are the same for models within the same quantum phase, whereas diﬀerent universal properties could distinguish diﬀerent quantum phases.

Universal properties and classiﬁcation: One important motivation to study and classify quantum phases is identifying and understand (potentially useful) physical properties. There are universal properties of phases that are insensitive to the details of the materials. For example,

8 superconductors can happen at a variety of materials; the quantization of Hall conductance is extremely accurate [14, 15], and the quantization value has nothing to do with the details of the material. The classiﬁcation of quantum phases and phase transitions is an important problem of modern theoretical physics. The focus of this thesis is on gapped phases.

1.2.3 Emergent laws

Some of the phenomena that happen in a many-body system can hardly be inferred from the property of the system on a small scale. These phenomena are emergent phenomena. The laws that describe these phenomena are sometimes called emergent laws. This idea is not limited to physics. In the context of many-body physics, a famous reference on this topic is by Anderson’s

“More is diﬀerent” [16]. See also the introduction of a book by Xiao-Gang Wen [17]. Laws that describe a system with a large number of particles may not be naively guessed from the laws obeyed by the constituent particles. Sometimes, it is further argued that the emergent laws cannot be deduced from the fundamental law of quantum mechanics. However, as we shall see, with some plausible assumptions, emergent laws can sometimes be derived. Here are some examples of emergent phenomena (not limited to physics):

consciousness (biology and neuroscience, poorly understood theoretically) •

spontaneous symmetry breaking (many-body physics) •

the second law of thermodynamics (derived in statistical mechanics) •

anyons and the physical law they obey (a general mechanism is explored in this thesis) •

entanglement area law in gapped quantum phases (conjectured to be generic) • 1.3 Anyon theory, 2D gapped phases, and entanglement

The emergence of anyon theory in gapped phases is the focus of this thesis. In this section, we brieﬂy review several phenomena related to 2D gapped phases. Focus is given to those

9 phenomena related to the anyon theory: the algebraic theory of anyon (sometimes called the unitary modular category (UMTC)) and the chiral central charge c−. Some recent ideas on the connection between the anyon theory to the entanglement of the many-body ground states are discussed.

Anyons [4, 5] are particles with exotic mutual braiding statistics. They are neither fermions nor bosons, and they are not an elementary excitation of the vacuum of our universe. Nonethe- less, anyons can emerge in interacting 2D gapped systems. These gapped phases possess a new type of order (topological order [18]). Topological order manifests on the ground state degeneracy. The degeneracy depends on the topology of the 2D manifold. Ever since the proposal for the concept of anyon, quite a few insightful theoretical works have been done in this direction.

Anyons are identiﬁed to exist in fractional quantum Hall (FQH) wave functions [19, 20]. The conservation of charge and many-body interactions in FQH states further make those anyons carrying a fraction of charge. However, the emergence of anyons does not have to invoke any symmetries. Anyons exist in topologically ordered systems and can cause the degeneracy on a nontrivial manifold (see Ref. [21] for a review); anyons can be non-Abelian [22]. Local indistinguishable states of anyons can prevent the system from undergoing decoherence. They are useful to perform fault-tolerant topological quantum computation [23]. Anyons, their fusion, braiding properties, and topological degeneracy are robust under any local perturbations.

Despite these developments, the underlying physical mechanism for anyons to emerge in 2D gapped phases is still an outstanding open problem. The algebraic theory of anyon, (summarized in Appendix E of [1]), is proposed as the most general anyon theory for gapped 2D systems without symmetry. While it is a well-known proposal, it has not been derived logically from plausible physical assumptions on 2D gapped phases. (The algebraic theory of anyon has axioms, but these axioms are a description of the anyon’s law rather than showing how the law emerges.)

This proposal is inspired by the framework of TQFT [24]. Therefore, one may argue that TQFT provides a justiﬁcation for this framework. However, questions remain on when and how the

10 gauge ﬁelds in TQFT emerge in 2D gapped phases, which has no symmetry requirement at the microscopic level.

This provides the motivation for the line of research discussed in this thesis. The goal of the thesis is to provide an answer to when and why we should expect the anyon theory to emerge in the 2D gapped phase. The underlying mechanics we identify is a version of entanglement area law.

1.3.1 Algebraic theory of anyons and chiral central charge

The algebraic theory of anyons (Appendix E of [1]) is proposed as a general framework to describe the fusion and braiding rules obeyed by anyons in 2D gapped phases without symmetries. (Fermionic models are excluded. This is because any local fermionic model preserves an extra Z2 symmetry generated by the fermionic parity.) It also speciﬁes the set of universal data of the anyon theory. These anyon data are relevant to the classiﬁcation of 2D gapped phases.

The algebraic theory of anyon has based on the mathematical framework of the unitary modular tensor category (UMTC). See [25] for a mathematical reference on tensor categories.

The framework of UMTC is abstract and intricate. Many data and consistency relations are needed to specify those rules. Interpreting these data and rules in many-body quantum systems is often hard for beginners. We encourage the readers to look into some exactly solvable models to collect concrete physical intuitions [23, 26]. Exactly solvable models with non-Abelian models are much harder than those with only Abelian anyons. However, one should carefully distinguish (universal) physical properties and (non-universal) accidental properties in exactly solvable Hamiltonians. Moreover, the theoretical framework presented in this thesis provides a Hamiltonian-independent way to think about these abstract data in quantum many-body systems. Here, we only give a very brief review of related data, which frequently being considered in modern literature. Then we present a brief discussion of these data.

superselection sectors = 1, a, b, • C { ···}

11 fusion rules a b = P N c c • × c ab

topological S-matrix S • { ab}

topological spins θ • { a}

F -symbols. •

R-symbols. •

Supersection sectors are particle types. Each particles can occupy a space of radius within a few times the correlation length. The sector 1 is the (unique) vacuum sector. Each particle a ∈ C has a unique antiparticlea ¯. The coeﬃcients N c are non-negative integers, and they satisfy a { ab} set of conditions: c c Nab = Nba

c Na1 = δa,c

1 = Nab δb,a¯ (1.1)

c c¯ Nab = N¯ba¯

X i d X d j NabNic = NajNbc. i j The set of quantum dimensions d can be deﬁned as the unique positive solution of d d = { a} a b P c c Nabdc. An anyon is non-Abelian if and only if da > 1. This happens precisely when the fusion outcome of a anda ¯ contain sectors other than the vacuum sector. The total quantum dimension of the theory is = pP d2. D a a

The topological spins can be grouped into the so-called T -matrix, where Tab = θaδa,b. S and

T matrices describe the mutual/self statistics of anyons; these data can uniquely determine a

UMTC if the total quantum dimension of the theory is small. More generally, there can be multiple inequivalent anyon theories with the same S and T -matrices [27]. The F -symbols and

R-symbols can completely determine a UMTC. However, they have a gauge dependence. In

12 other words, not every component of F and R-symbols are physical. For UMTC, the S-matrix is required to be unitary. This corresponds to the statement that the braiding is non-degenerate.

The chiral central charge c− is not included in UMTC. However, it is a physical quantity relevant to 2D gapped many-body systems with anyons. If c− is nonzero, we say the gapped phase is chiral. A chiral phase always has a gapless edge. The edge cannot be gapped out by adding perturbations to the edge. The energy current2 of the edge at (low) temperature [1] is given by π I = c T 2. (1.2) 12 − Here T is the temperature, and it is much smaller than the energy gap of the bulk. This non- vanishing energy current cannot exist for any 1D systems in thermal equilibrium [28]. Eq. (1.2) can be taken as the physical deﬁnition of c−. The following relation between UMTC data and the chiral central charge

1 X 2 2πic /8 daθa = e − (1.3) D a is expected if we assume certain chiral CFT description of the gapless edge. It is an open question whether the same condition holds under weaker assumptions.

Moreover, it is a well-known conjecture that the set of data (UMTC, c−) classiﬁes all the 2D gapped phases without symmetries.

Examples: For readers’ convenience, we present two examples of anyon data together with the chiral central charge. One is known as the toric code model, and the other is known as the

Ising anyon model. (For simplicity, F and R symbols are omitted. However, they can be found in Ref. [29], which also contains many more examples.)

The anyon data and chiral central charge for the toric code model:

Superselection sectors: = 1, e, m, . • C { }

Fusion rules: e e = m m = = 1, e m = , e = m, m = e. • × × × × × × 2Energy current can be compared to electric current. The former can be deﬁned for a system without symmetries whereas the latter is deﬁned for a system with U(1) charge conservation symmetry.

13 Quantum dimensions: d , d , d , d = 1, 1, 1, 1 . • { 1 e m } { }

Total quantum dimension: = 2. • D

Topological S-matrix: •  1 1 1 1  1  1 1 1 1  S =  − −  . (1.4) 2  1 1 1 1  1 −1 1− 1 − − Topological spins: θ , θ , θ , θ = 1, 1, 1, 1 . • { 1 e m } { − }

Chiral central charge: c− = 0. •

The anyon data and chiral central charge for the Ising anyon model:

Superselection sectors: = 1, σ, ψ . • C { }

Fusion rules: σ σ = 1 + ψ, σ ψ = σ, ψ ψ = 1. • × × ×

Quantum dimensions: d , d , d = 1, √2, 1 . • { 1 σ ψ} { }

Total quantum dimension: = 2. • D

Topological S-matrix: •  1 √2 1  1 S =  √2 0 √2  . (1.5) 2 − 1 √2 1 − i π Topological spins: θ , θ , θ = 1, e 8 , 1 . • { 1 σ ψ} { − }

1 Chiral central charge: c− = . • 2 1.3.2 Anyon theory and quantum entanglement

We summarize some recent ideas on the connection between the anyon theory and quantum entanglement. The ground states of gapped systems are expected to obey entanglement area law for subsystems at a length scale larger than the correlation length. The statement is that

14 the leading contribution to the von Neumann entropy, on a subsystem, is proportional to the length of the boundary of a subsystem. In 2D, the area law has a conjectured form [6, 7]

S(A) = α` γ, (1.6) − where γ is the same constant for any subsystem A of a disk topology. ` is the length of the boundary of A. The sub-leading correction, which vanishes in the ` limit, is suppressed → ∞ here. The coeﬃcient α is not universal; it depends on how many local degrees of freedom are located within a correlation length around the boundary of A. The constant piece γ is argued to be universal, and it is the topological entanglement entropy. It has been shown under diﬀerent assumptions that

γ = ln . (1.7) D The original paper [6] uses TQFT assumptions [24], whereas Ref. [7] calculated this result for the string-net models [26].

Note that, Eq. (1.6) and Eq. (1.7) is not sensitive to chiral central charge. They are expected to be generic for both chiral and nonchiral systems. One result presented in this thesis shows that Eq. (1.7) is implied by (a simpler version of) the area law formula Eq. (1.6) alone. (See

Chapter 6 and the original paper [8].) This brings Eq. (1.7) to a new level of generality.

It is worth mention that the 2D area law Eq. (1.6) can break. The sense of generality may need an extra condition to establish. This is a rather subtle point, and it is a topic under debate recently. Carefully study is needed in this direction. See Chapter 11 for a discussion.

Fruitful connections between quantum phases and entanglement go into diﬀerent aspects.

Here we mention some additional works. A 2D gapped phase that possesses anyons must be long-range entangled [30], namely that it cannot be turned to a product state by applying a

ﬁnite depth circuit. Note that all gapless systems are long-range entangled by this deﬁnition.

What is interesting is that the ground state of gapped phases can be nontrivial. The idea of

15 minimal entangled states [31] allows us to detect anyon sectors by looking at its entanglement entropy on an annulus region.

The development mentioned above starts with a physical system with certain universal properties (e.g., the existence of nontrivial anyonic excitations) and study its entanglement properties.

We further mention a few works which aim to turn the table around, namely to take the entanglement property as a starting point and use it to derive some general statement of the anyon theory. Pioneering thought to this direction should be credited to Isaac Kim. Here we mention two early results below. Based on the assumption of Eqs. (1.6) and (1.7), it is shown that

The value γ gives an upper bound of the ground state degeneracy on torus [32]. •

Anyons cannot exist if γ = 0 [33]. •

The key idea behind these results is the quantum Markov state structure of the ground state

[34, 12, 35, 36, 37]. Surprisingly, many stronger results follow from an even simpler assumption.

This development took years. From [12, 35] to [8, 9]. (See also [37, 11]). I have participated in some of these exciting developments. In fact, these developments are the main topic of the thesis. In the next chapter, we introduce the relevant quantum information theory background to understand this development.

16 Chapter 2: Entangled quantum states: background

A key feature of a quantum system is that subsystems can entangle. In this chapter, we review some theoretical tools to understand entangled quantum states. Many of these tools are developed by the quantum information theory community. Most of the statements in this chapter can be found in books and lecture notes [38, 39]. See also review papers on entanglement [40, 41].

This modern knowledge of entanglement enables us to study the nontrivial form of entanglement in the ground states of gapped physical systems and investigate the emergent physical law of anyons. Moreover, we establish relevant notations that will be used throughout the thesis.

2.1 Quantum entanglement

Bipartite pure state: The entanglement of bipartite pure states is the simplest. It is also the most relevant form of entanglement to our research. Consider a system which is divided into two subsystems. We call the system AB and the two subsystems A and B. We assume that the

Hilbert space factorizes as

= , (2.1) HAB HA ⊗ HB where and are the Hilbert spaces of subsystem A and B. We assume and are HA HB HA HB ﬁnite dimensional. We say a pure quantum state φ is entangled (separable) if it | ABi ∈ HAB cannot (can) be written as

φ = φ φ (2.2) | ABi | Ai ⊗ | Bi for any (some) φ and φ . | Ai ∈ HA | Bi ∈ HB

17 In general, a system can be divided into multiple pieces. If a system is divided into four pieces ABCD, we can talk about bipartite entanglement between A and BCD (or between AB and CD). We use AB as a shorthand notation of the disjoint union of A and B (i.e., A B) t when A B = . ∩ ∅ To quantify pure state entanglement, we need the notion of density matrices and entanglement entropy. We shall use Greek letters, e.g., ρ, σ for density matrices. Subsystems are speciﬁed in the subscript, e.g. ρA, σB. A density matrix (ρ) is an operator satisfying the following conditions:

Hermiticity: ρ† = ρ. •

Positivity: α ρ α 0 for any α . • h | | i ≥ | i

Normalization: Trρ = 1. •

A density ρ is a pure state if ρ2 = ρ or equivalently, ρ = α α for some α . Otherwise, ρ is a | ih | | i mixed state. We will sometimes call a density matrix as a state for short.

The reduced density matrix of a state ρAB on subsystem A is deﬁned as

ρA = TrBρAB, (2.3) where Tr is the partial trace on subsystem B. If ρ = ϕ ϕ , we say ρ is the reduced B AB | ABih AB| A density matrix of ϕ on subsystem A. We will sometimes refer to the reduced density matrix | ABi of a state as its marginal.

Here is a simple example. Consider a 2-qubit system. It has = , where HAB HA ⊗ HB dim = dim = 2. The basis vectors of ( ) are denoted as 0 , 1 ( 0 , 1 ). HA HB HA HB {| Ai | Ai} {| Bi | Bi} Let

θ 1 iθ Ψ = ( 0A0B + e 1A1B ), (2.4) | ABi √2 | i | i where θ [0, 2π]. The state Ψθ is an entangled state. Observe that an entangled state ∈ | ABi cannot be determined by its reduced density matrices. For this particular example, the global

The entanglement measure for bipartite pure quantum states is well-understood [38, 40, 41].

The standard choice is the von Neumann entropy

S(ρ) Tr(ρ ln ρ). ≡ − Let ρ be the reduced density matrix of pure state ϕ . Then S(ρ ) measures the amount A | ABi A of entanglement between subsystem A and its complement B. The von Neumann entropy is positive (zero) for any entangled (separable) pure state.

Depending on the context, we shall use the following shorthand notations to denote the von

Neumann entropy of the reduced density matrix over some subsystem: SA,(SA + SB)ρ. In the

ﬁrst case, the global state should be obvious from the context. In the second case, the global state is ρ.

Unlike the entropy of a classical system, which cannot decrease with the volume, the von

Neumann entropy can have S(ρAB) < S(ρA).

Two information-theoretic quantities will play an important role in this thesis: I(A : B) S + S S , ≡ A B − AB I(A : C B) S + S S S . | ≡ AB BC − B − ABC The ﬁrst quantity, known as the mutual information between A and B, quantiﬁes a correlation between A and B. The second quantity, known as the conditional mutual information between

A and C conditioned on B, quantiﬁes the correlation between A and C given a knowledge on

B. By the strong subadditivity of entropy [42], I(A : C B) 0 for any quantum state. We will | ≥ have more to say about these quantities in Sec. 2.4.

Bipartite mixed states: For a mixed bipartite state ρAB, one may still calculate S(ρA), where ρA is the reduced density matrix. However, this quantity does not measure the entanglement between A and B. The most general form of unentangled bipartite mixed state is a

19 separable state. It is a state that can be written as

X ρ = p ρi ρi , (2.6) AB i A ⊗ B i where p is a probability distribution and ρi and ρi are density matrices. We say a bipartite { i} A B mixed state is entangled if it is not separable. How to quantify the entanglement of a bipartite mixed state? The answer is more subtle than the entanglement measure for bipartite pure states.

The completely answer to this question is not known [40, 41, 43]. On the other hand, several quantities are proposed to quantify the entanglement for bipartite mixed states, e.g. [44, 45, 46].

2.2 Quantum state and convex set

Convex set is a natural tool in the study of quantum states and quantum information. We explain the basic intuition and necessary notations. We start by summarizing the basic facts and terminology of convex analysis [47].

Convex set: We consider a subset of a ﬁnite-dimensional real space RN closed under convex

N combinations, where N Z≥0. The convex set is compact if it is a compact subset of R . For ∈ 2 our purpose, for an N dimensional Hilbert space, the real space R2N could be identiﬁed as the 2N 2 real components of an operator acting on the Hilbert space.

We use conv( ) to denote the convex hull of a set RN , which is the smallest convex set X X ⊆ that contains set . In other words, it is the set of all convex combinations of elements in . X X An extreme point of a convex set is a point in , which does not lie in any open line S S segment joining two points of . We use ext( ) to denote the set of extreme point of a convex S S set . Finally, we mention the Minkowski-Caratheodory theorem for compact convex sets. S

Theorem 2.1 (Minkowski-Caratheodory). Let be a compact convex subset of RN of dimension S n. Then any point in is a convex combination of at most n + 1 extreme points. S

20 Thus, the extreme points can determine the whole compact convex set. Note that, without compactness, an element of a convex set sometimes cannot be written as a convex combination of extreme points.

Connection to information theory: For our purpose, we will consider convex sets of density matrices. The most important choice throughout this thesis is the information convex set; (see the various definitions: Definition 4.1, Definition 4.3 and Definition 10.2). Here, with two simple examples, we explain why convex sets naturally appear in both classical and quantum information theory.

(a) (b)

Figure 2.1: Examples of convex sets: (a) A simplex; (b) A solid ball.

The set of classical (convex) combination of n orthonormal quantum states i n can {| i}i=1 be written as = ρ ρ = P p i i , where p is a probability distribution. It is a convex { | i i| ih |} { i} set isomorphic to an (n 1)-dimensional simplex; it has n extreme points at the “corners” of − the simplex. Because the states are orthonormal, we can use the direct sum to rewrite as

= ρ ρ = L p i i . Objects involved in a direct sum have mutually orthogonal supports. { | i i| ih |} The set of all density matrices on Hilbert space is the state space of . We denote it as H H ( ). The state space is compact convex set for any ﬁnite dim . As an example, the state S H H space for a 2-dimensional Hilbert space is a solid ball, known as the Bloch ball. The set of extreme points of this convex set is the set of pure states; the extreme points are continuously

21 parameterized, and they form the Bloch sphere. Any mixed state can be written as a convex combination of 2 extreme points, and this is in agreement with Theorem 2.1.

Intuitively, quantum coherence manifest in compact convex sets with continuously parameterized extreme point (e.g., the Bloch ball) whereas classical information manifests in compact convex sets with a ﬁnite number of isolated extreme points (e.g., a simplex). See Fig. 2.1 for an illustration.

2.3 Quantifying the distance between quantum states

Given two density matrices, we often need to find out whether they are similar or not. We discuss two useful quantities, one is the trace distance, and another is fidelity. Both of them are monotonic under the application of quantum channels. We introduce quantum channels first.

Quantum channel: Quantum channel, also known as complete-positive trace-preserving

(CPTP) map, is a general form of physical operation that can be applied to a quantum state.

It is a linear map from bounded operators on to bounded operators on . It preserves HA HA0 positivity, even in the presence of any ancillary system, i.e., the operation is completely positive.

It also preserves trace and hermiticity. In particular, it maps density matrices to density matrices.

It can be written in an explicit form using a set of Kraus operators M : { a}

X † → (X ) = M X M , (2.7) EA A0 A a A a a where P M †M = 1 and 1 is the identity operator on . a a a A A HA Here are a few examples. Partial trace ρ ρ = Tr ρ is a quantum channel. Unitary AB → A B AB rotation ρ UρU † is a quantum channel. On the other hand, a projective measurement with → selected outcome P ρP ρ (2.8) → Tr(P ρ) is in general not a quantum channel, where P is a Hermitian projector (P 2 = P and P = P †).

22 Distance measure and trace distance: In order to quantify the distance between two quantum states, people often consider distance measures. Let D( , ) be a distance measure. It · · takes two density matrices (on the same Hilbert space) as the input. It is required that

1. D(ρ, σ) = D(σ, ρ) for any ρ and σ.

2. D(ρ, ρ) = 0 for any ρ.

3. It does not increase under CPTP maps: D(ρ, σ) D( (ρ), (σ)). ≥ E E

Trace distance is a commonly used distance measure. It satisﬁes all the conditions above. Trace distance is deﬁned as p ρ σ Tr (ρ σ)2 k − k1 ≡ − for any pair of density matrices ρ and σ. It is a reasonable notion of distance becasue two states close in trace distance cannot be distinguished well by any measurement.

Fidelity: Another way to quantify the distance of two states is fidelity. Fidelity is not a distance measure, but it quantifies the overlap between two states. Fidelity is defined as

2 q 1 1 F (ρ, σ) = Tr ρ 2 σρ 2 . (2.9)

It is a natural generalization of the absolute value of inner product into mixed state. F (ρ, σ) =

ψ ϕ 2 when ρ = ψ ψ and σ = ϕ ϕ . It is symmetric F (ρ, σ) = F (σ, ρ) and F (ρ, σ) [0, 1]. |h | i| | ih | | ih | ∈ F (ρ, σ) = 1 if and only if ρ = σ and F (ρ, σ) = 0 if and only if ρ and σ are orthogonal (ρ σ). ⊥ Why do we care about ﬁdelity? It is because ﬁdelity enjoys several useful properties.

ﬁdelity is non-decreasing under quantum channels: •

F ( (ρ), (σ)) F (ρ, σ). (2.10) E E ≥

Since partial trace is also a quantum channel, we have •

F (ρ , σ ) F (ρ , σ ). (2.11) A A ≥ AB AB 23 ﬁdelity factorizes on tensor product states: •

F (ρ ρ , σ σ ) = F (ρ , σ ) F (ρ , σ ). (2.12) A ⊗ B A ⊗ B A A · B B

Lastly, if two quantum states have unit ﬁdelity, their puriﬁcations are identical up to a unitary operator acting on the purifying space [48]. In other words, two states ψ and ϕ | ABi | ABi have the same reduced density matrix on subsystem B if only if there is a unitary operator UA such that

ϕ = U I ψ . (2.13) | ABi A ⊗ B| ABi 2.4 Properties of the von Neumann entropy

Let us begin with a few elementary facts about entropy. First, SA = SB for an arbitrary pure state ϕ . Secondly, suppose a set of density matrices ρi has mutually orthogonal supports, | ABi { } i.e. ρi ρj, i = j, then ⊥ ∀ 6 X X S( p ρi) = p (S(ρi) ln p ), (2.14) i i − i i i where p is a probability distribution. { i} For a bipartite quantum state, we have the following set of well-known inequalities:

I(A : B)ρ 0 ≥ (2.15) S + S S 0. BC C − B ≥ The ﬁrst inequality is known as the subadditivity of entropy, and the second inequality is known as Araki-Lieb inequality [49]. It is interesting to study the conditions under which these inequalities are satisﬁed with equality. The mutual information is 0 if and only if the underlying state is a product state over A and B. The condition for saturating the Araki-Lieb inequality is more subtle and interesting. We will revisit this condition later after we discuss inequalities for tripartite quantum states.

The most important inequality involving a tripartite quantum state is the strong subadditivity (SSA) of entropy [42]:

I(A : C B) 0. (2.16) | ≥ 24 This inequality is surprisingly powerful in that inequalities that may look “stronger” than this inequality are in fact implied by SSA. Here is a (incomplete) list of such inequalities: I(AA0 : BB0) I(A : B) ≥ I(AA0 : CC0 B) I(A : C B) | ≥ | I(AA0 : CC0 B) I(A : C A0BC0) | ≥ | S + S S I(A : C) BC C − B ≥ S + S S I(A : C B) BC C − B ≥ | S + S S S + S S BC C − B ≥ BB0C C − BB0 S + S S S I(A : C B) BC CD − B − D ≥ | S + S S S S + S S S . BC CD − B − D ≥ BB0C CDD0 − BB0 − DD0 Also, let ρi be a set of density matrices and pi is a probability distribution, then { AB} { } X p (S S ) i (S S )P i . (2.17) i AB − B ρ ≤ AB − B i piρ i To see why, let us introduce an auxiliary system C with an orthonormal basis i . Let {| C i} ρ P p ρi i i and notice I(A : C B) 0. ABC ≡ i i AB ⊗ | C ih C | | ρ ≥ Finally, for an arbitrary quantum state, the conditional mutual information I(A : C B) does | not increase under a quantum channel acting only on either A or C.

2.5 Quantum Markov states:

If a tripartite quantum state satisﬁes SSA with equality, i.e. I(A : C B) = 0, such a state | ρ

(ρABC ) is referred to as a quantum Markov state [50, 34]. In this section, we list a few useful properties of quantum Markov states.

Lemma 2.2. Let ρABC and σABC be density matrices such that (1) ρAB = σAB and ρBC = σBC ;

(2) I(A : C B) = I(A : C B) = 0. Then ρ = σ . | ρ | σ ABC ABC

The proof follows from Ref. [51]. Moreover, there is a CPTP map which can recover σABC from its reduced density matrices. This map is known as the Petz recovery map.

25 Lemma 2.3. (Petz recovery map [51]) For any tripartite state ρ , I(A : C B) = 0 if and ABC | ρ only if

ρ = ρ (ρ ) , (2.18) ABC EB→BC AB where the quantum channel ρ (the Petz recovery map) has the following explicit expression EB→BC 1 − 1 − 1 1 ρ (X ) = ρ 2 ρ 2 X ρ 2 ρ 2 . EB→BC B BC B B B BC

Let ρ be a quantum Markov state such that it has a vanishing conditional mutual information

I(A : C B) = 0. Following Ref. [34], there is a decomposition of the Hilbert space into a | ρ HB L direct sum of tensor products B = L R such that H j Hbj ⊗ Hbj M ρABC = pj ρ L ρ R , (2.19) Abj ⊗ bj C j where pj is a probability distribution, ρ L is a density matrix on A L and ρ R is a { } Abj H ⊗ Hbj Abj density matrix on R C . Eq. (2.19) implies that Hbi ⊗ H X Tr ρ = p ρj ρj . (2.20) B ABC j A ⊗ C j Note that it is separable and therefore subsystem A and C have only classical correlations (no quantum correlation).

For later usage, we make an important connection between the states saturating the Araki-

Lieb inequality and quantum Markov states. The density matrices which saturate Araki-Lieb inequality have the properties summarized in Lemma 2.4.

Lemma 2.4. The following conditions about density matrix ρBC are equivalent.

(1) (S + S S ) = 0, (saturated Araki-Lieb). BC C − B ρ

(2) Any state ρ which reduces to ρ on BC has I(A : C) = 0 and I(A : C B) = 0. ABC BC ρ | ρ

(3) For any expression of the form ρ = P q ρi , where q is a probability distribution BC i i BC { i} with q > 0, i and ρi is a set of density matrices, we have i ∀ { BC }

ρ = Tr ρi , i. (2.21) C B BC ∀ 26 (4) Let ρ = P q i i , with q > 0, i and i j = δ , i, j, we have BC i i| BC ih BC | i ∀ h BC | BC i i,j ∀

Tr i j = δ ρ , i, j. (2.22) B | BC ih BC | i,j C ∀

27 Chapter 3: Formulating quantum states with an area law

“The near-term goal of this program is to derive a general theory of anyons from plausible assumptions which do not invoke any symmetries.”

Isaac H. Kim (01/22/2015 slides [35])

Interacting gapped systems in 2D possess interesting emergent phenomena. A prominent class of such phases is topologically ordered phases, which goes beyond the symmetry breaking classiﬁcation. Anyons emerge in topologically ordered phases. The anyon theory is expected to be described by the algebraic theory of anyons [1]. See Sec. 1.3 for a brief review. Prior works have shown that aspects of these phenomena are, in fact, encoded in the entanglement properties of the ground state [6, 7].

The main goal of the research presented in this thesis is to turn the table around and derive the emergent laws of anyon from a plausible assumption on entanglement entropy. A key step in developing such a theoretical framework is to carefully formulate the ground state of physical

2D gapped systems with basic assumptions (i.e., axioms). Although the basic intuition is that the ground state(s) of a large class of 2D gapped phases possess an entanglement area law, it requires both scientiﬁc instinct and artistic taste to guess the minimal choice of the axioms.

Moreover, it was a bold conjecture that these entanglement-based assumptions might lead to nontrivial conclusions.

In this chapter, we introduce the axioms of our theoretical framework. These axioms (A0 and A1 below) are entropic conditions on bounded-sized disks. Credit should be given to Isaac

28 H. Kim for the early contemplation of this choice [12, 35]. The full power of this axiom set is still at the stage of being explored. However, the majority of the results of this thesis are derived from these axioms. This demonstrates their power. Due to the simplicity of the axioms, we state the axioms before a discussion of various physical arguments that can support this choice.

This chapter is organized as follows. In Sec. 3.1, we summarize the setup and the axioms; a physical discussion follows. In Sec. 3.2, we show that the reference state satisfying the axioms defines a renormalization group (RG) fixed point. In Sec. 3.3, we discuss additional properties of the RG fixed point that will be explored in later chapters.

3.1 Setup and axioms

In this section, we introduce the setup and the axioms of our framework.

Setup: Imagine that we have a 2D system V . The system is large enough, and we assume that the Hilbert space of the system has a tensor product structure = ∈ . Each local H ⊗v V Hv Hilbert space is ﬁnite-dimensional, and it is located in a certain position of the 2D system. Hv Suppose we have access to the density matrix (σ) of the system. We call the quantum state σ as the reference state.

Many interesting choices of the reference state come from the ground states of 2D gapped local

Hamiltonians. However, we only require the reference state to satisfy two entropy conditions

(A0 and A1) that we introduce below. This makes our framework completely Hamiltonian independent. Before we introduce the axioms, we describe a few length scales involved in this physical problem.

We will consider the set of all bounded-radius disks within the large disk region of length scale ∆. The length of these bounded-radius disks is less or equal to r. We will refer to this set of bounded-radius disks as the set of µ-disks. See Fig. 3.1 for a depiction. The speciﬁc scale of

∆ is not important. What is important is that it can be much smaller than the system size, and we do not need to access the global topology of the system. There is another length scale . We

29 are only interested in subsystems of thickness larger than . (Physically, we require that is a few times bigger than the correlation length of the system.) The scale of the µ-disk (r) should be a few times larger than . ∆ should be several times larger than the scale of the µ-disks.

Axioms: Now we are in the position to introduce our axioms (A0 and A1). They are two entropic conditions on the set of µ-disks, which we require the reference state σ to satisfy.

∆ B

C 2r

Figure 3.1: The reference state σ of a 2D quantum many-body system. Some of the µ-disks are shown. The zoomed-in depiction of µ-disk b with partition BC (BCD) relevant to Axiom A0 (A1). Three relevant length scales , r and ∆ are illustrated. All of them are larger than the correlation length and they can be much smaller than the system size.

Axiom A0. For any µ-disk b, for any conﬁguration of subsystems b = BC topologically equiv-

alent to the one described in Fig. 3.1,

(S + S S ) = 0. (3.1) BC C − B σ

Axiom A1. For any µ-disk b, for any conﬁguration of subsystems b = BCD topologically

equivalent to the one described in Fig. 3.1,

(S + S S S ) = 0. (3.2) BC CD − B − D σ

Several remarks are in order.

30 Some readers may prefer to have a more concrete formulation. On such formulation requires • the choice of a coarse-grained discrete lattice of the physical system. See Chapter 9 for

the details.

In our axioms, we assumed strictly zero on the right-hand side. This may seem unrealistic • because a real physical system always has corrections due to a ﬁnite (nonzero) correlation.

It would be desirable to relax this assumption to something that is less restrictive. We

expect our framework to have a natural extension to the case in which axiom A0 and

A1 holds approximately. This is because the tools we in this thesis have an analog for

such situations. For instance, there is an approximate recovery map [52], which generalizes

Lemma 2.3.

When do we expect the axioms to hold? The range of validity of the axioms is important. If the set of axioms hold in a wide variety of physical systems, the results derived from the framework will have a broad application. In contrast, if the axioms hold only on very special quantum systems, the proofs hold only for these limited types of systems. We would like to remark on the generality of the axiom.

The intuition behind the choice of axioms (A0 and A1) is the entanglement area law. More precisely, it is a conjectured form of entanglement area law for 2D gapped ground states [6, 7]:

S(A) = α` γ, (3.3) − where S(A) is the von Neumann entropy of a simply connected region A, ` is the perimeter of

A, and γ is a constant correction term that only depends on the topology of A. The sub-leading correction, which vanishes in the ` limit, is suppressed here. → ∞ This form of area law is expected to hold broadly for 2D gapped systems at length scales larger than the correlation length. It is tested for both exactly solved models, e.g. [23, 26] and in good agreement to a few numeric studies [53, 54]. The conjectured area law (3.3) implies

Eq. (3.1) and Eq. (3.2). This gives credence to our axioms.

31 On the other hand, it is worth pointing out that in 2D, proving the entanglement area law remains challenging. For 1D gapped systems, an entanglement is rigorously established by Hastings’ theorem [2]. No 2D analog of Hastings’ theorem is known. In fact, neither the conjectured area law Eq.(3.3) nor Eq. (3.1) and Eq. (3.2) hold in every ground state of 2D gapped systems. Examples of quantum states that violate Eq. (3.3) includes a state with non-Abelian anyons pinned down to ﬁxed positions [6, 55] a quantum state with non-Abelian topological defects [56, 57], and spurious contribution to TEE [58, 59, 60, 61]. (All of these examples violate A1 but not A0.) Therefore, the expected generality of the area law (3.3) may only be established under (hopefully broadly applicable) extra conditions. It is an open problem that deserves a careful study. Regarding this, one proposal is presented in Chapter 11. We conjecture that the axioms emerge at a stable renormalization group ﬁxed point; see Conjecture 11.1.

3.2 Renormalization group ﬁxed point

A quantum state wave function (reference state σ) satisfying Axiom A0 and A1 deﬁnes a renormalization group ﬁxed point. Establishing this fact is the goal of this section.

Saturation of entropy bound and quantum Markov state: First, we observe that the reference state σ is rather special in the space of all possible density matrices. For any state

ρ ( ), we must have ∈ S H

(S + S S ) 0, (S + S S S ) 0, (3.4) BC C − B ρ ≥ BC CD − B − D ρ ≥ for the partitions in Fig. 3.1. These inequalities follow from SSA. The inequalities saturate with

“=” for the reference state σ; see Eq. (3.1) and (3.2).

This saturation of entropy bound implies that the reference state forms a quantum Markov state for interesting partitions. We discuss several consequences of this observation. This line of thought was presented in Ref. [12]. Recall that quantum Markov states and related properties are reviewed in Sec. 2.5. A quantum Markov state is a tripartite state, say over subsystems

A, B, and C, such that the conditional mutual information I(A : C B) = 0. | 32 Suppose we know σAB and σBC for the partition in Fig. 3.2. Then A1 implies

I(A : C B) (S + S S S ) = 0. (3.5) | σ ≤ BC CD − B − D σ

Here, the “ ” follows from SSA. Lemma 2.2 implies that the state σ is uniquely determined ≤ ABC by its marginals (σAB and σBC ). By repeatedly using this argument, the reduced density matrix of the reference state on a large disk-like region can be determined by those on the µ-disks. This is similar to solving a jigsaw puzzle, albeit the µ-disks overlap with each other.

A B C D A

B C D

Figure 3.2: An illustration of the growth procedure of a disk from AB to ABC. Here A can be large and BCD is contained in a µ-disk in a manner similar to Fig. 3.1.

The reference state is an RG fixed point: Renormalization group (RG) fixed point is a broad idea [62]. It is a collection of related concepts instead of one concept. Its application range from gapped systems to gapless systems and from real space to momentum space. Therefore, it is important to be clear on what we mean by an RG fixed point.

What we discuss here is an RG ﬁxed point in real space. Running the RG ﬂow means forgetting the details on small length scales. This idea share similarities to the entanglement

RG [63, 64]. However, we do not apply local unitary operators to disentangle qubits. We always look at the same quantum state, e.g., the reference state σ. After each step of entanglement RG, the smallest length scale for which we care about the entanglement entropy increases. (More precisely, after a step of RG ﬂow, the length scales change as λ and r λr, where λ > 1, → →

33 B B B b b c c d d b c C C C d d0

D D

(a) (b) (c)

Figure 3.3: The extension of the axioms. A disk is divided into either BC or BCD.A µ-disk is on a smaller length scale, i.e., the small dashed circle surrounding the colored region. These ﬁgures represent three ways of enlarging C by a small step. (a) bc B and d C; (b) bc B and d C; (c) bc B, d C and d0 D. ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ e.g. λ = 2. The scale ∆ is not involved in the RG ﬂow; the only requirement is that it is large enough.)

We explicitly show that if the axioms (Axiom A0 and A1) hold on the set of bounded- radius disks (µ-disks), the same entropic conditions hold for disks of all larger length scales; see

Proposition 3.1. In this sense, the reference state σ is an RG ﬁxed point.

Proposition 3.1. For a reference state satisfying axioms A0 and A1, the entropic conditions

Eq. (3.1) and Eq. (3.2) are satisﬁed on all larger disk-like subsystems.

Proof. (Abridged.) By assumption, these axioms hold for µ-disks. The nontrivial part of the statement is that the axioms hold at a larger length scale. We shall extensively use the following two inequalities:

S + S S S + S S , (3.6) BC C − B ≥ BB0C C − BB0 S + S S S S + S S S . (3.7) BC CD − B − D ≥ BB0C CDD0 − BB0 − DD0

Both of them follow from SSA.

Let us show how to extend Axiom A0 to a larger scale. Without loss of generality, consider a disk and its subsystem B and C, shown in Fig. 3.3(a). Assume that BC is not contained in

34 any µ-disk. One can consider a sequence of (enlarged) subsystems to obtain this disk (BC) from a µ-disk.

For this purpose, it suﬃces to consider the following two moves. The ﬁrst move is to enlarge

B while keeping C fixed. The second move is to enlarge C while keeping BC fixed. Our goal is to show that, for both of these steps, the linear combination of entropy S + S S is BC C − B non-increasing. The first move preserves Eq. (3.1) because of Eq. (3.6) where we set BB0 to be the enlarged B. To understand why the second move preserves Eq. (3.1), consider a deformation depicted in Fig. 3.3(a). We need to show S + S S is non-increasing when we deform C BC C − B to include the colored region of Fig. 3.3(a). The condition we need to verify is simplified into

S + S S \ S S + S S S = 0, (3.8) B Cc − B c − C ≤ bc cd − b − d which is nothing but a variation of Eq. (3.7). The statement “= 0” follows from condition A1 on the µ-disk bcd. Therefore, both moves preserve Eq. (3.1). Because any disk can be enlarged from a µ-disk by applying a sequence of these moves, Axiom A0 holds for any disk.

The proof for the enlarged version of Axiom A1 is analogous, and it requires to consider the extension around the µ-disks in Fig. 3.3(b)(c). We leave it as an exercise to the readers.

Alternatively, the reader may ﬁnd the proof in Ref. [8].

One may further ask whether the RG fixed point is stable. For this purpose, it is beneficial to imagine a non-fixed point for which the two entropic conditions hold up to a mild correction.

If the entropy combinations always hold with a smaller error at larger length scales, the RG

fixed point is stable. Otherwise, the RG fixed point is unstable. It seems natural to expect this fixed point to be stable. Nevertheless, this expectation has not been rigorously justified.

It is suggested in Chapter 11 that proving Conjecture 11.1 (or ﬁnding a counterexample) may provide one step toward solving this problem.

35 3.3 Anyon data are the “order parameters” of topological orders

In this section, we argue that from the reference state (σ), one can extract plenty of data about the gapped phase. In particular, many anyon data can be extracted. The justiﬁcation of related claims is contained in later chapters. Moreover, we point out that the reference state σ can be chosen to be a pure state.

Here is some physical intuition. A ground state of a gapped phase often has a finite (but nonzero) correlation length. If the gapped phase has a symmetry-breaking order, one can define a local order parameter. Sometimes, the order parameter can be an operator acting on a single site. However, from the viewpoint of RG, a good choice is often an average of a quantity on a length scale larger than the correlation length. For example, in a 1D transverse-field Ising model, one can consider the expectation value of a “block spin” ψ Pr σz ψ , where r is greater than h | i=0 i | i the correlation length. If this expectation value is nonzero, the ground state has spontaneous symmetry breaking. (Note that the Z2 symmetry of the Ising model reverses the expectation value.) Therefore, in principle, the symmetry-breaking order can be detected by looking at a subsystem of length scale a few times the correlation length. It is a scale much smaller than the systems size in the thermodynamic limit. In this sense, we say this type of extraction of the properties of symmetry breaking order is “local”.3

How about a 2D gapped phase beyond the symmetry breaking classiﬁcation, e.g., a topologically ordered system with anyons? Early studies [65] suggest that topological order can be detected globally from the ground state degeneracy. The degeneracy depends on the topology of the manifold. Diﬀerent ground states are locally indistinguishable, but they can be distinguished by nonlocal string order parameters.

A more subtle question is whether we can detect the nontrivial properties of a topologically ordered system given the reduced density matrix of a single ground state on a subsystem of length

3This should be distinguished from another sense of locality. The length scale of the microscopic degrees of freedom is even smaller.

36 scale a few times the correlation length. Topological entanglement entropy (TEE) provides a partial answer to this question [6, 7]. By calculating a linear combination of the entanglement entropy on subsystems of size a few length scales, one can extract the total quantum dimension of the anyon theory = pP d2. Thus, the total quantum dimension can be extracted locally. D a a Nevertheless, the total quantum dimension is only a small piece of the anyon data. There are inequivalent anyon models which have identical total quantum dimension. For example, the Ising anyon model has 3 sectors, = 1, σ, ψ with quantum dimensions d , d , d = 1, √2, 1 ; CIsing { } { 1 σ ψ} { } the toric code model has 4 sectors = 1, e, m, with quantum dimensions d , d , d , d = CT.C. { } { 1 e m } 1, 1, 1, 1 . It is an interesting question whether one can extract additional anyon data locally. { } In the next few chapters, we will provide a positive answer to this question. On our way to derive the emergent laws of anyons, we demonstrate the fact that the related anyon data can, in principle, be extracted locally from a single quantum state. In Chapter 5, we show how to extract the sectors = 1, a, b, and the fusion multiplicities N c locally. In Chapter 6, C { ···} { ab} we rigorously prove that individual quantum dimensions d and the total quantum dimension { a} manifest in the entanglement entropy. This is in agreement with previous studies on TEE. In D Chapter 8, we show that the topological S-matrix can be extracted locally as well.

Finally, we point out that the (possibly mixed) reference state σ can always be replaced by a pure state. This replacement does not aﬀect the reduced density matrix on a local region (of length scale ∆).

Lemma 3.2. Consider a reference state σ which satisﬁes entropy condition

S + S S = 0 (3.9) EF E − F for the subsystem choice in Fig. 3.4. Let σ = P p ϕi ϕi , where p is a probability distribu- i i| ih | { i} tion and ϕi is normalized for any i. Then, | i

i i Tr ¯ ϕ ϕ = Tr ¯ σ, i. (3.10) E| ih | E ∀

Here E¯ is the complement of subsystem E.

37 Proof. It follows from Lemma 2.4.

Note that, Eq. (3.9) is an enlarged version of axiom A0. Lemma 3.2 implies that we may take a pure state ϕi as the reference state in place of σ. This does not aﬀect the reduced | i density matrix on a large enough local region (of length ∆). Thus, every anyon data that can be extracted locally will not be aﬀected by this replacement of the reference state.

E F

Figure 3.4: Subsystem E is a square of length scale ∆. F is an annulus around it. The thickness of F is at least .

38 Chapter 4: Information convex set and isomorphism theorem

For a theoretical framework to work, identifying the axioms is crucial. Almost equally important is finding the concepts/objects that play a central role. As a famous example, conformal symmetry and unitarity are the axioms of 1+1D conformal field theory (CFT), whereas the primary field is a key concept [3].

Having identiﬁed the axioms of our framework, we now introduce a central concept: information convex set. The information convex set Σ(Ω), or Σ(Ω σ) (Deﬁnition 4.1), is a convex set of | density matrix associated to a region Ω. The density matrices in the information convex set are locally indistinguishable from the reference σ. A natural requirement concern the correlation is imposed on the elements. It is also the main object of the discussion of Chapters 5, 9 and 10.

For most applications of this thesis, the choice of the region Ω is a subsystem. Nevertheless, the allowed choices of region Ω are not limited to subsystems.

We further prove that the structure of the information convex set is preserved under “smooth” deformations of the region. This is the content of the isomorphism theorem (Theorem 4.8). The key to the proof of isomorphism theorem is the merging technique (Lemma 4.6 and Proposi- tion 4.7). The merging technique not only allows us to deform the region smoothly but also allows us to make changes to the topology of regions. It will have other applications in later chapters (e.g. Chapter 5, 6, 7 and 8).

39 4.1 Information convex set

In this section, we introduce the concept of information convex set. We ﬁrst present an informal discussion. After that, we present a formal deﬁnition.

Informal discussion: Quantum system can store information in a highly non-local manner.

With quantum entanglement, it is possible to store a piece of information in the correlation of the subsystems, (say A and B). This fact is already manifest in the simple example in Eq. (2.4).

For the quantum state Ψθ , the information of θ can be seen from neither A nor B. The | ABi information of θ is encoded in the entanglement between A and B.

In quantum many-body systems, this intuition generalizes. In a topologically ordered system, multiple ground states can exist on topologically nontrivial manifolds. Therefore, one can store a piece of quantum information with these ground states. This piece of information cannot be extracted locally because all these ground states are locally indistinguishable. People who have access to all the reduced density matrices on the set of µ-disks cannot decode this quantum information. One needs to know the correlation among the µ-disks. As a consequence, noise

(modeled by local interaction between the quantum memory and the environment) can hardly destroy this piece of quantum information. This observation is the central idea of fault-tolerant topological quantum computation [23].

The concept of information convex set generalizes the set of indistinguishable global states to subsystems. Locally indistinguishable density matrices on a region Ω can store (classical or quantum) information as well. A subtlety here is that we want to have some control over the entanglement on the boundary of the subsystems. The expected quantum Markov state structure for a gapped system (above length scale ) solves this problem.

Formal deﬁnition: Let us begin with some deﬁnitions. We say that two density matrices ρ and ρ0 are consistent with each other if they have identical density matrices on the overlapping

0 0 support, i.e., ρA = ρA where A is the intersection of the support of ρ and that of ρ . If two

40 quantum states ρ and ρ0 are consistent, we write

ρ =c ρ0.

We will consider a thickening of a subsystem Ω. We say Ω0 is a thickening of Ω if Ω0 Ω ⊇ and it can be smoothly deformed into Ω. Here Ω is the minimal choice of thickening formed by Ω and an extra layer of thickness. This extra layer separates Ω and the rest of the system.

See Fig. 4.1 for an illustration. Note that Ω2 and Ω3 are valid choices of thickening as well.

(Readers interested in the mathematical rigor about the thickening are encouraged to look at the discrete version of thickening in Chapter 9.)

Ω0 Ω ⊇

b Ω

Figure 4.1: This ﬁgure is a schematic depiction of regions involved in the deﬁnition of information convex set Σ(Ω σ). Here Ω is the annulus between the black circles. Annulus Ω0 Ω is a thickening of Ω.| It is the region between the gray circles. Any element in Σ(Ω˜ 0 σ) is consistent⊇ with the reference state σ on every µ-disk b contained in Ω0. We choose Ω to be| an annulus for illustration purposes. Other topologies are allowed as well.

Now, we are in a position to deﬁne the information convex set.

Deﬁnition 4.1 (Information convex set). Let Ω V be a subsystem and let Ω0 be a thickening ⊆ of Ω. The information convex set of Ω, for reference state σ, is deﬁned as

˜ 0 Σ(Ω σ) ρ ρ = Tr \ ρ , ρ Σ(Ω σ) , (4.1) | ≡ { Ω| Ω Ω0 Ω Ω0 Ω0 ∈ | }

41 where Σ(Ω˜ 0 σ) is deﬁned as |

Σ(Ω˜ 0 σ) = ρ ρ =c σ , for any µ-disk b . | { Ω0 | Ω0 b }

We refer to [35, 8] for published materials that contain this definition. Note that, this definition appears to dependent on another subsystem Ω0. Therefore, it seems natural to denote the information convex as Σ(Ω, Ω0 σ). However, we decide to suppress Ω0 in the notation because | the end result is insensitive to the details of the thickening, e.g., Σ(Ω, Ω σ) = Σ(Ω, Ω σ). (This | 2| should not be obvious at the moment, but it is a consequence of our axioms.) Since we always consider the same reference state, we make an further simplification of the notation and denote the information convex set as Σ(Ω).

Remark. Two other deﬁnitions of information convex set can be found in this thesis. One is Σ(Ω)ˆ ; see Sec. 4.3.1. Under our axioms, we have Σ(Ω)ˆ = Σ(Ω). (See Deﬁnition 4.3 and

Proposition 4.12). Another related deﬁnition is based on a frustration-free local Hamiltonian

Σ(Ω, Ω0 H) (Deﬁnition 10.2). Under the conditions speciﬁed in Theorem 10.1, the dependence | of thickening Ω0 can be dropped and we can show Σ(Ω H) = Σ(Ω σ). | |

Readers may wonder why we need the extra layer at all. Involving an extra layer is a simple way to have the quantum Markov state structure extended to partitions of subsystems around the boundary of the subsystem. (This quantum Markov state structure is not guaranteed by local indistinguishability alone.) Intuitively, the extra layer absorbs irrelevant correlations.

Below are some simple properties of the information convex set. (Some proofs are omitted.

An interested reader may either take them as exercises or look up the poof in the original paper

[8].)

Proposition 4.1. For any nonempty Ω V , Σ(Ω) is a nonempty ﬁnite-dimensional compact ⊆ 0 0 convex set. Furthermore, if Ω Ω and ρ Σ(Ω ), then Tr \ ρ Σ(Ω). ⊆ Ω0 ∈ Ω0 Ω Ω0 ∈

42 The fact that information convex sets are compact convex sets implies that we can learn the full structure of these convex sets by studying the extreme points. (This fact follows from

Minkowski-Caratheodory theorem.)

Proposition 4.2. For any disk-like subsystem ω, we have

Σ(ω) = σ , (4.2) { ω} where σ Tr ψ ψ and ω¯ is the complement of ω. ω ≡ ω¯ | ih |

The key idea of the proof is illustrated in Fig. 4.2. More precisely, what Fig. 4.2 illustrates is a sequence of repeated usage of the growth procedure in Fig. 3.2. Recall that Lemma 2.2 implies that a quantum Markov state is uniquely determined by its marginals.

Figure 4.2: A µ-disk can grow until it covers a larger disk ω.

Proposition 4.3. Any state ρ Σ(Ω) satisﬁes Ω ∈

TrΩ\ω ρΩ = σω (4.3) on any disk-like subsystem ω Ω. ⊆

Proposition 4.3 implies that states in the information convex sets are indistinguishable with the reference state on every disk-like subsystem. This can be thought of as an enlarged version of local indistinguishability.

Closed manifold: The main results in this thesis do not rely on the global topology of the system. Those main results apply to both closed manifold or open surfaces and therefore has

43 high generality. (Recall that, what we need is the reduced density matrix of the reference state on the set of µ-disks. The µ-disks are located within a reasonable large disk of length scale ∆.

Our axioms are required to hold on these µ-disks. See Sec. 3.1.)

Nevertheless, some additional results can be derived if the 2D system is on a closed manifold.

Two simple results are presented below. In these derivations, we adopt a global version of the setup, namely that we require the set of µ-disks to cover the whole manifold. Axiom A0 and

A1 hold on all these µ-disks.

Theorem 4.4. Let M be a closed 2D manifold. Then

Σ(M) = (V) (4.4) S for some ﬁnite dimensional Hilbert space V . Moreover, V is nonempty. ⊆ H The proof requires a judicious usage of Lemma 2.4. An interesting special case is when the underlying manifold is a sphere(S2).

Proposition 4.5. Σ(S2) = ψ ψ . {| ih |} To prove this statement, one needs to cleverly cut the sphere into three pieces A, B, and

C, and use Lemma 2.2 to determine the uniqueness. Proposition 4.5 implies that Σ(S2) always contains a unique pure state. This result reproduces what is expected in TQFT [24, 66, 67].

The underlying assumption of our approach is diﬀerent.

4.2 Elementary steps and Isomorphism Theorem

In this section, we establish an isomorphism between two information convex sets (Theo- rem 4.8). This isomorphism exists if the subsystems associated with the two sets can be smoothly deformed into each other. (This requirement implies that the pair of subsystems involved are topologically equivalent. However, it is more restrictive than topological equivalence.) More precisely, to establish an isomorphism, we require that the two subsystems be connected by a path (Deﬁntion 4.2).

44 Heuristic discussions: We have already seen that the Markov state structure of the reference state allows us to deform the subsystems smoothly. See the examples in Fig. 3.2 and 3.3.

Recall that axiom A1 plays a key rule here.

How about the elements in an information convex set? Because states in the information convex set are locally indistinguishable from the ground state, one may expect that the region can be deformed smoothly while keeping a certain structure of the information convex set unchanged.

Can we justify this?

AB C D AB C D

(a) (b)

Figure 4.3: Smooth deformations of subsystems (ABC ABCD). The subsystem labels are chosen to make the (later) discussion of merging convenient.↔ (a) A smooth deformation of a disk. (b) A smooth deformation of an annulus. The annulus topological is chosen for illustration purposes. Other topologies are allowed as well.

An instructive but insuﬃcient observation may go as follows. We have seen that for every disk-like region ω, we have Σ(ω) = σ (Proposition 4.2). Therefore, under a smooth defor- { ω} mation of a disk, the structure of the information convex sets remains unchanged. For a small deformation of disk ABC ABCD shown in Fig. 4.3(a), we can formulate the isomorphism as →

σ (σ ) = σ , (4.5) EC→CD ABC ABCD where σ is the Petz recovery map. The Petz recovery map generates an isomorphism EC→CD between the information convex set Σ(ABC) and Σ(ABCD). (The proof follows immediately from the fact that I(AB : D C) = 0 and Lemma 2.3.) | σ 45 Then, one might attempt to apply the same logic for other topology types. For the annuli

ABC and ABCD shown in Fig. 4.3(b), one might expect that the same Petz map gives

σ (ρ ) = ρ , (4.6) EC→CD ABC ABCD and it maps ρ Σ(ABC) into an element ρ Σ(ABCD). If this is indeed the case, ABC ∈ ABCD ∈ one may further show that Eq. (4.6) generates an isomorphic map from Σ(ABC) to Σ(ABCD) and that the partial trace operator TrD generates the inverse map.

However, the above argument is insuﬃcient. The logic for the disk cannot be generated straightforwardly to the annulus. It is currently unknown whether every ρ Σ(ABC) has ABC ∈ an extension deﬁned on ABCD with the required quantum Markov state condition I(AB :

D C) = 0. (Note that both Lemma 2.2 and Lemma 2.3 require the existence of a quantum | Markov state.) We need to show the existence of a new quantum state rather than assuming its existence.

A new technique is needed. That is the merging technique we introduce below. (See

Lemma 4.6 and Proposition 4.7.) Roughly speaking, the end result says Eq. (4.6) is true if the subsystems are thick enough.

Merging technique: Let us consider the merging of quantum Markov states. Suppose we have two quantum states ρ and λ, which share overlapping support; the two states are consistent. The question is whether one can consistently “sew” them together. Namely, can we

find a state which is consistent with both ρ and λ? This is known as the quantum marginal problem. In general, even deciding whether there is such a state or not is known to be extremely difficult [68]. There are several nontrivial necessary conditions [32, 12], but sufficient conditions are rare. However, there is a nontrivial sufficient condition for quantum Markov states [37]. See the lemma below.

46 Lemma 4.6. (Merging Lemma [37]) Given two sets of quantum Markov states ρa and { ABC } λx such that { BCD} ρa =c λx, a, x (4.7) ∀ and

I(A : C B) a = I(B : D C) x = 0, a, x, (4.8) | ρ | λ ∀

x there exists a unique set of “merged” states τ ax = λ (ρa ) which satisfy the following { ABCD EC→CD ABC } properties.

(1) τ ax is consistent with ρa and λx, i.e.

ax a ax x TrDτABCD = ρABC and TrAτABCD = λBCD. (4.9)

(2) Vanishing of conditional mutual information,

I(A : CD B) ax = I(AB : D C) ax = 0, a, x. (4.10) | τ | τ ∀

(3) The conservation of von Neumann entropy diﬀerence:

S(τ ax ) S(τ by ) = S(ρa ) + S(λx ) S(ρb ) S(λy ). (4.11) ABCD − ABCD ABC BCD − ABC − BCD

Here λx is the Petz recovery map. EC→CD

a x ρABC A B C B C D λBCD

ax τABCD A B C D

Figure 4.4: A schematic depiction of the merging process (Lemma 4.6). A pair Markov states a x ax ρABC and λBCD are merged into yet another quantum Markov state τABCD.

47 Proof. Recall that ρa and λx are two sets of quantum Markov states. All of them are { ABC } { BCD} identical on subsystem BC. We denote the unique reduced density matrix on BC as ρBC . It follows from Lemma 2.3 that

a x ρa = ρ (ρ ), λx = λ (ρ ), (4.12) ABC EB→AB BC BCD EC→CD BC

a x where quantum channel ρ and λ are the Petz recovery maps. Let us deﬁne τ ax as EB→AB EC→CD ABCD

a x τ ax ρ λ (ρ ). (4.13) ABCD ≡ EB→AB ◦ EC→CD BC

a Note that, the two quantum channels ρ and λx commute. In the following, we check EB→AB EC→CD ax the three conditions for τABCD. For the ﬁrst condition, we have

a Tr τ ax = Tr ρ (λx ) D ABCD D ◦ EB→AB BCD ρa x = B→AB TrD(λBCD) E ◦ (4.14) a = ρ (ρ ) EB→AB BC a = ρABC . ax x A similar derivation shows TrAτABCD = λBCD. Therefore, the ﬁrst condition holds. To verify the second condition, the following general result is useful. Conditional mutual information I(X : Y Z) does not increase under a quantum channel acting only on either X or | a Y . Because τ ax = ρ (λx ), we must have ABCD EB→AB BCD

I(AB : D C) ax I(B : D C) x = 0. (4.15) | τ ≤ | λ

A similar derivation shows I(A : CD B) ax I(A : C B) a = 0. Therefore, the second condition | τ ≤ | ρ holds.

The third conditions holds because I(A : D BC) ax I(A : CD B) ax = 0 for any a and x. | τ ≤ | τ This completes the proof.

The signiﬁcance of this lemma lies in the fact that one can guarantee the existence of a global state from a (relatively) local information. What is given to us are the density matrices

48 over ABC and BCD, together with the conditions that can be verified on ABC and BCD. In particular, these conditions can be directly verified from the given states. Once the conditions are verified, one can guarantee the existence of a state over ABCD that is consistent with the given density matrices.

a x Suppose the given density matrices ρABC and λBCD belong to information convex set Σ(ABC)

ax and Σ(BCD) respectively. For our purpose, we want to know whether the density matrix τABCD generated in the merging process belongs to information convex set Σ(ABCD). This does not follow from Lemma 4.6 alone. We need the following proposition.

A B C D

A B0 C0 D

Figure 4.5: A schematic depiction of subsystem ABCD. The partition B0C0 = BC is chosen such that no µ-disk overlaps with both AB0 and CD. Note that, the subsystems A, B, C, D are allowed to take a variety of topologies.

Proposition 4.7. Consider two density matrices ρ Σ(ABC) and λ Σ(BCD). If ABC ∈ BCD ∈ the following conditions hold, ρABC and λBCD can be merged. Moreover, the resulting density matrix belongs to Σ(ABCD).

1. There exists a partition B0C0 = BC, such that no µ-disk overlaps with both AB0 and CD;

see Fig. 4.5.

2. ρ =c λ.

3. I(A : C B) = I(B : D C) = 0. | ρ | λ

4. I(A : C0 B0) = I(B0 : D C0) = 0. | ρ | λ

49 The proof requires some nontrivial considerations. Completing this proof is the purpose of

Sec. 4.3. More precisely, the proof follows from Proposition 4.10 and 4.12.

The merging technique (Lemma 4.6 and Proposition 4.7), together with our axioms(Axiom

A0 and A1) underpin the majority of our technical work. The interplay between the two is what allows us to start from strictly local information and conclude something nontrivial at a larger scale. Roughly speaking, such analysis is carried out as follows. Our axioms allow us to upper bound certain conditional mutual information by 0. We can then apply this fact to

Lemma 4.6 repeatedly to merge (many) density matrices. By Proposition 4.7, we can merge elements of multiple information convex sets into an element of yet another information convex set. This not only allows us to smoothly deform the boundary of a subsystem (Fig. 4.3) but also allows us to consider merging processes with nontrivial topology changes; see Sec. 5.3 and also

Chapter 6, 7 and 8.4

Elementary step: Now, we are in a position to concretely establish the “smooth” deformation of subsystems. What we mean is that one subsystem can be obtained from the other by either attaching or removing a region whose size is comparable to that of the µ-disks. (See

Fig. 4.3 for a depiction.) We refer to the process of subtracting/adding a disk-like region to a given subsystem as an elementary step of deformation.

As we mention above, we need the merging technique. Lemma 4.6 shows the existence of a global state, and if we can check the conditions in Proposition 4.7, we know the merging result is an element of the information convex set. The relevant partition is illustrated in Fig. 4.6.

Imagine zooming into the region in which this deformation occurs. Without loss of generality, we can consider two subsystems Ω = ABC and Ω0 = ABCD depicted in Fig. 4.6, where CD is contained in a µ-disk. (Note that Fig. 4.6 is essentially a zoomed-in depiction of Fig. 4.3.

The only diﬀerence is some minimum thickness we require.) By using the merging technique

4For the discrete version of the framework in Chapter 9, the same idea is used. A small improvement is made therein to further reduce the separation between A and D.

50 D D C C0 B B0 A A

Figure 4.6: The detailed partition relevant to the elementary step of deformation Ω Ω0 is illustrated, where Ω = ABC and Ω0 = ABCD. BCD is a disk, and CD is contained in a↔µ-disk. Only a part of A is shown. The topology of A can be arbitrary. B0C0 = BC. A and D are separated by at least 2r + , so that on µ-disk overlaps with both AB0 and CD.

and properties of quantum Markov states, we can establish an isomorphism between Σ(Ω) and

Σ(Ω0).

Let the domain of Tr and σ to be Σ(Ω0) and Σ(Ω) respectively. We are able to • D EC→CD verify that Im TrD Σ(Ω) ⊆ (4.16) Im σ Σ(Ω0). EC→CD ⊆ Moreover, for all ρ Σ(Ω) and ρ Σ(Ω0), Ω ∈ Ω0 ∈ σ (ρ ) =c ρ , (4.17) EC→CD Ω Ω σ Tr (ρ ) = ρ . (4.18) EC→CD ◦ D Ω0 Ω0 This implies that Tr and σ are bijections. In other words, they are isomorphic maps D EC→CD between Σ(Ω) and Σ(Ω0).

The resulting isomorphic maps only depend on Ω and Ω0. Although it may appear that • σ depends on the choice of C, it is possible to show that the isomorphic map (from EC→CD Σ(Ω) to Σ(Ω0)) that it generates is independent of the detailed choice of C. The uniqueness

is established by Lemma 2.2. Therefore, we give the isomorphic maps simpler names:

0 σ Φ → : Σ(Ω) Σ(Ω ) is the isomorphic map generated by . (4.19) Ω Ω0 → EC→CD 0 Φ → : Σ(Ω ) Σ(Ω) is the isomorphic map generated by Tr . (4.20) Ω0 Ω → D 51 Note that ΦΩ0→Ω is the inverse of ΦΩ→Ω0 .

Furthermore, it is possible to show that the isomorphic maps (ΦΩ→Ω0 and ΦΩ0→Ω) preserves three things.

1. The preservation of the structure as a convex set. Those convex set related structures

of information convex sets are preserved. For example, the dimension, compactness, the

number of extreme points are preserved. This is because Tr and σ are linear maps. D EC→CD

2. The preservation of distance measures. Let ρ , λ Σ(Ω). For any distance measure Ω Ω ∈ D( , ) between quantum states, · ·

D(ρΩ, λΩ) = D(ΦΩ→Ω0 (ρΩ), ΦΩ→Ω0 (λΩ)) (4.21)

The proof follows from the monotonicity of distance measures under a quantum chan-

nel. (Note that both Tr and σ are quantum channels and they are reversible on D EC→CD the information convex sets.) The same proof applies to the preservation of the ﬁdelity

p 2 F (ρ, τ) = (Tr √ρ τ√ρ) . Although ﬁdelity is not a distance measure, its behavior is

monotonic under the application of quantum channels.

3. The preservation of entropy. Let ρ , λ Σ(Ω). The von Neumann entropy satisfy Ω Ω ∈

S(ρ ) S(λ ) = S(Φ → (ρ )) S(Φ → (λ )) (4.22) Ω − Ω Ω Ω0 Ω − Ω Ω0 Ω

This property is a corollary of property (3) of Lemma 4.6.

Isomorphism Theorem: The isomorphism between two information convex sets can be established by repeating these elementary steps. However, we have to be careful on two points.

First, for two given topologically equivalent subsystems, there can be more than one way to deform one to the other. Second, even if the underlying subsystems are topologically equivalent, there may not be a smooth deformation between the two. As a trivial example, suppose we have two spheres. We can place two subsystems on each of these spheres. Even if these subsystems

52 are topologically equivalent to each other, there is no sequence of subsystems that smoothly deforms one to the other. Even on a connected space, one cannot make such a statement; see

Fig. 5.2.

Therefore, these (potentially diﬀerent) isomorphisms must be labeled by their paths. Let us formalize this notion below.

Deﬁnition 4.2. (Path) A ﬁnite sequence of subsystems Ωt with t = i/N and i = 0, 1, 2, ,N, { } ··· (N is a positive integer), is a path connecting Ω0 and Ω1 if each pair of nearby subsystems in the sequence are related by an elementary step of deformation, illustrated in Fig. 4.6.

Because a path is built up from elementary steps, we obtain the following theorem.

Theorem 4.8. (Isomorphism Theorem) If Ω0 and Ω1 are connected by a path Ωt , then there { } is an isomorphism

0 1 Φ{ t} : Σ(Ω ) Σ(Ω ) (4.23) Ω → uniquely determined by the path Ωt . Moreover, it preserves the distance and the entropy { } diﬀerence between elements

D(ρ, σ) = D Φ{Ωt}(ρ), Φ{Ωt}(σ) (4.24) S(ρ) S(σ) = S Φ{ t}(ρ) S Φ{ t}(σ) , (4.25) − Ω − Ω where D( , ) is any distance measure which is non-increasing under CPTP-maps. · ·

We omit the proof since it straightforwardly follows by applying the elementary step repeatedly. For any path Ωt , we can define an inverted path Ω1−t which reverses the sequence of { } { } 1 0 subsystems. This leads to the inverse isomorphism Φ 1 t : Σ(Ω ) Σ(Ω ). {Ω − } → Generally speaking, different paths may give rise to different isomorphisms. That is, under two different isomorphisms, an element of the information convex set may be mapped to two distinct elements. However, sometimes, we merely need the existence of an isomorphism. In

53 that case, we will use a notation

0 1 Σ(Ω ) ∼= Σ(Ω ) to indicate the existence of such an isomorphism. Under this condition, any distance measure and entropy diﬀerence is preserved; see Eqs. (4.24) and (4.25).

4.3 Merging of elements in the information convex sets

Our goal here is to justify Proposition 4.7. The proof involves an alternative formulation of the information convex set, which we denote as Σ(Ω).ˆ Under Axiom A0 and A1, Σ(Ω) becomes equivalent to Σ(Ω).ˆ

4.3.1 An alternative formulation of the information convex set

The set Σ(Ω),ˆ which we define in Definition 4.3, enjoys a number of properties which are not evident from the definition of Σ(Ω). We use these facts to prove Proposition 4.7.

The key difference between the definition of Σ(Ω)ˆ and Σ(Ω) is that the latter involves a thickened subsystem, whereas the former does not. (Reversely, the conditions on the latter definition can be checked locally, whereas the former cannot.)

Deﬁnition 4.3. Let Σ(Ω)ˆ be a set such that ρ Σ(Ω)ˆ ∀ ∈ c 1. ρ = σb for every µ-disk b.

2. I(A : C)ρ = 0 for the partition in Fig. 4.7.

3. I(A : C B) = 0 for the partitions in Fig. 4.8. | ρ This deﬁnition does not make it clear why Σ(Ω)ˆ is convex. The following proposition establishes this fact.

Proposition 4.9. Σ(Ω)ˆ is convex.

The proof can be obtained by using two facts: (1) the concaveness of conditional entropy; see Eq.(2.17); (2) The reduced matrices on the µ-disk BC are identical.

54 C

B A

Figure 4.7: A partition of the subsystem Ω for deﬁning Σ(Ω);ˆ see the second condition of Deﬁnition 4.3. Let Ω = ABC where BC is a subsystem contained in a µ-disk. The horizontal line is the boundary of Ω. Only part of A is shown for illustration purposes.

C C

B B A A

C C B B A A

Figure 4.8: A partition of the subsystem Ω for deﬁning Σ(Ω);ˆ see the third condition of Deﬁni- tion 4.3. Let Ω ABC where BC is a subsystem contained in a µ-disk. The horizontal line is the boundary of⊇ Ω. Only part of A is shown for illustration purposes.

4.3.2 Merging in the alternative formulation

In a variety of circumstances, elements in Σ(ˆ ABC) and Σ(ˆ BCD) can be merged, and the merging result is an element of Σ(ˆ ABCD). This is the content of the following proposition.

This proof contains some nontrivial ingredients. The key is the ability to deform the support of the quantum channel to avoid the overlap with any chosen µ-disk. Due to its importance, the proof of the original paper is repeated in glorious detail.

Proposition 4.10. Consider two density matrices ρ Σ(ˆ ABC) and λ Σ(ˆ BCD). If ABC ∈ BCD ∈ the following conditions hold, ρABC and λBCD can be merged. Moreover, the resulting density matrix is an element of Σ(ˆ ABCD).

55 1. There exists a partition B0C0 = BC, such that no µ-disk overlaps with both AB0 and CD;

see Fig. 4.5.

2. ρ =c λ.

3. I(A : C B) = I(B : D C) = 0. | ρ | λ

4. I(A : C0 B0) = I(B0 : D C0) = 0. | ρ | λ

Proof. It follows from the conditions of the proposition that ρABC and λBCD can be merged. In fact, there are two diﬀerent ways to merge these density matrices. Using the third condition, we

λ 0 ρ get τABCD → (ρABC ) and using the fourth condition, we get τ (λBCD). ≡ EC CD ABCD ≡ EB0→AB0 Note that τ = τ 0 . This is because both of them satisfy I(A : D BC) and they have ABCD ABCD | 0 the same marginal on ABC and BCD. Therefore, Lemma 2.2 implies that τABCD = τABCD. ˆ Below, we show that τABCD is an element of Σ(ABCD). In order to prove this claim, we need to show that τABCD satisﬁes the condition 1, 2, and 3 in Deﬁnition 4.3. Condition 1 is easy to check. Because no µ-disk overlaps with both AB0 and CD, the overlap between a µ-disk and

ABCD is either contained in ABC or BCD.

For the second and the third condition, the key observation is that we have the freedom to choose the quantum channel:

λ ρ τABCD → (ρABC ) = (λBCD). (4.26) ≡ EC CD EB0→AB0

For every µ-disk near the boundary of subsystem ABCD, we could pick a suitable quantum

λ ρ channel, (either → or ), which has no overlap with the µ-disk. EC CD EB0→AB0 For the second condition, we use the following fact. Consider the mutual information between two subsystems, say X and Y . The mutual information does not increase under a quantum channel acting only on either X or Y . Indeed, the channels we described above are instances of such quantum channels. Therefore, the mutual information in the second condition is upper bounded by 0. This subsequently implies that the second condition holds.

56 For the third condition, we use a similar fact. Consider the conditional mutual information

I(X : Y Z). This also does not increase under a quantum channel acting only on either X or | Y . Therefore, with the same logic, the third condition holds as well.

Remark. Proposition 4.10 is good enough for all usage of the thesis. It implies that merging can be done at a bounded length scale, and it can be much smaller than the size of the system. It is an open question whether the condition of merging can be further relaxed in a signiﬁcant way.

Concrete examples on discrete lattices are presented in Chapter 9, for which ABCD AB0C0D ↔ are shown explicitly. (Therein, the key idea is identical, but the detailed layer is slightly reduced.)

4.3.3 Equivalence of the deﬁnitions

Now we can show that Σ(Ω)ˆ is equivalent to Σ(Ω). This justiﬁes our choice of calling Σ(Ω)ˆ as the information convex set.

Proposition 4.11. Consider Ω = ABC and Ω0 = ABCD whose partitions are depicted in

Fig. 4.6, (note that A and D are assumed to be separated by at least 2r + ). Let σ be the EC→CD

Petz map constructed from the reference state density matrix σBCD. Then,

σ (ρ ) =c ρ , ρ Σ(Ω)ˆ , (4.27) EC→CD Ω Ω ∀ Ω ∈ σ (ρ ) Σ(Ωˆ 0), ρ Σ(Ω)ˆ . (4.28) EC→CD Ω ∈ ∀ Ω ∈

Proof. Eq. (4.27) follows directly from Lemma 4.6. To see why, let us observe that ρΩ and σBCD can be merged using Lemma 4.6. With condition 1, 2, 3 in Deﬁnition 4.3, one could verify the two conditions required in Lemma 4.6. First, ρΩ is consistent with the global state on any disk

ω Ω. In particular, ρ =c σ . Second, I(A : C B) = I(B : D C) = 0. ⊆ Ω BCD | ρ | σ Eq. (4.28) is essentially a corollary of Proposition 4.10. It is straightforward to construct the

B0C0 required in Proposition 4.10 and check all the conditions. The 2r + separation between

A and D is large enough for the construction. See Fig. 4.6. This completes the proof.

Proposition 4.12. Σ(Ω) = Σ(Ω)ˆ , Ω. ∀ 57 Proof. If Ω is a closed manifold, it is obvious that Σ(Ω) = Σ(Ω).ˆ If Ω has boundaries, it is easy to show Σ(Ω) Σ(Ω).ˆ On the other hand, Proposition 4.11 implies that any ρ Σ(Ω)ˆ can ⊆ Ω ∈ ˆ be written as ρ = Tr \ ρ for some element ρ Σ(Ω ). This is because of Eq. (4.28) and Ω Ω Ω Ω Ω ∈ that Ω and Ω are connected by a path which consists of a sequence of elementary extensions. It follows that ρ Σ(Ω) and therefore Σ(Ω) Σ(Ω).ˆ Thus, Σ(Ω) = Σ(Ω),ˆ Ω. This completes Ω ∈ ⊇ ∀ the proof.

58 Chapter 5: Fusion rules from entanglement

In this chapter, we discuss how our framework gives rise to a well-deﬁned notation of superselection sectors and the fusion rules.

The isomorphism theorem (Theorem 4.8) guarantees that the structure of the information convex sets are isomorphic under any smooth deformations of the subsystem. We now focus on how to extract the information of the superselection sectors and the corresponding fusion rules from the information convex set. We do this by studying the geometry of the information convex set, which depends only on the topology of the underlying subsystem. We then use the merging technique(Lemma 4.6 and Proposition 4.7) to relate subsystems with diﬀerent topologies and obtain several consistency equations. We then deﬁne the fusion rules and show that they satisfy all the constraints expected from the known algebraic theory of anyon [1].

Our result shows that the concept of superselection sectors and the fusion rules are logical consequences of an assumption of entanglement (axioms A0 and A1). These conditions can be checked on bounded-sized regions. Note that it is possible that the sector set only contains the vacuum sector, in which case no anyons exist. However, if another local calculation of entanglement entropy gives a nonzero value (i.e., the topological entanglement entropy is nonzero, see Chapter 6), it is guaranteed that nontrivial sectors exist in the system. A nontrivial sector is shown to have nontrivial mutual braiding statistics, and therefore it must be an anyon

(Chapter 7).

Our results are derived by the internal self-consistency of the information convex set. These consistency conditions come from SSA and the properties of quantum Markov states. This is

59 why we can derive nontrivial predictions even without using explicit forms of the wave function.

It further implies that the set of superselection sectors and the fusion multiplicities can be extracted locally from the reference state.

The result of this study is summarized in Table 5.1.

Physical data Number of holes Reference Superselection sectors 1 Theorem 5.1 Fusion multiplicities 2 Theorem 5.4 and 5.5 Axioms of the fusion theory 1, 2, 3, 4 (merging) Section 5.3

Table 5.1: Physical data that can be extracted from disks with diﬀerent number of holes.

5.1 Superselection sectors

Let us deﬁne a notion of superselection sectors.5 It is one of the key ingredients of the algebraic theory of anyon [1]. Historically, the notion of superselection sectors was introduced in the context of local ﬁeld theory; see [69, 70]. In the context of topologically ordered systems which is most relevant to our discussion, a nontrivial superselection sector corresponds to an anyon type that cannot be created by any local operator.

There are several recent attempts to rigorously formulate the superselection sectors based on operator algebra assumptions. One approach is based on the operator algebra on an annulus [71,

72], and another approach is based on the operator algebra on a cone-like subsystem of an inﬁnite lattice [73, 74]. Our approach to characterize the superselection sectors is similar to the one based on the operator algebra on annuli. However, these two approaches diﬀer in their assumptions and their range of validity.

We will identify a well-defined information-theoretic object and find that this object coincides with the conventional notion of superselection sectors in anyon theory. Importantly, we find that

5Superselection sectors are also known as topological charges.

60 the information convex set of an annulus forms a simplex (Theorem 5.1). The simplex has a

ﬁnite number of extreme points. Moreover, these extreme points are orthogonal to each other.

See Fig. 5.1(c) for an illustration. We will deﬁne these extreme points as the superselection sectors.

Theorem 5.1. (Simplex Theorem) For an annulus X, the information convex set is the convex hull of a ﬁnite number of orthogonal extreme points, σa , i.e. { X } ( ) M Σ( ) = = a (5.1) X ρX ρX paσX , a where a is a ﬁnite set of labels and p is a probability distribution. { } { a}

1 σX

Σ(X) = L M R ··· a σX b σX

(a) (b) (c)

Figure 5.1: (a) Division of an annulus X into three thinner annuli L, M, R. (b) A path (extensions-extensions-restrictions-restrictions) which generates an isomorphism Σ(L) ∼= Σ(X) ∼= Σ(R). (c) A schematic depiction of the simplex structure of Σ(X). The extreme points are the “corners” of the simplex. If the annulus X is contained in a disk, then one of the extreme points has the vacuum label “1”.

Here we show a sketch of the proof of Theorem 5.1. The orthogonality follows from the factor-

p 2 ization property of the ﬁdelity F (ρ, τ) = (Tr √ρ τ√ρ) . Let FX be the ﬁdelity of two extreme points in the information convex set of X = LMR in Fig. 5.1(a) (We use the same convention for subsystems). By using the fact that any extreme point has I(L : R) = 0 (Corollary 5.16.1),

61 we ﬁnd

FLR = FLFR. (5.2)

Because the ﬁdelity is non-decreasing under a partial trace, we have F F . Since LMR ≤ LR X and L, R are annuli connected by paths, see Fig. 5.1(b), the isomorphism theorem implies

F = FL = FR = FLMR and thus

F F 2. (5.3) ≤ F [0, 1]. Therefore, the two extreme points are either the same (F = 1) or orthogonal (F = 0). ∈ This derivation also shows that we can copy the information of the extreme point to L and R simultaneously. The ﬁniteness of the label set follows from the orthogonality and the setup that the Hilbert space is ﬁnite-dimensional.

Theorem 5.1 implies that Σ(X) forms a simplex in the state space. It has a ﬁnite number of extreme points σa , which can be perfectly distinguished from each other by a projective { X } measurement supported on the annulus. The simplex structure also implies that its elements can only store classical information in the probability distribution p . The isomorphism the- { a} orem 4.8 guarantees the universality of the label set, i.e., the fact that the same set of labels applies to all annuli, which could be connected to each other by a path. Note that there could be annuli not connected by any path, e.g., the X0 and X1 in Fig. 5.2. Theorem 5.1 is still applicable for both annuli, but the label sets for them can be diﬀerent. This is related to the existence of topological defects [56].

One of the extreme points is special. Consider a contractible annulus(see X0 in Fig. 5.2 for example). The information convex set of such annulus has a special extreme point which we label as “1”; physically, this corresponds to the vacuum sector.

Proposition 5.2. Let ω be a disk. For any annulus X ω ⊆

1 σ Tr \ σ , (5.4) X ≡ ω X ω is an extreme point of Σ(X).

62 푋1 푋0

Figure 5.2: A pair of annuli X0 and X1 on a torus. They cannot be connected by any smooth path because X0 is contractible and X1 is non-contractible.

Now we are ready to deﬁne the superselection sectors in our framework. When there is a pair of anyons, the topological charge can be measured by an Aharonov-Bohm type interferometry measurement by braiding the anyons [75]. Indeed, the projective measurement used for

a distinguishing diﬀerent σX corresponds to this interferometry measurement for several exactly solvable models. Based on this observation, we identify each label of the extreme points as a superselection sector of the system.

Deﬁnition 5.1. Let X be a contractible annulus. The set of superselection sectors is a set of extreme points in Σ(X).

Except for the vacuum sector, we label each extreme points with the lower-case Roman letters:

= 1, a, b, c, . (5.5) C { ···} Several authors have already made attempts to deﬁne superselection sectors in 2D gapped phases. A statement analogous to the simplex theorem was obtained recently in [72] for models with a local commuting parent Hamiltonian. This proof is based on the operator-algebraic framework of Haah [71].

We expect our derivation to hold more generally because we make no assumption about the parent Hamiltonian. If the area law(Eq.(1.6)) holds, our results follow. In particular, if we can prove approximate versions of our statements, we may be able to rigorously deﬁne a notion

63 of superselection sectors for models with non-zero Hall conductance or non-zero chiral central charge. These models cannot have a commuting projector parent Hamiltonian [76, 77].

The isomorphism theorem guarantees the label set of the superselection sector to be independent of the details of the annulus. However, this theorem in itself does not imply there is a well-deﬁned way to compare the sectors for two annuli.

In order to show that the isomorphisms associated with diﬀerent paths are identical, one necessarily has to invoke an extra condition on the paths. Lemma 5.3 establishes one such condition. Roughly speaking, it is possible to compare two annuli unambiguously independent of the path, as long as both paths lie in a single disk.

Lemma 5.3. Let X0 and X1 be two annuli contained in a disk C; see Fig. 5.3 for example.

Let Xt and Xt be two paths connecting X0 and X1 such that X0 = X0, X1 = X1 for { (1)} { (2)} (i) (i) i = 1, 2. Moreover, assume that Xt C, Xt C. Then, the isomorphisms ∪t (1) ⊆ ∪t (2) ⊆

0 1 Φ{Xt } : Σ(X ) Σ(X ) (1) → 0 1 and Φ{Xt } : Σ(X ) Σ(X ) (2) → are identical.

C X1 X˜ 0 X˜ 1 X0

Figure 5.3: Both C and BC are disks. B is an annulus. X0, X1, X˜ 0 and X˜ 1 are annuli. Note that X0 and X1 are subsets of C. In the proof of Lemma 5.3, we construct an extension Xt X˜ t with B X˜ t. → ⊆

64 a 1 Lemma 5.3 implies that we can always treat Φ t (σ 0 ) as the label a for X . The key {X(1)} X idea behind the proof is that one can copy the information about which superselection sector lies inside an annulus to a common annulus; see Fig. 5.3 for illustration.

Now that we have defined a notion of superselection sectors, we can define their quantum dimensions. We will do so by investigating a contribution to the entanglement entropy that depends on the choice of this sector. We will use the following definition. Later, we will be able to determine their value.

Deﬁnition 5.2 (Entropy contribution of superselection sector). For a contractible annulus X, we deﬁne the universal contribution to von Neumann entropy from superselection sector a as

S(σa ) S(σ1 ) f(a) X − X . (5.6) ≡ 2

The denominator 2 is introduced to take into account that X has two boundaries. For a connected 2D manifold, f(a) is a real number that does not depend on the choice of the contractible annulus. This is because the entropy diﬀerence is preserved by an isomorphism.

Furthermore, f(1) = 0 by deﬁnition.

Later, we shall study similar contributions for a n-hole disk with n 2. We will ﬁnd that f(a) ≥ appears generically, even for these more generic subsystems. The repeated appearance of these objects hints at a possibility that there may be nontrivial relations concerning f(a). Indeed, we will later see that f(a) = ln da, where da is the quantum dimension of an anyon/superselection sector a. We will also be able to derive the fusion axioms and expression for topological entanglement entropy.

5.2 Fusion rules and fusion spaces

The fusion rules determine the possible choice of the total composite superselection sectors of two given superselection sectors. In the algebraic theory of anyons, the fusion rule for sectors

65 a and b can be formally written as

X a b = N c c , × ab c

c 1 where N Z≥0 is the fusion multiplicity. This is analogous to the fact that two spin- particles ab ∈ 2 can fuse into spin-0 or spin-1 particle. Nevertheless, there is a fundamental diﬀerence between the fusion of spins and that of anyons. For the deﬁnition of particle spins and their fusion rules, rotational symmetry is often needed either in the Hamiltonian or the Lagrangian. In contrast, the notion of superselection sectors and their fusions rules can emerge from the collective properties of a many-body quantum system [16, 17, 1]. Indeed, we emphasize that our axioms (Axiom A0 and A1) are unrelated to any symmetry.

In our framework, the superselection sectors were identiﬁed from annuli. Thus, one may expect the fusion rules to be extracted from 2-hole disks. In this section, we show that this is indeed the case. Let us consider a 2-hole disk Y , which we depict in Fig. 5.4. Let B1, B2 and

B3 be the three annuli around the boundaries of Y . The information convex set of each annulus has the same simplex structure, and we can label the extreme points by the same label set. Let

c Σab(Y ) be a convex subset of Σ(Y ), deﬁned as

 Tr ρ = σa   Y \B1 Y B1  c b Σab(Y ) ρY Σ(Y ) TrY \B2 ρY = σB2 , (5.7) ≡  ∈ c  TrY \B3 ρY = σB3 where a is an extreme point of Σ( ), = 1 2 3. The convention of charge labeling among σBi Bi i , ,

c the diﬀerent annuli is ﬁxed by Lemma 5.3. Σab(Y ) may be empty if there is no state satisfying all the conditions. We call such a combination of (a, b, c) forbidden.

c We show that every extreme point of Σ(Y ) is contained in some Σab(Y ). Because Σ(Y ) is a convex set, the entire set can be characterized by Σc (Y ) . Each Σc (Y ) is isomorphic to the { ab } ab c state space of a ﬁnite-dimensional Hilbert space Vab. These two results are summarized below in Theorems 5.4 and 5.5.

66 Y 0

D1 D2

B1 B2 B3

0 Figure 5.4: A 2-hole disk Y = BY , with B = B1B2B3. B1, B2 and B3 are three annuli surrounding the three boundaries of Y . YD is a disk, where D = D1D2. D1 and D2 are the two disks surrounded by annuli B1 and B2.

Theorem 5.4. For a 2-hole disk Y , the information convex set Σ(Y ) is the following convex combination ( ) M Σ(Y ) = ρ = pc ρabc ρabc Σc (Y ) , (5.8) Y ab Y Y ∈ ab a,b,c∈C where pc is a probability distribution. { ab}

Proof. After taking a partial trace, the reduced density matrix of an extreme point of Σ(Y ) reduces to an extreme point of Σ(B1), Σ(B2) and Σ(B3). This fact follows from Lemma 5.18

c in Sec. 5.4.1. Therefore, every extreme point of Σ(Y ) is in Σab(Y ) for some a, b, and c. This implies Eq. (5.8).

This theorem implies that one can classify the extreme points by a triple of labels (a, b, c).

Furthermore, the convex combination in Eq. (5.8) is orthogonal, since one can perfectly distinguish these labels by projective measurements on the three distinct annuli.

c Now we study the geometric structure of each Σab(Y ). We should emphasize an important

c difference between Σab(Y ) and the information convex set of an annulus. On an annulus, the information convex set has a classical structure specified by a probability distribution p ∈C. { a}a In contrast, Σc (Y ) is coherent in the sense that it is isomorphic to the state space (Vc ) of a ab S ab c c certain finite-dimensional Hilbert space Vab. If the dimension of Vab is greater or equal to 2, the

67 c structure of Σab(Y ) allows the storage of quantum information. This structure is established by the following theorem.

Theorem 5.5. Consider a 2-hole disk Y . a, b, c , ∀ ∈ C

c c Σab(Y ) = (Vab), (5.9) ∼ S

c where Vab is a ﬁnite-dimensional Hilbert space.

c A particular choice of (a, b, c) is forbidden, when dim Vab = 0. See Sec. 5.5 for the proof of Theorem 5.5. The key idea is to show that there is a quantum channel which simultaneously puriﬁes every extreme point of Σc (Y ) into a state in Hilbert space (E is an auxiliary ab HEY system). We then show that any superposition of the puriﬁed states reduces to an extreme

c point of Σab(Y ) on Y . It follows that the quantum channel which achieves the purification provides an isomorphism between Σc (Y ) and (Vc ). ab S ab c We call the Hilbert space Vab defined in Theorem 5.5 as the fusion space. Physically, this Hilbert space is nonempty if the superselection sectors a, b has a total charge of c. We thus define the fusion rule using the dimension of the corresponding fusion space.

Deﬁnition 5.3. We deﬁne the fusion rule of labels a, b in by the formal product C X a b = N c c , (5.10) × ab c∈C where N c dim Vc . ab ≡ ab

The results in Theorem 5.4 and Theorem 5.5 generalize to n-hole disks with n 3. The ≥ same applies to the concepts of fusion spaces and fusion rules.

5.3 Derivation of the axioms of the fusion rules

In this section, we show how the axioms of the fusion rules emerge from our axioms (A0 and

A1). This derivation includes the existence of antiparticles and a set of rules that N c has to { ab} satisfy.

68 We have deﬁned the set of superselection sectors

= 1, a, b, c, C { ···} in terms of the extreme points of Σ(X), where X is a contractible annulus. is always a finite C set and there is a unique sector 1 which we refer to as the vacuum. We have also identified ∈ C a set of non-negative integers N c encoded in the structure of Σ(Y ) with a 2-hole disk Y . { ab} The following is a list of the results we will prove under our definitions.

c c 1. Nab = Nba. (Proposition 5.6).

c c 2. Na1 = N1a = δa,c. (Proposition 5.7.)

3. The existence of an anti-sectora ¯ for a such that N 1 = δ . (Proposition 5.9 ∈ C ∀ ∈ C ab b,a¯ and Deﬁnition 5.4.)

c c¯ 4. Nab = N¯ba¯. (Proposition 5.10.)

P i d P d j 5. i NabNic = j NajNbc. (Proposition 5.11.)

Together, these properties form a subset of the axioms of the algebraic theory of anyon summarized in Appendix E of [1], also known as the unitary modular tensor category (UMTC).

Concretely, what we derive in this section is the set of axioms of fusion rule algebra [78] which is also known under the name commutative fusion ring [25]. It contains slightly less axioms than a fusion category because we have not deﬁned the F -symbols.6

In the derivation of the axioms of the fusion rules, we will extensively use the merging technique to relate the subsystems of diﬀerent topologies. From our axioms, we can infer that elements in the information convex sets are quantum Markov states with respect to many relevant partitions of certain subsystems. Because quantum Markov states can be merged together

(Lemma 4.6), we can merge many of the elements of the information convex sets together.

6F -symbols will give further constraints to the fusion multiplicities, the simplest examples are studied in [79].

69 Moreover, this merging process can be repeated many times. With this process, we can generate elements of an information convex set over some regions from information convex sets of its subregions. We refer the readers to Proposition 4.7 for the technical details.

Proposition 5.6.

c c Nab = Nba. (5.11)

Proof. Let us consider a path which maps a 2-hole disk Y back to itself by exchanging the two internal holes. Associated with this path, there is an automorphism Σ(Y ) ∼= Σ(Y ). The automorphism permutes the labeling and induces an isomorphism Σc ( ) Σc ( ) for each ab Y ∼= ba Y c c a, b, c. Thus, Nab = Nba.

Proposition 5.7.

c c N1a = Na1 = δa,c. (5.12)

Figure 5.5: A hole with the vacuum charge can be merged with a disk. The case shown in this diagram involves an annulus and a disk, but the idea works for any n-hole disk with n 1. The left side shows the topology of the subsystems before they are merged. Also, the number≥ “1” is the vacuum sector. The merged subsystem is shown on the right. The three concentric lines partition the disk into the four subsystems used in the merging process.

Proof. Suppose ρ Σc (Y ) for 2-hole disk Y . Then the hole with the vacuum charge can be Y ∈ 1a merged with a disk, see Fig. 5.5. After the merging process, we obtain an annulus X. The density matrix obtained from the merging process belongs to Σ(X). The isomorphism theorem

70 c implies that the two boundaries of X detect the same topological charge. Therefore, N1a = δa,c.

c Then, Na1 = δa,c follows from Proposition 5.6.

1 1 One implication of this result is that Σ11(Y ) contains a unique element, which we call σY . This statement generalizes to n-hole disks with n 3. The following lemma, which is about ≥ the universal contribution to the von Neumann entropy, will be useful for the rest of the proofs.

Moreover, this lemma will be one of the key results that establish a connection between this contribution and the quantum dimension.

c 1 1 Lemma 5.8. Let ρY be an extreme point of Σab(Y ) and σY be the unique element of Σ11(Y ), then

S(ρ ) S(σ1 ) = f(a) + f(b) + f(c), (5.13) Y − Y where f( ) is the function deﬁned in Deﬁnition 5.2. ·

c The key idea is that for an extreme point of Σab(Y ), we can prove a condition similar to that in A0, which converts the entropy of a pair of 2-hole disks into that of the three annuli around the three disjoint boundaries of Y . The result generalizes easily to n-hole disks for any n 3. ≥ Compared to the previous proofs, the proofs of the rest of the properties requires a new technique. The key idea lies in deriving consistency equations of the entropy diﬀerence, obtained by the following four steps:

(i) Obtain an element of an information convex set by merging two (or three) extreme points of the information convex sets associated with the subsystems.

(ii) Compute the entropy of the merged element from the entropy formulas with respect to the pre-merged regions.

(iii) Compute the entropy of the merged element from the entropy formula with respect to the post-merged regions.

(iv) The entropy obtained from these two perspectives must yield the same result. This leads to a set of consistency equations, which leads to a set of nontrivial relations.

71 a a

b b

Figure 5.6: Merging two annuli and obtain a 2-hole disk. On the right side, there are two thin disk-like regions in the middle, which are chosen to be the B and C subsystem in the merging lemma(Lemma 4.6).

For a concrete example of the method, let us study the case shown in Fig. 5.6. The cases in Fig. 5.7, 5.8, 5.10, 5.11, 6.2 employ a similar logic. Let us explain the idea, which is broken down into four steps.

(i) We can merge the pair of annuli for any chosen charge pair a, b . This is possible ∈ C a×b because the conditions required for merging are satisﬁed. Let us call the merged state as σY . It follows that σa×b conv S Σc (Y ). Since the merged state exists, the set conv S Σc (Y ) Y ∈ c ab c ab is nonempty. This implies that P N c 1, a, b . Moreover, σ1×1 is equal to σ1 , which is c ab ≥ ∀ ∈ C Y Y 1 the unique element of Σ11(Y ). (ii) From the perspective of the two annuli, the von Neumann entropy diﬀerence can be expressed as:

S(σa×b) S(σ1 ) = 2f(a) + 2f(b). (5.14) Y − Y This result follows from the fact that merging preserves the entropy diﬀerence; see property (3) of Lemma 4.6. More explicitly, this result follows from conditional independence condition(I(A :

D BC) = 0) of the merged state; see Fig. 5.6. Here A (D) is the upper (lower) annuli, with | charge a (b); the two annuli are separated by disk-like region BC in the middle.

72 (iii) From the perspective of the 2-hole disk Y , the von Neumann entropy diﬀerence is

S(σa×b) S(σ1 ) Y − Y X c f(c) (5.15) = f(a) + f(b) + ln( Nabe ). c

S c To derive this result, note that the merged state is the maximal entropy element in conv c Σab(Y ) .

S c This is because the entropy of any state in conv c Σab(Y ) can be upper bounded by its marginals by the SSA and the merged state saturates this bound. Given the structure of Σ(Y ), it is easy to ﬁnd the maximal entropy in terms of N c and f( ). We calculated the maximal { ab} · entropy and obtained Eq. (5.15) [11].

(iv) By comparing the two perspectives in Eq. (5.14) and Eq. (5.15), we ﬁnd

f(a) f(b) X c f(c) e e = Nabe . (5.16) c

Readers well-versed in the fusion theory of anyon may have noticed the similarity between ef(a) and the quantum dimension [1]. This is not a coincidence. In Chapter 6, we shall see that they are, in fact, the same thing. To establish their equivalence, we will derive a few more identities involving N c . { ab} Moreover, we can calculate the probability of having charge c on the third boundary

N c ef(c) P = ab . (5.17) (a×b→c) ef(a)ef(b)

Its physical meaning is the probability to have an outcome c from the fusion of two independently created sectors a and b. In terms of the density matrices, P(a×b→c) is the coeﬃcient of the element

c a×b in the center of Σab(Y ) when writing σY in terms of a convex combination. It is worth noting that the same function f( ) appears in the entropy of the annulus and · the 2-hole disk. This is crucial for the comparing the two perspectives(Eq. (5.14) and (5.15)).

With this equivalence, we are in a position to derive more properties of N c . In deriving these { ab} properties, we will curtail our explanation a bit, because the argument is essentially the same.

73 1

1 a a

Figure 5.7: Merging a pair of annuli to obtain a 2-hole disk. We ﬁrst deform the annulus associated to 1 so that it becomes “longer” vertically. Then, the annulus associated to a is merged into the interior of this deformed annulus. The two thin U-shaped disk-like regions are chosen to be the subsystem B and C in the merging lemma(Lemma 4.6).

Proposition 5.9. For each charge sector a , there is a unique sector a¯ such that ∈ C ∈ C

1 Nab = δb,a¯. (5.18)

It further satisﬁes the following properties.

1¯ = 1, a¯ = a, f(¯a) = f(a). (5.19)

Proof. From the merging of two annuli with sectors 1 and a shown in Fig. 5.7, we can derive that P N 1 1, a because the merged state always exists. Furthermore, ef(a) = P N 1 ef(b). b ab ≥ ∀ b ab Let us pick a sector b such that N 1 1. We can see that ef(a) ef(b). However, since ab ≥ ≥ N 1 = N 1 , by repeating the same logic we obtain ef(b) ef(a). For both of them to be true, we ab ba ≥ 1 1 must have a unique sectora ¯ such that Nab = δb,a¯ and f(a) = f(¯a). Then it follows from N11 = 1 ¯ 1 1 ¯ that 1 = 1. Since Naa¯ = Naa¯ = 1, we have a¯ = a.

Deﬁnition 5.4 (Antiparticle). We deﬁne the antiparticle of a as the unique sectora ¯ ∈ C ∈ C established in Proposition 5.9.

The deﬁnition ofa ¯ is universal and insensitive to the choice of the subsystem.

Proposition 5.10.

c c¯ Nab = N¯ba¯. (5.20) 74 1 1

a a¯ a a¯

1 b ¯b b ¯b

Figure 5.8: Merging a pair of 2-hole disks to obtain a 4-hole disk.

a a¯

c c¯

b ¯b

¯ Figure 5.9: The fusion of (a, b) and (b, a¯), and matching the fusion probabilities P(a×b→c) and P(¯b×a¯→c¯).

Proof. We consider the merging process in Fig. 5.8. Before merging, the density matrices are

1 1 two extreme points from Σaa¯(Yu) and Σb¯b(Yd), if we call the pair of 2-hole disks as Yu and Yd.

a Since N11 = δ1,a, the outermost boundary of the merged subsystem must have charge 1. Now let us view the merged state in a diﬀerent way, as depicted in Fig. 5.9. We have derived that N 1 = δ , which implies that in the merged state, the fusion outcome of a b and that of cd d,c¯ × ¯b a¯ are perfectly correlated. Whenever we get the outcome c from the fusion of a and b, we × must getc ¯ from the fusion of ¯b anda ¯.

Furthermore, a and b in this state are “independently created” in the sense that fusion probability P(a×b→c) obeys Eq. (5.17). To see why, consider partial trace operations over (i) a region which connects the hole with chargea ¯ to the outer boundary and (ii) a region which

75 connects the hole with charge ¯b to the outer boundary. These regions are chosen so that the remaining subsystems are topologically equivalent to the ones appearing on the right side of

Fig. 5.6. Recalling the general inequality I(AA0 : CC0 B) I(A : C B), we observe that the | ≥ | annulus associated with a and the annulus associated with b are independent conditioned on a disk-like region in between them that separates the two annuli. As we have already discussed above, this conditional independence condition implies that P(a×b→c) obeys Eq. (5.17). Of course, an analogous argument can be applied to P(¯b×a¯→c¯).

Because c andc ¯ are completely correlated,

P(a×b→c) = P(¯b×a¯→c¯). (5.21)

c c¯ Then, noticing f(a) = f(¯a) from Eq. (5.19), we can derive Nab = N¯ba¯.

As mentioned earlier, the results in Theorem 5.4 and Theorem 5.5 generalize to n-hole disks with n 3, and the same applies to the concepts of fusion space and fusion rules. Let us ≥ introduce a few notations for n = 3 which are useful for the next proof. For a 3-hole disk Z,

d we use Σ(Z) to denote its information convex set, Σabc(Z) to denote the convex subset of Σ(Z)

d with ﬁxed sectors a, b, c, d on the boundaries (Fig. 5.10). The corresponding fusion space Vabc

d has a ﬁnite dimension N Z≥0. abc ∈

Proposition 5.11. The fusion rules are associative, i.e.,

d X i d X d j Nabc = NabNic = NajNbc. (5.22) i j

Proof. The key idea is to obtain a 3-hole disk Z in two diﬀerent ways; see Fig. 5.10 and 5.11.

d c The ﬁrst method gives us a lower bound of Nabc in terms of Nab, and the second method shows the bound saturates.

Let us consider the merging of a pair of 2-hole disks to obtain a 3-hole disk shown in Fig. 5.10.

We summarize the logic in a streamlined fashion in (i), (ii), (iii) below.

76 d d i j a b c a b c

Yl Yr YR YL

Figure 5.10: Merging a pair of 2-hole disks to obtain a 3-hole disk. Here Z = Yl YR = YL Yr. Here a, b, c, d, i, j are labels of the superselection sectors. ∪ ∪

(i) Let us consider the left side of Fig. 5.10, which describes the merging of Yl and YR. We

i i pick an orthonormal basis of Vab, which can be chosen to be the extreme points of Σab(Yl). The

i i number of such extreme points is equal to Nab, which is the dimension of the Hilbert space Vab.

d Applying the same logic to YR, we see that the number of these extreme points is Nic.

i (ii) Let us pick two arbitrary extreme points from the sets discussed above (one from Σab(Yl)

d d and another from Σic(YR)) and merge them. We get an element in Σabc(Z). It is an extreme point. This fact is veriﬁed by calculating the von Neumann entropy and making use of the 3-hole

i d version of Lemma 5.8. This way, we get NabNic number of extreme points, and any two of them are orthogonal. This follows from the fact that ﬁdelity is nondecreasing under a CPTP map.

P i d (iii) By applying the merging process for all i, we ﬁnd i NabNic mutually orthogonal extreme

d points of Σabc(Z). Therefore, we must have

X N d N i N d . (5.23) abc ≥ ab ic i

d d The reason is Nabc is the maximal number of mutually orthogonal extreme points in Σabc(Z). Similarly, from the right side of Fig. 5.10, we have

X N d N d N j . (5.24) abc ≥ aj bc j

We did not ﬁnd a way to turn “ ” into “=” from Fig. 5.10 alone. However, we can show ≥ “=” by considering a diﬀerent way of merging subsystems; see Fig. 5.11. The merged element,

a×b×c S d which we call σZ , is the maximal entropy element of conv d Σabc(Z) . Comparing with 77 a a a

b b b

c c c

Figure 5.11: Merging three annuli to obtain a 3-hole disk.

1 1 σZ , the unique element of Σ111(Z), it has an extra 2(f(a) + f(b) + f(c)) contribution to the von Neumann entropy. On the other hand, we use the structure of Σ(Z) to calculate the maximal entropy in the sector with charge a, b, c in terms of N d , we ﬁnd a contribution equals to { abc} P d f(d) f(a) + f(b) + f(c) + ln( d Nabce ). These two perspectives must provide a consistent answer. Thus,

f(a) f(b) f(c) X d f(d) e e e = Nabce . (5.25) d However, from Eq. (5.16) we know that

f(a) f(b) f(c) X X i d f(d) e e e = ( NabNic)e (5.26) d i and ef(·) is positive since f( ) is real. So the “ ” in Eq. (5.23) must be replaced by “=” and the · ≥ same replacement works for Eq. (5.24). Thus, we conclude that Eq. (5.22) holds.

The result and proof of proposition 5.11 generalizes to n-hole disks with n > 3.

5.4 Extreme points (details)

In this section, we prove various properties of the extreme points of the information convex set. Throughout this section, we shall often consider a slight “thickening” of a subsystem. Recall

78 that, thickening of a subsystem Ω is an enlarged subsystem Ω0, which is obtained by expanding the boundaries of Ω.

If the boundary of Ω is expanded by a thickness of δ, we shall refer to that subsystem as Ωδ.

For the ensuing analysis, it will be convenient to consider a length scale , which is greater than the correlation length. The thickenings Ω and Ω2 will be considered frequently.

Let us begin with the following lemma.

Lemma 5.12. Suppose ρ Σ(Ω ) can be written as Ω2 ∈ 2

X i ρΩ2 = qi ρΩ2 , (5.27) i where q is a probability distribution with q > 0, i and ρi is a set of density matrices. { i} i ∀ { Ω2 } Then,

i Tr \ ρ Σ(Ω). (5.28) Ω2 Ω Ω2 ∈

Proof. It suﬃces to show that every ρi reduces to σ on any µ-disk b Ω . In order to show Ω2 b ⊆ this fact, consider b Ω . Let C = b and choose B such that b = BC. Then, the topology of ⊆ 2 BC is identical to the one for axiom A0.

Because our axioms hold at a scale larger than the µ-disks(Proposition 3.1), and that ρBC =

σ (Proposition 4.3), we conclude that (S +S S ) = 0. Now, apply Lemma 2.4 (1) (3). BC BC C − B ρ ⇒ We conclude

i ρb = ρb = σb for any i and any µ-disk b Ω . This completes the proof. ∈

The following lemma shows that any element in the information convex set has zero conditional mutual information for onion-like partitions; see Fig. 5.12.

Lemma 5.13. Let Ω = ABC. Suppose B and C are concentric annuli described in Fig. 5.12.

Then,

I(A : C B) = 0, ρ Σ(ABC). (5.29) | ρ ∀ ABC ∈ 79 A B C

Figure 5.12: Here Ω = ABC is an arbitrary subsystem with a boundary. B and C are concentric annuli covering the boundary.

Lemma 5.14. Consider an extreme point σhei Σ(Ω ) written as Ω2 ∈ 2

h i X e = i (5.30) σΩ2 qi ρΩ2 i where q is a probability distribution with q > 0, i and ρi is a set of density matrices. { i} i ∀ { Ω2 } Then,

i hei Tr \ ρ = Tr \ σ , i Ω2 Ω Ω2 Ω2 Ω Ω2 ∀ is the same extreme point of Σ(Ω).

i Proof. It follows from Lemma 5.12 that TrΩ2\Ω ρΩ2 is an element of Σ(Ω) for all i. The nontrivial statement is that the reduced state on Ω is an extreme point and that the reduced state does not depend on i.

Suppose there is a dependences on . Then Tr hei must be a convex combination of i Ω2\Ω σΩ2 these distinct elements, so this density matrix cannot be an extreme point. This contradicts with the isomorphism theorem(Theorem 4.8), which implies that Tr hei is an extreme point of Ω2\Ω σΩ2 Σ(Ω). (Recall that any linear bijective map between two convex sets must map extreme points

i to extreme points.) Therefore, the density matrix TrΩ2\Ω ρΩ2 is independent of i and it follows that it is an extreme point of Σ(Ω).

Lemma 5.15. Consider an extreme point σhei Σ(Ω ) and let B = Ω Ω, then Ω2 ∈ 2 2\

(SBΩ + SΩ SB) e = 0. (5.31) − σh i 80 Proof. This result follows from Lemma 5.14 and (3) (1) of Lemma 2.4. ⇒

The signiﬁcance of Lemma 5.15 is that it applies to any subsystem. In particular, for 2-hole disks, we obtain the following corollary.

0 hei Corollary 5.15.1. Let Y be a 2-hole disk divided according to Fig. 5.4, i.e. Y = BY . Let σY be an extreme point of Σ(Y ), then

(SBY + SY SB) e = 0. (5.32) 0 0 − σh i

hei Lemma 5.16. Let Ω = ABC with a choice of subsystems described in Fig. 5.12. If σΩ is an extreme point of Σ(Ω):

I(A : C) e = 0. (5.33) σh i

Proof. Since σhei Σ(Ω), from Lemma 5.13, Ω ∈

I(A : C B) e = 0. (5.34) | σh i

Then, it follows from the explicit structure of quantum Markov state Eq. (2.19) and (2.20) that

h i X Tr σ e = p ρi ρi , (5.35) B Ω i A ⊗ C i where ρi and ρi are density matrices (which may or may not belong to Σ(A) and Σ(C)). p A C { i} is a probability distribution. We know from the isomorphism theorem 4.8 that σhei P p ρi A ≡ i i A is an extreme point of Σ(A).

0 0 Since A is thick enough, let A = A2. Here A has the same topology as A but it is thinner.

i hei 0 From Lemma 5.14 we know that TrA\A ρ = σ , i. Therefore, I(A : C)σ e = 0. Since we 0 A A0 ∀ h i could enlarge A0 (as that in Fig. 3.2) until it recovers A without changing the mutual information, we conclude that I(A : C) e = 0. σh i

As an immediate application of Lemma 5.16, we can prove the following factorization property between subsystem L and R described in Fig. 5.1. This plays an important role in the proof of the orthogonality of the extreme points.

81 Corollary 5.16.1. For the annulus X = LMR in Fig. 5.1(a), for any extreme point σa Σ(X), X ∈

Tr σa = σa σa , (5.36) M X L ⊗ R

a a a where σL and σR are the reduced density matrices of σX on L and R respectively.

Corollary 5.16.2. Consider the partition of a 2-hole disk Y in Fig. 5.4, i.e. Y = Y 0B and

hei B = B1B2B3. Let σY be an extreme point of Σ(Y ), then

(SB + SB + SB SB) e = 0. (5.37) 1 2 3 − σh i

Note that Eq. (5.37) is equivalent to saying that hei is a tripartite product state. σB1B2B3

5.4.1 Implication of the orthogonality

Based on the orthogonality of the extreme points, we can prove several new facts about the elements of the information convex set. In the remainder of this section, we use both the isomorphism theorem(Theorem 4.8) and the simplex theorem(Theorem 5.1). Let us begin with a succinct formula for the mutual information.

P a Proposition 5.17. Let ρX = a paσX be an element of Σ(X), written in terms of the orthogonal extreme points. Let X = LMR be a subsystem described in Fig. 5.1(a). Then,

X I(L : R) = p ln p . (5.38) ρ − a a a A similar result has been obtained in [36] using Chern-Simons theory. We obtained the same result as a consequence of A0 and A1.

As a consequence, we can identify the extreme points of Σ(X) as the elements with zero correlation between L and R.

Corollary 5.17.1. An element ρ Σ(X) is an extreme point if only if I(L : R) = 0 for the X ∈ ρ partition in Fig. 5.1(a).

hei Lemma 5.18. Let Ω = ABC with a choice of subsystems described in Fig. 5.12. If σΩ is an

hei extreme point of Σ(Ω), TrΩ\C σΩ is an extreme point of Σ(C).

82 5.5 Fusion space (details)

c Here, we prove that the convex set Σab(Y ) is isomorphic to the state space of a ﬁnite-

c dimensional fusion space Vab. Recall that a, b, and c are superselection sectors and Y is a 2-hole

c disk. We begin with the following lemma, which characterizes the extreme points of Σab(Y ).

c Lemma 5.19. Let Y be a 2-hole disk. Then, every extreme point of Σab(Y ) has the same von Neumann entropy.

0 Proof. Consider a partition of Y into Y = BY as described in Fig. 5.4. Let λY and ρY be two extreme points of Σc (Y ) and δ S(λ ) S(ρ ) be the entropy diﬀerence between the two ab ≡ Y − Y states. Eq. (5.32) and (5.37) imply that (SY + SY 0 )λ and (SY + SY 0 )ρ are identical. Also, by the isomorphism theorem (S + S ) (S + S ) = 2δ. Thus, δ = 0. Y Y 0 λ − Y Y 0 ρ

c We have seen that all the extreme points of Σab(Y ) have the same entropy. It follows that a non-extreme point, which is a convex combination of multiple extreme points, must have higher entropy. This fact follows from the general property of von Neumann entropy,

S(P p ρi) P p S(ρi), where p is a probability distribution with p > 0. The equality is i i ≥ i i { i} i achieved if and only if all ρi are identical. Thus, we have the following corollary.

Corollary 5.19.1. If the entropy of an element ρ Σc (Y ) is identical to that of an extreme Y ∈ ab c c point of Σab(Y ), ρY itself is an extreme point of Σab(Y ).

Proof of Theorem 5.5: In the following, we present the proof of Theorem 5.5.

Proof. Recall that we have already partitioned Y into BY 0; see Fig. 5.4. We shall consider two diﬀerent partitions of B. In Fig. 5.4, we have already considered a partition of B into

B = B1B2B3, which is a (disjoint) union of three annuli(B1,B2, and B3). We shall also consider a diﬀerent partition of B = BLBM BR. Here BL is a (disjoint) union of three “outermost” annuli and BR is a (disjoint) union of three “innermost” annuli; see Fig. 5.13.

83 In total, we are considering 9 disjoint subsets of B, Y 0, and E; see Table 5.2 for a detailed discussion on the partition of B. Here E is an auxiliary Hilbert space used to purify a density matrix supported on Y = BY 0.

Partitions B1 B2 B3 BR(Inner) B1R B2R B3R BM (Middle) B1M B2M B3M BL(Outer) B1L B2L B3L

Table 5.2: A partition of B used in the proof of Theorem 5.5.

E BL BM BR Y 0

Figure 5.13: A partition of B used in the proof of Theorem 5.5. This ﬁgure does not represent the actual underlying geometry. Rather, it represents the relative distance between the “inner” 0 0 part of Y (i.e., Y ) and the annuli surrounding Y (i.e., BR,BM , and BL). Auxiliary system E is introduced to purify the extreme points. Here BR is the “innermost” part of B that is directly 0 in contact with Y , BM is a disjoint union of annuli surrounding BR, and BL surrounds BM .

c c The statement is trivially true if Σab(Y ) is empty. In this case dim Vab = 0. For a nonempty Σc (Y ), we use σheix to denote the set of extreme points of Σc (Y ) and use λ for a generic ab { Y } ab Y c element of Σab(Y ). For the extreme points, the alphabet e signiﬁes that the density matrix is an extreme point. They are labeled by x, y, and z in this proof.

Pick an extreme point, say σheix . Let us purify σheix into ϕx and let ρ be its reduced Y Y | EY i EB density matrix on EB. From Corollary 5.15.1, one can verify I(E : B B B ) = 0. More- M R| L ρ over, for λ Σc (Y ) we have I(B : Y 0 B B ) = 0 and ρ =c λ . From the merging ∀ Y ∈ ab L | M R λ EB Y lemma(Lemma 4.6), there is a quantum channel ρ which deﬁnes a set of states EBL→EBL

ρ (λ ) λ Σc (Y ) (5.39) SEY ≡ {EBL→EBL Y | Y ∈ ab }

84 obtained from merging ρ with λ . It follows that Σc (Y ) = . Below, we will determine EB Y ab ∼ SEY the structure of . SEY Let us first show that the extreme points of are pure states. For the particular extreme SEY point we have already considered, i.e., σheix , the merged state is obviously the pure state ϕx . Y | EY i c Because all the extreme points in Σab(Y ) have the same entropy (Lemma 5.19) and because the entropy difference is preserved under the map ρ (property (3) of Lemma 4.6), all the EBL→EBL other extreme points are also mapped to pure states. This means the quantum channel ρ EBL→EBL c purifies all the extreme points of Σab(Y ) simultaneously. Now, we show is the state space of a finite dimensional Hilbert space. The nontrivial SEY statement is that any superposition of pure states in is again in . Once this statement is SEY SEY c verified, the finiteness of dimension follows straightforwardly from the fact that Σab(Y ) is finite dimensional. To prove this claim, we consider a normalized state

X ϕz z ϕyi , (5.40) | EY i ≡ i| EY i i

hei where ϕyi is the set of purifications (in ) of a finite (sub)set of extreme points σ yi {| EY i} SEY { Y ∈ Σc (Y ) . The complex numbers z can be arbitrary as long as ϕz is normalized. It is sufficient ab } i | EY i to show ϕz ϕz . A proof is done by the following steps. | EY ih EY | ∈ SEY

1. The reduced density matrix of ϕz on EB B is ρ . This is because (1) I(EB B : | EY i L M EBLBM L M Y 0) = 0 on ϕx and (2) ϕz = O ϕx for some operator O supported on Y 0. The | EY i | EY i Y 0 | EY i Y 0 ﬁrst equation follows from Corollary 5.15.1. The second equation follows from the fact

that ϕyi and ϕx have the same reduced density matrix ρ y . In fact, Eq. (7.1) | EY i | EY i EB ∀ i implies an explicit choice O = P z U i , where U i are unitary operators. Y 0 i i Y 0 Y 0

z 0 heiz 2. The reduced density matrix of ϕ on BM BRY , which we denote as σ , is an | EY i BM BRY 0 extreme point of Σc (B B Y 0). To see this, we ﬁrst observe that ϕz ϕz =c σ for ab M R | EY ih EY | b any µ-disk b (B B Y 0) . The logic to establish this fact is similar to that leads to ⊆ M R the point made above: (1) ϕx has vanishing correlation between b and EY b and (2) | EY i \ 85 z x heiz 0 ϕ = OEY \b ϕ for some OEY \b . Thus, σ Σ(BM BRY ). Its reduced density | EY i | EY i BM BRY 0 ∈ heiz matrix ρB determines the charge sectors a, b, c. The entropy of σ is identical to M BM BRY 0

c 0 heix that of known extreme points of Σ (BM BRY ), e.g., σ . Therefore, according to ab BM BRY 0

heiz c 0 Corollary 5.19.1, σ is an extreme point of Σ (BM BRY ). BM BRY 0 ab

3. The state ϕz has vanishing conditional mutual information I(B : B Y 0 B ) = 0. | EY i L R | M Therefore, its reduced density matrix on Y is uniquely determined from its reduced density

heiz z z matrices ρB B and σ (by Lemma 2.2). Therefore, TrE ϕ ϕ is the extreme L M BM BRY 0 | EY ih EY | point of Σc (Y ) obtained from an extension of σheiz . We denote this extreme point as ab BM BRY 0

heiz σY .

4. From the discussion above one can see, for any ϕz of the form (5.40), there exists an | EY i heiz c extreme point σY of Σab(Y ) such that

ϕz ϕz = ρ (σheiz ). (5.41) | EY ih EY | EBL→EBL Y

Thus, ϕz ϕz . | EY ih EY | ∈ SEY

We have proved that the set in Eq. (5.39) is the state space of some finite dimensional Hilbert SEY space. The Hilbert space depends on the purification, but its dimension cannot depend on this detail. The reason is that the state spaces of two finite-dimensional Hilbert spaces are isomorphic if and only if the dimension of the Hilbert spaces are the same and that = Σc (Y ). Therefore, SEY ∼ ab c c c we can assign an abstract finite dimensional Hilbert space V with dim V = N Z≥0, such ab ab ab ∈ that

c c Σab(Y ) = EY = (Vab). (5.42) ∼ S ∼ S

Here (Vc ) is the state space of Vc . This completes the proof. S ab ab

86 Chapter 6: Topological entanglement entropy

In this chapter, we calculate a certain linear combination of the entropies of the reference state. Its value only depends on the topology of the subsystem choice, and the relevant subsystems can be chosen to have a bounded size. This value is known as the topological entanglement entropy (TEE). If we assume the area law formula7

S(A) = α` γ, (6.1) − then the TEE is the sub-leading term γ. We independently show that the value γ for a disk is given by the well-known formula [6, 7]:

γ = ln , (6.2) D where is the total quantum dimension defined from our definition of the fusion multiplicities D N c . Our derivation is rigorous and it is based on axiom A0 and A1. We show this result { ab} by calculating two different linear combinations of subsystem entropies proposed by Kitaev-

Preskill [6] and Levin-Wen [7] respectively, see Fig. 6.1.

There were two known methods for deriving TEE: assuming an underlying ﬁeld theory description or explicitly calculating entropy in an exactly solvable model. Our method, on the other hand, shows that the area law formula itself implies the formula for TEE γ = ln . This D approach may apply to a larger class of systems. The ingredients behind this proof are scattered in literature [32, 33, 37, 10]. Recently, it is shown that the quantum dimension must show up

7This formula has been written down in Chapter 1 and 3. See Eq. (1.6) and Eq. (3.3). Therefore, we save the explanation of notation here.

87 A A B B B C C (a) (b)

Figure 6.1: (a) The Kitaev-Preskill partition; (b) the Levin-Wen partition.

in the von Neumann entropy if the fusion space is coherently encoded in the 2-hole disk [11].

Finally, Ref. [8], the accumulation of this line of eﬀorts, reduces the assumption to the minimal set Axiom A0 and A1.

6.1 The derivation of TEE

We begin by deﬁning the quantum dimensions in our framework.

Deﬁnition 6.1. We deﬁne the set of quantum dimensions d as the unique positive solution { a} of the equation set

X c dadb = Nab dc , (6.3) c where N c is defined in Definition 5.3. The total quantum dimension is defined as = ab D D pP 2 a da.

Note that given the results in Sec. 5.3, the uniqueness of Eq. (6.3) is guaranteed by the

Perron-Frobenius theorem, see e.g. appendix of [11] for a self-contained derivation. Furthermore,

d = d , d 1, d = 1. (6.4) a¯ a a ≥ 1 Recall that we have deﬁned a universal entropy contribution f( ) of a superselection sector · (Def. 5.2). We have the following proposition.

Proposition 6.1. The universal contribution to the von Neumann entropy from superselection sector a is given by

f(a) = ln da. (6.5)

88 The proof follows from Eq. (5.16) which is obtained from the merging process in Fig. 5.6.

Note that ef(a) is positive.

In Ref. [7], it is proposed that the conditional mutual information I(A : C B) for the partition | in Fig. 6.1(b) matches to 2 ln . Therein, it is calculated for a class of exactly solvable model D called the Levin-Wen model (also known as the string-net model) [26]. Here we show that the same formula also holds in our framework.

Proposition 6.2. For the Levin-Wen partition (Fig. 6.1(b)), it holds that

I(A : C B) 1 = 2 ln . (6.6) | σ D

Figure 6.2: Merging a pair of disks to obtain an annulus. Two disks are deformed so that, once merged together, they form an annulus.

Proof. Let us consider the merging process in Fig. 6.2, which obtains an annulus X from a pair of disks. Letσ ˜ Σ(X) be the element obtained from merging. It is in the center of Σ(X), i.e., X ∈ the maximal entropy element. Dividing X according to the Levin-Wen partition in Fig. 6.1(b), gives I(A : C B) = 0 because of the property of the merged state; see Lemma 4.6. | σ˜ Because of the simplex structure of Σ(X)(see theorem 5.1) and the fact that f(a) is equal to ln da (see Eq. (6.5)), we can expressσ ˜X as a convex combination of extreme points

X d2 ˜ = a a (6.7) σX 2 σX . a D

89 This formula is obtained by maximalizing the von Neumann entropy. From it, one derives

S(˜σ ) S(σ1 ) = 2 ln . It follows that Eq. (6.6) is true. X − X D

Proposition 6.3. For the Kitaev-Preskill partition,

γ (S + S + S S S S S ) (6.8) ≡ AB BC CA − A − B − C − ABC σω where ω = ABC, see Fig. 6.1(a), then γ = ln . D

The idea of the proof is to relate the Levin-Wen combination with two copies of Kitaev-

Preskill combinations.

Our logic to derive the formula γ = ln also applies to other entropy combinations. As an D exercise, we encourage the readers to derive the formula

(S + S S S ) = 2 ln (6.9) BC CD − B − D σ D for the partition BCD in the following ﬁgure:

B C B

6.2 Implications

Our derivation of the TEE shows that the “quantized” value γ = ln follows from our D axioms A0 and A1. Note that these axioms can be thought of as a simpliﬁed version of the area law formula Eq. (6.1). They can be derived from Eq. (6.1). They are simpler in the sense that they are only required to be satisﬁed for bounded-radius disks. Moreover, we only need the set of bounded-radius disks within a large enough disk region of length ∆, e.g., square E in Fig. 3.4.

Due to the small sizes of the involved subsystems, axiom A0 and A1 can be checked with much less computational power than the original area law formula Eq. (6.1). For these reasons, we

90 advocate the usage of axioms A0 and A1 over the original area law formula (6.1). When we say γ is quantized, what we refer to is the fact that not every real number is allowed for γ; if it is nonzero, it must be at least ln √2.

Do our results imply that the area law Eq. (6.1) and γ = ln holds for any ground state D of 2D gapped systems? No, they do not. We can derive the results if the ground state satisﬁes axiom A0 and A1. However, there are ground states of gapped systems which break at least one of our axioms. If a ground state has non-Abelian anyons pinned down to some ﬁxed positions, the quantum state can violate axiom A1. (It also violate the area law Eq. (6.1).) This violation of A1 can modify the identity (S +S S S ) = 0 to (S +S S S ) = 2 ln d BC CD − B − D σ BC CD − B − D σ a for bounded-sized disks that contain the anyon a. Note that 2 ln da is greater or equal to ln 2 if it is greater than zero. A very similar type of contribution applies to topological defects [56, 57].

(However, suppose that the ground state has isolated regions that violate A1, which is typically the case when we have isolated anyons or topological defects. In this case, we can still deﬁne the anyon theory and extract the anyon data using a large enough region that does not contain any non-Abelian anyons/defects. This is possible because our method extracts things locally. See

Chapter 3.3.)

Furthermore, a quantum state related to our reference state by a finite depth quantum circuit may have spurious contributions to the area law [58, 59, 60, 61]. For a carefully chosen circuit, the new quantum state can break A1 at all length scales. These violations may be pathological unless certain symmetries are imposed. However, they can persist in certain subsystem symmetry- protected phases [60]. Reconciling our framework with these systems remains an open problem for future investigation.8 Nonetheless, our result does shed some light on a related issue: if we check the quantum state on a finite length scale and verify A0 and A1, then it is guaranteed that TEE will not suffer from any spurious contribution on all larger length scales.

8One may attempt to show that the fusion and braiding data of anyon is invariant under a ﬁnite depth circuit. Suppose that the reference state satisﬁes both of the axioms before acting the circuit.

91 Despite of the subtlety of the 2D area law Eq. (6.1) (and axioms A0 and A1), they are conjectured to apply to very general 2D gapped physical systems at large enough length scales.

To our knowledge, the justiﬁcation remains challenging. The counterexamples mentioned above indicate that one of the challenges is the proper way to formulate the problem. In Chapter 11, we attempt to provide some thought on it.

Finally, our derivation of TEE says something concrete about the relation of TEE and the anyon theory. Previously, it is known that a topological order with anyons can have nontrivial

TEE (γ = ln ) [6, 7]. Subtlety remains on whether and when TEE can have a contribution D from things other than anyons for some systems and whether a nontrivial entanglement signature always implies that the system has anyons. Our results imply that we can tell the existence of anyons by checking several entropy conditions on a ground state (reference state). If the reference state satisﬁes axioms A0 and A1, then a (possibly trivial) anyon theory emerge. If, in addition, the TEE (for either partition in Fig. 6.1) is nonzero, a nontrivial anyon theory must emerge in such a quantum system.

92 Chapter 7: String operators and circuit depth

In this chapter, we establish the existence of unitary string operators that create an anyon pair and study the circuit depth. The derivation is based on axiom A0 and A1 in Chapter 3.

The result implies that a non-vacuum sector cannot be created by any local operator. Instead, it can be created by a deformable unitary string operator (Proposition 7.2). This fact justiﬁes the name “superselection sector” for the label set = 1, a, b, c, . C { ···} We will ﬁnd that the unitary string operator is useful in the study of the braiding properties, e.g., the topological S-matrix; see Chapter 8. It may also be useful in the extraction of topological spins, the F -symbols, and the R-symbols.

7.1 Heuristic discussions

In the context of the anyon theory, why are we allowed to consider string operators rather than operators of other shapes? Moreover, why should we expect that the string operators can be taken to be unitary? An astonishingly simple answer can be derived from our axioms (Axiom

A0 and A1). Before diving into the details, we would like to present a heuristic discussion. The key intuition comes from the Uhlmann’s theorem.

Uhlmann’s theorem [48]: Two pure quantum states ψ and ϕ have the same | ABi | ABi reduced density matrix on subsystem B if only if there is a unitary operator UA such that

ϕ = U I ψ . (7.1) | ABi A ⊗ B| ABi

We will use this result to show the existence of unitary operators in various settings.

93 Unitarity: An operator can be replaced by a unitary version of it when it acts on a quantum state that lacks a certain type of correlation. To see this, we consider the following lemma.

Lemma 7.1. Let ψ be a pure state that satisﬁes I(A : C) = 0. For any operator O , there | ABC i C exists a unitary operator UBC and a complex number c such that

1 O ψ = c 1 U ψ . (7.2) AB ⊗ C | ABC i A ⊗ BC | ABC i

Here 1AB and 1A are identity operators.

Suppose we are given a pure quantum state ψ , which may be a ground state of a 2D gapped | i many-body system. Let ABC be the partition of the 2D system shown below.

A B C

Here C is a string-like region; B separates A and C by at least a few correlation lengths; A can be large. One may think of BC as a thickening of C. One should expect9 the correlation between

A and C vanishes on the physical ground (I(A : C) = 0). It follows from Lemma 7.1 that for any operator (1 O ) supported on C, there is a unitary version of it (1 U ) which is AB ⊗ C A ⊗ BC supported on the thickened region BC. For this reason, one can consider unitary operators in a wide variety of settings.

When do we have string operators, and why are they deformable? Local indistinguishability is a key feature of the low energy states in topologically ordered systems. It provides the key intuitive answer to the question:

9The vanishing of the correlation I(A : C) = 0 follows from our axioms. More precisely, the enlarged version of A0, S + S S = 0, implies I(A : C) = 0. BC C − B 94 When do we have string operators, and why are they deformable?

Here we explain the basic intuition for a sphere. Let us consider low energy excitations of a sphere.

The number of excitations are 1, 2 and 3 respectively. In the spirit of local indistinguishability, let us assume that the excited state is identical to that of the ground state on any disk-like subsystem ω that does not contain any excitation.10

By considering large enough disk-like regions and use Uhlmann’s theorem, one derives powerful constraints to the support of the string operators. Speciﬁcally, we may choose the disk to be the blue region in the ﬁgure above. In this way, the support of the string operator is restricted to the complement of the disk.

This analysis suggests: (1) An isolated excitation on a sphere can be created by a local unitary operator around the excitation. Therefore, the excitation must carry the vacuum sector.

(2) A pair of excitations on a sphere can be created by a unitary string operator with two endpoints. (3) Three excitations on a sphere can be created by a unitary string operator with three endpoints. Moreover, because we have the freedom to deform the disk region, the support of the string operators can be deformed smoothly. A similar analysis can be carried out on other manifolds and anyon numbers as well.

10Local indistinguishability only require the states to agree on disks up to a bounded length scale. Nevertheless, if axiom A0 and A1 hold, one can show the indistinguishability holds on larger disks as well.

95 7.2 String operators from entanglement

In this section, we present a rigorous result based on axiom A0 and A1. Some ingredients of the proof has been explained in Sec. 7.1 already. Note that the result below works for any system topology. Recall that the reference state can be chosen to be a pure state; see Lemma 3.2.

Proposition 7.2. Given a pure reference state ψ , two holes within a disk and a , there | i ∀ ∈ C exists a deformable unitary string operator U (a,a¯) supported within the disk, such that the state

ϕ(a,a¯) U (a,a¯) ψ , (7.3) | i ≡ | i has topological charges a and a¯ in the two holes.

Here, deformable means the support of the string operator U (a,a¯) can be smoothly deformed while keeping its endpoints ﬁxed. It is easy to see, for a = 1, the string can be chosen to be the identity operator while for a = 1, the string cannot break apart. 6

Y W a a¯ U (a,a¯)

Figure 7.1: Disk ω is the union of 2-hole disk Y and its two holes. W is the complement of ω. The topological charges a anda ¯ within the two holes are created by unitary string operator U (a,a¯). The support of U (a,a¯) is the union of the deformable gray area and the two holes shown in red.

Proof. Let ω be the disk required in the proposition and W is its complement, see Fig. 7.1. One can verify that σ Tr ψ ψ Σ(W ) and that σ is an extreme point. The 2-hole disk W ≡ ω| ih | ∈ W Y ω is obtained by erasing the two holes from disk ω. From Proposition 5.9, Σ1 (Y ) contains ⊆ aa¯ 2

96 aa¯1 a unique element which we denote as σY2 , where Y2 is a thickening of Y . Here the subscript 2 means that Y is expanded along its boundary by 2.

aa¯1 The elements σW and σY2 , whose supports are overlapping around the boundary of ω, can be merged and the resulting state is an extreme point of Σ(WY2), where WY2 is again the

(a,a¯) (a,a¯) (a,a¯) thickening of WY by 2. Let ϕ = ϕ ϕ \ , where ϕ is an eigenvector (with | i | WY2 i⊗| V WY2 i | WY2 i nonzero eigenvalue) of the merged state and ϕ \ is an arbitrary pure state. According to | V WY2 i Lemma 5.14, the reduced density matrix of ϕ(a,a¯) on WY is identical to that of the merged | i state. Therefore, ϕ(a,a¯) has topological charges a anda ¯ within the two holes, and ϕ(a,a¯) is | i | i identical to the reference state ψ on any subsystem W 0 WY which is connected to W by | i ⊆ a path, where the path is within WY . In particular, we can choose W 0 to be the complement of disk ω0, where ω0 is the union of the gray string and the two holes in Fig. 7.1. Since ψ | i and ϕ(a,a¯) are identical on W 0, by applying Eq. (7.1), we have ϕ(a,a¯) = U (a,a¯) ψ for a unitary | i | i | i operator U (a,a¯) supported on ω0. Because we may deform W 0 and ω0, the support of U (a,a¯) can be deformed smoothly. This completes the proof.

In the proof above, we assumed that the µ-disks cover the whole manifold. In fact, this assumption is not necessary. We only need the set of µ-disks to cover a large enough disk region that contains ω, (e.g., ω10). The proof is essentially the same as the proof above, but they diﬀer slightly in the details. We leave the proof to the readers.

7.3 Circuit depth of the string operators

In this section, we study the circuit depth of the string operator shown to exist in Proposi- tion 7.2. The circuit depth of a unitary operator is a measure of how complex a unitary operator is, from the viewpoint of a ﬁxed basis (i.e., a lattice in real space). Therefore, by studying the circuit depth, we can get an estimation of the computational cost of creating an anyon anti-anyon pair and separate them by a certain distance.

97 The spirit of this discussion is similar to the study of minimal circuit depth to convert a topologically ordered ground state into a product state [30, 71]. Because two states within the same gapped phase may (approximately) be related by a ﬁnite-depth quantum circuit, the circuit depth of unitary string operators (either ﬁnite depth or a depth scale with some length scale) is a universal property of a gapped phase.

In many exactly solvable models, an Abelian anyon string is a ﬁnite depth quantum circuit.

The depth is independent of the distance separations of the anyons. (In fact, usually, the depth equals one.) However, these exactly solvable models are non-chiral. To our knowledge, there is no proof of whether Abelian anyon strings are ﬁnite-depth quantum circuits in general.

We present a concrete result for non-Abelian anyons. The circuit depth of a non-Abelian anyon string is at least linear to the distance separation of the anyon pair. (See Ref. [80] for a derivation of a closely related result.)

As a side note, in the presence of gapped boundaries, this statement generalizes and changes.

Essentially, a similar statement applies to non-Abelian boundary excitations, whereas the gapped boundary may lower the computational cost for separating some of the non-Abelian bulk anyons.

We refer the interested reader to [11].

(a) (b)

l A B C A0 C0 L M R a¯ a b ¯b a¯ a b A0 C0

¯b

Figure 7.2: (a) Anyon pairs (a, a¯) and (b, ¯b) are independently created on the reference state. (b) The annulus X = LMR detects the fusion result of a and b. L, M and R are concentric annuli and L and R are separated by distance l.

98 Proposition 7.3. The unitary string operator U (a,a¯) in Fig. 7.2(b) has a circuit depth at least

(l )/2. −

Proof. The key idea is to show if the circuit depth of U (a,a¯) is smaller, there will be a contradiction. A small depth circuit cannot generate mutual information between distant annuli L and

R in Fig. 7.2(b) when it acts on the reference state.

¯ a×b First, we notice that the quantum state with anyon pairs (a, a¯) and (b, b) is identical to σY on the 2-hole disk Y = ABC in Fig. 7.2(a). The argument goes as follows. The reference state

σ has conditional independence I(AA0 : CC0 B) = 0 for the partition in Fig. 7.2(a). We apply | the unitary string operators U (a,a¯) and U (b,¯b) to the reference state. The support of the string operators and the locations of the anyons are depicted in Fig. 7.2(a). Let the new quantum state beσ ˜(a,b). Explicitly, we have

(a,b) (a,a¯) (b,¯b) (a,a¯) (b,¯b) † σ˜ABC = TrA0C0 (U U )σAA0BCC0 (U U ) (7.4)

(a,b) a×b On the 2-hole disk Y = ABC, stateσ ˜Y is nothing but the merging result σY in Fig. 5.6. This

0 0 is because (1) I(A : C B) (a,b) = 0, which follows from SSA, I(AA : CC B) = 0 and Eq. (7.4). | σ˜ | σ (a,b) a×b (2) I(A : C B) a b = 0. (3) The reduced density matrices ofσ ˜ABC and σABC are identical on | σY× annuli AB and BC.

Therefore, on the outer boundary of the 2-hole disk, the probability of ﬁnding the fusion result c, adapted from Eqs. (5.17) and (6.5), is

c Nabdc P(a×b→c) = . (7.5) dadb

This probability can be interpreted as the probability for a pair of “independently created” anyons a and b to fuse into c. Here, we say two anyons are independently created if they are created by diﬀerent string operators. The support of these string operators are separated by a length larger than the correlation length.

99 Now, consider a non-Abelian anyon a and take b =a ¯. For an non-Abelian anyon, the fusion result with its antiparticle must have multiple possible outcomes, namely a a¯ = 1 + ; at × ··· least one non-vacuum sector exists on the right-hand side.

Then, it follows from Proposition 5.17 that the stateσ ˜(a,b) must have I(L : R) > 0. The string operator U (a,a¯) must have circuit depth at least (l )/2 because an operator with a lower − circuit depth cannot generate nonzero mutual information between L and R when acting on the reference state.

100 Chapter 8: Verlinde formula from entanglement

The main result of this chapter is the definition of a quantity for a pure reference state ψ , | i which is identified with the topological S-matrix of the underlying anyon theory. Under our assumption (axioms A0 and A1), we show the S-matrix we define is unitary, and it recovers the fusion multiplicities through the Verlinde formula

c X SaxSbxScx¯ Nab = , (8.1) S1x x∈C where the components of the S-matrix, S with a, b , has S = da and the following ab ∈ C a1 D symmetries

∗ Sab = Sba,Sab = Sab¯ . (8.2)

∗ Here Sab¯ is the complex conjugation of Sab¯ . If a mixed reference σ is given, then according to Lemma 3.2, it is possible to choose a pure reference state ψ , which is consistent with σ. There are multiple choices of ψ . However, it | i | i is easy to show that the topological S-matrix we deﬁne is independent of the choice. What we need is the reduced density matrix on a local region (e.g., region E in Fig. 3.4).

This establishes the modularity of the theory. It is tied to the braiding nondegeneracy, which is the statement that a nontrivial sector must braid nontrivially with at least one other sector.11

It corresponds to an independent axiom of the algebraic theory of anyon 12. The result in

11As a reminder, modularity and braiding nondegeneracy are expected to be true for bosonic models. Fermionic models can behave diﬀerently. Our setup and axioms are expected to capture bosonic models but not fermionic ones. 12It does not follow from the pentagon equation or the hexagon equations.

101 this chapter shows that the mutual braiding statistics of a non-vacuum sector in must be C nontrivial. These sectors have not only nontrivial fusion rules but also have nontrivial mutual braiding statistics. This fact justiﬁes calling them anyons.

It is intriguing to compare our derivation with a previous well-known derivation in the framework of 1+1D unitary CFT [3, 81, 82]. In the framework of CFT, the general form of Verlinde formula (8.1) is derived by Erik Verlinde in Ref. [81], wherein the modular S-matrix is shown to fully determine the fusion rules of the primary ﬁelds. The underlying assumption is conformal symmetry, and the modular S-matrix is related to the modular transformations of a torus. In the context of 2D gapped phases, the same algebraic relation comes out, but here the sectors

(anyons) and topological S-matrix have diﬀerent physical interpretations than CFT. We derive the same formula from an assumption on entanglement (axioms A0 and A1). Our assumption is arguably natural for 2D gapped phases, whereas the conformal symmetries are expected to describe very general classes of critical systems.

8.1 Our deﬁnition of the S-matrix

We deﬁne Sab as follows: d d S a b f , (8.3) ab ≡ ab D f Tr(U a†U a σb ). (8.4) ab ≡ L R X

a a Here X is an annulus, UR is an operator which creates a pair of anyons (a, a¯), see Fig. 8.1. UL is obtained from U a by a deformation on the reference state, namely we require that U a ψ = R L| i U a ψ . (Recall that these deformable unitary string operators are shown to exist in Chapter 7.) R| i b The density matrix σX is an extreme point of the information convex set Σ(X). By deﬁnition, Tr(U a†U a σ1 ) = 1. This implies that f = 1, a. L R X a1 ∀

This Sab is well-deﬁned in the sense that it is invariant under the deformation of three things: the annulus X, the support of the strings, the positions of the anyons. To establish this fact, we

b ﬁrst notice that the extreme point σX can be obtained by acting string operators on the reference

102 a¯ a¯

a a (a) (b)

a Figure 8.1: An annulus X and string operators supported within it. (a) String operator UR a which creates a pair of anyons a anda ¯ on the reference state. (b) String operator UL is obtained a by deforming UR on the reference state.

state. Therefore, one can rewrite Eq. (8.4) as an expectation value of four string operators. First, for generic deformable unitary strings (see Fig. 8.2), we deﬁne

f(U, V ) ψ U † V †U V ψ . (8.5) ≡ h | L R R L| i

It recovers fab when the strings carry ﬁxed sectors, i.e.

f = f(U a,V b) = ψ U a†V b†U a V b ψ . ab h | L R R L| i

Because these string operators act directly on the reference state (either to the left on ψ or to h | the right on ψ ), small deformation of any one of them will leave f(U, V ) invariant. Moreover, | i modifying the string operators by a slight change of the position of an anyon (without passing the anyon through another string) will not aﬀect the value of f(U, V ). This is because applying this new string operator on the reference state is equivalent to applying the original one and then applying an additional unitary operator supported on the union of two disk-like regions.

In the expectation value (8.5), these additional unitary operators are canceled.

Using the trick of deforming the string operators, taking a partial trace and making use of the aforementioned invariant property, one ﬁnds

∗ fab = fba, fab = fab¯ . (8.6)

103 VL VR

UR UL

(a) (b)

Figure 8.2: Two distinct ways to create four sectors: (a) with UR and VL, (b) with UL and VR. Here UL ψ = UR ψ and VL ψ = VR ψ . Depending on the context of the discussion, an operator may| eitheri corresponds| i to| i a string| carryingi a ﬁxed sector or a string bundle.

In more detail, to verify these identities, one can diagrammatically represent both sides of the identity and then smoothly deform one to another. The deformation involves both the strings and the anyon positions. These identities imply that our deﬁnition of S-matrix obeys the requisite symmetries (8.2).

8.2 The proof of the Verlinde formula

To facilitate the proof, we remark on the approach of deriving (8.6). First, the deformation of string operators and taking a partial trace allows us to obtain a quantity in a few diﬀerent ways. By matching these results, one can derive a constraint. Second, the deformation of a string operator is a rather general property. It works not only for a string which carries a ﬁxed sector but also for a string bundle, which is a product of multiple string operators with disjoint supports (see Fig. 8.3). By applying the idea above to string bundles, we obtain the following proposition.

Proposition 8.1. The S-matrix we deﬁne satisﬁes

X SaxSbx N c S = . (8.7) ab cx S c 1x

104 (a) (b)

Figure 8.3: (a) A single string. (b) A string bundle. In this particular ﬁgure, the string bundle consists of two strings.

c P Nabdc Proof. We show that P × → f = f f , then Eq. (8.7) follows. Here P × → = . c (a b c) cx ax bx (a b c) dadb Let us consider f(U ab,V x), where U ab is a string bundle consists of two strings with sector a and b, and V x is a string with sector x. We calculate f(U ab,V x) in two ways.

First, we have ab x ab† ab x f(U ,V ) = Tr(UL UR σX ) (8.8)

= faxfbx. x† In the ﬁrst line, we have deformed VR and done a partial trace such that the remaining subsystem

ab ab X is an annulus containing UL and UR . In the second line, we have used the fact that the

x extreme point σX is “factorizable.” Speciﬁcally, in Corollary 5.16.1, it was shown that the extreme points of an annulus, restricted to disjoint sub-annuli, have a tensor product form. Therefore,

ab ab a a b b we can split UL and UR into two families (UL,UR) and (UeL, UeR) so that the string operators for a and b are supported on disjoint sub-annuli. Because operators belonging to diﬀerent families commute with each other, the expectation value becomes:

ab† ab x a† a b† b x Tr(UL UR σX ) = Tr(UL URUeL UeRσX )

a† a x b† b x = Tr(UL URσX1 )Tr(UeL UeRσX2 ), where X ,X X are disjoint sub-annuli of X. 1 2 ⊆ ab ab Second, we deform the string UR in the same manner. Because the string bundle U can

c Nabdc produce sector c on an annulus surrounding a and b with probability P × → = , see (a b c) dadb Eq. (7.5), we obtain the following expression:

ab x X f(U ,V ) = P(a×b→c)fcx. (8.9) c By matching the two expressions (8.8) and (8.9) one obtains Eq. (8.7).

105 Note that Proposition 8.1 in itself does not imply modularity (i.e., that the S-matrix is unitary). For example, a solution like f = 1, a, b is consistent with Eq. (8.7) but it leads to ab ∀ a noninvertible S-matrix. We need a concrete statement on the nontrivial braiding. The key is the following lemma.

2 P da a Lemma 8.2. Let σ˜X = a D2 σX be the maximal-entropy element of Σ(X), then

a† a Tr(UL URσ˜X ) = δa,1. (8.10)

The proof Lemma 8.2 is presented in the latter part of this section. Based on Lemma 8.2, we show that the S-matrix is unitary, and we further derive the Verlinde formula.

Proposition 8.3. The S-matrix is unitary and the Verlinde formula (8.1) holds.

Proof. We only need to show that the S-matrix is unitary. The Verlinde formula follows from P unitarity and Eq. (8.7). Eq. (8.10) implies that x S1xSax = δa,1. Multiplying S1x to both

1 P sides of Eq. (8.7), do the sum of x, and use Nab = δb,a¯, one derives that x SaxSbx = δb,a¯. This, together with the symmetry properties (8.2), implies that the S-matrix is unitary. This completes the proof.

The same logic applies to a generic string bundle, and the end result is the Verlinde formula for a generic number of sectors.

The proof of Lemma 8.2:

Proof. For a = 1, Eq. (8.10) is trivially true. In order to derive Eq. (8.10) for the case of a = 1, we consider the merging process described in Fig. 8.4(a). We merge two reduced density 6 1 matrices of the reference state, namely σABC and σCD, where C = C1C2. We call the density matrix obtained by this merging as τABCD. (The two states can be merged because they are identical on C and the conditional mutual information I(AB : C C ) = I(C : D C ) = 0 for 2| 1 1 | 2 the reference state.) Note that, while ABC and CD are subsystems of the original physical system, the support of the merged state is not. This is because A and D on the original physical

106 A

C2 C1 a¯ a¯ B

C1 C2 a a (a) (b) (c)

1 Figure 8.4: (a) The merging of σABC and σCD, where C = C1C2. ABCD is not a subsystem of the original system, and it has a topology equivalent to a torus with one hole. (b) The unitary a a string operator UR is supported on BC. (c) The unitary string operator UL is supported on CD. a It is obtained from the deformation of UR.

system overlaps nontrivially, yet in the merged state, they do not share any common region.

For the state τABCD, A and D belong to diﬀerent Hilbert spaces. What is important here is that the merged state τABCD exists, even though one cannot obtain such a state by tracing out subsystems from the original physical system.

Let us consider the reduced density matrices of τABCD on annuli ABC and BCD. TrDτABCD =

1 σABC carries the vacuum sector. TrAτABCD =σ ˜BCD is the maximal-entropy element of Σ(BCD).

a a After applying UR or UL onto τABCD, see Fig. 8.4, the sectors seen on AB are

a a† 1 TrCD(ULτABCDUL ) = σAB,

a a† a¯ TrCD(URτABCDUR ) = σAB.

Thus, the two density matrices U aτ U a† and U a τ U a† are orthogonal for a = 1. This L ABCD L R ABCD R 6 fact follows from that σ1 σa¯ for a = 1 and the monotonicity of ﬁdelity. In more details, for AB ⊥ AB 6 a = 1 we have 6

F (U aτ U a†,U a τ U a†) F (σ1 , σa¯ ) = 0 U aτ U a† U a τ U a†. L ABCD L R ABCD R ≤ AB AB ⇒ L ABCD L ⊥ R ABCD R

Therefore,

Tr(U a†U a σ˜ ) = Tr(U a†U a τ ) = 0, a = 1. L R BCD L R ABCD ∀ 6 107 Since BCD can be any annulus, Lemma 8.2 is justiﬁed.

We would like to remark on a counterintuitive aspect of the proof. A careful reader may

1 1 imagine another quantum state, ρABCD, which reduces to σABC and σBCD. Then, by applying

a† a 1 the same logic, one seems to get a contradiction, namely Tr(UL URσBCD) = 0. It is not a real contradiction. Instead, it means that such a state ρABCD cannot exist. This phenomenon can be understood from the entropic “uncertainty principle” in Ref. [31, 36], which implies that annuli

ABC and BCD cannot both obtain the vacuum sector. Note that the ABCD in Fig. 8.4(a) is topologically equivalent to the 1-hole torus considered in [31]. In comparison, our method does not make use of the global topology of the system, and τABCD is constructed given the reduced density matrices within a disk region. It makes our method applicable to a broader context.

From our discussion, the topological S-matrix can, in principle, be extracted locally in a single quantum many-body wave function. (See Sec. 3.3 for a discussion on what we mean by local extraction.) This result should be contrasted with [31], which makes use of multiple ground states on a torus. We would like to compare our result with another recent attempt [71] to deﬁne the S-matrix from one single ground state. It makes assumptions concerning the

Hamiltonian and operator algebra. As the author remarks, the method therein requires the assumption of unitary modular tensor category description to complete the argument that the invariant constructed matches the S-matrix. In comparison, with axiom A0 and A1, we are able to deﬁne the S-matrix and derive the Verlinde formula it obeys.

108 Chapter 9: Discrete version of the framework

In previous chapters, we have discussed a theoretical framework that aims to provide a derivation of a general theory of anyon for gapped many-body systems without symmetries.

Both the fusion rules and mutual braiding statistics of anyons are studied. The underlying input is a reference state (σ), which satisﬁes two local entropic conditions (axiom A0 and A1).

The subsystem choices and their topologies are formulated in an abstract way.

In this chapter, we work on a particular lattice. We explicitly write down our axioms for the honeycomb lattice. Note that the fact that we can choose the honeycomb lattice does not mean the underlying physical system has the particular symmetry of the lattice.13 What we do is to group the local degrees of freedom into coarse-grained “supersites”. (The idea of choosing a coarse-grained lattice and study the entropy conditions on it has been explored in literature; see Ref. [12, 13].)

We state our results on this lattice. The veriﬁcation of these statements is left as exercises for the readers. However, solving these exercises are essentially translating the results of Chapter 3,

4, 5 and 6 into this lattice. No additional technique is necessary. It is worth noting that we make it clear that a minimal thickness (layers) of the subsystem is needed in our proof of the isomorphism theorem and the structure theorems. This is a technical point which might easily be overlooked in the previous abstract formulation. However, it is essential to understand our proofs. Furthermore, the string operators (Chapter 7) and the possible choices of Ω, which are

13Whether the system is embedded in strictly Euclidean space should not matter either. However, there can be exceptions. For example, it is not clear if the honeycomb lattice can apply to a hyperbolic plane when the correlation length is larger than the radius of curvature. In that case, however, it is expected that a hyperbolic tiling works.

109 not subsystems (Chapter 8) also have discrete formulations. However, we choose to omit related discussions.

The discussion in this section might be useful for numerical studies because discrete lattice provides explicit subsystem choices and reduces such choices to a small set. However, to proceed, one may further develop a method for approximately satisﬁed axioms.

9.1 Setup and axioms on the lattice

We work with the honeycomb lattice. Each hexagon is of a size larger than the correlation length of the physical system. It contains a collection of physical spins; see the following ﬁgure.

(9.1)

We require the µ-disks to be the disk-like regions containing seven hexagons. Axioms A0 corresponds to a unique entropic condition shown below. Axiom A1 corresponds to the three entropic conditions below together with those with subsystems rotated.

(9.2)

We assume there is reference state σ for which the discrete version of the axioms hold.

Now we are ready to compare this discrete version with Fig. 3.1. We explain what the length scales , r and ∆ correspond to in the discrete lattice. The length scale corresponds to one lattice spacing. Let us write = 1. Then, the length scale r 1.5, and this can be seen from ' the size of the µ-disks. How large ∆ needs to be for the honeycomb lattice? From the subsystem

110 choices discussed in this chapter, it is not diﬃcult to convince oneself that ∆ = 50 is large enough. Therefore, the length scale ∆ is several times larger than and r. (We do not care about the minimal value of ∆ in this work. The important thing is that it can be chosen to be a ﬁnite number times the correlation length.)

9.2 Reference state properties

From the discrete version of axiom A0 and A1, we can derive the following properties of the reference state. (These results are the discrete versions of the discussion in Chapter 3.2.)

Conditional independence: We have the conditional independence I(A : C B) = 0 for | the reference state for the following partitions.

(9.3)

In this ﬁgure, two choices of ABC are shown, where ABC is a ﬁnite-sized subsystem. A and C are separated by B.

Enlarged version of the axioms: The enlarged version of axiom A0 is satisﬁed for the following conﬁgurations:

(9.4)

The enlarged version of axiom A1 is satisﬁed for the following conﬁgurations:

(9.5)

111 9.3 The information convex set

We provide explicit subsystem choices for the information convex set. To familiar the readers with our deﬁnition, both valid choices and invalid choices of subsystems are discussed. (The discussion in this section is a discrete version of the discussion in Chapter 4.1, which concerns the deﬁnition of the information convex set.)

Valid choices of subsystems: Below are some valid choices of Ω for the information convex set Σ(Ω).

Below are some valid choices of disk-like subsystems ω. The thickenings of the disks are shown.

(9.6)

0 Observe that both the dark blue region ω and its thickening ω = ω are disks. We can write

ω = ω ∂ω, where ∂ω is the light blue region along the boundary of ω. t Here is a valid choice of subsystem X which is topologically equivalent to an annulus. It is

0 a valid choice for the same reason: the subsystem X and its thickening X = X both have an annulus topology.

(9.7)

112 Invalid choices of subsystems: Below is an invalid choice of disk. It is an invalid choice because the thickening of Ω is not deﬁned. (Recall that we require the thickening to have the same topology as the original region.)

(9.8)

Below is an invalid choice of annulus because it is impossible to pick a thickening that has the same topology as the original region.

(9.9)

9.4 The isomorphism theorem

There can be choices of subsystem Ω for which Σ(Ω) is deﬁned whereas the isomorphism theorem (Theorem 4.8) and structure theorems (e.g., Theorem 5.1, 5.4 and 5.5) do not apply.

The intuitive reason is that only for subsystems that are “thick enough,” can we prove the isomorphism theorem.

Once the isomorphism theorem is proved, we know these subsystems can be “smoothly” deformed, and the structure of the information convex set remains unchanged. Our proofs of the structure theorems need some minimal thickness as well because that the isomorphism theorem is used in these proofs.

The goal of this section is to provide some explicit examples.

113 We explicitly point out that for some choices of the subsystems below, we do not have a • proof of the isomorphism theorem. (These subsystems are “too thin”.)

We provide some explicit choices of the subsystems for which the isomorphism theorem • applies. (These subsystems are “thick enough”.)

Note that, it is an open problem whether the isomorphism theorem actually extends to some of the thin subsystems. (We do not have proof, but we do not have a counterexample either at the moment.)

Subsystem choices to which the isomorphism theorem does not apply: The following two choices of annuli are too thin for the isomorphism theorem to apply. Although the information convex set Σ(Ω) is well-deﬁned for both of them.

(9.10)

In this figure, the single hexagon attached to the boundary of Ω (shown in a different color) is the place where an elementary step of extension is attempted. We do not have a justification for this elementary step. However, if the annulus is thicker, the isomorphism theorem applies.

See below for an example.

114 Subsystem choices to which the isomorphism theorem applies: Below is a choice of annulus subsystem to which the isomorphism theorem (Theorem 4.8) applies.

(9.11)

Recall that the proof of the isomorphism theorem for Σ(Ω) is subtle. It involves a sequence of nontrivial considerations in Sec. 4.2 and 4.3. In particular, we need to consider a deformation of subsystems (ABCD AB0C0D), as is shown in Fig. 4.5. We also need an alternative deﬁnition ↔ of information convex set, i.e. Σ(Ω)ˆ (Deﬁnition 4.3). The proof is both important and subtle.

Therefore, it is beneﬁcial to see a more concrete discussion of it. Here we provide such a concrete illustration.

We derive the isomorphism theorem for another object Σ(Ω)ˆ (Deﬁnition 4.3). Then, one can follow the logic in Proposition 4.12 to derive the isomorphism theorem of Σ(Ω). This is because we can show Σ(Ω)ˆ and Σ(Ω) are equivalent for suitable choices of Ω, including the

3-layer annulus shown above.

How do we establish the isomorphism theorem for Σ(Ω)?ˆ We can consider simpler steps.

In particular, we can consider an elementary step of extension. For the lattice we consider, an elementary step of extension involves the attachment of a hexagon into Ω without changing its

115 topology. See the ﬁgure below.

(9.12)

The extra hexagon represents the place where an elementary step of extension takes place. (More explicitly, the partition in the middle ﬁgure is the discrete version of the ABCD of Fig. 4.6.)

The two partitions (ABCD and AB0C0D), shown in the middle and right ﬁgure, are relevant to the merging technique. As is observed in Proposition 4.10, the alternative partition B0C0 = BC

ˆ ˆ 0 is crucial to justify the isomorphism Σ(Ω) ∼= Σ(Ω ). By repeatedly using the elementary steps and the inverse steps, one can generate more general deformations of Ω.

Alternatively, one may attach several hexagons simultaneously; see below. Although the same end result can be achieved by a sequence of elementary steps, this method deforms the subsystem in bigger steps. Therefore, it can be more eﬃcient.

(9.13)

In the left figure, the hexagons in a different color are attached to the annulus. The middle and right figure shows the partitions (ABCD and AB0C0D) needed in the proof of isomorphism theorem.

116 9.5 Structure theorems

Below, we discuss subsystem choices for which the structure theorems can be proved.

For a disk-like region ω, which has a well-deﬁned information convex set, the structure • theorem Σ(ω) = σ holds. { ω}

For an annulus subsystem, we can prove the simplex theorem (Theorem 5.1) only if the • annulus is thick enough. This is because the proof of the simplex theorem needs the

isomorphism theorem. As an example, the information convex set of the 3-layer annulus

in (9.11) obeys the simplex theorem.

For the 2-hole disk shown below, the information convex set is well-deﬁned. Moreover, the • information convex set for this 2-hole disk obeys the isomorphism theorem (Theorem 4.8)

and structure theorems (Theorem 5.4 and 5.5).

(9.14)

9.6 Merging with a change of topology

We provide several examples of nontrivial merging processes. These merging processes induce topology changes in the subsystems. (Recall that these merging processes have played a key role in the derivation of the axioms of the fusion rules in Chapter 5.3. Also see Chapters 6, 7 and 8.)

In more detail, we consider subsystems ABC, BCD and ABCD, for which the information convex set is deﬁned, and the structure theorems hold. Certain elements of Σ(ABC) and

Σ(BCD) can be merged and the merging result is an element in Σ(ABCD). Usually, a key step

117 in determining whether the merging result is an element of Σ(ABCD) is the construction of a partition B0C0 = BC, which satisﬁes a set of requirements. See Proposition 4.10 and 4.7. We provide explicit illustrations of such partitions ABCD and AB0C0D.

Patching a hole of annulus: If an annulus is in the vacuum sector, we can patch the hole.

Below is a the subsystems involved in this merging process.

(9.15)

This ﬁgure is a discrete version of the merging process in Fig. 5.5.

The merging result is the reference state σω, the unique element of Σ(ω), where ω = ABCD is the disk. Note that, for this example, there is no need to construct a diﬀerent partition

B0C0 = BC.

Merging a pair of disks into an annulus: The merging result in the ﬁgure below is the “center” of the simplex Σ(X). More precisely speaking, the center is the maximal-entropy

2 P da a element of Σ(X). It has the explicit formσ ˜X = a D2 σX . (See Eq. 6.7.)

(9.16)

This ﬁgure is a discrete version of the merging process in Fig. 6.2. The subsystem partitions

X = ABCD = AB0C0D are explicitly shown.

Merging a pair of annuli into a 2-hole disk: Below is the merging of a pair of annuli into a 2-hole disk. In this process, any pair of elements of Σ(ABC) and Σ(BCD) can be merged.

118 This is because any choice possesses the required conditional independence.

(9.17)

This ﬁgure is a discrete version of the merging process in Fig. 5.6. The subsystem partitions

ABCD and AB0C0D are explicitly shown.

9.7 Topological entanglement entropy

Let us start by asking a question:

Question: Given a reference state on the discrete lattice, for what subsystem size can we calculate the topological entanglement entropy and learn the total quantum dimension of the anyon theory?

There can be two possible answers to this question: (i) The ln contribution holds only for D “thick enough” subsystems. (ii) The ln contribution already shows up at the smallest lattice D scale on the discrete lattice.

Our ﬁnding in this section is that the latter answer holds. The TEE ln already shows up D at the smallest lattice scale, either it is the Kitaev-Preskill partition or the Levin-Wen partition.

We present a sketch of the logic which leads to this result. (The subsystems shown below are discrete versions of those in Fig. 6.1.)

119 Levin-Wen partitions: First, we notice that below is a valid Levin-Wen Partition.

(9.18)

What we mean is that if we calculate the Levin-Wen combination of entropy on the reference state, we ﬁnd I(A : C B) = 2 ln . | D Because axioms A0 and A1 and the enlarged versions hold on the reference state, we can deform the subsystems “smoothly” without changing the value of the entropy combination. In fact, it is possible to shrink the subsystems to the minimal choice below:

(9.19)

For this partition, we still have I(A : C B) = 2 ln on the reference state. | D Kitaev-Preskill partitions: Similarly, one can verify that valid Kitaev-Preskill partitions range from very small ones (containing 3 hexagons) to very large ones. See the ﬁgure below for examples.

(9.20)

For all these partitions, S + S + S S S S S = ln for the reference state. AB BC CA − A − B − C − ABC D For the idea of the proof, we simply recall that we are able to convert a Levin-Wen partition into two copies of Kitaev-Preskill partitions.

120 Chapter 10: Information convex set and Hamiltonians

The main topic of the thesis is an attempt to understand the general anyon theory in 2D gapped many-body systems. By deﬁnition, a gapped many-body system is described by a

Hamiltonian with an energy gap. On physical ground, the Hamiltonian is often assumed to be local (or with the interaction strength decay fast with distance). In the theoretical framework described in previous chapters, we used a ground state as the input. We formulated the ground state with a plausible assumption on the entanglement (axioms A0 and A1). The rest of the derivation is completely Hamiltonian-independent.

More precisely, every statement in Chapter 3, 4, 5, 6, 7 and 8 are Hamiltonian-independent.

(The same for the statements in the discrete version of the framework, which we discussed in

Chapter 9.) Making use of the key concept information convex set, we can deﬁne the superselection sectors, their fusion rules, and the mutual braiding statistics with the sole input: the reference state. We need a mild property of the Hamiltonian only when we want to claim that the anyons (e.g., the red dots in Fig. 7.1) are excitations localized in space; the relevant property is that the Hamiltonian is (quasi) local.

The goal of this chapter is to define and calculate the information convex set for certain given Hamiltonians. A Hamiltonian-based definition of information convex set is presented for frustration-free Hamiltonians (Definition 10.2). Under relatively general conditions, we prove that the Hamiltonian-based definition of the information convex set matches the previous quantum state-based definition (Definition 4.1). See Theorem 10.1.

121 The Hamiltonian-based deﬁnition of the information convex set can be used to do calculations in exactly solvable models based on a Hamiltonian. We brieﬂy discuss one such calculation.

The exactly solvable model we consider is a class of commuting projector model known as the quantum double models [23]. In fact, by comparing the explicit calculation with the previously known UMTC description of the quantum double models, the structure theorems of the information convex sets can be veriﬁed explicitly. In Ref. [10], this analysis is carried out for both the bulk and the (untwisted) gapped boundaries of the quantum double models.

We have seen that the deﬁnition based on the quantum state can give a powerful constraint for the structure theorem of the information convex set (Chapter 5). That line of thinking gives further constraints to the underlying anyon theory. In comparison, we do not have a way to derive structure theorems of the information convex sets with merely the frustration-free requirement of the Hamiltonian.

Nevertheless, the Hamiltonian approach can be useful to carry out concrete calculations and to study the universal properties of quantum phases. In fact, among the numerous of exactly solvable models, many models are beyond 2D bosonic gapped phases and the 3D analog.

Therefore, they are not covered by the framework we develop previously. Examples includes 2D gapped phases with fermionic local degrees of freedoms [83], fractons [84, 85], 3D gapped phases which has anyons on its 2D surface [86, 87], etc.

10.1 Information convex set for frustration-free Hamiltonian

We consider a general setup of lattice models. Similar to the setup of Chapter 3, we assume that the total Hilbert space is the tensor produce of local Hilbert spaces = ; each local H ⊗vHv Hilbert space is ﬁnite-dimensional. We require the Hamiltonian to be local. In other words, it P is possible to write the Hamiltonian as H = i hi, where each term hi is supported within a bounded-sized region of the lattice.

122 Frustration-free local Hamiltonians: The Hamiltonian for most physical systems are frustrated. The terms in the Hamiltonian do not obtain their minimal eigenvalues simultaneously on a ground state. Nevertheless, many nontrivial quantum phases have a representative frustration-free local Hamiltonian. In fact, several important classes of exactly solvable models of topologically ordered systems are frustration-free local Hamiltonians, e.g., the quantum double models [23] and the string-net models [26].

We use the following deﬁnition of frustration-free local Hamiltonian; one may refer to Ref. [88] for this deﬁnition.

Deﬁnition 10.1 (Frustration-free local Hamiltonian). A frustration-free local Hamiltonian is a P Hamiltonian written as H = i hi, which satisﬁes

1. Each hi is a Hermitian operator supported within a bounded-sized disk. The minimal

eigenvalue of each hi is 0.

2. h P = 0, i, where P is the projector onto the subspace of ground states of H. i 0 ∀ 0

Remark. The requirement that the minimal eigenvalue of hi to be 0 is only for simplicity. One can always add a constant to h to satisfy this condition. The condition h P = 0, i implies that i i 0 ∀ every h obtains its minimal eigenvalue 0 on a ground state ψ . i | i

For later convenience, we deﬁne HΩ0 be the operator obtained by keeping terms of H that are supported on subsystem Ω0. It follows that the ground state ψ minimizes the Hamiltonian | i ¯ 0 0 HΩ , i.e. HΩ 1¯ ψ = 0. Here Ω is the complement of Ω . 0 0 ⊗ Ω0 | i Hamiltonian-based deﬁnition of information convex set: We consider the following

Hamiltonian-based deﬁnition of the information convex set introduced in Ref. [10]. (For a closely related deﬁnition formulated for Hamiltonians with operator algebra assumptions, see Ref. [72].)

Deﬁnition 10.2 (Information convex set). Consider a frustration-free local Hamiltonian H, a subsystem Ω and its thickening Ω0. See Fig. 10.1 for an illustration of the subsystems. We deﬁne

123 the following information convex set

0 Σ(Ω, Ω H) ρ ρ = Tr \ ρ , where Tr(H ρ ) = 0 . (10.1) | ≡ { Ω | Ω Ω0 Ω Ω0 Ω0 Ω0 }

We would like to require Ω0 to contain all terms in H which overlap with Ω. For certain models, the set is insensitive to the detailed thickness of Ω0. In that context, we can drop Ω0 and take a simpler notation Σ(Ω H) or Σ(Ω). | As a side note, this definition is flexible enough to accommodate a subsystem Ω0 which satisfies Ω0 Ω but has a different topology than Ω. In other words, Ω0 can be more general ⊇ than the thickening of Ω.

Ω0 Ω ⊇

hi Ω

Figure 10.1: This ﬁgure is a schematic depiction of regions involved in the deﬁnition of information convex set Σ(Ω, Ω0 H). Here Ω is the annulus between the black circles. Annulus Ω0 Ω is a thickening of Ω. It| is the region between the gray circles. The small disk-like region is⊇ the support of a term hi of the Hamiltonian HΩ0 . We choose Ω to be an annulus for illustration purposes. Other topologies are allowed as well.

General properties of the information convex set: We do not have a way to derive the structure theorems of the information convex set merely using Deﬁnition 10.2. (Unlike the framework deﬁned in Chapter 3, we do not expect to derive anything about anyon theory merely based on the frustration-free condition.) However, a few general statements can be proved. We summarize these general properties below.

124 Σ(Ω, Ω0 H) is a compact convex set if the subsystem Ω has a ﬁnite size. • |

If Ω0 Ω00, then Σ(Ω, Ω00 H) Σ(Ω, Ω0 H). • ⊆ | ⊆ |

A ground state of H, reduced onto Ω, is an element of Σ(Ω, Ω0 H). • | 10.2 Equivalence between Σ(Ω σ) and Σ(Ω H) under conditions | |

In this section, we discuss a set of general conditions under which the two deﬁnitions of information convex set are equivalent. The two deﬁnitions are the one based on a Hamiltonian

(Deﬁnition 10.2) and the one based on a quantum state (Deﬁnition 4.1).

We will consider the following two conditions:

We say a frustration-free Hamiltonian satisﬁes the topological quantum order (TQO) con- •

dition [88] if for any µ-disk b, Tr(H ρ ) = 0 implies Tr \ ρ = σ . Here, σ = ψ ψ is a b b b b b b | ih | ground state of H. (Recall that the notion of µ-disk is introduced in Chapter 3.1.)

We say H is compatible with axiom A0 and A1 if every ground state of H satisﬁes these • axioms.

If these two conditions hold, we can prove the equivalence between diﬀerent deﬁnitions of information convex set.

Theorem 10.1. Consider a frustration-free local Hamiltonian H which satisﬁes TQO and is compatible with the axiom A0 and A1. σ is a ground state of H. Then, we have

Σ(Ω H) = Σ(Ω σ), (10.2) | | where Σ(Ω H) Σ(Ω, Ω H) and Σ(Ω σ) Σ(Ω, Ω σ). | ≡ 2| | ≡ |

Proof. Because the state σ satisﬁes axiom A0 and A1, the information convex set Σ(Ω, Ω0 σ) | does not depend on the thickness of Ω0. For example, Σ(Ω, Ω σ) = Σ(Ω, Ω σ). We are allowed | 2|

125 to define Σ(Ω σ) Σ(Ω, Ω σ). (See Definition 4.1.) Moreover, it follows from Definition 10.2 | ≡ | that Σ(Ω, Ω H) Σ(Ω, Ω σ) and Σ(Ω, Ω H) Σ(Ω, Ω σ). 2| ⊆ | 2| ⊇ 2| Therefore, with the definition Σ(Ω H) Σ(Ω, Ω H), we can verify Eq. (10.2). This com- | ≡ 2| pletes the proof.

In this context, we can take the simpler notation Σ(Ω). If TQO condition, axiom A0 and

A1 are known to hold for a certain model, the equivalence hold in this model as well.

Corollary 10.1.1. Σ(Ω H) = Σ(Ω σ) for the quantum double models and the string-net models. | | 10.3 Explicit calculation for quantum double models

In this section, we discuss how to calculate the structure of the information convex set for the quantum double model. The quantum double model is a class of exactly solved models for topologically ordered systems introduced by Kitaev in Ref. [23]. (See also Ref. [89] for a review of the quantum double model.) The model is based on a ﬁnite group G. When the group G is an Abelian group, we obtain a model with only Abelian anyons. (For the simplest choice

G = Z2, we obtains the famous toric code model.) When the group G is non-Abelian, there are non-Abelian anyons in the model.

As an aside, the quantum double models are non-chiral and have gapped boundaries. Models of these gapped boundaries have been studied [90, 91]. There are interesting (deconﬁned) gapped excitations along the gapped boundaries. The emergent law describing them has been proposed in literature [92, 93, 94, 95, 96].

The quantum double model: A quantum double model on a 2D lattice is deﬁned for any

ﬁnite group G. Let us consider a square lattice for simplicity. (See Fig. 10.2.) The generalization to other lattices is straightforward. The total Hilbert space is a tensor product of the local

Hilbert spaces on each link. The Hilbert space for each link (labeled by e) is G dimensional: | | = span g g G , where g g G is an orthonormal basis. We denote a vertex as He { | ie | ∈ } { | ie | ∈ } v and denote a face as f. See Fig. 10.2. A bulk site s = (v, f) is a pair containing a face f and

126 f e = g g¯ | i | i v

Figure 10.2: The quantum double model on a square lattice. Notation: v is a vertex, e is an edge and f is a face. When we ﬂip the arrow on an edge, the local Hilbert space has a basis change g g¯ , whereg ¯ is the inverse of g. | i → | i an adjacent bulk vertex v. The Hamiltonian for quantum double model can be written as

X X H = (1 A ) + (1 B ). (10.3) − v − f v f

Constants are added into the Hamiltonian to keep the minimal eigenvalue to be zero. Here,

1 X A Ag; B B1. (10.4) v ≡ G v f ≡ s | | g∈G Each operator Ag, Bh with g, h G acts on the links around a vertex v or a face f. They are v s ∈ deﬁned as c gc g d b gd gb Av = a ga c c h Bs d b = δh,abcd d b a a (10.5)

The Hamiltonian (10.3) is a local commuting projector Hamiltonian.14 This is because h 1 A , 1 B are commuting projectors. Moreover, the Hamiltonian is frustration-free, i∈ { − v − f } 14 P 2 A Hamiltonian H = i hi is a commuting projector Hamiltonian if, for all i, j,[hi, hj] = 0 and hi = hi.

127 because a ground state minimizes all terms in the Hamiltonian. Below is a ground state:

Y ψ = A 1, 1, , 1 . (10.6) | i v| ··· i v Here, 1 represents the identity element of group G. Intuitively, B fixes the fluxes to zero, { f } and A makes the group elements on the links fluctuate. The resulting ground state ψ is a { v} | i superposition of “zero flux” configurations. One verifies that

A ψ = B ψ = ψ , v, f. (10.7) v| i f | i | i ∀

It is known that the quantum double model satisﬁes the TQO condition and that ψ satisﬁes | i axiom A0 and A1. Therefore, Σ(Ω H) = Σ(Ω σ) by Theorem 10.1. Below we give a sketch of | | the calculation of the information convex set structures.

Calculation with “minimal diagrams”: Because the quantum double model is based on a ﬁnite group G, every result calculated will be written in terms of group-theoretic quantities. A challenge is how to derive these results from a quantum state. The actual calculation is technical.

We decide to left out the calculation details and explain the basic intuition.

First, given a subsystem Ω, we need to ﬁnd a thickening Ω0. It is shown in Theorem 10.1 that the information convex set does not depend on Ω0 as long as Ω0 is thick enough, e.g.,

0 Ω = Ω2 on the lattice. This, however, does not directly show what is the minimal choice of

Ω0. In practice, we ﬁnd that for the quantum double model, it is possible to choose Ω0 to be the smallest subsystem that contains all the terms in the Hamiltonian that overlap with Ω.

Here we summarize the basic intuition of the “minimal diagram” calculation. (More details can be found in Ref. [10].) Recall that, for the quantum double model, local Hilbert space is associated with the links. On each link, the state is labeled by the group element g , g G. | i ∈ Once we ﬁx a subsystem Ω, there are three types of links: (1) links outside Ω; (2) links inside

Ω; (3) links at the boundary of Ω.

By requiring B = 1 for each plaquette supported on Ω, we find that the state ρ Σ(Ω) f Ω ∈ must only involve configurations with “zero flux”. Furthermore, the group element of a link

128 around the boundary of Ω is a piece of classical information; this classical information can be seen on both inside Ω and at Ω0 Ω. The density matrix ρ can be written as a convex \ Ω combination of states labeled by the group elements (ha) on the boundary of Ω and some extra degrees of freedom (λ). X ρ = pλ h ; λ h ; λ . (10.8) Ω {ha}|{ a} ih{ a} | Finding all the possible states that involve in this convex combination corresponds to solving the structure theorem of the information convex set Σ(Ω).

A subsystem Ω may contain many links. In practice, it is possible to simplify the lattice to a diagram with just a few links. This change of the lattice preserves the structure of the information convex set. We call the resulting diagram as the minimal diagram.

Below are the calculation results for three basic topology types: disk ω, annulus X, and

2-hole disk Y . The minimal diagrams are illustrated in Fig. 10.3.

µ ν t 1 g g ga gb

gc (a) (b) (c)

Figure 10.3: The minimal diagrams for three topology types: a disk, an annulus and a 2-hole disk. We have ﬁxed the group elements for the links on the boundary of the subsystems.

Disk: The minimal diagram for a disk has only one link. This link lies on the boundary of • the disk. Because of the ﬂux constraint, the group element of this link is the identical group

element (1 G). There is no extra degree freedom. The minimal diagram calculation gives ∈ Σ(ω) = σ . { ω}

129 Annulus: The minimal diagram for an annulus has three links. Two links are at the • boundary of Ω and one link lies inside Ω. The two links on the boundary are in a classical

superposition of diﬀerent choices of group elements. For an extreme point, the classical

superposition is within a conjugacy class C (G) . In Fig. 10.3(b), we have ﬁxed the ∈ cj value of group element g to a speciﬁc element of C. Other choices are can be obtained

by a unitary rotation on the density matrix. By the ﬂux constraint, we have t E(C), ∈ where E(C) is the centralizer of C, deﬁned as E(C) = t G g = tgt¯ , where g is a { ∈ | } representative of C. The additional degree of freedom λ is the irreducible representation

R (E(C)) . The minimal diagram calculation veriﬁes the simplex theorem: ∈ ir X Σ(X) = ρ ρ = p σa , (10.9) { X | X a X } a where p is a probability distribution. The anyon types labeled by a = (C,R), where { a} C (G) and R (E(C)) . It can be checked easily that σa σb for a = b. The ∈ cj ∈ ir X ⊥ X 6 quantum dimension are ﬁxed by the value of the von Neumann entropy as d = C dim R.15 a | |· Indeed, these data matches the known anyon data for the quantum double model.

2-hole disk: The minimal diagram for a 2-hole disk has ﬁve links. Three of them are • on the boundary of the subsystem and two of them are inside the subsystem. One can

verify that, for an extreme point, the three boundaries can be labeled by three anyon types

a = (Ca,Ra), b = (Cb,Rb) and c = (Cc,Rc). The group elements in the minimal diagram

are ﬁxed as follows, g C , g C and g C . The extra degree of freedom comes a ∈ a b ∈ b c ∈ c from the decomposition of a Hilbert space

∗ µ, ν µ, ν G and g =µg ¯ µνg¯ ν . (10.10) H ≡ {| i| ∈ c a b }

into irreducible representations of E(C ) E(C ) E(C ). As Eq. (10.10) indicates, the a × b × c Hilbert space ∗ is the collection of all possible µ, ν that survive the zero-flux constraint. H | i 15The entanglement spectrum for the quantum double model is flat and all the Rényi entropies are identical. Nevertheless, this property does not generalizes to more general classes of models, e.g., the string-net models.

130 The fusion multiplicity N c is given by the copies of irreducible representation R R R ab a × b × c in the representation ∗. The group action given by H

Γ(t , t , t ) µ, ν = t µt¯ , t νt¯ , (t , t , t ) E(C ) E(C ) E(C ). (10.11) a b c | i | a c b ci a b c ∈ a × b × c

The result of the calculation is consistent with structure theorems for 2-hole disks (Theo- rem 5.4 and 5.5).

131 Chapter 11: On the generality of area law: an RG point of view

In previously chapters, we have seen that entanglement-based axioms (A0 and A1) are powerful statements. We have derived the fusion rules and nontrivial braiding statistics of anyons from these simple assumptions. Since everything is based on these assumptions, it would be important to know how general these assumptions apply. Moreover, putting aside the relation to our theoretical framework, the generality of entanglement area law has been a focus of attention for quite a few years; see Ref. [97] for a review of entanglement area law. Hastings’ theorem [2] implies that any gapped 1D system obeys an area law. For the conjectured form of

2D area law

S(A) = α` γ, (11.1) − the stability has been studied under local perturbations [98]. Overall speaking, however, the generality of the area law assumptions, either Eq. (11.1) or axioms A0 and A1, are much less understood than the 1D counterpart.

As we have commented previously, despite the conjectured generality, the area law can break in several ways in 2D gapped systems. See e.g., the discussion in Chapter 6.2. Due to these counterexamples, any eﬀort to prove the conjectured area law must formulate the problem properly. The goal should not be showing the area law (11.1) or axioms A0 and A1 hold for every 2D gapped phase. The goal should be showing that the area law emerges at large length scales under some broadly applied conditions.

In this chapter, we attempt to formula this problem from a renormalization group (RG) point of view. This approach is motivated by the observation that a reference state satisfying axioms

132 A0 and A1 deﬁnes a RG ﬁxed point (Chapter 3.2). In Sec. 11.1 we present a conjecture. In

Sec. 11.2, we test the conjecture with a few known examples.

11.1 A conjecture

We conjecture that the reference state satisfying axioms A0 and A1 is a stable RG fixed point. Roughly speaking, the conjecture says that if A0 and A1 are satisfied up to some (mild) errors, then they are satisfied with smaller errors at larger length scales. Below is an attempt to turn this idea into a rigorous statement.

We will keep the exact form of A0 but relax A1. We will use A1(δ) to denote a relaxed version of A1 that allows corrections less than δ, i.e.

S + S S S δ (11.2) BC CD − B − D ≤ for any µ-disk that is divided into the BCD partition used in A1.

Conjecture 11.1. Suppose axiom A0 and A1(δ) hold for all µ-disks at a length scale r on a reference state. δ is below a certain threshold δ∗. For µ-disks on a larger length scale λr, A0 and A1(δ0) are satisﬁed, where δ0 decays faster than any power of λ.

As we discussed in Chapter 3.2, after a step of RG ﬂow, the length scales change as λ → and r λr. (We can either assume the scale ∆ is large enough or take it to be inﬁnity.) → 11.2 Testing the conjecture with known examples

In this section, we test the conjecture with known examples. These examples put a constraint on the threshold δ∗. We must have

0 < δ∗ < ln 2 (11.3) for Conjecture 11.1 to be consistent with these examples. On the other hand, any value of

δ∗ (0, ln 2) have not been excluded. Intuitively, all of the examples can break A1, but these ∈ breaking have a “gap” ln 2.

133 Topological defects: Axiom A1 can break in a certain natural context of 2D gapped systems. If there is a topological defect [56], denoted by x, within the C subsystem of the µ-disk

(see Fig. 11.1), then we have

S + S S S = 2 ln d , (11.4) BC CD − B − D x where dx is the quantum dimension of topological defect x. If the defect is Abelian, dx = 1.

Otherwise, d ln √2. Anyons are a speciﬁc type of topological defect, but topological defects x ≥ are more general [56, 57]. (Sometimes defects are studied for systems with extra symmetries [99]; those are symmetry defects. However, topological defects can exist for systems without symmetries. In fact, if we relax A1 in a certain region, our framework can be generalized. This approach can be used to study the defect fusion rules and provide an independent derivation of

Eq. (11.4) without invoking any symmetries.)

Equation (11.4) implies that A1 can break down in this case, and there is a minimal amount of violation ln 2. We have S + S S S ln 2 when A1 is broken by an isolated BC CD − B − D ≥ topological defect. Axiom A0 is not aﬀected.

Figure 11.1: With a topological defect: A µ-disk is divided into BCD. An isolated topological defect x is in C.

Spurious contribution to TEE: A quantum state related to our reference state by a ﬁnite depth quantum circuit may have spurious contributions to the area law [58, 59, 60, 61]. For a carefully chosen circuit, the new quantum state can break A1 at all length scales. Axiom A0

134 holds on length scales above the circuit depth. For all of these examples, A1 is violated by at least ln 2.

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