The Mathematical Structure of Tensor Networks

Gert Vercleyen

Promotor: Prof. dr. Frank Verstraete

Masters Thesis Submitted in Partial Fulﬁlment of the Requirements for the Degree of

Master of Science in Mathematics

Academic Year: 2017-2018

I dedicate this thesis to Zempo, Spock, James Tiberius Kirk, Leonard ‘bones’ McCoy, Yanick, Jenny, Chris, Febe, and the other ten ﬁsh swimming gracefully in our pond. Acknowledgements

One of the great Zen masters had an eager disciple who never lost an opportunity to catch whatever pearls of wisdom might drop from the master's lips, and who followed him about constantly. One day, deferentially opening an iron gate for the old man, the disciple asked, `How may I attain, enlightenment?' The ancient sage, though withered and feeble, couldbe quick, and he deftly caused the heavy gate to shut on the pupil's leg, breaking it. It would seem to the unenlightened as though the master, far from teaching his disciple, had left him more perplexed than ever by his cruel trick. To the enlightened, the anecdote expresses a deep truth.

(Carl E. Linderholm, Mathematics Made Difficult)

To me the quote above contains a big message when it comes to research. An important part of research consists of doing things yourself, even if this implies bashing your head against a wall until either the wall or your head breaks. Research consists of stepping in the dark. Research consists of spending weeks of time inventing stuff only to discover the next day that the stuff you’ve invented not only already exists, but is also better formulated, deeper in structure and is moreover contained in one of the references you’ve overlooked at the start of the project. Ironically those wasted weeks might turn out to be the most productive weeks, or at least in my case they were. To stay motivated would be a challenge were it not for the fact that I received a lot of support. Therefore I would like to thank everyone I could count on to make this masters thesis the document it is now. In particular I would like to thank Frank Verstraete, without whom none of this would have been possible. I am very grateful for the lectures on tensor networks, for the conferences and references on the subject, for the hospitality in the quantum group, for the freedom in the content of the research, but most of all for the endless conﬁdence given to me. Of course I would also like to thank all members of the quantum group in Ghent for making me feel like part of the research group. Special thanks goes to

i Sam, Tommeke, Spoon and Nadia who proofread the pile of errors that I presented as being a thesis. Of course many thanks goes to my family for all the support along the way and the same holds for everyone attending the weekly mindfulness sessions with me. Last but not least I would like to thank Natasha and her little wonder Eden for the truly wonderful time spent together.

ii Terms of use

The author grants permission to make this master’s thesis available for consultation and to copy parts of the master’s thesis for personal use. Any other use falls under the limitations of copyright, in particular with regard to the obligation to explicitly mention the source when citing results from this master’s thesis.

Date: 31/05/2018

iii iv Contents

1 Quantum Mechanics 5 1.1 Introduction ...... 5 1.2 States and Observables ...... 6 1.2.1 Observables ...... 6 1.2.2 States ...... 7 1.3 Measurement ...... 8 1.4 Composite Systems and Entanglement ...... 10 1.5 Entanglement Entropy ...... 11

2 Graphical Language of Category Theory 13 2.1 Categories, Functors and Natural Transformations ...... 13 2.1.1 Categories ...... 13 2.1.2 Functors between Categories ...... 14 2.1.3 Natural Transformations between Functors ...... 15 2.1.4 Graphical Language ...... 16 2.2 Monoidal Categories ...... 18 2.2.1 Defintition ...... 18 2.2.2 Graphical Language ...... 19 2.3 Braided Monoidal Categories ...... 22 2.3.1 Definition ...... 22 2.3.2 Graphical Language ...... 23 2.4 Symmetric Monoidal Categories ...... 24 2.4.1 Definition ...... 24 2.4.2 Graphical Language ...... 24 2.5 Compact Closed Categories ...... 25 2.5.1 Definition ...... 25 2.5.2 Graphical Language ...... 26 2.6 Dagger Categories ...... 28 2.6.1 Definition ...... 28 2.6.2 Graphical Language ...... 29

v 2.7 Dagger Compact Categories ...... 29 2.7.1 Deﬁnition ...... 29 2.7.2 Graphical Language ...... 30

3 Category Theory Redefined 31 3.1 Categories, Functors and Natural Transformations ...... 31 3.1.1 Categories ...... 31 3.1.2 Functors between Categories ...... 33 3.1.3 Natural Transformations between Functors ...... 35 3.2 Strict Monoidal Categories ...... 36 3.3 Symmetric Monoidal Categories ...... 37 3.4 Compact Closed Categories ...... 38 3.5 Dagger Compact Categories ...... 40

4 Categorical Quantum Mechanics 43 4.1 Hilbert Space Formalism ...... 43 4.2 Density Matrix Formalism ...... 46

5 Topological Quantum Field Theory 53 5.1 Cobordisms ...... 53 5.2 Axioms of TQFT ...... 56

6 Finite Spin Lattices and Quantum Phases 59 6.1 Finite Spin Lattices ...... 59 6.1.1 Deﬁnitions ...... 59 6.1.2 Locality ...... 61 6.2 Quantum Phases ...... 64

7 Matrix Product States 65 7.1 Introductory Example ...... 65 7.2 Deﬁnitions ...... 68 7.3 Basic Properties ...... 70 7.3.1 Span of generalized MPS ...... 70 7.3.2 Entanglement ...... 71 7.3.3 Expectation Values ...... 71 7.4 Canonical Form and Injectivity ...... 72

Appendices 77

A Dutch summary 77 A.1 Spin Roosters ...... 77

vi A.2 Categorie Theorie ...... 78

B Rigorous Graphical Calculus 79 B.1 Monoidal Categories and Their Diagrams ...... 79 B.1.1 Monoidal Categories ...... 79 B.1.2 Graphs ...... 82 B.1.3 Progressive Plane Diagrams ...... 84 B.1.4 Free Monoidal Categories ...... 89 B.2 Symmetric Tensor Categories and Progressive Polarised Diagrams .. 96 B.2.1 The Value of a Progressive Polarised Diagram ...... 96 B.2.2 Free Symmetric Tensor Categories ...... 102

1 2 Introduction

This thesis has two goals.

The ﬁrst goal is to translate concepts in category theory to diagrams, in the hope to facilitate communication between mathematicians and physicists.

Category theory or “abstract nonsense seems” to be taking over mathematics and this trend affects the language used in papers on mathematical-and theoretical physics. Often categorical concepts are used where a traditional set-based approach would sufﬁce. In such cases no insight in category theory is required, although it might seem to be the case merely because of the language used. This paper hopes to resolve this issue by translating some of the common categorical notions to a simple diagrammatic language. It turns out that the language developed, simple as it may be, is both sound and coherent for many important categories.

The second goal is to provide an introduction to tensor networks for those people with a mathematical background.

Category theory, and the corresponding diagrammatic language, can then be used to rephrase the postulates of quantum mechanics. This would seem overkill were it not that the, otherwise abstract, notion of a topological quantum field theory also fits in the same scheme. The graphical language used to define quantum mechanics is also the same as the one used for constructing tensor networks. This way we can truly regard tensor networks as being built up of maximally entangled pairs. The structure of this thesis is as follows. In chapter 1 a brief introduction to quantum mechanics is given. People familiar with the concept of entanglement might wish to skip this chapter. In chapter 2 various categorical concepts are defined together with their corresponding diagrammatic representations. People traumatised by category theory are invited to skip this chapter, only to return to it when a deeper understanding is necessary. In chapter 3 the categorical definitions are given using diagrams only. Various tables that translate the diagrams to conventional categorical nomenclature are included. Chapter 4 translates the postulates of finite dimensional quantum mechanics to the graphical language. This is done both for the ‘Hilbert space formalism’ as the ‘density matrix formalism’. Chapter 5 introduces the notion of a topological

3 quantum field theory. It serves as an example of the power of categorical (read: diagrammatic) techniques. Chapter 6 contains a brief summary of some central definitions from the theory of quantum phases. These definitions are central to the problem of classifying quantum phases of matter, topological phases being the prime example. Chapter 7 introduces matrix product states. The special properties of these states can be used to classify ground states of local, gapped Hamiltonians. A sketch of how to achieve such a classification is presented. Translating formal category theory to the more loose notion of a diagram is not an easy task. Therefore only formal proofs that capture the nature of the translation process are included in this paper and they can be found in the appendix. Uitleggen wat van mij is en niet van mij

4 Chapter 1

Quantum Mechanics

In this chapter we will briefly review the mathematical machinery of quantum mechanics. This will be done in a more informal way and is meant to provide an introduction to the basics of quantum mechanics needed in chapters further on. A more powerful machinery based on category theory will be provided in chapter 2. In chapter 4 the various concepts introduced in this chapter will be translated in terms of categories and a corresponding diagrammatic language. Most of the material in this section is a summary of the excellent lecture notes of Michael M. Wolf[22]. From now on we assume that all quantities are dimensionless and ℏ = 1. We will restrict ourselves to finite dimensional systems. All sesquilinear forms will be linear in the second argument and anti-linear in the first. Sometimes it is useful to use ‘pure functions’ for which the possible arguments are not named. We will use the sign for an unnamed argument of such functions.

1.1 Introduction

According to The Stanford Encyclopedia of Philosophy: “Quantum mechanics is, at least at ﬁrst glance and at least in part, a mathematical machine for predicting the behaviours of microscopic particles — or, at least, of the measuring instruments we use to explore those behaviours — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had.” Given such a ‘mathematical machine’, a physical theory of quantum mechanics is then given by a set of correspondence rules telling how to interpret the mathematical structures as physical entities. This allows one to perform experiments and test whether the underlying machinery is able to provide consistent predictions. Ideally a physical theory should predict the outcome of any measurement, given all the information about a prepared system, i.e. the initial conditions of the system. This is not the case when doing quantum experiments and there are very good reasons to

5 believe that the unpredictiveness is not due incompleteness of the theory but rather a fundamental part of it (see [22] for extra information on this topic). In the end it turns out that the best we can do in quantum mechanics is to predict probabilities of outcomes. Quantum experiments can be divided into two main parts: preparation and measurement. By a preparation of a system we mean a set of actions such that:

(i) the preparation determines all probability distributions of the outcomes of any possible measurement performed on the system and

(ii) when applied to a statistical ensemble, leads to converging relative frequencies so that we can talk about probabilities.

Many different preparations can give rise to the same system, i.e. the same set of probability distributions. Therefore it is interesting to introduce the concept of a state of a system. A state specifies the outcome of, an equivalence class, of preparations and is thus independent of the specific preparation that gave rise to its existence. It is important to note that a state does not correspond to a single system but rather a statistical ensemble. It tells us how the relative frequencies of the outcomes of any experiment converge when performing the experiment on identical states at infinitum. In classical theoretical physics, measurements are often not incorporated in the structure of a theory. Quantum mechanics differs in the sense that it cannot be described, even theoretically, without incorporating the notion of measurements. There is no consensus on what it means to measure an observable quantity but for our goals it suffices to regard a measurement as reception of a certain amount of information from a system. As the famous Professor Jan Ryckebusch puts it: “You dial the system, and the system responds”. We won’t go into the details of how one can dial a system since it does matter for what follows.

1.2 States and Observables

The division of physical experiments into preparation of a state and measurement of an observable is reﬂected in the mathematical structure of quantum mechanics. Ob- servables in quantum mechanics are deﬁned the following way.

1.2.1 Observables

Consider a complex Hilbert space ℋ with associated non-degenerate sesquilinear form ⟨, ⟩ and deﬁne the map

∗ Φℋ ∶ ℋ → ℋ ; 휓 ↦ Φℋ(휓) ∶= ⟨휓, ⟩, ∀휓 ∈ ℋ. (1.1)

6 It is clear that Φℋ is an anti-linear map that maps vectors in ℋ to elements of its ∗ −1 algebraic dual space ℋ . It is easy to see that Φℋ is bijective and Φℋ is anti-linear.

Definition 1.2.1. Let ℋ, 풦 be ﬁnite dimensional complex Hilbert spaces with dual ∗ ∗ spaces ℋ , 풦 and anti-linear forms Φℋ,Φ풦 respectively. The Hermitian conjugate operator ()† is recursively deﬁned by

푎† ∶= 푎, ∀푎 ∈ C, (1.2) † 휓 ∶= Φℋ(휓), ∀휓 ∈ ℋ, (1.3) † −1 ∗ 휓 ∶= Φℋ (휓), ∀휓 ∈ ℋ , (1.4) (휓 ⊗ 휑)† ∶= 휓† ⊗ 휑† ∀휓 ∈ ℋ, ℋ∗, ∀휑 ∈ 풦, 풦∗, (1.5)

† where in the last line Φ풦 is used to calculate 휑 . A linear operator 퐴 ∶ ℋ → ℋ is called Hermitian or self-adjoint if 퐴† = 퐴.

Note 1.2.2. In general there is no canonical relation between |휓⟩ and ⟨휓|. It is the sesquilinear form ⟨, ⟩ that allows a deﬁnition of the †-operator.

Notation 1.2.3. Given a Hilbert space ℋ then we set

|휓⟩ ≡ 휓 ∈ ℋ, (1.6) ⟨휓| ≡ 휓† ∈ ℋ∗, (1.7) † ⟨휓1|휓2⟩ ≡ 휓1(휓2) = ⟨휓1, 휓2⟩ ∀휓1, 휓2 ∈ ℋ, (1.8) ⟨휓⟩ ≡ ⟨휓|휓⟩ ∀휓 ∈ ℋ. (1.9)

We are now ready to deﬁne the concept of observables.

Definition 1.2.4. Let ℋ be a Hilbert space and denote the set of bounded linear Her- mitian operators from ℋ to ℋ by ℬ(ℋ). Using the composition of maps as a product C ℬ(ℋ) can be upgraded to an algebra over which we denote by 풜ℋ. 풜ℋ represents the set of observables of a system and is called the observable algebra.

1.2.2 States

∗ A state belongs to the algebraic dual of the observable algebra 풜ℋ. States are thus linear functionals that map observables to elements of C: their expectation values. In order to get a nice interpretation in terms of probabilities only certain linear functionals are suitable to be deﬁned as states:

1 Definition 1.2.5. Given an algebra of observables 풜ℋ with identity and algebraic ∗ ∗ dual 풜ℋ then an element 휎 ∈ 풜ℋ is called a state if it is

7 R (i) real-valued: ∀퐴 ∈ 풜ℋ ∶ 휎(퐴) ∈ ,

(ii) normalized: 휎(1) = 1 and

† (iii) positive: ∀퐴 ∈ 풜ℋ ∶ 휎(퐴 퐴) ≥ 0.

∗ We call the algebra of states 풜ℋ state space and its elements will be called density operators.

Note 1.2.6. Since we only consider systems of ﬁnite dimension 푑 we can represent observables and states as 푑×푑 matrices acting on a Hilbert space 퐻 = C. In this paper we will embrace the distinction between an algebra and its image under a certain representation. The former, being more abstract, allows for elegant basis independent proofs. The latter is useful when considering concrete examples, visualisation and a fruitful style of reasoning based on block structures in matrices. An inner product on the observable algebra can be deﬁned as follows.

Definition 1.2.7. Let ℋ be a Hilbert space and 풜 its corresponding observable algebra then we deﬁne the Hilbert-Schmidt inner product as

C † ⟨ , ⟩퐻푆 ∶ 풜 × 풜 → ; (퐴, 퐵) ↦ ⟨퐴, 퐵⟩퐻푆 = Tr[퐴 퐵], (1.10)

The Hilbert space 퓐 ∶= (풜, ⟨ , ⟩퐻푆) is called the space of observables.

1.3 Measurement

Definition 1.3.1. Given a space of observables 퓐 and let 휌 ∈ 풜∗ be a density operator. The expectation value of an observable 푂 is given by

휌(푂) = Tr[휌푂]. (1.11)

The following property of density operators allows us to incorporate classical probabilities into the formalism of quantum mechanics.

푛 Lemma 1.3.2. The set of density matrices is convex, i.e., if {휌푖}푖=1 is a ﬁnite set of density operators and {휆푖|푖 = 1, …, 푛 & 0 ≤ 휆푖 ≤ 1} a set of probabilities such that 푛 ∑푖=1 휆푖 = 1 then the mixture

푛

휌 = ∑ 휆푖휌푖 (1.12) 푖=1 is also a density matrix.

8 Proof. Clearly

푛 푛 푛 1 휌( ) = Tr[휌] = Tr [∑ 휆푖휌푖] = ∑ 휆푖 Tr [휌푖] = ∑ 휆푖 = 1 (1.13) 푖=1 푖=1 푖=1 and for all 퐴 ∈ 풜

푛 † † 휌(퐴 퐴) = ∑ 휆푖휌푖(퐴 퐴) ≥ 0. 푖=1 ■

The mixture

휌 = ∑ 휆푖휌푖 (1.14) 푖 describes a state which can be prepared by generating the states 휌푖 with probability

휆푖. This leads to the following deﬁnition.

Definition 1.3.3. If a state 휌 has no non-trivial convex decomposition

휌 = ∑ 휆푖휌푖, (1.15) 푖 with {휆푖} a probability distribution and each 휌푖 a density operator then we call 휌 a pure state. Otherwise we call 휌 a mixed state.

If 휌 is pure then there exists a vector |휓⟩ ∈ ℋ such that 휌 = |휓⟩⟨휓|. Such a vector is traditionally called a state vector or wave function. Every mixed state can be decomposed into a convex composition of pure states. One interesting way to achieve this is by using the spectral decomposition but this is by no means the only decomposition: there are an inﬁnite amount of decompositions of a tensor product as a combination of tensor monomials. This implies that there are an inﬁnite amount of possible preparations of a mixed state from pure states. The use of canonical structures, i.e. tensors, for describing states incorporates the fact that once a state is prepared there is no way of knowing how it was prepared. If we label the distinct measurement outcomes corresponding to an observable 퐴 by an index 훼 and denote their values by 푣훼 and the probabilities of measuring those values by 푝훼 then the expectation value of a measurement reads

⟨퐴⟩ = ∑ 푎훼푝훼. (1.16) 훼

9 1 The probabilities 푝훼 are calculated by assigning a positive operator 0 ≤ 푃훼 ≤ to each outcome in such a way that

푝훼 = Tr[휌푃훼], (1.17) and

1 ∑ 푃훼 = . (1.18) 훼

A set of positive operators {푃훼} fulﬁlling (1.17) and (1.18) is called a positive operator- valued measure (POVM). The connection between POVMs and self adjoint operators is given by the spectral decomposition

퐴 = ∑ 푎훼푃훼. (1.19) 훼 where the 푃훼 are orthogonal projectors onto the eigenspaces of 퐴 with corresponding eigenvalues 푎훼. These projectors form a POVM and the set of eigenvalues then corresponds to the set of possible measurement outcomes.

1.4 Composite Systems and Entanglement

In classical theories systems are composed by taking direct products of the parame- ter spaces describing those systems. In quantum theory, systems are composed by taking tensor products of the corresponding Hilbert spaces and algebras. The tensor product structure is responsible for one of the most fascinating phenomena that arises in physics: entanglement. Say we have two systems 퐴 and 퐵 with underlying Hilbert spaces ℋ퐴 and ℋ퐵 respectively. The composite system is then described by observables and states on the tensor product ℋ퐴 ⊗ ℋ퐵. In such case we say that ℋ퐴 corresponds to a subsystem of ℋ and likewise for ℋ퐵.

Some states can be written as a tensor product of a state in ℋ퐴 and one in ℋ퐵. We call such states product states. States that do not factorize are said to exhibit correlations. For states that exhibit correlations there always exists observables say 퐴 and 퐵 such that ⟨퐴 ⊗ 퐵⟩ ≠ ⟨퐴 ⊗ 1⟩⟨1 ⊗ 퐵⟩. One type of correlations that could arise are those of classical origin: if a state is given by a convex combination of product states

(퐴) (퐵) 휌 = ∑ 푝푛휌푛 ⊗ 휌푛 , (1.20) 푛 then one could prepare such a state by using a machine that prepares a product state

10 (퐴) (퐵) prepares the product state 휌푛 ⊗ 휌푛 with probability (푝푛). A state of this form is then called separable or unentangled. If a state is not separable it is called entangled. Note that the notion of entanglement is independent of choice of local bases, yet it is dependent on subspace decomposition of the composite Hilbert space. Any state in the composite Hilbert space is trivially a product state. Only when considering subspaces and describing a composite state using these subspaces a notion of entanglement can be deﬁned. Therefore one can only talk about entanglement of a state with respect to a subspace decomposition. Often this decomposition is given and we just talk about entanglement of a state. A pure state is separable if and only if it is a product state. This means that every kind of correlation comes from entanglement. These are therefore the most interesting states to study when one is looking for entanglement properties. Every pure state admits a particularly neat form thanks to the following proposition.

Proposition 1.4.1 (Schmidth decomposition). Consider a composite Hilbert space ℋ with subspace decomposition ℋ = ℋ퐴 ⊗ ℋ퐵. Then for every vector |휓⟩ ∈ ℋ퐴 ⊗ ℋ퐵 there exist orthonormal bases {|푖⟩ ∈ ℋ퐴} and {|푗⟩ ∈ ℋ퐵} such that

푑

|휓⟩ = ∑ √휆푘|푘⟩ ⊗ |푘⟩ (1.21) 푘=1 with 휆푘 ≥ 0, ∑ 휆푘 = ⟨휓|휓⟩ and 푑 = min(dim ℋ퐴, dim ℋ퐵). 푘

The coefﬁcients {√휆푘} are called the Schmidt coefﬁcients and the 푑 is called the Schmidt rank of |휓⟩.

1.5 Entanglement Entropy

Given a Hilbert space ℋ with a subspace decomposition ℋ = ℋ퐴 ⊗ℋ퐵, then we can deﬁne the notion of entanglement entropy of a state, between subsystems 퐴 and 퐵. There are various measures of entropy but the most important ones are special cases of the ‘Renyi entropy’.

Definition 1.5.1. Consider a Hilbert space ℋ = ℋ퐴 ⊗ ℋ퐵 and a state 휌. Let

휌퐴 = Tr퐵[휌], (1.22) where Tr퐵 denotes the partial trace. The Renyi entropy of the state 휌 with respect to

11 the subspace decomposition ℋ = ℋ퐴 ⊗ ℋ퐵 is deﬁned as

1 푅 (휌 ) = log Tr [휌훼 ] 훼 ∈ [0, 1[∪]1, +∞[ (1.23) 훼 퐴 1 − 훼 퐴 and the limit

푆(휌퐴) ∶= lim 푅훼(휌퐴) = −푇 푟[휌퐴 log(휌퐴)] (1.24) 훼→1 is called the Neumann entropy.

For pure states we have that

푑 푑

휌 = |휓⟩⟨휓| = ∑ ∑ √휆푖√휆푗(|푖⟩⟨푗|)퐴 ⊗ (|푖⟩⟨푗|)퐵 (1.25) 푖=1 푗=1 from which it follows that

푑

휌퐴 = ∑ 휆푖(|푖⟩⟨푖|)퐴 (1.26) 푖=1 and therefore the Neumann entropy of a pure state is given by

푑

푆(휌퐴) = ∑ 휆푖 log(휆푖). (1.27) 푖=1

12 Chapter 2

Graphical Language of Category Theory

The goal of this chapter is to introduce several definitions from category theory parallel with a diagrammatic language and the rules for using such language. In the next chapter we take the opposite viewpoint and present an account of category theory using nothing but diagrams. By doing so, I hope that the reader with a minor background in category theory might find some clarification of the abstract nonsense that seems to grow in popularity in the world of mathematics. As a matter of fact, the impatient reader might wish to skip this chapter altogether. The next chapter is written such that the rules for the graphical language, and almost all definitions, from this chapter are included. This chapter starts by introducing the basic notion of a category and as the chapter continues more specific categories with the appropriate diagrammatic language are introduced. The definitions in this chapter can be found in any introductory book on category theory. The structure and most of the content of this section is based on a paper by Selinger [18]. Selinger’s paper is more ambitious by translating much more categorical structures to diagrams than we do.This chapter only contains the relevant structures for treating quantum mechanics. Most of the theorems come from the paper [8], on which I also based a lot of the informal paragraphs concerning the rules for deforming diagrams. The full proofs can be found in the appendix.

2.1 Categories, Functors and Natural Transformations

2.1.1 Categories

Definition 2.1.1. A category 풞 consists of the following data:

• a collection |풞| of objects;

13 • for any two objects 푉 , 푊 ∈ |풞|, a set 풞(푉 , 푊 ) of morphisms with domain 푉 and codomain 푊 ;

• for each object 푉 ∈ |풞|, an identity morphism ퟣ푉 ∈ 풞(푉 , 푉 ), also written as ퟣ if it is clear on which object it acts;

• for any objects 푈, 푉 , 푊 ∈ |풞|, a sequential composition operator ∘ for morphisms

∘ 풞(푈, 푉 ) × 풞(푉 , 푊 ) 풞(푈, 푊 ), (2.1)

sending (햡, 햠) to 햡 ∘ 햠 = 햡햠, called the composition of 햡 and 햠.

The above data is required to satisfy the following two axioms.

(i) Associativity: Let (햢, 햡, 햠) ∈ 풞(푉 , 푊 ) × 풞(푈, 푉 ) × 풞(푇 , 푈) then

햢 ∘ (햡 ∘ 햠) = (햢 ∘ 햡) ∘ 햠 in 풞(푇 , 푊 ). (2.2)

(ii) Unity: For all objects 푉 , 푊 ∈ |풞| and all morphisms 햠 ∈ 풞(푉 , 푊 )

햠 ∘ ퟣ푉 = 햠 = ퟣ푊 ∘ 햠 in 풞(푉 , 푊 ). (2.3)

The collection of all morphisms in 풞 is written as 햬허헋(풞).

2.1.2 Functors between Categories

It is difﬁcult to do much with categories without discussing the maps between them. A map between categories is called a functor.

Definition 2.1.2. Let 퐹 ∶ 풞 → 풟 be a map from a category 풞 to a category 풟 such that 퐹 maps

• objects to objects

퐹 ∶ |풞| ⟶ |풟|, 푉 ⟼ 퐹 푉 ; (2.4)

• and morphisms to morphisms such that for objects 푉 , 푊 ∈ |풞|, and 햠 ∈ 풞(푉 , 푊 )

퐹 ∶ 풞(푉 , 푊 ) ⟶ 풟(퐹 푉 , 퐹 푊 ), 햠 ⟼ 퐹 햠. (2.5)

If 퐹 satisﬁes the following two axioms

14 (i) Preservation of Identity: ∀푉 ∈ |풞|

퐹 (ퟣ푉 ) = ퟣ퐹푉 (2.6)

and

(ii) Preservation of Composition: ∀(햡, 햠) ∈ 풞(푉 , 푊 ) × 풞(푈, 푉 )

퐹 (햡 ∘ 햠) = 퐹 햡 ∘ 퐹 햠, (2.7) then we say that 퐹 is a functor from 풞 to 풟.

Functors can have many properties, some of which we will need later.

Definition 2.1.3. A functor 퐹 ∶ 풞 → 풟 is called

(i) the identity functor, also written 1풞 ∶ 풞 → 풞, if it is the identity function on both objects and morphisms of 풞,

(ii) full if for any objects 푉 , 푊 ∈ 풞, the function on morphism sets (2.5) is surjective,

(iii) faithful if for any objects 푉 , 푊 ∈ 풞, the function on morphism sets (2.5) is injective, and

(iv) an isomorphism if there exists a functor 퐹 −1 ∶ 풟 → 풞 such that

−1 −1 퐹 퐹 = 1풟 and 퐹 퐹 = 1풞. (2.8)

Note that the inverse functor 퐹 −1 is unique if it exists.

2.1.3 Natural Transformations between Functors

Definition 2.1.4. Suppose 퐹 , 퐺, 퐻 ∶ 풞 → 풟 are functors from 풞 to 풟.

(i) A natural transformation 휃 ∶ 퐹 ⇒ 퐺 consists of a set of structure morphisms ′ ′ {휃푉 ∈ 풟(퐹 푉 , 퐺푉 )|푉 ∈ |풞|} such that ∀푉 , 푉 ∈ |풞|, ∀햠 ∈ 풞(푉 , 푉 ) the following diagram commutes

퐹 푉 퐹햠 퐹 푉 ′

휃푉 휃푉 ′ (2.9) 퐺푉 퐺햠 퐺푉 ′

15 (ii) If 휃 ∶ 퐹 ⇒ 퐺 and 휂 ∶ 퐺 ⇒ 퐻 are natural transformations, their composition is the

natural transformation 휂휃 ∶ 퐹 ⇒ 퐻 with structure morphisms (휂휃)푉 = 휂푉 ∘ 휃푉 , ∀푉 ∈ |풞|.

(iii) A natural isomorphism is a natural transformation in which every structure morphism is an isomorphism.

2.1.4 Graphical Language

The foundations of the graphical languages used to describe categories are based on graph theory. We quickly revise some notions necessary for the introduction of the graphical languages. First of all we introduce the topological deﬁnition of a graph.

Definition 2.1.5. A graph with boundary Γ = (Γ, Γ0, 휕Γ) consists of

(i) a compact Hausdorff topological space Γ,

(ii) a discrete closed subset Γ0 ⊆ Γ such that the complement Γ1 = Γ ∖ Γ0 is a topological sum of open intervals (called edges) and circles, and

(iii) a subset 휕Γ ⊆ Γ0 such that each 푥 ∈ 휕Γ has sufﬁciently small connected neighbourhood 푉 in Γ for which 푉 − {푥} has a single connected component.

The elements of Γ0, 휕Γ, and Γ0 ∖휕Γ are called the nodes, outer nodes, and inner nodes of Γ respectively.

Note 2.1.6. Each outer node can and will be identiﬁed with the unique edge whose closure it is in. As usual mathematical structures are only interesting if accompanied by a proper notion of a map between them.

Definition 2.1.7. An isomorphism 햠 ∶ Γ → Γ′ of graphs with boundary is a homeomorphism inducing bijections on the inner nodes and on the outer nodes.

To connect this abstract deﬁnition to the real pictures drawn on a piece of paper we must embed the graph in some subset of R2. Recall that an injective continuous map 푓 ∶ 푋 → 푌 between topological spaces is a topological embedding if 푓 yields a homeomorphism between 푋 and 푓(푋). We will immediately impose some extra structure on this embedding so that the diagrams are “well behaved” in the sense of theorem (3.1.3).

Definition 2.1.8. Let 푎 < 푏 be real numbers. A progressive plane graph between the levels 푎 and 푏 is a graph Γ with boundary, embedded in R × [푎, 푏] such that

16 (i) 휕Γ = Γ ∩ (R × {푎, 푏}) and

(ii) for each connected component of Γ ∖ Γ0, the projection

R2 R pr2 ∶ → , (푥, 푦) ↦ 푦 (2.10)

is injective.

It follows that a progressive plane graph has no circles or circuits. Note that condition (ii) means that the projection maps on each edge separately should be injective and not that one projection operator working on all edges should be injective. The edges and nodes of a progressive plane graph Γ represent objects and morphisms in a category as follows. Objects are represented as wires (the edges of Γ) and morphisms are represented as boxes (the nodes of Γ). An identity morphism is represented as a continuing wire and composition is represented by connecting the outgoing wire of one diagram to the incoming wire of another. Composition of any wire with an identity wire should have no effect and therefore lengths of wires have no meaning. This is the only invariant property of the graphical language for general categories. Adding more structure to categories will enable us to bend wires, rotate boxes, etc, but at the moment this is all there is.

푊 푊 햡 푉 햠 푉 푉 푉 햠 푈

푉 햠 ∶ 푉 → 푊 ퟣ푉 ∶ 푉 → 푉 햡 ∘ 햠 Object Morphism Identity Composition

Maps must always be read from bottom to top and in general there is no mirror symmetry between the two. The same is true about the left and right of the diagram, except when it comes to the labels of objects. Labels might be on the left of the wire or on the right and there is no difference in meaning between the two. Notes 2.1.9. (i) A morphism variable, or box, can be viewed as a placeholder for an arbitrary, possibly composite, diagram.

(ii) We have equipped wires with a bottom-to-top arrow, and cut off a piece of the boxes to break rotation-and mirror symmetry. These markings are of no use at the moment, but will become important as we extend the language in what follows.

17 The following theorem is one of many coherence theorems we will encounter. They are of central importance to the sections on categories, for without them one can not be sure that diagrams can be unambiguously interpreted, nor that they are capable of providing the full categorical picture.

Theorem 2.1.10 (Coherence of the Graphical Language for Categories). A well-formed equation between two morphism terms in the language of categories follows from the axioms of categories if and only if it holds in the graphical language up to isomorphism of diagrams.

Proof. See [18]. ■

Definition 2.1.11. An equivalence between categories 풞 and 풟 consists of a pair of functors 퐹 ∶ 풞 → 풟 and 퐺 ∶ 풟 → 퐶 and a pair of natural isomorphisms

1풞 → 퐺퐹 and 1풟 → 퐹 퐺. (2.11)

We call the categories 풞 and 풟 are equivalent via the functors 퐹 and 퐺.

2.2 Monoidal Categories

2.2.1 Defintition

A monoidal category (also sometimes called tensor category) is a category with an associative unital tensor product. More speciﬁcally:

Definition 2.2.1. A monoidal category (MC) is a six-tuple (풞, ⊗, 퐼, 푎, 푙, 푟) where

• 풞 is a category;

• 퐼 is an object of 풞, called the unit object of 풞;

• ⊗ ∶ 풞 × 풞 → 풞 is a functor, called the tensor product;

• 푎 ∶ ⊗ ∘ (⊗ × 1) → ⊗ ∘ (1 × ⊗) is a natural isomorphism;

• 푙 ∶ ⊗ ∘ (푘 × 1) → 1 and 푟 ∶ ⊗ ∘ (1 × 푘) → 1 are natural isomorphisms;

18 such that for all 푇 , 푈, 푉 , 푊 ∈ 풞 the following diagrams commute:

((푇 ⊗ 푈) ⊗ 푉 ) ⊗ 푊

푎푇⊗푈,푉 ,푊 푎푇,푈,푉 ⊗ퟣ

(푇 ⊗ 푈) ⊗ (푉 ⊗ 푊 ) (푇 ⊗ (푈 ⊗ 푉 )) ⊗ 푊 (2.12)

푎푇,푈,푉 ⊗푊 푎푇,푈⊗푉 ,푊

푇 ⊗ (푈 ⊗ (푉 ⊗ 푊 )) 푇 ⊗ ((푈 ⊗ 푉 ) ⊗ 푊 ) ퟣ⊗푎푈,푉 ,푊

푎 (푉 ⊗ 푘) ⊗ 푊 푉 ,푘,푊 푉 ⊗ (푘 ⊗ 푊 ) 푟푉 ⊗푊 . (2.13) 푉 ⊗푙푊 푉 ⊗ 푊

Equation (2.12) is called the pentagon equation and equation (2.13) is called the triangle equation. The map 푎 is called the associativity constraint of the MC and the maps 푙 and 푟 are called the left and right unit constraints of the SM.

Definition 2.2.2. A strict monoidal category is a monoidal category (풞, ⊗, 퐼, 푎, 푙, 푟) where 푎, 푙, 푟 are identity maps. In this case we will identify (풞, ⊗, 퐼) with the six-tuple (풞, ⊗, 퐼, 푎, 푙, 푟)

2.2.2 Graphical Language

We extend the graphical language of categories as follows. A tensor product of objects is represented by drawing the corresponding wires in parallel. The unit object 퐼 is represented by an empty diagram. A scalar 푠 ∶ 퐼 → 퐼 is represented by the symbol

푠 which is not surrounded by a box. A morphism 햠 ∶ 푉1 ⊗ ⋯ ⊗ 푉푚 → 푊1 ⊗ ⋯ ⊗ 푊푛 is represented as a box with 푛 input wires and 푚 output wires. A tensor product of morphisms is represented by horizontally stacking the corresponding diagram. All of this must be done such a way that the blocks have exactly the same 푦-coordinate.

19 푊1 ⋯ 푊푛 푊1 푊2

푉 푊 (empty) 푠 햠 햠1 햠2

푉1 ⋯ 푉푚 푉1 푉2

푉 ⊗ 푊 퐼 푠 ∶ 퐼 → 퐼 햠 ∶ 햠1 ⊗ 햠2

푉1 ⊗ ⋯ ⊗ 푉푚 →

푊1 ⊗ ⋯ ⊗ 푊푛 Product of Unit object Scalar General Product of objects morphism morphisms

It is important to note that an embedding, by deﬁnition, is an injective map. This implies that all graphs are assumed to be planar graphs. Wires are not allowed to cross each other, nor are they allowed pass through boxes. The same holds for boxes. For strict monoidal categories we are allowed to move the boxes and wires within the plane as long as we respect several rules. First it is important to remind ourselves that a diagram has a distinguished bottom and top, namely the levels 푦 = 푎 and 푦 = 푏 which are part of the deﬁnition of a plane graph. We can also attach horizontal labels 푥 = 훼 and 푥 = 훽 which are such that all nodes and edges of the plane graph belong to the rectangle [훼, 훽] × [푎, 푏].

푦 = 푏 푊1 ⋯ 푊푛 햠

푉1 ⋯ 푉푚 푦 = 푎 푥 = 훼 푥 = 훽 We call two plane graphs, belonging to [훼, 훽] × [푎, 푏], with incoming and outgoing wires attached to the top and bottom of the rectangle, planar isotopic if it is possible to transform one to the other by continuously moving around boxes in the rectangle, such that

(i) no boxes or wires cross each other,

(ii) no boxes or wires are detached from the boundary of the rectangle.

If it is possible to deform one diagram to the other while preserving the progressive structure of the graph, we call such a deformation a deformation of progressive plane graphs. The following coherence theorem was ﬁrst proven by Joyal and Street [8].

Theorem 2.2.3 (Coherence for Monoidal Categories). A well-formed equation between morphism terms in the language of strict monoidal categories follows from the axioms

20 of strict monoidal categories if and only if it holds, up to deformation of progressive plane graphs, in the graphical language.

Proof. See B. ■

Because of the coherence theorem, it is not actually necessary to memorize the axioms of strict monoidal categories. Indeed, one could use the coherence theorem as the deﬁnition of the strict monoidal category. For non-strict monoidal categories one should insert brackets when taking multiple tensor products since the tensor product is not associative per se. Also the empty diagram should not be empty anymore. Since combining horizontal brackets with vertical composition is a mess, no graphical language exists for non-strict tensor categories. It turns out, however, that this is not as big an issue as it looks. If we want to understand why, we need a few extra deﬁnitions.

Definition 2.2.4. Let (풞, ⊗, 퐼, 푎, 푙, 푟) and (풞′, ⊗′, 퐼′, 푎′, 푙′, 푟′) be monoidal categories. A strong monoidal functor, also called tensor functor, between 풞 and 풞′ is a functor 퐹 ∶ 풞 → 풞′ together with natural isomorphisms 휙2 ∶ 퐹 푈 ⊗ 퐹 푉 → 퐹 (푈 ⊗ 푉 ) and 휙0 ∶ 퐼 → 퐹 퐼, such that the following diagrams commute:

휙2⊗ퟣ 휙2 (퐹 푈 ⊗ 퐹 푉 ) ⊗ 퐹 퐶 퐹퐶 퐹 (푈 ⊗ 푉 ) ⊗ 퐹 퐶 퐹 ((푈 ⊗ 푉 ) ⊗ 푊 )

푎퐹푈,퐹푉 ,퐹푊 퐹(푎푈,푉 ,푊 ) (2.14) ퟣ ⊗휙2 휙2 퐹 푈 ⊗ (퐹 푉 ⊗ 퐹 푊 ) 퐹푈 퐹 푈 ⊗ 퐹 (푉 ⊗ 푊 ) 퐹 (푈 ⊗ (푉 ⊗ 푊 ))

퐹 푈 ⊗ 퐼 푟 퐹 푈 퐼 ⊗ 퐹 푈 푙 퐹 푈

0 0 ퟣ퐹푈⊗휙 퐹(푟) 휙 ⊗ퟣ퐹푈 퐹(푙) (2.15) 휙2 휙2 퐹 푈 ⊗ 퐹 퐼 퐹 (푈 ⊗ 퐼) 퐹 퐼 ⊗ 퐹 푈 퐹 (퐼 ⊗ 푈)

Definition 2.2.5. Let (풞, ⊗, 퐼, 푎, 푙, 푟) and (풞′, ⊗′, 퐼′, 푎′, 푙′, 푟′) be monoidal categories, and let 퐹 , 퐺 ∶ 풞 → 풞′ be strong monoidal functors. A natural transformation 휏 ∶ 퐹 → 퐺 is called monoidal, or a tensor transformation, if the following diagram commutes for all 푉 , 푊 ∈ |풞|:

휙2 퐹 푉 ⊗ 퐹 푊 퐹 (푉 ⊗ 푊 )

휏푉 ⊗휏푊 휏푉 ⊗푊 (2.16) 휙2 퐺푉 ⊗ 퐺푊 퐺(푉 ⊗ 푊 )

The following theorem resolves the issue of ‘not having a graphical language for

21 non-strict categories’.

Theorem 2.2.6 (Mac Lane's Coherence Theorem). For every monoidal category (풞, ⊗, 퐼, 푎, 푙, 푟) there exists a strict monoidal category (풞′, ⊗′, 퐼′, 푎′, 푙′, 푟′) together with an equivalence of categories 퐹 ∶ 풞 → 풞′, where 퐹 is a strong monoidal functor.

Proof. See [10]. ■

2.3 Braided Monoidal Categories

2.3.1 Definition

Definition 2.3.1. Let (풞, ⊗, 퐼, 푎, 푙, 푟) be a monoidal category, and consider the switch functor

휏 ∶ 풞 × 풞 → 풞 × 풞, 휏(푉 , 푊 ) = (푊 , 푉 ), 휏(푓, 푔) = (푔, 푓). (2.17)

A braiding on 풞 is a natural isomorphism 휎 ∶ 1 ⇒ 휏 such that ∀푉 ∈ 풞

휎퐼,푉 = 휎푉 ,퐼 = ퟣ푉 (2.18) and the following diagrams commute:

푎 (푉 ⊗ 푈) ⊗ 푊 푉 ,푈,푊 푉 ⊗ (푈 ⊗ 푊 )

휎푈,푉 ⊗ퟣ ퟣ⊗휎푈,푊

(푈 ⊗ 푉 ) ⊗ 푊 푉 ⊗ (푊 ⊗ 푈) (2.19)

푎푈,푉 ,푊 푎푉 ,푊,푈 푈 ⊗ (푉 ⊗ 푊 ) (푉 ⊗ 푊 ) ⊗ 푈 휎푈,푉 ⊗푊

푎 (푉 ⊗ 푈) ⊗ 푊 푉 ,푈,푊 푉 ⊗ (푈 ⊗ 푊 ) −1 −1 휎푉 ,푈⊗ퟣ ퟣ⊗휎푊,푈

(푈 ⊗ 푉 ) ⊗ 푊 푉 ⊗ (푊 ⊗ 푈) (2.20)

푎푈,푉 ,푊 푎푉 ,푊,푈 푈 ⊗ (푉 ⊗ 푊 ) (푉 ⊗ 푊 ) ⊗ 푈 −1 휎푉 ⊗푊,푈

22 These equations are also called the hexagon axioms. If

−1 휎푉 ,푊 ≠ 휎푊,푉 (2.21) for all 푉 , 푊 ∈ 풞, we call the monoidal category (풞, ⊗, 퐼, 푎, 푙, 푟, 휎) a braided monoidal category.

Definition 2.3.2. A braided monoidal functor between braided monoidal categories is a monoidal functor 퐹 that is compatible with the braiding, i.e. the following diagram commutes:

휙2 퐹 푉 ⊗ 퐹 푊 퐹 (푉 ⊗ 푊 )

휎 퐹푉 ,퐹푊 퐹휎푉 ,푊 . (2.22) 휙2 퐹 푊 ⊗ 퐹 푉 퐹 (푊 ⊗ 푉 )

2.3.2 Graphical Language

The graphical language of monoidal categories is extended with the following braiding

푊 푉

푉 푊

휎푉 ,푊 Braiding

If 푉 and/or 푊 are composite objects, then 휎푉 ,푊 braids all wires of 푉 over all wires of 푊 . The hexagon axioms translate into the following graphical equations.

푉 푊 푈

푉 푈 푊 = (2.23)

푈 푉 푊

Example 2.3.3. The Yang-Baxter equation

(휎푉 ,푊 ⊗ ퟣ) ∘ (ퟣ ⊗ 휎푈,푊 ) ∘ (휎푈,푉 ⊗ ퟣ) = (ퟣ ⊗ 휎푈,푉 ) ∘ (휎푈,푊 ⊗ ퟣ) ∘ (ퟣ ⊗ 휎푉 ,푊 ) (2.24)

23 is graphically depicted as

푊 푉 푈 푊 푉 푈

푉 푊 푈 푊 푈 푉

= (2.25)

푉 푈 푊 푈 푊 푉

푈 푉 푊 푈 푉 푊

A coherence theorem for braided categories is proven in [8], but both the statement and the proof of the theorem are very technical. The reason we included braided categories is because they allow symmetric categories to be deﬁned in a natural man- ner. The coherence theorem for symmetric categories is much simpler. The statement can be found in the next section and the proof can be found in the appendix.

2.4 Symmetric Monoidal Categories

2.4.1 Definition

Definition 2.4.1. Let (풞, ⊗, 퐼, 푎, 푙, 푟, 휎) be a monoidal category with braiding 휎. We call 휎 a symmetry, and (풞, ⊗, 퐼, 푎, 푙, 푟, 휎) a symmetric monoidal category if

−1 휎푉 ,푊 = 휎푊,푉 (2.26) for all 푉 , 푊 ∈ 풞.

2.4.2 Graphical Language

The symmetry 휎 is graphically represented by a crossing

푊 푉

푉 푊 .

휎푉 ,푊 Symmetry

24 In the same paper where coherence for monoidal categories was proven the following theorem was proven as well.

Theorem 2.4.2 (Coherence for Symmetric Monoidal Categories). A well-formed equation between morphisms in the language of symmetric monoidal categories follows from the axioms of symmetric monoidal categories if and only if it holds, up to isomorphism of diagrams, in the graphical language.

Proof. See B ■

2.5 Compact Closed Categories

2.5.1 Definition

Definition 2.5.1. A compact closed category is a symmetric monoidal category (풞, ⊗, 퐼, 휎, 휀, 휂, ∗) such that for all objects 푉 ∈ |풞|, there exists an object 푉 ∗ ∈ |풞| and morphisms

∗ 휀푉 ∶ 푉 ⊗ 푉 → 퐼 (2.27) ∗ 휂푉 ∶ 퐼 → 푉 ⊗ 푉 (2.28) such that the following diagrams commute

−1 1 ⊗휂 푎 ∗ 휀 ⊗1 푉 ⊗ 퐼 푉 푉 푉 ⊗ (푉 ∗ ⊗ 푉 ) 푉 ,푉 ,푉 (푉 ⊗ 푉 ∗) ⊗ 푉 푉 푉 퐼 ⊗ 푉

≅ ≅ , (2.29) 1 푉 푉 푉

휂 ⊗1 ∗ 푎 ∗ ∗ 1 ∗ ⊗휀 퐼 ⊗ 푉 ∗ 푉 푉 (푉 ∗ ⊗ 푉 ) ⊗ 푉 ∗ 푉 ,푉 ,푉 푉 ∗ ⊗ (푉 ⊗ 푉 ∗) 푉 푉 푉 ∗ ⊗ 퐼

≅ ≅ . (2.30)

1 ∗ 푉 ∗ 푉 푉 ∗

휂푉 is called the unit and 휀푉 is called the counit (of 푉 ).

Definition 2.5.2. Let 햠 ∶ 푉 → 푊 be a morphism in a category 풞 then 햠∗ ∶ 푊 ∗ → 푉 ∗ is called the adjoint mate of 햠.

Since ()∗ is now deﬁned both for objects and morphisms we can talk about the

25 ∗-functor. In general we have the following properties

(푉 ⊗ 푊 )∗ ≅ 푉 ∗ ⊗ 푊 ∗ (2.31) 퐼∗ ≅ 퐼 (2.32) 푉 ∗∗ ≅ 푉 . (2.33)

2.5.2 Graphical Language

If 푉 is an object variable, then 푉 ∗ is represented by a wire running from top to bottom. The unit and counit are represented as half turns. The adjoint mate 햠∗ of a morphism 햠 is represented by the diagram for which the input and output wires are bent around.

푉

푉 푉 푉 햠 푉 푉

푊 ∗ ∗ ∗ ∗ 푉 휂푉 휀푉 햠 ∶ 푊 → 푉 Dual of object Unit Counit Adjoint mate

One of the reasons we used non-symmetric boxes is to simplify the representation of the adjoint mate by setting

푉 푉 햠 = 햠 . (2.34) 푊 푊

This deﬁnition is ambiguous though. Consider for example the morphism 햠 ∶ 푉1 ⊗푉2 →

푊1 ⊗ 푊2

푊1 ⊗ 푊2 푊1 푊2 햠 = 햠 (2.35)

푉1 ⊗ 푉2 푉1 푉2

26 then we could say that either

푉1 푉2

푉2 푉1 햠 ∶= 햠∗ or 햠 ∶= 햠∗. (2.36)

푊2 푊1

푊1 푊2

The second deﬁnition is the one that is used when dealing with matrices and we will call it the algebraic transpose and denote it by 햠푇 :

푉1 푉2

햠 ∶= 햠∗ =∶ 햠푇 . (2.37)

푊1 푊2

The ﬁrst deﬁnition is also useful and we will just call it the transpose and denote it as 햠푡:

푉2 푉1 햠 ∶= 햠푡 (2.38)

푊2 푊1

Axioms (2.29) and (2.30) are then graphically depicted as

푉 푉 푉 = 푉 , 푉 = 푉 . (2.39) 푉 푉

The coherence theorem for compact closed categories was proven by Kelly and Laplaza [11] but only for ‘simple signatures’, i.e. for a diagrammatic language where labels are not allowed to stand for objects or morphisms composed by tensor products. For example, the coherence theorem makes no statements about diagrams such as the left hand side of equation (2.35). We already encountered problems when consid-

27 ering such constructions but managed to resolve them. Whether other ambiguous situations could arise when using composed labels is not clear. Some authors [18, 3] believe that no other problematic situations exist and the following conjecture should be considered as true.

Conjecture 2.5.3 (Coherence for Compact Closed Categories). A well-formed equation between morphisms in the language of compact closed categories follows from the axioms of compact closed categories if and only if it holds, up to isomorphisms of diagrams, in the graphical language.

2.6 Dagger Categories

2.6.1 Definition

The concept of a dagger category is motivated by the category of Hilbert spaces, where each morphism 햠 ∶ 푉 → 푊 has an adjoint 햠† ∶ 푊 → 푉 .

Definition 2.6.1. A dagger category is a category (풞, †) equipped with an functor † ∶ 풞 → 풞 such that

(i) for all objects 푉 ∈ 풞

푉 † = 푉 (2.40) † ퟣ푉 = ퟣ푉 (2.41)

(ii) for every morphism 햠 ∶ 푉 → 푊 in 풞 there exists a 햠† ∶ 푊 → 푉 such that for all 햠 ∶ 푈 → 푉 and 햡 ∶ 푉 → 푊

(햡 ∘ 햠)† = 햠† ∘ 햡†, (2.42) (햠†)† = 햠. (2.43)

We call 햠† the adjoint map of 햠.

Note 2.6.2. In the previous deﬁnition the term “adjoint” is used in a way analogous to the linear-algebraic sense, not in the category-theoretic sense.

We then have the usual notions of unitary and Hermitian operators.

Definition 2.6.3. In a dagger category, a morphism 햠 ∶ 푉 → 푊 is called

(i) unitary if it is an isomorphism and 햠−1 = 햠†, and

28 (ii) self-adjoint or Hermitian if 햠 = 햠†.

Functors between dagger categories are called dagger functors if they are compatible with the dagger structure.

Definition 2.6.4. A functor 퐹 ∶ (풞, †) → (풞′,†′) between two dagger categories is called a dagger functor if it satisﬁes

′ 퐹 (햠†) = (퐹 햠)† (2.44) for all 햠 ∈ mor 풞.

2.6.2 Graphical Language

The graphical language of dagger categories extends that of categories. The adjoint of a morphism variable 햠 ∶ 푉 → 푊 is depicted as follows.

† 푊 푉 ⎜⎛ ⎟⎞ 햠† = ⎜ 햠 ⎟ = 햠 (2.45) ⎜ ⎟ ⎝ 푉 ⎠ 푊

The adjoint of any diagram is obtained by ﬁrst mirroring the diagram to a horizontal axis and then mirroring each arrow again such that the orientation of the arrow remains the same as before.

2.7 Dagger Compact Categories

2.7.1 Definition

Definition 2.7.1. A dagger compact category is a compact closed category (풞, ⊗, 퐼, 휎, 휀, 휂, ∗, †) with a dagger functor such that for all objects 푉 , 푊 , for all morphisms 햠, 햡 the following equations hold

(햠 ⊗ 햡)† = 햠† ⊗ 햡† (2.46) † 휎푉 ,푊 = 휎푊,푉 (2.47) † 휂푉 = 휎푉 ,푉 ∗ 휀푉 (2.48)

Note 2.7.2. Even though the term dagger compact category is used, a dagger compact category is closed as well, and thus the name dagger compact closed category is equally valid.

29 2.7.2 Graphical Language

The graphical language of dagger compact categories is the same as for compact closed categories except that we now also have the following identity

푉 푉 푉 푉 = (2.49)

If conjecture (2.5.3) is assumed to be true, one can prove the following theorem which plays a central role in this paper.

Theorem 2.7.3 (Coherence for Dagger Compact Categories). Given that conjecture (2.5.3) holds, a well-formed equation between morphisms in the language of dagger compact closed categories follows from the axioms of compact closed categories if and only if it holds, up to isomorphisms of diagrams, in the graphical language.

Proof. See [19], page 12. ■

30 Chapter 3

Category Theory Redefined

In this section we will deﬁne various notions of category theory using diagrams.

3.1 Categories, Functors and Natural Transformations

3.1.1 Categories

Definition 3.1.1. A category 풞 consists of the following data:

(i) a collection |풞| of labelled edges

푉 ;

(ii) for any two edges 푉 , 푊 ∈ |풞|, a set 풞(푉 , 푊 ) of directed progressive graphs, also called morphisms,

푊 햠 푉

with one labelled node and two labelled edges 푉 and 푊 ;

(iii) for all morphisms

푉 푊 햠 and 햡 (3.1) 푈 푉

31 푊 햡 푉 . 햠 푈

is also a valid morphism.

Here a progressive graph is to be interpreted as a graph whose edges point up- wards and do not turn back at any point. We make a clear distinction between the bottom of a picture and the top. Maps must always be read from bottom to top and in general there is no mirror symmetry between the two. The same is true about the left and right of the diagram, except when it comes to the labels of objects: labels might be on the left of the edge or on the right and there is no difference in meaning between the two. So what freedom does one have when drawing such a diagram? In general one is allowed to move nodes, bend and stretch wires as long as the composite diagram stays a progressive graph. Adding more structure to categories will enable us to bend wires more freely, rotate boxes, etc, but at the moment this is all there is. In general different nomenclature is used when reading about categories. The following table compares the diagrammatic notions with the conventional deﬁnitions.

푊 푊 햡 푉 햠 푉 푉 푉 햠 푈

푉 햠 ∶ 푉 → 푊 ퟣ푉 ∶ 푉 → 푉 햡 ∘ 햠 Object Morphism Identity Composition

Table 3.1: The different structures that constitute a category. From bottom to top: conventional name, conventional notation, diagrammatic notation

Notes 3.1.2. (i) A morphism variable, or box, can be viewed as a placeholder for an arbitrary, possibly composite, diagram.

32 Because of the following theorem one can merely use diagrams without having the fear of ‘missing out’ or ending up with contradictions.

Theorem 3.1.3 (Coherence for Categories). A well-formed equation between two morphism terms in the language of categories follows from the axioms of categories if and only if it holds in the graphical language up to isomorphism of diagrams.

Traditionally, mathematics has been founded on the category Set, where the objects are sets and the morphisms are functions. Therefore it is tempting to specify a physical system by giving its set of states, and a process by giving a function from states of one system to states of another. However, it turns out that the category Set is not the most interesting one for modelling quantum mechanics. For example, if we want to combine systems set-wise we do so by taking a direct product of sets. The direct product of two sets, 푆 = 푆1 × 푆2, has the property that there exists projections

푝1 ∶ 푆 → 푆1 and 푝2 ∶ 푆 → 푆2 such that by considering the images 푝1(푠) and 푝2(푠) of any element 푠 ∈ 푆 we can reconstruct the element 푠. There exist no projections on the tensor product of Hilbert spaces with such structure. Of course, one can always construct the tensor product of Hilbert spaces using a set-theoretic construction, but this makes it harder to understand the phenomena originating from the absence of projectors. Entanglement is a prime example. In chapter 4 we will introduce a better alternative to Set, namely FdHilb, the category with finite dimensional Hilbert spaces as objects and linear maps as morphisms. In physics other categories are used as well. Take for example a category where the objects represent choices of space, and the morphisms represent choices of spacetime. The simplest is nCob, where the objects are (푛 − 1)-dimensional manifolds, and the morphisms are 푛-dimensional cobordisms. Informally a cobordism 푓 ∶ 푋 → 푌 is an 푛-dimensional manifold whose boundary is the disjoint union of (푛−1)-dimensional manifolds 푋 and 푌 . Cobordisms are composed by ‘gluing the output of the first cobordism to the input of the second’. We will see that both nCob and FdHilb are examples of dagger compact categories and that suitable, structure preserving, maps from nCob to FdHilb define so called topological quantum field theories. Maps between categories that ‘preserve structure’ are called functors.

3.1.2 Functors between Categories

We call 퐹 ∶ 풞 → 풟 a functor if 퐹 maps edges to edges

퐹 ∶ 푉 ⟼ 퐹 푉 ; (3.2)

33 and morphisms to morphisms

푊 퐹 푊 퐹 ∶ 햠 ⟼ 퐹 햠 (3.3) 푉 퐹 푉 in such a way that

푊 퐹 푊 ⎜⎛ ⎟⎞ ⎜ 햡 ⎟ 퐹 햡 ⎜ ⎟ 퐹 ⎜ 푉 ⎟ = 퐹 푉 . (3.4) ⎜ ⎟ ⎜ 햠 ⎟ 퐹 햠 ⎜ ⎟ ⎝ 푈 ⎠ 퐹 푈

In some sense a functor captures the idea of changing context. Consider for example two categories 풞 and 풟 where 풞 is very abstract and 풟, in contrast, is a very concrete category. We can think of a functor 퐹 ∶ 풞 → 풟 as giving a ‘representation’ of the structure of 풞 in 풟. The following is a standard example. Example 3.1.4. For example, consider an abstract group 퐺. This group can be described by a category 햦 which has the following structure.

Objects: 햦 has one object, say ∗.

Morphisms: The set of morphisms of 햦 is given by

{푓푔 ∶ ∗ ↦ ∗| 푔 ∈ 퐺}, (3.5) where the composition of morphisms satisﬁes

푓푔 ∘ 푓ℎ = 푓푔ℎ (3.6) and

푓푒 = ퟣ∗. (3.7)

We then have that

A representation of 퐺 on a ﬁnite dimensional Hilbert space is a functor 퐹 ∶ 햦 → FdHilb.

An action of 퐺 on a set is a functor 퐹 ∶ 햦 → Set.

Both notions are ways to make an abstract group more concrete.

34 Note that the use of the word ‘representation’ in category theory is not necessarily associated with a matrix representation. It is just a way of representing object and morphisms in a more concrete way. When one is interested in matrix representation theory then one should look for functors from a category to the category of 푘-modules, a generalization of the category of vector spaces. A complex matrix representation of a category 풞 is then a functor 풞 → FdHilb. The concept of a ‘theory’ in quantum physics can be seen as a functor 퐹 ∶ 퐶 → 퐷 where the default choice for 퐷 is often either the category FdHilb, of finite dimensional Hilbert spaces, or InfHilb, the category of infinite dimensional Hilbert spaces. Topo- logical quantum field theories then neatly fit this description.

3.1.3 Natural Transformations between Functors

If functors are regarded as theories, then the notion of natural transformations could be regarded as maps between theories.

Definition 3.1.5. Given two functors 퐹 , 퐹 ′ ∶ 풞 → 풟, a natural transformation 휃 ∶ 퐹 → 퐹 ′ ′ maps assigns to every object 푉 in 풞 a morphism 휃푉 ∶ 퐹 (푉 ) → 퐹 (푉 ) such that for any morphism 햠 ∶ 푉 → 푊 in 풞 the following equation holds

′ 휃푊 ∘ 퐹 (햠) = 퐹 (햠) ∘ 휃푉 . (3.8)

Equations of the form of (3.8) are often rephrased as: ”the following diagram commutes for all 푉 , 푊 ∈ 풞, 햠 ∈ 풞(푉 , 푊 )”

퐹 푉 퐹햠 퐹 푊

휃푉 휃푊 (3.9) ′ 퐹 ′푉 퐹 햠 퐺푊

Commuting diagrams often reveal a lot of structure that would otherwise be hidden by traditional linear notation such as the notation used in (3.8). If we look at the top half of the diagram and ignore the 퐹 symbols we see the basic structure of a morphism in the category 풞. Since the diagram commutes for all 푉 , 푊 ∈ 풞, 햠 ∈ 풞(푉 , 푊 ) we see (while still ignoring the 퐹 symbols) all possible objects and morphisms in the category 풞. The 퐹 symbols now tell us that we see all possible objects and morphisms coming from the category 풞, but interpreted in the category 풟. The same is true for the bottom part of the diagram. The connection between the two theories or interpretations provided by 퐹 and 퐹 ′ is given by the natural transformation 휃. The commutativity of the diagram then tells us that we can safely work with the theory provided by the functor 퐹 and translate those results to the theory provided by the functor 퐹 ′ at any time. A concrete

35 example might clarify things further. Example 3.1.6. Let 퐹 ∶ 햦 → FdHilb and 퐹 ′ ∶ 햦 → FdHilb be two representations of a group 햦 (seen as a one element category). Then 퐹 (∗) and 퐹 ′(∗) are Hilbert spaces and a natural transformation 훼 ∶ 퐹 ⇒ 퐹 ′ is then an intertwining operator from one representation to another, i.e a linear operator

푋 ∶ 퐹 (∗) → 퐹 ′(∗) (3.10) such that

푋퐹 (푔) = 퐹 ′(푔)푋, ∀푔 ∈ 햦. (3.11)

3.2 Strict Monoidal Categories

Definition 3.2.1. A strict monoidal category (MC) is a triple (풞, ⊗, 퐼) where

(i) 풞 is a category;

(ii) ⊗ ∶ 풞 × 풞 → 풞 is a functor and

(iii) 퐼 is an object of 풞 such that ∀퐵 ∈ |풞|

퐼 ⊗ 푉 = 푉 = 푉 ⊗ 퐼. (3.12)

The following table compares the diagrammatic notions with the conventional definitions.

푊1 ⋯ 푊푛 푊1 푊2

푉 푊 (empty) 푠 햠 햠1 햠2

푉1 ⋯ 푉푚 푉1 푉2

푉 ⊗ 푊 퐼 푠 ∶ 퐼 → 퐼 햠 ∶ 햠1 ⊗ 햠2 푉1 ⊗ ⋯ ⊗ 푉푚 → 푊1 ⊗ ⋯ ⊗ 푊푛 Product of Unit object Scalar General Product of objects morphism morphisms

Table 3.2: The different structures that constitute a strict monoidal category. From bottom to top: conventional name, conventional notation, diagrammatic notation

For strict monoidal categories we are allowed to move the boxes and wires within the plane as long as we respect several rules. First it is important to remind ourselves

36 that a diagram has a distinguished bottom and top, namely the levels 푦 = 푎 and 푦 = 푏 which are part of the deﬁnition of a plane graph. We can also attach horizontal labels 푥 = 훼 and 푥 = 훽 which are such that all nodes and edges of the plane graph belong to the rectangle [훼, 훽] × [푎, 푏].

푦 = 푏 푊1 ⋯ 푊푛 햠

(i) no boxes or wires cross each other,

(ii) no boxes or wires are detached from the boundary of the rectangle.

Theorem 3.2.2 (Coherence for Monoidal Categories). A well-formed equation between morphism terms in the language of strict monoidal categories follows from the axioms of strict monoidal categories if and only if it holds, up to deformation of progressive plane graphs, in the graphical language.

3.3 Symmetric Monoidal Categories

Definition 3.3.1. A symmetric monoidal category (풞, ⊗, 퐼) is a monoidal category such that for all 푉 , 푊 ∈ |풞| there exists a

푊 푉

휎푉 ,푊 (3.13) 푉 푊 for which

−1 푊 푉 푉 푊 ⎜⎛ ⎟⎞ ⎜ 휎푉 ,푊 ⎟ = 휎푊,푉 (3.14) ⎜ ⎟ ⎝ 푉 푊 ⎠ 푊 푉

37 A symmetry tensor 휎푉 ,푊 will often be depicted by

푊 푉

. (3.15)

푉 푊

Theorem 3.3.2 (Coherence for Symmetric Monoidal Categories). A well-formed equation between morphisms in the language of symmetric monoidal categories follows from the axioms of symmetric monoidal categories if and only if it holds, up to isomorphism of diagrams, in the graphical language.

Note that graph isometry does not allow us to remove the crossing of wires introduced in (3.15), since this is shorthand notation for the tensor 3.13.

3.4 Compact Closed Categories

Definition 3.4.1. A compact closed category is a strict symmetric monoidal category (풞, ⊗, 퐼, 휀, 휂, ∗) such that for all edges 푉 ,

(i) there exists an edge

∗ ( 푉 ) ≡ 푉 ≡ 푉 ∗ (3.16)

and

(ii) morphisms

푉 푉 and . (3.17) 푉 푉

Here 푉 푉 is called the unit and is called the counit (of 푉 ). 푉 푉

Each morphism then has an adjoint mate

∗ 푉 푊 푉 ⎜⎛ ⎟⎞ ⎜ 햠 ⎟ = 햠 ≡ 햠 . (3.18) ⎜ ⎟ 푉 푊 ⎝ ⎠ 푊

This deﬁnition is ambiguous though. Consider for example the morphism 햠 ∶ 푉1 ⊗

38 푉2 → 푊1 ⊗ 푊2

푊1 ⊗ 푊2 푊1 푊2 햠 = 햠 = 햠 (3.19)

푉1 ⊗ 푉2 푉1 푉2 then we could say that either

푉1 푉2

푉2 푉1 햠 ∶= 햠∗ or 햠 ∶= 햠∗. (3.20)

푊2 푊1

푊1 푊2

In the ﬁrst case the adjoint of a diagram is simply a rotation of the diagram over 휋 radians. In the second case the new diagram is constructed from the original one by bending in-and outgoing wires, while keeping the order of the tensor product, and then yanking the wires. The second deﬁnition is the one that is used when dealing with matrices and we will call it the algebraic transpose and denote it by 햠푇 :

푉1 푉2

햠 ∶= 햠∗ =∶ 햠푇 . (3.21)

푊1 푊2

The ﬁrst deﬁnition is also useful and we will just call it the transpose and denote it by 햠푡:

푉2 푉1 햠 =∶ 햠푡 (3.22)

푊2 푊1

Table 3.4 compares the diagrammatic notions with the conventional deﬁnitions. In contrast to the previous diagrammatic notions we do not have a proper coherence theorem but only a conjecture.

Conjecture 3.4.2 (Coherence for Compact Closed Categories). A well-formed equation

39 푉

푉 푉 푉 햠 푉 푉

푊 ∗ ∗ ∗ ∗ 푉 휂푉 휀푉 햠 ∶ 푊 → 푉 Dual of object Unit Counit Adjoint mate

Table 3.3: The different structures that constitute a compact closed category. From bottom to top: conventional name, conventional notation, diagrammatic notation between morphisms in the language of compact closed categories follows from the axioms of compact closed categories if and only if it holds, up to isomorphisms of diagrams, in the graphical language.

A ‘weaker’ version of the above theorem has been proven by Kelly and Laplaza. They proved the above conjecture for the same graphical language as introduced above, but with the assumption that no substitution of diagrams, composed of tensor products, is allowed. We already encountered an ambiguity when considering the transpose of a diagram with multiple legs. Whether there are other ambiguous situations is not known, so the only thing we can say is that one should be careful with substituting diagrams composed of tensor products.

3.5 Dagger Compact Categories

The concept of a dagger compact category is motivated by the category of Hilbert spaces, where each morphism 햠 ∶ 푉 → 푊 has an adjoint 햠† ∶ 푊 → 푉 .

Definition 3.5.1. A dagger compact category is a compact closed category with an additional † functor that

(i) mirrors a graph to a horizontal axis;

(ii) mirrors the arrows again to point in the original direction

† 푊 푉 ⎜⎛ ⎟⎞ ⎜ 햠 ⎟ = 햠 (3.23) ⎜ ⎟ ⎝ 푉 ⎠ 푊

and

40 (iii) the following equation holds

푉 푉 푉 푉 = . (3.24)

Examples 3.5.2. (i) The category FdHilb of ﬁnite dimensional Hilbert spaces and linear maps is a dagger compact category. Here the morphisms are linear operators between Hilbert spaces, the product is the usual tensor product and the dagger is the Hermitian conjugate. Next chapter is devoted to this example.

(ii) The category nCob of cobordisms is a dagger compact category. The objects are smooth oriented manifolds and the morphisms 푛-dimensional cobordisms. The tensor product is the disjoint union and the dagger reverses the orientation of objects. A topological quantum ﬁeld theory can be deﬁned as a functor from nCob to FdHilb. We will expand on this example in chapter 5.

(iii) Infinite dimensional Hilbert spaces are not dagger compact. They are described by dagger symmetric monoidal categories. This is not surprising since the dual of a dual of an infinite dimensional Hilbert space is in general not isomorphic to the space itself. Hence the name compact: compact categories carry some form of ‘finiteness’.

41 42 Chapter 4

Categorical Quantum Mechanics

In this chapter the postulates of ﬁnite dimensional quantum mechanics are introduced using a graphical notation. The ﬁrst section deals with the Hilbert space formalism and the second section extends this formalism to the density matrix formalism. The section on Hilbert spaces is based on a combination of papers and books by Abramsky, Co- ecke and Kissinger [1, 6, 5, 3]. Even though the book [3] discusses the density matrix approach to quantum mechanics as well, most of the material on that section comes from [19].

4.1 Hilbert Space Formalism

Most people will be familiar with the following axiomatization of quantum mechanics.

Axioms 4.1.1. Axiom 1. Each physical system is associated to a complex Hilbert space 푉 .

Axiom 2. A state of a system is associated to a ray in 푉 , i.e. a subspace of 푉 of complex dimension 1.

Axiom 3. The Hilbert space of a composite system is the tensor product of the Hilbert spaces of the individual systems.

Axiom 4. Physical observables are represented by Hermitian matrices on 푉 .

Axiom 5. The expectation value of the observable 퐴 for a system in state represented by the unit vector |휓⟩ is

⟨휓|퐴|휓⟩ (4.1)

43 Axiom 6. The time evolution of a state is determined by the Schrödinger equation

푑 푖 |휓(푡)⟩ = 퐻|휓(푡)⟩ (4.2) 푑푡

Now consider the category FdHilb of ﬁnite dimensional Hilbert spaces. This is a, monoidally strict, dagger compact closed category that has the following structure.

Objects Finite dimensional Hilbert spaces 푉 푉

푊 Morphisms Linear maps between these Hilbert spaces 햠 푉

푊 햡 Composition Composition of linear maps 푉 햠 푈

Identity morphism Identity map on vectors 푉

Tensor product Tensor product between ﬁnite dimensional 푉 푊 between objects Hilbert spaces

Tensor product 푊1 푊2

between Tensor product between linear maps 햠1 햠2

morphisms 푉1 푉2

Monoidal unit The ﬁeld C (empty)

푉1 푉2 휎 ∶ 푉 ⊗ 푉 → 푉 ⊗ 푉 Symmetry 푉1,푉2 1 2 2 1

푉1 푉2

Dual object Conjugate Hilbert space 푉 ∗ = 푉̅ 푉

푉 Conjugate Algebraic transpose 햠 morphism 푊

dim 푉 퐴 퐴 C ∗ Unit 휂푉 ∶ → 푉 ⊗ 푉 ∶ 1 ↦ ∑ |푒푖⟩ ⊗ ⟨푒푖| 푖=1

44 ∗ Counit 휀푉 ∶ 푉 ⊗ 푉 → 퐼 ∶ |휙⟩ ⊗ ⟨휓| ↦ ⟨휓|휙⟩ 퐴 퐴

푉 Dagger Hermitian adjoint 햠 푊

Here the conjugate Hilbert space 푉 ∗ ∶= 푉̅ is the Hilbert space with the same elements as 푉 but with inner product and scalar multiplication conjugated. This special mul- tiplicative structure of the conjugate space is actually neatly encoded in the bra-ket notation:

휆(⟨휓|) = ⟨휆휓|,̄ ∀휆 ∈ C, ∀|휓⟩ ∈ 푉 ∗ (4.3) (⟨휓|) ⋅ (⟨휙|) = ⟨휓|휙⟩. (4.4)

The existence of the conjugate Hilbert space is not often mentioned in text books on quantum mechanics. The reason is that, when dealing with Hilbert spaces, the conjugate space is isomorphic to the algebraic dual space of 푉 , often also denoted by 푉 ∗. We will also identify these two spaces. We can rewrite the axioms of quantum mechanics as follows

Axioms 4.1.2. Axiom 1. Each physical system is associated to an object of FdHilb, i.e. a ﬁnite dimensional complex Hilbert space 푉 .

Axiom 2. A state of a system 푉 is associated to a morphism

푉 (휓 ∶ 퐼 → 푉 ) = (4.5) 휓

This is consistent with the idea of a state being a ray in 푉 . Since 휓 is linear, the image of 휓 is indeed a one dimensional subspace. Often the notation |휓⟩ is used and we shall adopt this convention.

Axiom 3. Composite objects are obtained as tensor products of individual objects and therefore the Hilbert space of a composite system is the tensor product of the Hilbert spaces of the individual systems.

45 Axiom 4. Physical observables are represented by Hermitian maps 햠:

푉 햠 (4.6) 푉

Axiom 5. The expectation value of the observable 햠 for a system in state represented by the unit vector |휓⟩ is given by

휓

푉 ⟨휓|햠|휓⟩ = 햠 (4.7) 푉 휓 where ⟨휓| ≡ (|휓⟩)†.

Axiom 6. The time evolution of a state is determined by the Schrödinger equation

푑 푖 |휓(푡)⟩ = 퐻|휓(푡)⟩ (4.8) 푑푡 푟 푟 where |휓(푡)⟩푟 is a representative for |휓⟩.

Note that so far we have not mentioned the idea of a basis or a direct sum of spaces. We could always choose an object in FdHilb and use the properties of Hilbert spaces to construct a basis and direct sum decompositions. It turns out that those concepts can be deﬁned on a categorical level as well. We will not expand on these structures for they cannot be presented properly by diagrammatic structures.

4.2 Density Matrix Formalism

The axiomatization given in the previous section is not the only one that is used to describe quantum mechanics. Often an approach based on density matrices is better suited to talk about probabilities and entanglement, for it enables one to easily distin- guish between classical correlations and entanglement correlations. In [19] Selinger provided a construction to implement density matrices and completely positive maps using the diagrammatic language above. This construction is called a CPM (read: completely positive map) construction and in general it takes a compact dagger category as input and returns a new category with the same objects

46 but where the morphisms are given by completely positive maps. To do so we need a few extra deﬁnitions.

Definition 4.2.1. (i) A morphism

푊 햠 (4.9) 푊

is called positive if there exists an object 푉 and a morphism

푊 햡 (4.10) 푉

such that

푊 푊 햡 햠 = 푉 . (4.11) 푊 햡 푊

In that case we also use the notation

푊 햠 . (4.12) 푊

(ii) By the matrix ⌈햠⌉ ∶ 퐼 → 푉 푡 ⊗ 푊 of a generic morphism

푊 햠 (4.13) 푉

47 we mean the following diagram

푊 푉

햠 (4.14)

(iii) A positive matrix is a morphism that is the matrix of a positive map.

Any positive matrix can be brought into the following form

푊 푊 푊 푊 푊 푊 햡 햡 = 햪 햪 = . (4.15) 햧 푉 푉

We then have the following deﬁnition.

Definition 4.2.2. We say that a morphism

푊 푊 햠 (4.16) 푉 푉 is completely positive if for all edges 푈 and all positive matrices 햡 the morphism

푊 푊 푈 햠 푈 (4.17) 푉 푉 햡 is a positive matrix.

Now we arrive at what Selinger calls the CPM construction in [19].

Definition 4.2.3. Given a dagger compact category 풞, we deﬁne a new category CPM(풞) whose objects are the same as the objects of 풞 and whose morphisms are completely positive maps in 풞. The composition of morphisms is the same as in 풞.

In the same paper the following theorem is proven.

Theorem 4.2.4. CPM(풞) is a dagger compact category.

48 The structure of CPM(FdHilb) is the following

Objects Finite dimensional Hilbert spaces 푉 푉

푊 푊 Completely positive maps (CPMs) on Morphisms 햠 Hilbert spaces 푉 ⊗ 푉 ∗ 푉 푉

푊 푊 햡 Composition Composition of CPMs 푉 푉 햠 푈 푈

Identity morphism Identity map on vectors 푉 푉

Tensor product Tensor product between ﬁnite dimensional 푉 푊 between objects Hilbert spaces

푊2 푊1 푊1 푊2

Tensor product between ‘Ordered product’ 햠1 ⊗ 햠2 햠1 햠2 morphisms

푉2 푉1 푉1 푉2 Monoidal unit The ﬁeld C (empty)

푊 푉 푉 푊 Symmetry 휎푉 ⊗푊,푊⊗푉 ∶푉⊗푊⊗푊⊗푉 →푊⊗푉⊗푉⊗푊 푉 푊 푊 푉

Dual object Conjugate Hilbert space 푉 ∗ = 푉̅ 푉

푉 푉

Conjugate Algebraic transpose 햠 morphism

푊 푊

49 dim 푉 퐴 퐴 C ∗ Unit 휂푉 ∶ → 푉 ⊗ 푉 ∶ 1 ↦ ∑ |푒푖⟩ ⊗ ⟨푒푖| 푖=1

∗ Counit 휀푉 ∶ 푉 ⊗ 푉 → 퐼 ∶ |휙⟩ ⊗ ⟨휓| ↦ ⟨휓|휙⟩ 퐴 퐴

푉 푉 Dagger Hermitian adjoint 햠 푊 푊

The axioms for quantum mechanics based on the density matrix formalism are the following.

Axioms 4.2.5. Axiom 1. Each physical system is associated to an object of CPM(FdHilb), i.e. a ﬁnite dimensional complex Hilbert space 푉 .

Axiom 2. A state 휌 of a system is a positive matrix

푉 푉 (4.18) 휌 such that

푉 = 1. (4.19) 휌

Axiom 3. Composite system correspond to tensor products of objects of CPM(FdHilb).

Axiom 4. Physical observables are represented by maps

푂 (4.20) 푉 푉 that decompose as

푉

푂 = . (4.21) 푉 푉 햠 햠 푉 푉

50 Axiom 5. The expectation value of the observable 푂 for a system with state 휌 is given by

푂 푉 푉 . (4.22) 휌

Axiom 6. The time evolution of a state is determined by a unitary map

푉 햴 (4.23) 푉 acting on a state in the following way

푉 푉

푉 푉 푒푖푡햴 푒푖푡햴 = . (4.24) 휌(푡) 푉 푉 휌(0)

51 52 Chapter 5

Topological Quantum Field Theory

In this chapter we will define what topological quantum field theories (TQFTs) are. The definition originally stems from a paper by Atiyah [2]. At that time the definition seemed to capture most of the known TQFTs but even then Atiyah proposed some possible modifications to the given definition. We will only give the original definition. To do so, we need a few concepts from differential topology.

5.1 Cobordisms

A cobordism is an equivalence relation on the class of compact topological manifolds of the same dimension, using the concept of the boundary of a manifold. We will assume the manifolds to be smooth differentiable manifolds and just call them manifolds for simplicity. Note that this is by no means the most general setting. Formally we have the following deﬁnitions.

Definition 5.1.1. (i) An (푛 + 1)-dimensional cobordism is a quintuple (햠, 푉 , 푊 , 푖, 푗) consisting of an (푛+1)-dimensional compact differentiable manifold with boundary, 햠; closed 푛-manifolds 푉 , 푊 ; and embeddings 푖 ∶ 푉 ⊂ 휕햠, 푗 ∶ 푊 ⊂ 휕햠 with disjoint images such that

휕햠 = 푖(푉 ) ⊔ 푗(푊 ). (5.1)

We will often abbreviate (햠, 푉 , 푊 , 푖, 푗) by (햠, 푉 , 푊 ).

(ii) If a cobordism (햠, 푉 , 푊 ) exist then we call 푉 and 푊 cobordant. The set of all manifolds cobordant to a ﬁxed manifold 푉 is called the cobordism class of 푉 .

(iii) If a manifold 푉 is the boundary of a compact manifold 푉 , then we call 푉 a boundary manifold.

53 (iv) An oriented cobordism (햠, 푉 , 푊 ) is a cobordism where 햠, 푉 and 푊 are oriented manifolds such that

휕햠 = 푉 ⊔ 푊 ∗, (5.2)

where 푊 ∗ denotes 푊 with the opposite orientation.

Some confusion may arise as to why the opposite orientation of the second boundary manifold is chosen in the deﬁnition of an oriented cobordism. This is mainly due the way an orientation of a manifold induces the orientation of the boundary. The following example should clarify this. Example 5.1.2. Let 푉 and 푊 be the oriented manifolds parametrized by

푉 ↔ (sin 휃, cos 휃, 0) ∈ R3, 휃 ∈ [0, 2휋[ (5.3) 푊 ↔ (sin 휃, cos 휃, 1) ∈ R3, 휃 ∈ [0, 2휋[ (5.4) and let their orientation be determined by their parametrization. An oriented cobordism (햠, 푉 , 푊 ) is given by the cylinder

햠 ↔ (sin 휃, cos 휃, 푧) ∈ R3, 휃 ∈ [0, 2휋[, 푧 ∈ [0, 1], (5.5)

where the orientation on 햠 is chosen such that 푉 ⊂ 휕햠1 has the same orientation as

휕햠1. Note that the orientations of the boundaries of the manifold 햠 are not determined by the parametrization but are induced by the orientation that 햠 caries. Therefore the ∗ border 휕햠2 ∶= 푊 has the opposite orientation of 푊 and 햠 = 휕햠1 ⊔ 휕햠2 = 푉 ⊔ 푊 . The structure of the category nCob is contained in the following table. The images in the rightmost column are examples of images, traditionally used to depict the corresponding structures.

Figure 5.1: Left: 푉 and 푊 embedded in R3. Center: 햠 embedded in R3 with induced orientation on the boundaries. Right: (햠, 푉 , 푊 ) embedded in R3.

The 푛-dimensional oriented cobordisms form a category nCob described as follows.

54 (푛 − 1)-dimensional compact Objects 푉 oriented manifolds 푉

푊 푛-dimensional oriented Morphisms 햠 cobordisms (햠, 푉 , 푊 ) 푉

푊 햡 (햠, 푈, 푉 ) ∘ (햡, 푉 , 푊 ) = Composition 푉 (햠 ∪ 햡, 푉 , 푊 ) 햠 푈

Identity 1 Cilinder 푉 = (푉 × [0, 1], 푉 , 푉 ) 푉 morphism

Tensor product Disjunct union 푉1 ⊔ 푉2 푉1 푉2 between objects

Tensor product 푊1 푊2 (햠, 푉1, 푊1) ⊔ (햠2, 푉2, 푊2) = between 햠1 햠2 (햠1 ⊔ 햠2, 푉1 ⊔ 푉2, 푊1 ⊔ 푊2) morphisms 푉1 푉2

Monoidal unit ∅ (empty) (empty)

푉1 푉2 휎 ∶ 푉 ⊔ 푉 → 푉 ⊔ 푉 Symmetry 푉1,푉2 1 2 2 1

푉1 푉2

55 푉 ∗ is the manifold 푉 with Dual object 푉 opposite orientation

푉 ∗ Conjugate (햠∗, 푊 ∗, 푉 ∗) 햠 morphism 푊 ∗

∗ 푉 푉 Unit 휂푉 ∶ ∅ → 푉 ⊔ 푉 , ∀푉 ∈ | nCob |

∗ Counit 휀푉 ∶ 푉 ⊔ 푉 → ∅, ∀푉 ∈ | nCob | 푉 푉

푉 Dagger (햠, 푉 , 푊 )† = (햠∗, 푊 , 푉 ) 햠 푊

5.2 Axioms of TQFT

Definition 5.2.1. Consider the categories nCob and FdHilb.A topological quantum field theory (TQFT) in dimension 푛 over C is a tensor functor 퐹 ∶ nCob → FdHilb that associates a ﬁnite dimensional Hilbert 퐹 Σ space to each compact oriented smooth 푛-dimensional manifold Σ such that

(i) there exists an element 퐹 푀 ∈ 퐹 휕푀 associated to each oriented smooth (푛+1)- dimensional manifold 푀;

(ii) 퐹 (Σ∗) = (퐹 Σ)∗ and

(iii) 퐹 (Σ†) = (퐹 Σ)†

56 Physically the manifolds Σ should be interpreted as manifolds of physical space. For every Σ we have a unit cobordism Σ×[0, 1] where the extra dimension is to be interpreted as imaginary time. The space 퐹 (Σ) is the Hilbert space of the quantum theory. In general a physical theory comes with a Hamiltonian 퐻 and a corresponding evolution operator exp(푖푡퐻). What distinguishes TQFTs is that 퐻 = 0, which implies that there are no real time dynamics along the cylinder Σ × [0, 1]. There can be non-trivial ∗ ‘propagation’ from manifolds Σ0 to Σ1 through a cobordism (푀, 휕푀0 = Σ0, 휕푀1 = Σ1). Also note that even though the Hamiltonian is trivial, the Lagrangian need not be trivial. TQFTs are very rich in structure and even the simplest cases turn out to be interesting as shown in the following example. Example 5.2.2. Zero dimensional TQFT. In the zero dimensional case the space Σ, being compact, consists of ﬁnitely many points, say 푛. To a single point 푝푖 the functor 퐹 associates a vector space 푉 = 퐹 (푝푖) and to 푚 points the functor associates the 푚-fold tensor product 푉 ⊗푚. Cobordisms are lines connecting points. By using the composition of morphisms in 0Cob, the set of cobordisms becomes a group, isomorphic to the symmetric group 푆푛. Zero dimensional TQFT thus corresponds to the theory of representations of the symmetric group. There is also a connection between 0D TQFTs and classical representation theory of Lie groups. For more information about this connection and examples in higher dimensions see [2].

57 58 Chapter 6

Finite Spin Lattices and Quantum Phases

In this chapter we introduce spin lattices for, which the Hamiltonians are local and gapped, and the concept of quantum phases. All of the material in this chapter comes from the PhD. thesis of Marien [13]. Most of the deﬁnitions have been rewritten for clarity and only the theory of relevance to us is presented.

6.1 Finite Spin Lattices

6.1.1 Definitions

Definition 6.1.1. (i) A finite lattice is an undirected graph ℒ = (풱, ℰ) where 풱 denotes the set of vertices of ℒ, ℰ denotes the set of edges (couples of vertices) of ℒ, and 풱 has ﬁnite cardinality.

(ii) Given a ﬁnite lattice ℒ = (풱, ℰ), a finite spin lattice is a map 푆ℒ ∶ 풱 → FdHilb

which maps each vertex 푣 to a ﬁnite dimensional Hilbert space 푝푣. The tensor product of all Hilbert spaces in the image of 푆,

푃ℒ = ⨂ 푃푣 (6.1) 푣∈ℒ

is called the lattice space and elements of the lattices space are called lattice states.

(iii) A periodic spin lattice is a spin lattice with a periodic graph.

(iv) A finite spin chain is a map 푆ℒ ∶ 풱 → FdHilb such that ℒ can be embedded on a circle. The length of the ﬁnite spin chain is deﬁned as |풱|

We will only be concerned with periodic ﬁnite spin lattices deﬁned by planar graphs. This implies that the graphs can be embedded in a circle or torus.

59 If we want to talk about correlations between spins, area laws, boundaries, and other essential notions for the theory of entanglement, then we will need to endow the lattice with a metric structure. In this paper the shortest distance metric will be used.

Definition 6.1.2. Let ℒ = (풱, ℰ) be a lattice and 푥, 푦 ∈ ℒ.

N (i) Let 푆(푥, 푦) denote the set of paths from 푥 to 푦 and let | |푥,푦 ∶ 푆(푥, 푦) → be the map that takes a path and returns it’s length. The shortest path metric d(, ) is deﬁned by

N d( , ) ∶ 풱 × 풱 → , (푥, 푦) ↦ 푑(푥, 푦) = min |푝|푥,푦. (6.2) 푝∈푆(푥,푦)

(ii) The distance between two subsets 푋, 푌 ⊂ 풱 is deﬁned as

D(푋, 푌 ) = min d(푥, 푦). (6.3) 푥∈푋,푦∈푌

(iii) The diameter of a subset 푋 ⊂ ℰ is deﬁned as

diam(푋) = max d(푥1, 푥2). (6.4) 푥1,푥2∈푋

(iv) The ball centered at 푣0 ∈ 풱 with radius 푅 is deﬁned as

퐵푅(푣0) = {푣 ∈ 풱| d(푣0, 푣) ≤ 푅}. (6.5)

(v) Let 푋 ⊂ 풱 then the boundary 휕푋 of 풱 is deﬁned as

휕푋 = {푣 ∈ 푋|퐵1(푣) ∩ (풱 ∖ 푋) ≠ ∅}. (6.6)

(vi) Let 푋1, 푋2 be a bipartition of 풱, i.e. 풱 = 푋1 ⨆ 푋2. The size of the area of the boundary 퐴 of this bipartition is deﬁned as

퐴 = max(|휕푋1|, |휕푋2|). (6.7)

Note 6.1.3. Other metrics are feasible as well as long as there exists a polynomial 푃 such that

max |퐵푅(푣)| ≤ 푃 (푟). (6.8) 푣∈풱

Here 퐵푅(푣) stands for the ball centered at 푣 with radius 푅 deﬁned by the metric in question.

60 Physical models are constructed from ﬁnite spin lattices by endowing the lattice with a Hamiltonian. This is a function which takes a lattice state and returns a real number, the energy associated to that state. To do so we need a few extra deﬁnitions.

Definition 6.1.4. Consider a ﬁnite spin lattice 푆ℒ.

(i) The algebra of observables associated to a site 푣 ∈ 풱 = dom(푆ℒ) is deﬁned as

풜 = (푆 (푣)) ≅ (C). 푣 hom ℒ Matdim 푆ℒ(푣) (6.9)

(ii) The algebra of observables of a subset 푋 of 풱 is deﬁned as

풜푋 = ⨂ 풜푣 (6.10) 푣∈푋

(iii) The support supp(퐴) of an operator 퐴 ∈ 풜풱 is deﬁned as the smallest set of sites on which 퐴 acts non-trivially.

It is often useful to deﬁne a lattice independent potential that generates the Hamil- tonian of spin lattice.

Definition 6.1.5. Given a ﬁnite spin lattice 푆ℒ.

풱 (i) A lattice potential is a map Φ ∶ 2 → 풜풱, 푋 ↦ Φ(푋) which maps subsets 푋 of 풱

to the operator algebra 풜푋 such that Φ(푋) is Hermitian for all 푋 ⊂ 풱.

(ii) The Hamiltonian 퐻Φ corresponding to Φ on 푆ℒ is deﬁned as

퐻Φ = ∑ Φ(푋). (6.11) 푋⊂풱

6.1.2 Locality

Not all Hamiltonians are good candidates to study with tensor networks techniques. Certain locality conditions and the appearance of a spectral gap are required. The following deﬁnitions capture these ideas

Definition 6.1.6. Let 푆ℒ ∶ 풱 → FdHilb be a ﬁnite spin lattice.

(i) A strictly local Hamiltonian 퐻 is a Hamiltonian for which there exists an 푅 ∈ N and a 퐶 ∈ R such that

퐻 = ∑ ℎ푣, and supp(ℎ푣) ⊂ 퐵푅(푣)∀푣 ∈ ℒ. (6.12) 푣∈ℒ

61 (ii) 퐻 is called quasi-local with decay function 푓 if it can be written as

퐻 = ∑ ∑ ℎ푣(푟), supp(ℎ푣(푟)) ⊂ 퐵푅(푣) and ‖ℎ푣(푟)‖ ≤ 푓(푟) ∀푣 ∈ ℒ. (6.13) 푣∈ℒ 푟∈N

(iii) A lattice potential is called strictly local (resp. quasi local) if it generates a strictly local (resp. quasi local) Hamiltonian.

(iv) A lattice potential Φ is called translation-invariant if

Φ(휎(풱)) = Φ(풱) (6.14)

for all permutations 휎 of the set of vertices 풱 of any lattice ℒ.

If a potential is translation invariant then we can use it to generate a set of Hamilto- nians {퐻ℒ|ℒ a finite spin lattice}. A Hamiltonian encodes the way spins interact with each other. To deﬁne the notion of a gapped Hamiltonian we will need spin lattices with the same interaction that grow in system size. This is the reason why we introduced translation invariant Hamiltonians, for there is no natural way increase the lattice size and maintain ‘the same interactions’ if the Hamiltonian is not translation invariant. Now we have all the necessary tools to deﬁne the notion of a gapped Hamiltonian.

Definition 6.1.7. Let Φ be a translation invariant lattice potential that generates a set of translation invariant Hamiltonians {퐻ℒ}. We call the Hamiltonians 퐻ℒ and the potential Φ gapped with ground state degeneracy 푞 if

1 2 푞 (i) 푞 is the largest number such that the 푞 lowest eigenvalues 퐸0 (퐻ℒ), 퐸0 (퐻ℒ), …, 퐸0(퐻ℒ)

of 퐻ℒ satisfy

푖 푗 lim max |퐸0(퐻ℒ) − 퐸0(퐻ℒ)| = 0 (6.15) |풱|→∞ 푖,푗

and

(ii) there exists a Δ ∈]0, +∞[ such that for each lattice ℒ and each eigenvalue 퐸(퐻ℒ)

that is not one of the smallest 푞 eigenvalues of 퐻ℒ

푖 min |퐸0(퐻ℒ) − 퐸(퐻ℒ)| > Δ. (6.16) 푖

The constant Δ is called the spectral gap.

It turns out that, given a lattice ℒ with metric d and a strictly local Hamiltonian 퐻 a strictly local observable 퐴 under the time evolution generated by 퐻 remains ‘approxi- mately local’ after a ﬁnite time 푡. The precise statement is the following

62 Theorem 6.1.8 (Lieb-Robinson). Let ℒ be a ﬁnite spin lattice with metric d and lattice potential Φ for which there exist 휇, 푠 ∈]0, +∞[ such that for all 푣 ∈ 풱 and subsets

풳푣 ⊂ 풱 containing 푣

휇 diam 풳푣 ∑ ‖Φ(풳푣)‖|풳푣|푒 ≤ 푠 < ∞, (6.17)

풳푣

Let 풴 ⊂ ℒ, 퐻 be the Hamiltonian generated by Φ on 풴 and let 푂풜, 푂ℬ be local operators supported on disjoint ﬁnite sets 풜, ℬ ⊂ 풴. Then

2푠|푡|−휇 d(풜,ℬ) ∥[ exp(−푖퐻푡)푂풜 exp(푖퐻푡), 푂ℬ]∥ ≤ 2 ∥푂풜∥ ∥푂ℬ∥ |풜| 푒 . (6.18)

There is also a more general theorem that holds for quasi-local Hamiltonians.

Theorem 6.1.9. Let ℒ be a ﬁnite spin lattice with metric d and lattice potential Φ for which there exists a function 푄 ∶ R →]0, +∞[ and a constant 휆 ∈]0, +∞[ such that for all 푣, 푤 ∈ 풱 and all subsets 풳푣,푤 ⊂ 풱 that contain 푣, 푤 we have the following inequalities:

∑ 푄(d(푣, 푥))푄(d(푥, 푤)) ≤ 휆푄(d(푣, 푤)), (6.19) 푥∈ℒ

∑ ‖Φ(풳푣,푤‖) ≤ 푄(d(푣, 푤)). (6.20)

풳푣,푤

Let 풴 ⊂ ℒ, 퐻 be the Hamiltonian generated by Φ on 풴, 푂풜, 푂ℬ be local operators supported on disjoint ﬁnite sets 풜, ℬ ⊂ 풴. Then

푒2휆|푡| ∥[ exp(−푖퐻푡)푂 exp(푖퐻푡), 푂 ]∥ ≤ 2 ∥푂 ∥ ∥푂 ∥ |풜||ℬ| 푄(d(풜, ℬ)) . (6.21) 풜 ℬ 풜 ℬ 휆

We can then deﬁne the speciﬁc potentials that we will be dealing with when talking about exact quasi-adiabatic continuation.

Definition 6.1.10. (i) Consider a ﬁnite spin lattice ℒ equiped with metric d and let Φ be a potential on ℒ. We call the quatruple (ℒ, d, Φ, 푄) with 푄 ∶ R →]0, +∞[ an LR-local system with decay function 푄 if 푄 satisﬁes (6.19) and (6.20). If there for all 훼 ∈ R there exists an 푥 ∈]0, +∞[ such that

푄(푦) < 푦−훼 ∀푦 > 푥 (6.22)

then we say that 푄 decreases superpolynomially and we call the quatruple (ℒ, d, Φ, 푄) an LR-local system.

63 6.2 Quantum Phases

We will only present the basic definitions we need to get a feeling for the concept of gapped quantum phases. At the moment there is still a lot of research going on concerning the theory of quantum phases and not every author uses the same definitions. However, most of the definitions are based on the same principles that we will present in this section. To define quantum phases we need the concept of a quasi-adiabatic path.

Definition 6.2.1. Consider a ﬁnite spin lattice ℒ and a 푑-dimensional differentiable manifold ℳ of lattice potentials Φ. If there exists a piecewise differentiable path 훾 ∶ 푠 → Φ(푠), 푠 ∈ [0, 1] of LR-local, gapped potentials Φ(푠) such that there exists a super- polynomial decay function 푄 that dominates the decay functions of all Φ(푠) and 휕푠Φ푠 then we call 훾 a quasi-adiabatic path.

This definition induces an equivalence relation on the manifold ℳ of lattice potentials Φ, which in turn defines an equivalence relation on the set of ground states of Hamiltonians generated by those lattice potentials. This allows us to define the concept of a gapped quantum phase.

Definition 6.2.2. Let ℒ be a lattice and 퐻(0), 퐻(1) be Hamiltonians generated by LR- local gapped potentials Φ(0) resp. Φ(1). If there exists a quasi adiabatic path 훾 ∶ 푠 → Φ(푠), 푠 ∈ [0, 1] of LR-local gapped potentials Φ(푠) then we say that the elements of the ground state subspaces 푉퐺푆(퐻(0)) and 푉퐺푆(퐻(1)) are in the same gapped quantum phase.

Note that a strictly local Hamiltonian is always LR-local.

64 Chapter 7

Matrix Product States

In this chapter matrix product (MPS) states will be introduced. Matrix product have many interesting properties, both computationally and theoretically. Matrix product states and higher dimensional analogues are often described by drawing graphs where the vertices are tensors, open edges tensor indices, and connected edges for contraction of those indices. By seeing a contraction as a form of maximal entanglement we thus obtain a network of entanglement. In the first section MPS are introduced by considering a concrete example. In the second section the diagrams, constructed in the previous chapters, will be used to define MPS. A brief description of some basic properties is given in the third section as well. In the fourth section we take a look at a how to bring MPS in a particularly useful form, called the block injective canonical form, and we state the fundamental theorem of matrix product states. We end the thesis by giving references to papers that connect the theory of matrix product states and the classification on gapped quantum phases. One of these papers provides a connection with TQFTs. The main structure of this chapter is based on the works by Verstraete, Wolf, Perez- Garcia, Schuch, and Cirac [21, 4]. The introduction to tensor networks by Roman [14] and the PhD Thesis of Marien [13] provided the material for the subsection on properties of MPS.

7.1 Introductory Example

We will introduce matrix product states using a simple example.

1 Example 7.1.1. Consider a spin 2 chain of length 푁 and the following Hamiltonian

푁 1 1 2 2 3 3 퐻 = ∑ 휎푖 휎푖+1 + 휎푖 휎푖+1 + 휎푖 휎푖+1 (7.1) 푖=1

1 1 where 휎푖휎푖+1 ≡ ⊗ ⋯ ⊗ 휎푖 ⊗ 휎푖+1 ⊗ ⋯ ⊗ denotes the action of the 푚th Pauli matrix

65 on the 푖th and (푖 + 1)st spin. If 푁 = 2, i.e. if we are working with a chain of two spins, then the Hamiltonian has the following form

1 0 0 0 ⎡ ⎤ ⎢0 −1 2 0⎥ 퐻 = ⎢ ⎥ . (7.2) ⎢0 2 −1 0⎥ ⎣0 0 0 1⎦

퐻 has 1 lowest eigenvalue equal to −3 with corresponding normalised eigenvector

|01⟩ − |10⟩ |Φ ⟩ = √ . (7.3) 퐺푆 2

So the ground state for the two-spin system with Hamiltonian 퐻 is a maximally entangled state. Graphically this ground state is depicted by

|Φ⟩ = |Φ⟩ . (7.4)

How would one construct the ground state for a spin chain of length 푁? Clearly diag- onalisation of the Hamiltonian is not an option for the system size grows exponentially in 푁. Looking at the two-spin system we might propose a translationally invariant state where each spin is maximally entangled to both neighbouring spins. The problem is that the principle of monogamy of entanglement [20] dictates that if two spins are maximally entangled, their entanglement with any other spin must be 0. The solution to this problem comes from the clever observation that tensor contraction can actually be regarded as some form of maximal entanglement ‘between the tensor indices’. This comes from the fact that the matrix of the identity operator 1 dim ℋ = ∑푘=1 |푖⟩⟨푖| on a Hilbert space ℋ can be regarded as a maximally entangled state itself. The clever trick is now to regard each spin as being composed of two smaller particles, each living in a ‘virtual’ Hilbert space of dimension 퐷 and relate the state of the real spin to those of the virtual spins by a tensor 햠 ∶ 푉 ⊗ 푉 ∗ → 푃 , where 푉 denotes a virtual Hilbert space and 푃 denotes the physical (real world) Hilbert space. ′ Since we now have two constituent particles, say 푝푖 and 푝푖 for each 푖th site we can safely build maximally entangled pairs between virtual particles in neighbouring physical spins. By introducing periodic boundary conditions, which should not alter the systems behaviour for large 푁, we can also entangle the two spins at the end points. We will assume periodic boundary conditions from now on. The ansatz for the ground state then becomes

dim 푃 푖1 푖2 푖푛 |휓퐺푆⟩ = ∑ Tr[햠 햠 ⋯햠 ]|푖1⟩ ⊗ |푖2⟩ ⊗ ⋯ ⊗ |푖푛⟩ (7.5) 푖1,푖2,…,푖푛=1

66 ↓

↓ |Φ⟩ |Φ⟩ |Φ⟩ |Φ⟩ |Φ⟩

Figure 7.1: In a generic spin chain we split the spins in subparticles which can then be maximally entangled with each other

푖 푖푗 where 햠 푗 is shorthand for 햠푚푛|푚⟩⟨푛|, i.e. the 푗th 퐷 × 퐷 matrix in the representation C 휙 ∶ 햠 → 푃 ⊗Mat퐷( ). A shorter notation is obtained by setting 푖1 ≡ ퟏ, 푖2 ≡ ퟐ, …, 푖푛 ≡ 퐍 and |ퟏ, ퟐ, ⋯, 퐍⟩ ≡ |푖1⟩ ⊗ |푖2⟩ ⊗ ⋯ ⊗ |푖푛⟩:

dim 푃 ퟏ ퟐ 퐍 |휓퐺푆⟩ = ∑ Tr[햠 햠 ⋯햠 ]|ퟏ, ퟐ, ⋯, 퐍⟩. (7.6) ퟏ,ퟐ,⋯퐍=1

Finding the ground state now corresponds to determining the matrix coefﬁcients of 푖 푃 {햠푚푛}푖=1. Any state can be written down as a matrix product state if we allow the matrices to be site dependent and allow the 퐷 to grow exponentially in system size. However, it turns out that if we are looking for ground states of strictly local gapped Hamiltonians, the size of the matrices only grows polynomially in system size for a ﬁxed error threshold. On top of that, we can use transfer matrix techniques to optimize the search for the best parameters.

In the following sections we will see that there are also good theoretical reasons for working with matrix product states. As a matter of fact, we will not be bothered too much with computational complexity. The eventual goal will be to present an informal idea of how to approach a classification of ground states of translation invariant strictly local gapped Hamiltonians defined on finite spin lattices.

Only statements for which the proofs are constructive and might give some insight in the nature of the theory will be proven. In the other cases a reference is provided to the paper(s) in which the result is proven.

In what follows 푃 , 푉 and 푉푖 will always denote Hilbert spaces with dim 푃 = 푑, dim 푉 = 퐷 and dim 푉푖 = 퐷푖.

67 7.2 Definitions

Definition 7.2.1. Consider a ﬁnite spin chain of length 푁. The matrix product state (MPS) |휓(푁)(햠)⟩ of length 푁 generated by the tensors

푃푖 햠 = 햠 , 푖 = 1, …, 푁 (7.7)

푉푖 푉푖+1

with 푉푁+1 ≡ 푉1 is deﬁned as

푃1 푃2 푃푁−1 푃푛 |휓(푁)(햠)⟩ = 햠 햠 ⋯ 햠 햠 , (7.8) 푉푁

푉1 푉2 푉푁−2 푉푁−1 where the diamond shapes at the outer edges imply that they should be connected. We write 핍(햠) ∶= {|휓(푁)(햠)|푁 ∈ N} for the set of MPS generated by 햠 and call 햠 the generating tensor of 핍(햠).

Notation 7.2.2. Often MPS are depicted as follows

푃1 푃2 푃푁−1 푃푛 . (7.9) 햠1 햠2 ⋯ 햠푁−1 햠푁 푉1 푉2 푉푁−2 푉푁−1 푉푁

We will adopt this convention for situations where no ambiguity arises.

Examples 7.2.3. AKLT. Consider a spin chain where each lattice site has spin 1 and 1 let {|퐢⟩}퐢=−1 be the standard orthonormal basis for each 푝푖. Let

0 1 0 0 −푖 0 1 0 0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 푆1 ∶= ⎢ ⎥ , 푆2 ∶= ⎢ ⎥ , 푆3 ∶= ⎢ ⎥ , 푖 ⎢1 0 1⎥ 푖 ⎢−푖 0 −푖⎥ 푖 ⎢0 0 0 ⎥ (7.10) ⎣0 1 0⎦ ⎣ 0 −푖 0 ⎦ ⎣0 0 −1⎦ be matrices that work on the 푖푡ℎ lattice site and set

1 1 2 2 3 3 퐒푖 ⋅ 퐒푖+1 ∶= 푆푖 푆푖+1 + 푆푖 푆푖+1 + 푆푖 푆푖+1. (7.11)

The AKLT-Hamiltonian is deﬁned as

1 2 퐻퐴퐾퐿푇 ∶= ∑ 퐒푖 ⋅ 퐒푖+1 + (퐒푖 ⋅ 퐒푖+1) . (7.12) 푖 3

68 The ground state |휓(푁)(햠)⟩ of this system is generated by the tensor 햠 where

2 0 0 1 −1 0 2 0 1 햠−1 = −√ [ ] , 햠0 = √ [ ] , 햠1 = √ [ ] (7.13) 3 1 0 3 0 1 3 0 0

Notice that

2 2 (햠−1) = (햠1) = 0, (7.14) and √ 2 햠0햠−1 = −햠−1햠0 = 햠−1, (7.15) 3√ 2 햠1햠0 = −햠0햠1 = 햠1. (7.16) 3

Since the coefﬁcients of the ground state are obtained by a trace operator we immediately see from (7.15) that the ground state contains no terms with only one spin different from zero. By combining (7.14) with (7.15) we see that it is also impossible for the ground state to have two spins different from 0 next to each other or separated only by spin 0 states. This implies that a spin −1 state on the chain is only possible if it is accompanied by a neighbouring (modulo spin 0 states) spin 1 state and vice versa. Indeed, for a chain of length 4 we get

2 |0000⟩ − |00 ↓↑⟩ + |0 ↓ 0 ↑⟩ − |0 ↓↑ 0⟩ − | ↓ 00 ↑⟩ 4 ⎛ ⎞ |휓(4)(햠)⟩ = ⎜+| ↓ 0 ↑ 0⟩ − | ↓↑ 00⟩ + | ↓↑↓↑⟩ − |00 ↑↓⟩ + |0 ↑ 0 ↓⟩⎟ , 9 ⎜ ⎟ ⎝−|0 ↑↓ 0⟩ − | ↑ 00 ↓⟩ + | ↑ 0 ↓ 0⟩ − | ↑↓ 00⟩ + | ↑↓↑↓⟩ ⎠ where | ↓⟩ ≡ | − 1⟩ and | ↑⟩ ≡ |1⟩, and the brackets denote the bracketing of a large number of summations, not a matrix.

Definition 7.2.4. The map

∗ ⊗푁 (푁) 휑푛 ∶ (푉 ⊗ 푉 ) ⊗ 푃 → 푃 , 햠 ↦ |휓 (햠)⟩ (7.17) will be called the lattice generating map of length 푁. The map

Φ ∶ (푉 ∗ ⊗ 푉 ) ⊗ 푃 → ⋃ {푝⊗푁 } , 햠 ↦ 핍(햠) (7.18) 푁∈N will be called the total lattice generating map.

Note 7.2.5. Let 햠 and 햡 be generating tensors of a MPS, and 햷 ∶ 푉 → 푉 a non-

69 degenerate tensor such that

퐵 = 햷햠햷−1, (7.19) then, ∀푁 ∈ N,

휑푛(햠) = 휑푛(햡) (7.20) i.e. none of the 휑푛 are injective.

Definition 7.2.6. Consider two tensors 햠, 햡 that generate a MPS. If there exist a non- degenerate tensor 햷 ∶ 푉 → 푉 such that

퐵 = 햷햠햷−1 (7.21) then we say that the tensors 햠 and 햡 are related by a gauge tensor 햷 and that the MPS are invariant under the gauge transformation induced by 햷.

7.3 Basic Properties

7.3.1 Span of generalized MPS

The MPS in this paper are all translation invariant. If we drop this assumption and allow the tensors that generate the MPS to be depend on the lattice site then any state can be represented by a MPS.

N Theorem 7.3.1. Let |휓⟩ ∈ 푃1 ⊗ ⋯ ⊗ 푃푛, 푛 ∈ . Then there exists a basis {|ퟏ⟩ ⊗ ⋯ ⊗ 푑 |퐧⟩}ퟏ,…,퐧 of 푃1 ⊗ ⋯ ⊗ 푃푛 and (site dependent) tensors {햠[푖]|푖 = 1, …, 푛} such that

푑 |휓⟩ = ∑ Tr(퐴ퟏ[1]퐴ퟐ[2]⋯퐴퐧[푛])|ퟏ⟩ ⊗ ⋯ ⊗ |퐧⟩ (7.22) ퟏ,…,퐧=1

The proof of this statement can be found in most introductory texts on MPS and basically consists of splitting the vector space, applying a singular value decomposition for the bipartition obtained and recursively continuing this process. The states of interest are then these for which 퐷 is bounded or 퐷 ∼ poly(푛). Most papers on the subject reserve the term MPS for such states. Typically by ”states that can be approximated efﬁciently as MPS” it is meant that it is possible to keep the error

‖휓푡푟푢푒 − 휓푀푃 푆‖ (7.23)

70 within an certain error threshold 휀 for increasing size 푁 if we allow the bond dimension 퐷 to grow polynomially: 퐷 ∼ poly(푛). The reason MPS are so interesting is because low energy states of gapped strictly local translation invariant Hamiltonians in 1푑 can be efﬁciently approximated by an MPS with with a constant value of 퐷 [7]. For the rest of the paper we will work with translation invariant states.

7.3.2 Entanglement

Consider an MPS

푑 |휓(푁)(햠)⟩ = ∑ Tr[햠ퟏ햠ퟐ⋯햠퐍]|ퟏ, ퟐ, ⋯, 퐍⟩ (7.24) ퟏ,ퟐ,⋯퐍=1 and let 푃 ⊗푚 ⊗ 푃 ⊗푛 = 푃 ⊗푁 be a bipartition corresponding to two connected regions on the spin chain. By performing a Schmidt decomposition we ﬁnd that

퐷 ( ′ |휓 푁)(햠)⟩ = ∑ √휆푖|흓푖⟩ ⊗ |흓푖⟩, (7.25) 푖=1 where some of the 휆푖 are allowed to be zero. The Neumann entropy is then independent of the system sizes 푚, 푛 and therefore translation invariant MPS satisfy an area law.

7.3.3 Expectation Values

Let 푂 ∈ 푃푖 ⊗ 푃푖+1 ⊗ ⋯ ⊗ 푝푗 → 푃푖 ⊗ 푃푖+1 ⊗ ⋯ ⊗ 푝푗 be an operator acting on spins with indices 푖, 푖 + 1, …, 푗 and then its expectation value ⟨푂⟩ for the state |휓(푁)(햠)⟩ is given by

푉1 푉푖 푉푖−1 푉푗−1 푉푗 푉푁−1 푉푛 햠1 ⋯ 햠푖 ⋯ 햠푗 ⋯ 햠푁

푃푖 푃푖 ⟨푂⟩ = , (7.26) 푃1 푂 푃푛

푃푖 푃푗 푉1 푉푖−1 푉푖 푉푗−1 푉푗 푉푁−1 푉푛 햠1 ⋯ 햠푖 ⋯ 햠푗 ⋯ 햠푁 i.e. by sandwiching the operator between the MPS and its Hermitian conjugate.

71 7.4 Canonical Form and Injectivity

At the end of section 7.2 we have seen that different tensors can generate the same MPS. A straightforward example being two tensors related by a gauge tensor. Picking the right tensor to generate a MPS can have great advantages. The following theorem shows that given a tensor 햡, we can always construct a new tensor 햠 such that Φ(햠) = Φ(햡) and 햠 has a particularly nice form.

Lemma 7.4.1. Consider a tensor 햡 ∈ (푉 ∗ ⊗ 푉 ) ⊗ 푃 generating a MPS. There exist 푟 ∗ tensors {햠(푘)}푘=1 and a tensor 햠 ∈ (푉 ⊗ 푉 ) ⊗ 푃 such that

핍(햠) = 핍(햡) (7.27)

C and (with 휇푘 ∈ )

푟 햠 = ⨁ 휇푘햠(푘) (7.28) 푘=1 where none of the 햠(푘) have a proper invariant subspace of 푣.

Proof. Either

(i) 햡 has no invariant subspace in 푣 or

(ii) there exist one or more subspaces of 푣 invariant under the action of 햡.

In the ﬁrst case we can set

햠 ∶= 햡 (7.29) and the lemma is proven. Assume that the second case holds and denote by 푆1 one of the subspaces of 1 푣 with dimension 퐷1 such that 푆 contains no proper invariant subspace. Denote by ∗ 1 ∗ ⟂ 햯1 ∈ 푣 ⊗ 푣 (햰1 = − 햯1 ∈ 푣 ⊗ 푣 ) an orthogonal projector onto 푆1 (푆1 ). We then have that

햡햯1 = 햯1햡햯1, (7.30) 햰 햡 = 햰 햡햰 = ퟘ ⊕ (햱) , 1 1 1 퐷1×퐷1 1 (7.31)

ퟘ ∶ 퐻 → 퐻 ; 휓 ↦ 0 휓 = 0 where 퐷1 퐷1 .

Now consider (햱)1 and apply the same proces: ﬁnd a 퐷2 dimensional invariant ⟂ subspace 푆2 ⊂ 푆1 that does not contain any proper invariant subspace, denote by 햯2

72 1 an orthogonal projector on 푆2 and set 햰2 = − 햯1 − 햯2 then we have

햡햯2 = 햯2햡햯2, (7.32) 햰 햡 = 햰 햡햰 = ퟘ ⊕ (햱) . 2 2 2 (퐷1+퐷2)×(퐷1+퐷2) 2 (7.33)

Repeating this proces we ﬁnd 푟 ∈ N proper invariant subspaces. Setting

푟

햠 = ∑ 햯푘햡햯푘 (7.34) 푘=1 we ﬁnd that

푟 햠 = ⨁ 휇푘햠(푘) (7.35) 푘=1 where the 햠(푘) have no proper subspace of 푣 and the 휇푘 represent the redundancy in ■ choice of projectors 햯푘.

We can look at the scalar ⟨휓(푁)(햠(푘))|휓(푁)(햠(푘))⟩ as being two MPS stacked on top of each other but also as the trace of 푁 maps built by the contraction of the physical indices of 햠(푘) with (햠(푘))†. By sliding the bottom tensor 햠(푘) to the right of (햠(푘))† we obtain a completely positive map.

Definition 7.4.2. Given a tensor

푃푖 햠 (7.36)

푉푖 푉푖+1

푟 such that 햠 = ⨁푘=1 휇푘햠(푘), then we deﬁne the completely positive map ℰ햠(푘) by

푉푖+1(푘) 푉푖+1(푘)

푉푖+1 푉푖+1

ℰ햠(푘) = 햠(푘) 햠(푘) (7.37) 푉푖(푘) 푉푖(푘)

푉푖(푘) 푉푖(푘)

푃푖

The scalar ⟨휓(푁)(햠(푘))|휓(푁)(햠(푘))⟩ can then be seen as the trace of the stacking of 푁 completely positive maps.

∗ 푟 Definition 7.4.3. We call a tensor 햠 푉 ⊗ 푉 → 푃 with decomposition 햠 = ⨁푘=1 휇푘햠(푘) a normal tensor (NT) if

73 (i) there exists no non-trivial projector 햯 ∶ 푉 → 푉 such that 햠햯 = 햯햠햯

(ii) for all 푘 = 0, ..., 푟, ℰ햠(푘) has a unique eigenvalue 휆푘 such that 휆푘 = |휆푘| = 1.

We call |휓푁 (햠)⟩ a normal matrix product state of length 푁 (NMPS) if it is generated by a normal tensor.

Definition 7.4.4. We say that a tensor 햠 has a canonical form (CF) if

푟 햠 = ⨁ 휇푘햠(푘) (7.38) 푘=1

푟 and the tensors 햠(푘) are NT. The decomposition ⨁푘=1 휇푘햠(푘) is called a canonical decomposition of 햠.

푟 It is clear that if 햠 = ⨁푘=1 휇푘햠(푘) then

푟 (푁) (푁) |휓 (햠)⟩ = ∑ 휇푘|휓 (햠(푘))⟩. (7.39) 푘=1

A canonical decomposition is not unique and the vectors |휓(푁)(햠(푘))⟩ need not be independent per se. Linear dependence happens, for instance, if two of the tensors, say 퐴(푘) and 퐴(푗)̃ , satisfy

̃ 푖휑푗 −1 퐴(푗) = 푒 푋푗퐴(푘)푋푗 (7.40) for some invertible 푋푗 and phase 휑푗. We can tweak the set of block tensors such that a new canonical form is obtained where the different vectors appearing in (7.39) are linearly independent.

푟 Definition 7.4.5. Let 햠 = ⨁푘=1 휇푘햠(푘) then the tensors {햠(푗)|푗 = 1, ..., 푔, 푔 ≤ 푟} form a basis of normal tensors (BNT) of a tensor 햠 if

(i) the 햠(푗) are NT;

(ii) for each 푁, |휓(푁)(햠)⟩ can be written as a linear combination of |휓(푁)(햠(푗))⟩;

(푁) (iii) there exists some 푁0 such that for all 푁 > 푁0 |휓 (햠(푗))⟩ are linearly independent.

Proposition 7.4.6. The tensors 퐴(푗) (푗 = 1, ..., 푔) form a BNT of 퐴 iff

(i) for all NT appearing in its CF 7.38 퐴(푘)̃ there exist a 푗, a non-degenerate tensor

74 푋푘, and some phase 휑푘 such that

̃ 푖휑푘 −1 퐴(푘) = 푒 푋푘퐴(푗)푋푘 ; (7.41)

(ii) for any element 퐴(푗), there is no other 푗′ for which this is possible.

It turns out that for any tensor, there always exists a BNT [4]. Now we introduce a deﬁnition of fundamental importance in tensor network theory.

Definition 7.4.7. (i) A NT 햠 ∶ 푉 ∗ ⊗ 푉 → 푃 is called injective if ∀햡 ∈ 푉 → 푉 , there exists a 푣 ∈ 푃 such that

햡 = 햠(푣). (7.42)

(ii) If 햠 is a tensor with BNT {햠(푘)|푘 = 1, …, 푔}, then we say that 햠 has a block injective 푟 ∗ canonical form (BICF) if for each 푋 ∈ ⨁푗=1(푉푘 ⊗ 푉푘 ) there exists a 푣푋 ∈ 푃 such that

푟 푋 = [⨁ 햠(푘)] (푣푋). (7.43) 푘=1

Notes 7.4.8. (i) The deﬁnition of injectivity basically says that we can build any tensor 햡 ∶ 푉 → 푉 by acting on the physical index. The deﬁnition of BICF then says that by acting on the physical index we have access to every element of every BNT.

(ii) Having a BICF is a very strong property and given that in general 퐷 ≫ 푑 one may wonder whether tensors with a BICF actually exist. However by taking 푚 tensors together, also called blocking, we can create generating tensors that have the same bond dimension 퐷 and a physical dimension 푑푚. It turns out that after blocking at most 3퐷5 sites, every tensor has a BICF

The reason we call 햠 injective if it satisﬁes (7.42) is because of the following lemma

Lemma 7.4.9. The map

푑 햠 ∗ ⊗퐿 ퟏ 퐋 Γ퐿 ∶ 푉 ⊗ 푉 → 푃 , 햷 ↦ ∑ Tr(햷햠 ⋯햠 )|ퟏ, ⋯, 퐋⟩ (7.44) ퟏ,…,퐋 is injective if and only if

ퟏ 퐋 C Span{햠 ⋯햠 |1 ≤ ퟏ, …, 퐋 ≤ 푑} = Mat퐷( ). (7.45)

75 Proof. This follows directly by considering the Hilbert-Schmidt inner product. We are now ready to state the main result of this section, that we call The Fundamental Theo- rem of Matrix Product Vectors. It states which freedom is left when considering canonical forms of a tensor to generate a state. ■

Theorem 7.4.10 (Fundamental Theorem of Matrix Product States). Let 햠 and 햡 be two 푔햠 푔햡 N tensors in CF, with BNTs {햠(푚)}푚=1 and {햡(푛)}푛=1, respectively. If for all 푁 ∈ , 햠 and 햡 generate MPS that are proportional to each other, then:

(i) 푔햠 = 푔햡 =∶ 푔;

(ii) for all 푘 there exist a 푗푘, phases 휑푘, and non-singular tensors 햷푘 ∶ 푉 (푘) → 푉 (푘) such that

푖휑푘 −1 햡(푘) = 푒 햷푘햠(푗푘)햷푘 (7.46)

Proof. See [4]. ■

So basically for for tensors with a BICF not only does a gauge transform leave the MPS invariant, the reverse is also true: if two tensors generate MPS that are proportional to each other then they must be related by a (generalized) gauge transform. The fundamental theorem allows us to express physical symmetries in terms of virtual symmetries. The action of a symmetry group on a MPS can be translated to a virtual conjugation combined with adding a phase factor. Since adding a phase factor leaves states invariant the different symmetries that matrix product states can have are classiﬁed by considering projective complex representations. More information of how to apply the theory of MPS to the classiﬁcation of gapped quantum phases of matter quantum can be found in [17]. Since MPS deal with ground states of gapped Hamiltonians, i.e. states without energy one may wonder whether MPS can be used to describe TQFTs. The connection between these two approaches to gapped phases of matter has been worked out to some extent in [9].

76 Appendix A

Dutch summary

A.1 Spin Roosters

Spin-roosters zijn vereenvoudigingen van echte systemen in de natuur, maar fun- damenteel voor ons begrip van het collectieve gedrag van interagerende deeltjes. Rooster spin-modellen worden vaak gedefinieerd op basis van een rooster, een reeks configuraties die de verschillende spins kunnen aannemen, en een Hamiltoniaan die een roosterconfiguratie neemt, ook wel een ‘staat’ genoemd, en de energie terug geeft die zo’n configuratie heeft. Zelfs als men gebruik maakt van periodieke roosters en eenvoudige Hamiltonianen, is het systeem als geheel vaak buitengewoon complex. De fysica in deze paper bespreekt de lage energie-en temperatuurfysica van kwantum spin-ketens. Lage energie betekent dat we ons bezig houden met die staten, de grondtoestanden genoemd, waarvoor de Hamiltoniaan de kleinste waarden van energie teruggeeft. Lage temperatuur betekent dat we er altijd van uitgaan dat de temperatuur 0 Kelvin is. Klassiek worden die toestanden als oninteressant beschouwd, omdat wordt aangenomen dat geen communicatie tussen deeltjes mogelijk is bij een temperatuur gelijk aan het absolute nulpunt. Het blijkt dat bij het overwegen van kwan- tumsystemen, door verstrengeling, dit niet meer het geval is en er veel interessant gedrag ontstaat. Typische benaderingen voor het begrijpen van de lage energiefysica van statis- tische modellen (met name spinroosters) zijn gemiddelde veldentheorie, perturbatietheorie, monte carlo-methoden en in sommige specifieke gevallen kan het probleem analytisch worden aangepakt. De meeste van deze methoden hebben ernstige nadelen als het gaat om het anal- yseren van verstrengelde systemen. De gemiddelde veldentheorie benadert een systeem door een groot deel van de interacties tussen de deeltjes te negeren, waardoor het niet mogelijk is de meeste van de interessante eigenschappen die voortkomen uit verstrengeling te verklaren. Hetzelfde kan gezegd worden voor de perturbatietheorie die niet in staat is om met sterk interagerende systemen om te gaan. Quantum Monte

77 Carlo-methoden vereisen enorme rekenhulpbronnen die niet gunstig schalen met de systeemafmetingen en natuurlijk bestaan erin het algemeen slechts zeer weinig analytisch oplosbare modellen. Tensornetwerken bieden een alternatieve taal voor het begrijpen van sterk ver- warde toestanden. In plaats van te werken met de volledige Hilbert-ruimte, biedt tensor-netwerktheorie staten aan die slechts een exponentieel klein deel van die ruimte innemen. Daarom is slechts een exponentieel kleine hoeveelheid parameters vereist om deze tensornetwerktoestanden te modelleren. Een belangrijke observatie is dat de grondtoestanden van gekloofde, lokale Hamiltonianen ook een exponentieel klein deel van de volledige Hilbert-ruimte innemen. In feite blijkt het dat de staten afkom- stig van tensornetwerken in hetzelfde kleine deel van de Hilbert-ruimte te leven als de grondtoestanden van gekloofde lokale Hamiltonianen. Daarom is het mogelijk om dergelijke grondtoestanden te beschrijven met tensornetwerkstaten door slechts een klein aantal parameters af te stemmen.

A.2 Categorie Theorie

Bij het werken met tensornetwerken is men meer geïnteresseerd in het samenstellen van tensoren dan in de argumenten die eraan worden gegeven. Dit komt vooral omdat tensornetwerken werken door samentrekking van indices over ’virtuele ruimtes’. De structuur van deze ruimtes is niet van belang, alleen de afbeeldingen ertussen en de samenstelling van die afbeeldingen. Dit is een bekend fenomeen voor mensen die met categorietheorie werken. Cat- egorietheorie bestaat uit objecten en kaarten tussen die objecten. De inhoud van de objecten zelf is vaak niet van belang. In deze thesis bouwen we de brug tussen de abstracte categorietheorie en de toegepaste tensor netwerken. In het bijzonder zullen we vele termen in de categorietheorie vertalen naar diagrammen die overeenstemmen met de typische diagrammen die men tegenkomt als men met tensor netwerken werkt. Daarenboven zullen we de theorie van de kwantum mechanica vertalen naar een diagramatische theorie en aantonen dat deze theorie heel dicht aan leunt tegen de theorie van de zogenaamde topologische kwantum velden. De theorie van topologische kwantum velden beschrijft fysische velden die geen energie bevatten. Dit zijn ook excact dezelfde velden waar we mee bezig zijn als we grondtoestanden van Hamiltonianen bestuderen. Het blijkt dat, ondanks de afwezigheid van energie en temperatuur, er toch verschillende fasen van materie kunnen voorkomen, kwantum fasen genoemd. Deze fasen worden kort geintroduceerd en het idee van hoe we bepaalde kwantum fasen kunnen classiﬁceren aan de hand van tensornetwerken wordt kort uitgelegd.

78 Appendix B

Rigorous Graphical Calculus

Introduction

The goal of this chapter is to provide a rigorous theory that formalises the diagrammatic language introduced in chapter 2. Penrose [15, 16] was the first to use the graphical notation for calculating with tensors. This appendix consists of two sections. In the first section, we deal with monoidal categories and prove that the diagrammatic language introduced is sound and complete. In the second section we consider the case of symmetric tensor categories. This requires some extra effort since the graphs might not be planar anymore. We will prove that the graphical language for symmetric monoidal categories is sound and complete, and allows more freedom when considering graph deformations. The reason for including this rather sharp analysis is twofold. First, many diagrammatic definitions and proofs may look intuitive but turn out to be incorrect in the end. Second, the proofs given below are technical but not hard to do. One may find that by reading and understanding the proofs, more insight is gained in category theory and the associated diagrammatic language.

B.1 Monoidal Categories and Their Diagrams

B.1.1 Monoidal Categories

In this section we will introduce topological graphs and define the concept of ‘valuation’, which labels the nodes of a graph with arrows from a tensor category 풞 and labels the edges with objects of 풞. Then we will extend this definition to define the ‘value of a graph’ 푣(퐺) in 풞 which stands for the interpretation of a labelled graph as a morphism in 풞. The main result of this section is that the value of a graph is invariant under ‘continuous deformation of plane diagrams’.

79 Recall the deﬁnition of a monoidal category

Definition B.1.1. A monoidal category (MC) is a six-tuple (풞, ⊗, 퐼, 푎, 푙, 푟) where

• 풞 is a category;

• 퐼 is an object of 풞, called the unit object of 풞;

• ⊗ ∶ 풞 × 풞 → 풞, (퐴, 퐵) ↦ 퐴 ⊗ 퐵 is a functor, called the tensor product;

• 푎 ∶ ⊗ ∘ (⊗ × ퟣ) → ⊗ ∘ (ퟣ × ⊗) is a natural isomorphism;

• 푙 ∶ ⊗ ∘ (푘 × ퟣ) → ퟣ and 푟 ∶ ⊗ ∘ (ퟣ × 푘) → ퟣ are natural isomorphisms; such that for all 푀, 푁, 푃 , 푄 ∈ 풞 the following diagrams commute:

((푀 ⊗ 푁) ⊗ 푃 ) ⊗ 푄 1 푎푀⊗푁,푃,푄 푎푀,푁,푝⊗

(푀 ⊗ 푁) ⊗ (푃 ⊗ 푄) (푀 ⊗ (푁 ⊗ 푃 )) ⊗ 푄 (B.1)

푎푀,푁,푃⊗푄 푎푀,푁⊗푃,푄

푀 ⊗ (푁 ⊗ (푃 ⊗ 푄))1 푀 ⊗ ((푁 ⊗ 푃 ) ⊗ 푄) ⊗푎푁,푃,푄

푎 (푀 ⊗ 푘) ⊗ 푁 푀,푘,푁 푀 ⊗ (푘 ⊗ 푁) 푟푀⊗푁 . (B.2) 푀⊗푙푛 푀 ⊗ 푁

Equation (B.1) is called the pentagon equation and equation (B.2) is called the triangle equation. The map 푎 is called the associativity constraint of the MC and the maps 푙 and 푟 are called the left and right unit constraints of the MC.

We have the following important theorem.

Theorem B.1.2 (MacLane's Coherence Theorem). For every monoidal category (풞, ⊗, 퐼, 푎, 푙, 푟) there exists a strict monoidal category (풞′, ⊗′, 퐼′, 푎′, 푙′, 푟′) together with an equivalence of categories 퐹 ∶ 풞 → 풞′, where 퐹 is a strong monoidal functor.

The coherence theorem of MacLane states that all diagrams built up from 푎, 푙, 푟 by tensoring, substituting, and composing, commute. It follows that all the objects obtained by computing the tensor product of a sequence 퐴1 ⊗ ⋯ ⊗ 퐴푛, by bracketing it differently, and by cancelling units are coherently identiﬁed with each other. More

80 precisely, the different ways of computing the tensor product 퐴1 ⊗ ⋯ ⊗ 퐴푛 produce a what is called a clique. A clique is a non-empty family {퐶푖|푖 ∈ 퐼} of objects together with a family {푢푗푖 ∶ 퐶푖 → 퐶푗|(푖, 푗) ∈ 퐼 × 퐼} of maps such that 푢푖푖 = ퟙ and 푢푘푖 = 푢푘푗푢푗푖.

This implies that for every two different ways to compute a tensor product 퐴1 ⊗⋯⊗퐴푛, there exists a map that maps one result to the other. The cliques in 풞 are the objects of a category clq 풞 in which a map 푓 ∶ {퐶푖|푖 ∈ 퐼} → {퐷푘|푘 ∈ 퐾} is a family of maps

푓푘푖 ∶ 퐶푖 → 퐷푘 such that

푓푘푖 퐶푖 퐷푘

푢푗푖 푢푚푘 (B.3)

푓푚푗 퐶푗 퐷푚 commutes for every (푖, 푗) ∈ 퐼2 and (푘, 푚) ∈ 퐾2. It is sometimes convenient to think of the 푛-fold tensor product as a functor

풞푛 → clq 풞. (B.4)

The functor 풞 → clq 풞, which associates to each 퐴 ∈ 풞 the singleton clique {퐴} ∈ clq 풞, is full and faithful. Since any clique is isomorphic to the singleton clique of any one of its members, this functor is an equivalence. This equivalence between 풞 and clq 풞 shows that the ambiguity which exists in computing the 푛-fold tensor product is not a real one. Furthermore, any tensor category is equivalent to a strict one st 풞; that is, one in which each constraint is an identity morphism. The objects of st 풞 are words 푤 = ′ ′ 퐴1퐴2⋯퐴푚 in objects of 풞. A morphism 푓 ∶ 푤 → 푤 of st 풞 is a morphism 푓 ∶ [푤] → [푤 ] in 풞, where

[∅] = 퐼, [퐴] = 퐴, and [퐴1⋯퐴푖+1] = [퐴1⋯퐴푖] ⊗ 퐴푖+1. (B.5)

The tensor ⊗̄ for st(풞) is given by 푣⊗푤̄ = 푣푤 and

푓⊗푔 [푣] ⊗ [푤] [푣′] ⊗ [푤′]

≅ ≅ . (B.6) 푓⊗푔̄ [푣푤] [푣′푤′]

In principle, most results obtained with the hypothesis that a tensor category is strict can be reformulated and proved without this condition. In what follows we will only use non-strict monoidal categories where differences in theories for strict and non- strict categories arise. The reason for focusing on strict categories is that, without the

81 need for pentagon and triangle equations, the exposition is much clearer. In a tensor category there are two operations for constructing new arrows from old ones: composition 푓 ∘ 푔 and tensor product 푓 ⊗ 푔. Using ordinary algebraic notation, we could be faced with expressions like

(ퟣ퐵 ⊗ 푐 ⊗ 푑) ∘ (ퟣ퐵 ⊗ ퟣ퐵 ⊗ 푏 ⊗ ퟣ퐶) ∘ (푎 ⊗ ퟣ퐵 ⊗ ퟣ퐶) = 푤1 (B.7) and

(ퟣ퐵 ⊗ ퟣ퐶 ⊗ 푑) ∘ (ퟣ퐵 ⊗ 푐 ⊗ ퟣ퐷 ⊗ ퟣ퐶) ∘ (푎 ⊗ 푏 ⊗ ퟣ퐶) = 푤2. (B.8)

Using this linear notation it may be unclear whether two words like 푤1 and 푤2 are equivalent. Graphical notation makes it easier to detect such equalities. Figure (B.1) contains the graphical notation we will use for the words 푤1 and 푤2. It is obvious that

퐶 퐷 퐷 푐 푑 푑 퐶 퐷 퐶 퐵 퐵 푏 퐵 퐷 퐶 푐 퐶 퐵 퐶 퐵 푎 푎 푏 퐴 퐴 퐵

푤1 푤2

Figure B.1: Graphical notation for tensors 푤1 and 푤2 as deﬁned in (B.7) and (B.8). the diagrams are deformations of one another. This will, after the results below, enable us to deduce the equality 푤1 = 푤2.

B.1.2 Graphs

The following is a list of deﬁnitions which are used in the theorems. Most of these are elementary and one may skip this subsection, only to return to it if some concepts, used later on, are unclear.

Definition B.1.3. (i) A generalized (topological) graph 퐺 = (퐺, 퐺0) consists of a

Hausdorff space 퐺 and a discrete closed subset 퐺0 ⊂ 퐺 such that the com-

plement 퐺 ∖ 퐺0 is a 1-dimensional manifold without boundary. That is, 퐺 ∖ 퐺0 is the topological sum of open intervals and circles.

(ii) An element of 퐺0, is called a vertex or node.

82 (iii) A connected component of 퐺 ∖ 퐺0 is called an open edge if it is homeomorphic to an open interval. Otherwise it is called a circle.

(iv) A closed edge ̂푒 is an open edge 푒 that is compactiﬁed by adjoining its two end- points.

(v) The degree of a node 푛 is the number of connected components of 푉 ∖{푛} where 푉 is a neighbourhood of 푛 that contains no other nodes.

(vi) An edge 푒 is called pinned when the inclusion 푒 → 퐺 can be extended to a continuous map ̂푒→ 퐺, called the structure map. When the inclusion 푒 → 퐺 extends only to ̂푒 minus one end-point, we call 푒 half-loose. An edge is called loose when it is neither pinned nor half-loose.

(vii) A graph is a generalised graph in which all the edges are pinned. It is called an ordinary graph when it has no circles.

(viii) We say that a graph 퐺 = (퐺, 퐺0) is finite generalised if 퐺0 and the set 휋0(퐺∖퐺0)

of connected components of 퐺 ∖ 퐺0 are ﬁnite.

(ix) We write 퐺̂ for the compactification of the generalised graph; adjoining one end- point to each half-pinned edge and two end-points to each loose edge. These ̂ ̂ ̂ extra points, along with 퐺0, are the nodes of a graph 퐺 = (퐺, 퐺0); the elements ̂ of 퐺0 ∖ 퐺0, are called the outer nodes of G.

(x) If 퐺 = 퐺̂ then we say that 퐺 is a compact graph.

(xi) A graph with boundary Γ = (Γ, 휕Γ) is a compact graph Γ together with a distinguished set of nodes 휕Γ such that each 푥 ∈ 휕Γ is of degree one. The graph 퐺,̂ together with the set of outer nodes of 퐺, is a graph with boundary.

(xii) Consider the generalised graph 퐺 = Γ ∖ 휕Γ whose compactiﬁcation 퐺̂ is Γ and the set of outer nodes of 퐺 is 휕Γ. The elements of 휕Γ are called the outer nodes of (Γ, 휕Γ), and the nodes of Γ ∖ 휕Γ are called the inner nodes of (Γ, 휕Γ).

(xiii) An isomorphism 푓 ∶ (Γ, 휕Γ) → (Γ′, 휕Γ′) of graphs with boundary is a homeomorphism 푓 ∶ Γ → Γ′ inducing bijections on the inner nodes and on the outer nodes.

(xiv) An oriented edge of Γ is an edge 푒 equipped with an orientation; or equivalently, with a linear order on 휕. ̂푒 The source 푒(0) of an oriented edge 푒 is the image of the ﬁrst element of 휕 ̂푒 under the structure map ̂푒→ Γ; the target 푒(1) is the image of the last element; the opposite edge 푒0 is obtained by taking the opposite orientation of 푒.

83 (xv) An oriented graph is a graph together with a choice of orientation for each of its edges and circles. For an oriented graph Γ, the input in(푥) of an inner node

푥 ∈ Γ0 is deﬁned to be the set of oriented edges with target 푥; the output out(푥) of 푥 is the set of oriented edges with source 푥.

(xvi) A polarised graph is an oriented graph together with a choice of linear order on each in(푥) and out(푥).

(xvii) A progressive graph is an oriented ordinary graph with no circuits. The domain dom Γ of a progressive graph Γ consists of the edges which have outer nodes as sources; the codomain cod Γ consists of the edges which have outer nodes as targets. In what follows we often identify dom Γ and cod Γ with their corresponding set of outer nodes.

(xviii) A parametrisation of an oriented edge 푒 is an orientation preserving homeomorphism [0, 1] →. ̂푒 The composition of this homeomorphism with the structure map ̂푒→ Γ is a function 훾 ∶ [0, 1] → Γ which is called a parametrised edge with source 훾(0) and target 훾(1). Given a parametrised edge 훾 then we call the function 훾(푡) ∶= 훾(1 − 푡) the opposite parametrisation.A parametrisation of a circle is a homeomorphism with the unit cicle 푆1 ⊂ C.

(xix) A parametrised graph is a graph together with a choice of parametrisation for each oriented edge and each oriented circle such that the opposite edges and cicles have the opposite parametrisations.

(xx) A graph is called smooth when each closed edge and each circle is equipped with a 퐶∞ structure. Parametrisations of smooth graphs will always be taken as smooth.

B.1.3 Progressive Plane Diagrams

The purpose of this subsection is

(i) to describe the diagrams appropriate for calculations in an arbitrary tensor category,

(ii) to deﬁne an interpretation of these diagrams in terms of morphisms in a monoidal category, and

(iii) to prove that each morphism belonging to a diagram is invariant under deformation of the diagram.

Definition B.1.4. Let 푎 < 푏 be real numbers. A progressive plane graph between levels 푎 and 푏 is a graph Γ with boundary that is embedded in R × [푎, 푏] such that

84 (i) 휕Γ = Γ ∩ (R × {푎, 푏}), and

R (ii) the second projection pr2 ∶ × [푎, 푏] → [푎, 푏] is injective on each connected component of Γ ∖ Γ0.

Notes B.1.5. (i) Each progressive plane graph Γ is both progressive and polarised in the sence of subsection 2.

(ii) Each edge 푒 of a progressive plane graph can be given an orientation by setting

푒(0) < 푒(1) ⟺ pr2푒(0) < pr2푒(1). (B.9)

(iii) Condition (ii) basically says that circles and circuits are excluded from the deﬁ- nition. For a general inner node it will be useful to order the sets in(푥) and out(푥). Recall the following deﬁnition

Definition B.1.6. A linear order on a set 푋 is a binary relation ≤ such that all 푎, 푏, 푐 ∈ 푋 the following statements hold.

• If 푎 ≤ 푏 and 푏 ≤ 푎 then 푎 = 푏.

• If 푎 ≤ 푏 and 푏 ≤ 푐 then 푎 ≤ 푐.

• 푎 ≤ 푏 or 푎 ≤ 푐.

For any inner node 푥, in(푥) and out(푥) can be linearly ordered as follows. Choose

푢 ∈ [푎, 푏] smaller than but close enough to pr2(푥).Then each edge 푒 ∈ in(푥) intersects the line R×{푢} in one point. We then let the set of intersection points inherits the order from (R, <) just as in ﬁgure (B.2).

훿1 훿2 훿푚 R ⋯ 푥 ⋯ R 훾1 훾2 훾푚 Figure B.2: The linear ordering on in(푥) and out(푥) can be inherited from R

The order on out(푥) is deﬁned similarly by intersecting with R × {푢} for 푢 larger than but close to pr2(푥). Notice that dom Γ and cod Γ are naturally linearly ordered as subsets of R × (푎) and R × (푏), respectively.

85 Graphs consisting of multiple nodes and edges should be decomposed in elementary pieces if one wants to deﬁne notions such as “deformation” and “valuation” of a graph. Therefore we introduce the following deﬁnitions.

Definition B.1.7. Consider a graph Γ.

(i) A number 푢 ∈ [푎, 푏] is called a regular level for Γ when the line R × {푢} contains no inner nodes.

(ii) If Γ is a progressive plane graph and 푐 < 푑 are regular levels of Γ, we write Γ[푐, 푑] R R for the graph Γ∩( ×[푐, 푑]) whose set of inner nodes is (Γ0 ∖휕Γ)∩( ×[푐, 푑]) and whose set of outer nodes is Γ ∩ (R × {푐, 푑}). The graph Γ[푐, 푑] is a progressive plane graph between levels 푐 and 푑 and is called a layer of Γ.

(iii) Suppose Γ is the disjoint union of two subgraphs Γ1 and Γ2. If there exists a 휉 ∈ R such that

Γ1 ⊆] − ∞, 휉[×[푎, 푏] and Γ2 ⊆]휉, +∞[×[푎, 푏], (B.10)

we say that the pair (Γ1,Γ2) is a tensor decomposition of Γ and write Γ = Γ1 ⊗Γ2. This notion extends in the obvious way to 푛-fold tensor decompositions

Γ = Γ1 ⊗ ⋯ ⊗ Γ푛. (B.11)

The main power of graphical languages is that certain amount of deformation is allowed. Different tensor categories allow for different kinds of deformations. For general monoidal categories only “deformations of progressive plane graphs” (see deﬁnition below) are allowed.

Definition B.1.8. Let (Γ, 휕Γ) denote a graph with boundary. A deformation of progressive plane graphs between levels 푎 and 푏 is a continuous function

ℎ ∶ Γ × [0, 1] → R × [푎, 푏] (B.12) such that, for all 푡 ∈ [0, 1], the function

ℎ(, 푡) ∶ Γ → R × [푎, 푏] (B.13) is an embedding whose image is a progressive plane graph (Γ(푡), 휕Γ(푡)) between levels 푎 and 푏.

The central concept that we are interested in is the meaning of a graph in terms of

86 objects and morphisms in a monoidal category. We call this the value of the graph in that category and the map that assigns this value is called a valuation.

Definition B.1.9. Consider a progressive plane graph Γ and let Γ1 be the set of edges of Γ and let Γ0 ∖ 휕Γ be the set of inner nodes of Γ. (i) A valuation 푣 ∶ Γ → 풞 of a progressive plane graph Γ in a tensor category 풞 is a pair of functions

푣0 ∶ Γ1 → |풞|, 푣1 ∶ Γ0 ∖ 휕Γ → 햧허헆(풞), (B.14)

where, for all inner nodes 푥 of Γ the map 푣1 has the following structure:

푣1(푥) ∶ 푣0(훾1) ⊗ ⋯ ⊗ 푣0(훾푚) → 푣0(훿1) ⊗ ⋯ ⊗ 푣0(훿푛), (B.15)

where 훾1 < ⋯ < 훾푚, 훿1 < ⋯ < 훿푛 are the ordered lists of elements of in(푥) and out(푥), respectively.

(ii) The pair (Γ, 푣) is called a progressive plane diagram in 풞 and is denoted by Γ when the context is clear.

(iii) The domain and codomain of a diagram (Γ, 푣) are the families of objects

dom(Γ, 푣) = {푣0(푧)|푧 ∈ dom Γ}, cod(Γ, 푣) = {푣0(푧)|푧 ∈ cod Γ} (B.16)

indexed by the linearly ordered sets dom Γ, cod Γ

If 푐 < 푑 are regular levels for a diagram Γ = (Γ, 푣), the valuation 푣 restricts to a valuation on the layer Γ[푐, 푑] and we also denote this by 푣. Similarly, if Γ = Γ1 ⊗ Γ2, the valuation 푣 restricts to valuations on Γ1 and Γ2 and we also denote this by 푣. If ℎ ∶ Γ × [0, 1] → R × [푎, 푏] is a deformation of progressive plane graphs then a valuation deﬁned on one Γ(푡0) for some 푡0 ∈ [0, 1] can be transported along the isomorphisms Γ(푡0) ≅ Γ ≅ Γ(푡) to a valuation on Γ(푡) for all 푡 ∈ [0, 1]. In this way ℎ becomes a deformation of diagrams. Our intention now is to assign a value 푣(Γ) ∈ 햧허헆(풞) to each progressive plane diagram (Γ, 푣) in 풞 and prove it invariant under deformation of diagrams. To do this we must subdivide the diagram into elementary building blocks and combine these by composing and tensoring.

Definition B.1.10. (i) A diagram Γ is called prime when it is connected and has precisely one inner node 푥. In this case, we deﬁne the value of Γ by the equality

푣(Γ) = 푣1(푥). (B.17)

87 Figure B.3: The top layer would not be elementary in the sense of note (B.1.12).

(ii) A diagram Γ is called invertible when it has no inner nodes.

In an invertible diagram we have bijections

dom Γ ≅ 휋0(Γ) ≅ cod Γ (B.18) between the domain, connected components, and codomain of the graph Γ such that the composite is order-preserving. Thus we obtain a linear order 푒1 < 푒2 < ⋯ < 푒푛 on the set 휋0(Γ). In this case, the value 푣(Γ) of Γ is deﬁned to be the identity arrow of

푣0(푒1) ⊗ ⋯ ⊗ 푣0(푒푛).

Definition B.1.11. A diagram Γ is called elementary when it has a tensor decomposition Γ = Γ1 ⊗ ⋯ ⊗ Γ푛 with each Γ푖(1 ≤ 푖 ≤ 푛) either prime or invertible. In this case we deﬁne

푣(Γ) = 푣(Γ1) ⊗ ⋯ ⊗ 푣(Γ푛). (B.19)

That (B.19) is independent of the choice of tensor decomposition follows from the facts that

• prime diagrams are tensor indecomposable, and

1 2 1 2 1 2 • if Ω ⊗ Ω is invertible then 휋0(Ω ⊗ Ω ) is the ordered sum of 휋0(Ω ) and 휋0(Ω ).

Note B.1.12. One might feel that a more restrictive notion of “elementary diagram” should have been used, namely, those diagrams which admit a tensor decomposition into primes and connected invertibles. It is true that every diagram decomposes into layers of this kind. However, the restricted notion is not inherited by taking further layers (see Fig (B.3)).

88 Proposition B.1.13. If 푢 is a regular level for an elementary diagram Γ between levels 푎 and 푏 then Γ[푎, 푢], Γ[푢, 푏] are elementary, and

푣(Γ) = 푣(Γ[푢, 푏]) ∘ 푣(Γ[푎, 푢]). (B.20)

For any progressive plane diagram Γ between levels 푎 and 푏, we can now deﬁne its value by

푣(Γ) = 푣(Γ[푢푛−1, 푢푛]) ∘ ⋯ ∘ 푣(Γ[푢0, 푢1]), (B.21) where 푎 = 푢0 < 푢1 < ⋯ < 푢푛 = 푏 are regular levels for Γ such that each layer

Γ[푢푖−1, 푢푖] is elementary for 1 ≤ 푖 ≤ 푛. The existence of such regular levels can be seen by choosing 푢1, …, 푢푛−1, to be numbers close enough to, and on both sides of, each critical (= non-regular) level. The independence of the deﬁnition under different choices amounts to independence under a reﬁnement, which follows from Proposition (B.1.13). We now state the main theorem of this subsection which amounts to the statement that a deformation of progressive plane diagrams does not alter the value of such a diagram. One says that “the graphical language is sound with respect to monoidal categories”

Theorem B.1.14 (Monoidal Equivalence of Diagrams). If ℎ ∶ Γ × [0, 1] → R × [푎, 푏] is a deformation of progressive plane diagrams then

푣(Γ(0)) = 푣(Γ(1)) (B.22)

B.1.4 Free Monoidal Categories

Previous subsection we considered the valuation of a plane graph in a category and showed that the value of any plane graph is independent under planar deformation. In this section we go one step further: we will build a graphical calculus, show that the axioms for this calculus make it a monoidal category and furthermore show that any monoidal category is equivalent to a category built from these graphical axioms. To compare monoidal categories and deﬁne a notion of monoidal equivalence we introduce following deﬁnitions.

Definition B.1.15. Let (풞, ⊗, 퐼) and (풞′, ⊗′, 퐼′) be strict monoidal categories. A strong monoidal functor, also called tensor functor, between 풞 and 풞′ is a functor 퐹 ∶ 풞 → 풞′ together with natural isomorphisms 휙2 ∶ 퐹 퐴 ⊗ 퐹 퐵 → 퐹 (퐴 ⊗ 퐵) and 휙0 ∶ 퐼 → 퐹 퐼, such

89 that the following diagrams commute:

휙2⊗ퟣ (퐹 퐴 ⊗ 퐹 퐵) ⊗ 퐹 퐶 퐹퐶 퐹 (퐴 ⊗ 퐵) ⊗ 퐹 퐶

2 2 ퟣ퐹퐴⊗휙 휙 (B.23) 휙2 퐹 퐴 ⊗ 퐹 (퐵 ⊗ 퐶) 퐹 (퐴 ⊗ 퐵 ⊗ 퐶)

퐹 퐴 ⊗ 퐼 푟 퐹 퐴 퐼 ⊗ 퐹 퐴 푙 퐹 퐴

0 0 ퟣ퐹퐴⊗휙 퐹(푟) 휙 ⊗ퟣ퐹퐴 퐹(푙) . (B.24) 휙2 휙2 퐹 퐴 ⊗ 퐹 퐼 퐹 (퐴 ⊗ 퐼) 퐹 퐼 ⊗ 퐹 퐴 퐹 (퐼 ⊗ 퐴)

The natural isomorphisms 휙0 and 휙2 belong to a bigger class of useful natural N isomorphisms {휙푛|푛 ∈ }. These natural isomorphisms

휙푛 ∶ 퐹 퐴1 ⊗ ⋯ ⊗ 퐹 퐴푛 → 퐹 (퐴1 ⊗ ⋯ ⊗ 퐴푛) (B.25) are deﬁned inductively as follows:

(i) 휙0 is given,

(ii) 휙1 is the identity,

(iii) 휙2 is given and

(iv) 휙푛+1 is the composite

ퟣ ⊗휙 퐹퐴1 푛 퐹 퐴1 ⊗ ⋯ ⊗ 퐹 퐴푛+1 퐹 퐴1 ⊗ 퐹 (퐴2 ⊗ ⋯ ⊗ 퐹 퐴푛+1) (B.26)

휙2 퐹 (퐴1 ⊗ ⋯ ⊗ 퐴푛+1). (B.27)

The following triangle then commutes

휙푚+푛 퐹 퐴1 ⊗ ⋯ ⊗ 퐹 퐴푚 ⊗ 퐹 퐵1 ⊗ ⋯ ⊗ 퐹 퐵푛 퐹 (퐴1 ⊗ ⋯ ⊗ 퐴푚 ⊗ 퐵1 ⊗ ⋯ ⊗ 퐵푛) (B.28)

휙푚⊗휙푛 휙2

퐹 (퐴1 ⊗ ⋯ ⊗ 퐴푚) ⊗ 퐹 (퐵1 ⊗ ⋯ ⊗ 퐵푛)

Definition B.1.16. Let (풞, ⊗, 퐼) and (풞′, ⊗′, 퐼′) be strict monoidal categories, and let 퐹 , 퐺 ∶ 풞 → 풞′ be strong monoidal functors. A natural transformation 휏 ∶ 퐹 → 퐺 is called monoidal, or a tensor transformation, if the following diagrams commute for all

90 퐴, 퐵 ∈ |풞|:

퐹 퐼 휙2 퐹 퐴 ⊗ 퐹 퐵 퐹 (퐴 ⊗ 퐵) 휙0

휏퐴⊗휏퐵 휏퐴⊗퐵 , 퐼 휏퐼 . (B.29) 휙2 휙0 퐺퐴 ⊗ 퐺퐵 퐺(퐴 ⊗ 퐵) 퐺퐼

We write Ten(풞, 풞′) for the category whose objects are tensor functors from 풞 to 풞′ and whose morphisms are tensor transformations.

To show that a graphical language is complete with respect to monoidal categories we will introduce the concept of a tensor scheme (also called monoidal signature [18]). This is an abstract concept that allows us to use the notion of variables. To get a feeling for such concepts consider the familiar language of arithmetic expressions. This language deals with terms such as

(푥 + 푦 + 2) × (푥 + 3) (B.30) which are built up from variables such as 푥 and 푦, constants such as 2 and 3 and operations such as addition + and multiplication ×.

Variables can be viewed in several different ways.

(i) They can be viewed as abstract symbols that can be compared. One can say that (B.30) contains two variables 푥 and another different variable 푦. Since we use different symbols and only interpret the symbols as abstract entities 푥 and 푦 are indeed different.

(ii) They can be viewed as placeholders for arbitrary constants, e.g. 푥 = 3 and 푦 = 5. In contrast with the previous viewpoint where 푥 and 푦 were different, they now might be equal to each other in certain situations.

(iii) They can also be viewed as placeholders for arbitrary terms built from a combination of variables, constants and operations. For example 푥 = 푎 + 푏 and 푦 = 푧2

For many mathematical models one set of variables sufﬁces. The formal language of category theory is different for it requires two sets of variables: object variables (for

91 labeling edges) and morphism variables (for labeling edges). Each morphism variable is also equiped with a speciﬁed domain and codomain. The following deﬁnition captures this idea.

Definition B.1.17. A tensor scheme 풟 consists of two sets obj 풟 and mor 풟 together with a function which assigns to each element 푑 ∈ mor 풟 a pair (푑(0), 푑(1)) of words in the elements of obj 풟. Write

푑 ∶ 푋1⋯푋푚 → 푌1⋯푌푛 (B.31) for 푑 ∈ mor 풟 with 푑(0) = 푋1⋯푋푚, 푑(1) = 푌1⋯푌푛.

Each tensor scheme 풟 and tensor category 풞 determine a category [풟, 풞] described as follows.

An object 퐾 is a pair of functions

퐾 ∶ obj 풟 → obj 풞, 퐾 ∶ mor 풟 → mor 풞 (B.32)

such that, for all 푑 ∶ 푋1⋯푋푚 → 푌1⋯푌푛, we have

퐾푑 ∶ 퐾푋1 ⊗ ⋯ ⊗ 퐾푋푚 → 퐾푌1 ⊗ ⋯ ⊗ 퐾푌푛. (B.33)

A morphism 휅 ∶ 퐾 → 퐿 is a family of arrows

휅푋 ∶ 퐾푋 → 퐿푋, 푋 ∈ obj 풟, (B.34) in 풞 such that, for all 푑, the following square commutes:

퐾푑 퐾푋1 ⊗ ⋯ ⊗ 퐾푋푚 퐾푌1 ⊗ ⋯ ⊗ 퐾푌푛

휅푋1⊗⋯⊗휅푋푚 휅푌1⊗⋯⊗휅푌푛 . (B.35) 퐿푑 퐿푋1 ⊗ ⋯ ⊗ 퐿푋푚 퐿푌1 ⊗ ⋯ ⊗ 퐿푌푛

A composition functor

Ten(풞, 풞′) × [풟, 풞] → [풟, 풞′], (퐹 , 퐾) ↦ 퐹 ∘ 퐾, (휏, 휅) ↦ 휏 ∘ 휅 (B.36)

92 exists, where (퐹 ∘ 퐾)푋 = 퐹 퐾푋, (퐹 ∘ 퐾)푑 is the following composite

휙푚 퐹 퐾푋1 ⊗ ⋯ ⊗ 퐹 퐾푋푚 퐹 (퐾푋1 ⊗ ⋯ ⊗ 퐾푋푚) (B.37) 퐹퐾푑 퐹 (퐾푌1 ⊗ ⋯ ⊗ 퐾푌푛) (B.38) −1 휙푛 퐹 퐾푌1 ⊗ ⋯ ⊗ 퐹 퐾푌푛, (B.39) and (휏 ∘ 휅)푋 = 휏퐿푋 ∘ 퐹 휅푋. Notes B.1.18. The above remark about the different roles of variables in arithmetic also holds for the diagrammatic language of categories. • On the one hand, the labels can be viewed as formal symbols. This is the abstract view we will use to prove that all categories admit a graphical language.

• The labels can also be viewed as placeholders for speciﬁc objects and morphisms in an actual category. This is exactly what the objects 퐾 ∶ obj 풟 → obj 풞 of [풟, 풞] encode. They map abstract words and relations between these words to speciﬁc objects resp relations between these objects in a monoidal category. Such maps are therefore also called “interpretations”. The notion of a free category on a tensor scheme encodes the idea of a category that has the most general interpretation of 풟.

Definition B.1.19. A tensor category ℱ is said to be free on the tensor scheme 풟 when there exists an object 푁 of [풟, ℱ] such that the functor

∘ 푁 ∶ Ten(ℱ, 풞) → [풟, 풞] (B.40) is an equivalence of categories for all tensor categories 풞.

If ℱ′ is also a free tensor category on 풟 then there exists an equivalence of tensor categories ℱ → ℱ′. It can be proven that for any category 풟 there exists a strict tensor category 픽(풟) that is exactly the category build from operations on progressive plane graphs deﬁned below. First we introduce some notation. Notation B.1.20. (i) In deﬁning operations on boxed graphs we shall use the functions 훾, 휇 ∶ R2 → R2, given by

훾(푥, 푡) = (푥, 푡/3), 휇(푥, 푡) = (푥/2, 푡) (B.41)

R2 and the points 푒1 = (1, 0), 푒2 = (0, 1) ∈ .

93 R2 (ii) Formulas such as 훾(푆 + 푒2), for 푆 ⊂ denote the set

{(푥, (푡 + 1)/3)|(푥, 푡) ∈ 푆}. (B.42)

(iii) The following parametrised lines allow for lighter notation

푎 − 푏 푎 + 푏 푙(푎 , 푏 ) = {( 1 1 푢 + 1 1 , 푢) |푢 ∈ [−1, 1]} . (B.43) 1 1 2 2

Definition B.1.21. (i) A plane graph Γ will is called boxed when it is between levels −1 and +1, and is contained in ] − 1, 1[×[−1, 1]. We write Γ ∶ 푚 → 푛 when 푚 and 푛 are the cardinalities of dom Γ and cod Γ respectively.

(ii) The tensor product Γ1⊗Γ̃ 2 of two boxed plane graphs Γ1,Γ2 is the space 휇((Γ1 − 2 1 2 푒1) ∪ (Γ + 푒1)) with 휇((Γ0 − 푒1) ∪ (Γ0 + 푒1)) as the set of nodes.

(iii) Suppose Γ ∶ 푚 → 푛, Ω ∶ 푛 → 푝 are boxed plane graphs. Let 푎1 < 푎2 < ⋯ < 푎푛 be the elements of the codomain of Γ, and let 퐼 be the set of inner nodes of

the graph 훾(Γ − 2푒2). Let 푏1 < 푏2 < ⋯ < 푏푛 be the elements of the domain of

Ω, and let 퐽 be the set of inner nodes of the graph 훾(Ω + 2푒2). The composite Ω ∘ Γ ∶ 푚 → 푝 is the plane graph consisting of the space

Ω ∘ Γ = 훾((Γ − 2푒2) ∪ 푙(푎1, 푏1) ∪ ⋯ ∪ 푙(푎푛, 푏푛) ∪ (Ω + 2푒2)) (B.44)

with 퐼 ∪ 퐽 as the set of inner nodes.

The tensor product ⊗̃ is graphically depicted as

Γ1 ⊗̃ Γ2 = 휇Γ1 휇Γ2 , (B.45)

while composition of boxed graphs is depicted by

훾Ω

Ω ∘ Γ = . (B.46)

훾Γ

The concept of a valuation in a tensor category 풞 needs neither composition nor

94 tensor products of morphisms in 풞.

Definition B.1.22. (i) A valuation 푣 ∶ Γ → 풟 of a progressive plane graph Γ in a tensor scheme 풟 is deﬁned just as in 풞 except that the tensor products in the

domain and codomain of 푣1(푥) must be replaced by words in the elements of obj 풟.

(ii) A progressive plane diagram in a tensor scheme 풟 is a couple (Γ, 푣) where Γ is a progressive plane graph and 푣 ∶ Γ → 풟 a valuation. The domain (codomain)

of such a diagram is deﬁned to be the word 푣0(푧1)⋯푣0(푧푛), where 푧1 < ⋯ < 푧푛 are the elements of dom Γ (cod Γ).

The tensor product and composite of boxed progressive plane diagrams in a tensor scheme are deﬁned in the obvious way. A tensor product Γ1 ⊗ Γ2 has a tensor decomposition into subgraphs isomorphic to Γ1,Γ2; so valuations on Γ1,Γ2 transport to the subgraphs and these together give a valuation on Γ1,Γ2. If the codomain of (Γ, 푣) agrees with the domain of (Ω, 푤) then there is a unique valuation on Ω∘Γ whose restriction to (Ω∘Γ)[−1, −1/3] transports to 푣 under the canonical isomorphism with Γ, and whose restriction to (Ω ∘ Γ)[1/3, 1] transports to 푤 under the canonical isomorphism with Ω (note that the layer (Ω ∘ Γ)[1/3, 1] has no inner nodes). For each tensor scheme 풟, there exists a strict tensor category 픽(풟) deﬁned as follows.

• The objects are words in elements of obj 풟.

• The morphisms are deformation classes of boxed progressive plane diagrams in 풟.

• The domain, codomain and composition of morphisms are induced on deformation classes by the corresponding operations for the diagrams.

• Identity morphisms are deformation classes of diagrams with invertible graph.

• The tensor product on objects is given by juxtaposition of words

• The tensor product on morphisms is induced by the tensor product of boxed diagrams.

We have arrived at the main theorem of this subsection.

Theorem B.1.23. 픽(풟) is the free tensor category on the tensor scheme 풟.

95 So it happens that indeed for every tensor scheme there exists an interpretation of 풟 as a category with objects abstract labels and morphisms equivalence classes of progressive plane diagrams. On top of that, any other interpretation of 풟 as a category is equivalent to an interpretation of 풟 as this category. Indeed, the above shows that ∘푁 gives an isomorphism of categories between the full subcategory of Ten(픽(풟), 풞) consisting of the strict tensor functors (i.e., those with 휙0 and 휙1 identities) and the category [풟, 풞].

B.2 Symmetric Tensor Categories and Progressive Polarised Diagrams

B.2.1 The Value of a Progressive Polarised Diagram

A symmetry for a strict tensor category 풞 is a natural family of isomorphisms

휎퐴,퐵 ∶ 퐴 ⊗ 퐵 → 퐵 ⊗ 퐴 (B.47) such that the following diagrams commute:

휎 퐴 ⊗ 퐵 퐴,퐵 퐵 ⊗ 퐴 퐴 ⊗ 퐵 ⊗ 퐶 휎퐴⊗퐵,퐶 휎퐵,퐴 , ퟣ퐴⊗휎퐵,퐶 . (B.48) ퟣ퐴⊗퐵 퐴 ⊗ 퐵 퐴 ⊗ 퐶 ⊗ 퐵 퐶 ⊗ 퐴 ⊗ 퐵 휎퐴,퐶⊗ퟣ퐵

A symmetric tensor category is a tensor category with a distisnguished symmetry 휎. Suppose Γ is a progressive polarised graph and suppose 풞 is a symmetric tensor category. We would like to define a valuation 푣 ∶ Γ → 풞 of Γ in 풞 analogously to definition (B.1.9) but the problem is that we need the domain dom Γ and codomain cod Γ to be linearly ordered. These linear orders are not part of the definition of a diagram because we need to consider different layers of diagrams which do not have natural linear orders on their domains and codomains. This is due the demand that a progressive plane graph is an imbedding and the diagrams for symmetric monoidal categories contain “crossings of edges”. Our strategy is to define the value ̄푣(Γ) in an extension 풞̄ of the category 풞. This will not require artificial choices of linear orders. ̄ The category 풞 has object families {퐴푠|푠 ∈ 푆} of objects of 풞 indexed by finite sets

96 푆. Each such object gives rise to a clique

푚 (⨂ 퐴휙(푘)|휙 ∶ [1, 푚] → 푆) (B.49) 푘=1 in 풞 indexed by bijections 휙 ∶ [1, 푚] → 푆, where [1, 푚] = {1, 2, …, 푚}; the maps for the clique are

푚 푚 −1 ⟨휓 휙⟩ ∶ ⨂ 퐴휙(푘) → ⨂ 퐴휓(푘) (B.50) 푘=1 푘=1 where ⟨휄⟩ for a permutation 휄 will now be described. If 휄 is the simple transposition interchanging 푖 and 푖 + 1 then ⟨휄⟩ = ퟣ ⊗ ⋯ ⊗ 휎 ⊗ ⋯ ⊗ ퟣ ((푚 − 1) terms with a 휎 in the 푖th position). For a general permutation 휏, decompose it as a product of simple transposition 휄, and obtain ⟨휏⟩ as the composite of the corresponding (휄). That this is well-deﬁned and yields a clique follows from MacLane’s coherence theorem for symmetry [12]. ̄ The morphisms 푓 ∶ {퐴푠|푠 ∈ 푆} → {퐵푡|푡 ∈ 푇 } in 풞 are precisely maps between the associated cliques. Another description of the morphisms of 풞̄ will be useful. In general, for cliques {퐶푖|푖 ∈ 퐼}, {퐷푗|푗 ∈ 퐽} in any category 풜, the set of clique maps from the ﬁrst to the second is isomorphic to the quotient set

∑ 풜(퐶푖, 퐷푗)/ ∼, (B.51) (푖,푗)∈퐼×퐽 where {푓 ∶ 퐶푖 → 퐷푗} ∼ {푔 ∶ 퐶푘 → 퐷푚} when 푔 ∘ 푢푘푖 = 푢푚푗 ∘ 푓. Thus a morphism

[휙, 푓, 휓] ∶ {퐴푠|푠 ∈ 푆} → {퐵푡|푡 ∈ 푇 } (B.52) in 풞̄ is an equivalence class of triples (휙, 푓, 휓) consisting of linear orderings 휙 ∶ [1, 푚] → 푚 푛 푆, 휓 ∶ [1, 푛] → 푇 of 푆, 푇 and a morphism 푓 ∶ ⨂ 퐴휙(ℎ) → ⨂ 퐵휓(푘) in 풞, where (휙, 푓, 휓) ℎ=1 푘=1 is equivalent to (휙′, 푓′, 휓′) if and only if the following square commutes

푚 푓 푛 ⨂ 퐴휙(ℎ) ⨂ 퐵휓(푘) ℎ=1 푘=1

⟨휙′−1휙⟩ ⟨휓′−1휓⟩ (B.53)

푚 푓′ 푛 ⨂ 퐴휙′(ℎ) ⨂ 퐵휓′(푘) ℎ=1 푘=1

The functor 풞 → 풞,̄ obtained by regarding objects of 풞 as singleton families, is an equivalence of categories.

97 For any valuation 푣 ∶ Γ → 풞, we put 푣(푠) = 푣0(푠) ∈ obj 풞 for 푠 ∈ dom Γ ∪ cod Γ. The intended value ̄푣(Γ) of Γ will be an arrow

̄푣(Γ) ∶ {푣(푠)|푠 ∈ dom Γ} → {푣(푡)|푡 ∈ cod Γ} (B.54) in 풞.̄ Obviously, if dom Γ and cod Γ are linearly ordered, we obtain an arrow 푣(Γ) in 풞. As in the first section, our definition of value will involve some choices. To show the independence of these choices we need some formal properties of iterated tensor products. For these we introduce a category sp(푁, 풞) for each finite set 푁. 휙 Objects of sp(푆, 풞) have the form {퐴푠|푠 ∈ 푆 푁}, where {퐴푠|푠 ∈ 푆} is a familiy of objects of 풞 indexed by the finite set 푆 and 휙 ∶ 푆 → 푁 is a function equipped with 휙 −1 a linear order on each fibre 휙 (푛), 푥 ∈ 푁. A map {퐴푠|푠 ∈ 푆 푁} → {퐵푡|푡 ∈ 휓 푇 푁} in sp(푁, 풞) is a family

{푢푖 ∶ ⨂ 퐴푠 → ⨂ 퐵푡|푛 ∈ 푁} (B.55) 휙(푠)=푛 휓(푡)=푛 of maps 푢푖 in 풞 indexed by 푁. We now deﬁne a functor

̄ ⨂ ∶ sp(푁, 풞) → 풞̄ (B.56) 푛∈푁 which is given on objects by

̄ 휙 ⨂{퐴푠|푠 ∈ 푆 푁} = {퐴푠|푠 ∈ 푆}. (B.57) 푛∈푁

휙 휓 Suppose {푢푛|푛 ∈ 푁} ∶ {퐴푠|푠 ∈ 푆푁} = {퐴푠|푠 ∈ 푆} → {퐵푡|푡 ∈ 푇 푁} is a map in sp(푁, 풞). Choose a linear order on 푁; this together with the linear orders on the ﬁbres of 휙, 휓 determines linear arders on 푆, 푇 (ordinal sums of the ﬁbres). The morphism

̄ ⨂푢푛 ∶ {퐴푠|푠 ∈ 푆} → {퐵푡|푡 ∈ 푇 }, (B.58) 푛∈푁 in 풞̄ is the map of cliques determined by these orders on 푆 and 푇 , and the map

⨂ 푢푛 ∶ ⨂ ⨂ 퐴푠 → ⨂ ⨂ 퐵푡 (B.59) 푛∈푁 푛∈푁 휙(푠)=푛 푛∈푁 휓(푡)=푛 in 풞. Different choices of linear order on 푁 lead to the same map of cliques. Given a function 휅 ∶ 푁 → 푀 equipped with a linear order on each ﬁbre 휅−1(푗),

98 there is a functor

휅∗ ∶ sp(푁, 풞) → sp(푀, 풞) (B.60) given on objects by

휙 휅휙 휅∗{퐴푠|푠 ∈ 푆 푁} = {퐴푠|푠 ∈ 푆 푀} (B.61) where the linear order on the ﬁbres of 휅휙 is such that 푠 ≤ 푠′ in (휅휙)−1(푚) when either ′ −1 ′ ′ −1 휙(푠) < 휙(푠 ) in 휅 (푚), or 휙(푠) = 휙(푠 ) = 푛 and 푠 ≤ 푠 in 휙 (푖). On maps, 휅∗ is given by

휅∗{푢푛|푛 ∈ 푁} = { ⨂ 푢푛|푚 ∈ 푀} . (B.62) 푘(푛)=푚

The required associativity of iterated tensor products can be expressed as a com- mutative triangle of functors

sp(푁, 풞) ⨂̄ 푛∈푁

휅∗ 풞̄ (B.63)

⨂̄ 푚∈푀 sp(푀, 풞)

This is trivial on objects, while on maps it amounts to the equality

̄ ̄ ⨂푢푛 = ⨂ ⨂ 푢푛 (B.64) 푛∈푁 푚∈푀 푘(푛)=푚 which can be proven by choosing a linear order on 푀, and then obtaining linear orders on 푆, 푇 .

The value of a diagram will be deﬁned by cutting into layers just as in the plane case. Since in this case we do not have the aid of the embedding in the plane to give us levels, here we need a different approach to layers.

Suppose Γ is a progressive graph with boundary. Let Γ̊ denote the set of inner nodes of Γ as a partially ordered set (푥 ≤ 푦 when there is a directed path from 푥 to 푦 in Γ). Deﬁne a level 푎 of Γ to be an initial segment of Γ̊ (i.e., 푥 ≤ 푦 ∈ 푎 ⇒ 푥 ∈ 푎). The smalledst level is ∅ and the largest level is Γ̊.

An edge 푒 is said to be cut by level 푎 when 푒(0) ∈ 푎∪dom Γ and 푒(1) ∈ (Γ∖푎)∪̊ cod Γ.

99 Let cut 푎 denote the set of edges cut by 푎. Note that we have bijections

cut ∅ ≃ dom Γ, cut Γ̊ ≃ cod Γ. (B.65)

An interval in Γ is a pair of levels 푎 ⊆ 푏. Deﬁne the layer Γ[푎, 푏] to be the generalised oriented graph whose nodes are the elments of 푏 − 푎, whose pinned edges are those edges 푒 of Γ with source and target in 푏 − 푎, and whose loose and half-loose edges are the elements of cut 푎 ∪ cut 푏. We have bijections

cut 푎 ≃ dom Γ[푎, 푏], cut 푏 ≃ cod Γ[푎, 푏]. (B.66)

We now proceed to deﬁne the value ̄푣(Γ) ∈ mor 풞̄ of a progressive polarised diagram Γ. Call Γ elementary when the inner nodes are incomparable; that is, when the order ̊ on Γ is discrete. Put 퐼 = 휋0(Γ), the set of connected components of Γ. For each 푖 ∈ 퐼, deﬁne a morphism 푢푖 of 풞 by

{⎧푣 (푠) when 푖 contains a single inner node 푥, 푢 = 1 푖 ⎨ (B.67) {ퟣ when 푖 is a single edge. ⎩ 푣표(푖)

The functions dom Γ → 퐼, cod Γ → 퐼 (taking the outer nodes to the components in which they lie) have ﬁbre over 푖 either a singleton or in natural bijection with in 푥, out 푥, respectively; so the ﬁbres are linearly ordered. Thus we have a map

{푢푖|푖 ∈ 퐼} ∶ {푣(푠)|푠 ∈ dom Γ → 퐼} → {푣(푡)|푡 ∈ cod Γ → 퐼} (B.68) in sp(퐼, 풞). Deﬁne

̅ ̄푣(Γ) = ⨂푢푖 ∶ {푣(푠)|푠 ∈ dom Γ} → {푣(푡)|푡 ∈ cod Γ} (B.69) 푖∈퐼 in 풞.̄

Proposition B.2.1. If 푎 is a level for an elementary diagram Γ then Γ[∅, 푎], Γ[푎, Γ]̊ are elementary, and

̄푣(Γ) = ̄푣(Γ[푎, Γ])̊ ∘ ̄푣(Γ[∅, 푎]). (B.70)

For any progressive polarised diagram Γ, choose a maximal chain

̊ ∅ ⊂ 푎1 ⊂ 푎2⋯ ⊂ 푎푛 = Γ (B.71)

100 of levels of Γ. Then each layer Γ[푎푖, 푎푖+1] has precisely one inner node and so is elementary. Deﬁne

̄푣(Γ) = ̄푣(Γ[푎푛−1, 푎푛]) ∘ ⋯ ∘ ̄푣(Γ[∅, 푎1]). (B.72)

We must see that this definition is independent of the choice of maximal chain. It is possible to move from any maximal chain to any other by a finite sequaence of steps each of which involves replacement of a single 푎푖 by anothe level. To see that ̄푣(Γ) is well defined then comes to showing that, for levels 푎 ⊂ 푏 ⊂ 푐 and 푎 ⊂ 푏′ ⊂ 푐 where 푏 − 푎 = {푤} = 푐 − 푏′, 푐 − 푏 = {푦} = 푏′ − 푎, we have

̄푣(Γ[푏, 푐]) ∘ ̄푣[Γ[푎, 푏]] = ̄푣(Γ[푏′, 푐]) ∘ ̄푣(Γ[푎, 푏′]). (B.73)

Since 푥, 푦 are incomparable, the diagram Γ[푎, 푐] is elementary and proposition (B.2.1) applies to show that both sides of the last equation are equal to ⟨푣| (Γ[푎, 푐]). An isomorphism 푓 ∶ Γ → Γ′ of progressive polarised diagrams Γ, Γ′ is an isomorphism of graphs with boundary which preserves orientation and the orderts on each input and output, and is compatible with the valuations.

Theorem B.2.2. If 푓 ∶ Γ → Γ′ is an isomorphism of progressive polarised diagrams in a symmetric tensor category 풞 then the square

≅ {푣(푠)|푠 ∈ dom Γ} {푣(푠′)|푠′ ∈ dom Γ′}

̄푣(Γ) ̄푣(Γ′) (B.74) ≅ {푣(푡)|푡 ∈ cod Γ} {푣(푡′)|푡′ ∈ cod Γ′} commutes in 풞̄ where the horizontal isomorphisms are determined by the bijections dom Γ ≃ dom Γ′, cod Γ ≃ cod Γ′ induced by 푓.

Definition B.2.3. (i) A progressive graph Γ is called anchored when it is equipped with linear orders on dom Γ and cod Γ. In this case, ̄푣(Γ) determines a map

푣(Γ) ∶ ⨂ 푣(푠) → ⨂ 푣(푡) (B.75) 푠∈dom Γ 푡∈cod Γ

in 풞.

(ii) An isomorphism 푓 ∶ Γ → Γ′ of progressive graphs is anchored when the bijections dom Γ ≃ dom Γ′, cod Γ ≃ cod Γ′ induced by 푓 are order preserving.

Corollary B.2.4. If 푓 ∶ Γ → Γ′ is an anchored isomorphism of progressive polarised diagrams then 푣(Γ) = 푣(Γ′).

101 B.2.2 Free Symmetric Tensor Categories

Recall the deﬁnition of a braided functor.

Definition B.2.5. A braided monoidal functor between braided monoidal categories 풞 and 풞′ is a monoidal functor 퐹 that is compatible with the braiding, i.e. the following diagram commutes:

휙2 퐹 퐴 ⊗ 퐹 퐵 퐹 (퐴 ⊗ 퐵)

휎 퐹퐴,퐹퐵 퐹휎퐴,퐵 (B.76) 휙2 퐹 퐵 ⊗ 퐹 퐴 퐹 (퐵 ⊗ 퐴)

A braided functor 퐹 ∶ 풞 → 풞′ that maps a symmetric category to a symmetric category is called a symmetric functor. We write STen(풞, 풞′) for the category with objects symmetric tensor functors and morphisms the tensor transformations between them.

Definition B.2.6. A symmetric tensor category 퐹 is said to be free on the tensor scheme 풟, when there exists an object 푁 of [풟, ℱ] such that the functor

∘ 푁 ∶ STen(ℱ, 풞) → [풟, 풞] (B.77) is an equivalence of categories for all symmetric tensor categories 풞.

The notion of a valuation 푣 ∶ Γ → 풟 of a progressive polarised graph Γ in a tensor scheme 풟 is deﬁned just as for a progressive plane graph. Call (Γ, 푣) a progressive polarised diagram in 풟. When Γ is anchored, the domain (codomain) of (Γ, 푣) is de-

ﬁned to be the word 푣0(푧1)⋯푣0(푧푛) where 푧1 < ⋯ < 푧푛 are the elements of dom Γ (cod Γ). The tensor product Γ1⊗Γ̃ 2 of two progressive polarised diagrams Γ1 and Γ2 is the diagram whose graph is the disjoint union Γ1 + Γ2 (with inner nodes those of Γ1 and those of Γ2) and whose valuation restricts to Γ1,Γ2 to give their valuations. Suppose Γ, Γ′ are anchored progressive polarised diagrams in 풟 with cod(Γ, 푣) = dom(Γ′, 푣) (as wordts in the elements of obj 풟). This produces an order-preserving bijection cod Γ ≅ dom Γ′. The composite Γ′ ∘ Γ is the diagram deﬁned as follows. The graph is obtained from the disjoint union Γ′ + Γ by identifying the outer nodes which correspond under cod Γ ≅ dom Γ′. The inner nodes and edges are those of Γ and of Γ′, except for the edges of Γ which have target an outer node and the edges of Γ′ which have source an outer node; these edges pair up via corresponding outer nodes, each pair contributing an edge to Γ′ ∘Γ. As paired edges have equal values, we obtain

102 a valuation on Γ′ ∘ Γ.

For each tensor scheme 풟, there is a symmetric strict tensor category 픽푠(풟) de- ﬁned as follows. The objects are words in elements of obj 풟. The morphisms are anchored isomorphism classes of anchored progressive polarised diagrams in 풟. The domain, codomain, composition, and tensor product are induced on anchored isomorphism classes by the corresponding operations on diagrams. The symmetry

휎푣,푤 ∶ 푉 푊 → 푊 푉 , where 푉 , 푊 are the words 푋1⋯푋푚 and 푌1⋯푌푛, respectively, is the anchored isomorphism class of the anchored diagram (Γ, 푣) described as follows. The graph Γ is the union of the 푚+푛 disjoint closed intervals [푎푖, 푏푖], [푐푗, 푑푗] (푖 = 1, …, 푚; 푗 = 1, …, 푛) with their natural orientation and with no inner nodes. The anchoring of Γ is given by

dom Γ = {푎1 < ⋯ < 푎푚 < 푐1 < ⋯ < 푐푛} (B.78)

cod Γ = {푑1 < ⋯ < 푑푛 < 푏1 < ⋯ < 푏푚}. (B.79)

The valuation is given by

푣0(푎푖, 푏푖) = 푋푖, 푣0(푐푗, 푑푗) = 푌푗. (B.80)

Theorem B.2.7. 픽푠(풟) is the free symmetric tensor category on the tensor scheme 풟.

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