Introduction to Categorical Quantum Mechanics. Chris Heunen and Jamie Vicary

Transcription

1 Introduction to Categorical Quantum Mechanics Chris Heunen and Jamie Vicary February 20, 2013

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3 Preace Physical systems cannot be studied in isolation, since we can only observe their behaviour with respect to other systems, such as a measurement apparatus. The central idea o this course is that the ability to group individual systems into compound systems should be taken seriously. We take the action o grouping systems together as a primitive notion, and build models o quantum mechanics rom there. The mathematical tool we use or this is category theory, one o the most wide-ranging parts o modern mathematics. It has become abundantly clear that it provides a deep and powerul language or describing compositional structure in an abstract ashion. It provides a uniying language or an incredible variety o areas, including quantum theory, quantum inormation, logic, topology and representation theory. These notes will tell this story right rom the beginning, ocusing on monoidal categories and their applications in quantum inormation. Much o this relatively recent ield o study is covered only ragmentarily or at the research level, see e.g. [18]. We eel there is a need or a sel-contained text introducing categorical quantum mechanics at a more leisurely pace; these notes are intended to ill this space. cknowledgement Thanks to the students who let us use them as guinea pigs or testing out this material! These notes would not exist were it not or the motivation and assistance o Bob Coecke. We are also grateul to leks Kissinger, lex Merry and Daniel Marsden or careul reading and useul eedback on early versions. iii

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5 Contents 0 Background material Category theory Hilbert spaces Monoidal categories Monoidal categories Graphical calculus Examples States Braiding and Symmetry Exercises bstract linear algebra Scalars Superposition djoints and the dagger The Born rule Exercises Dual objects Dual objects Functoriality Dagger compact categories * Interaction with linear structure Traces and dimensions Inormation low v

6 vi CONTENTS 3.7 Exercises Classical structures Monoids and comonoids Frobenius algebras Normal orms Phases State transer Modules and measurement Exercises Complementarity Bialgebras Hop algebras and complementarity Strong complementarity pplications Exercises Copying and deleting Closure Uniorm deleting Uniorm copying Products Exercises Complete positivity Complete positivity The CP construction Environment structures Exercises Bibliography 165 Index 173

7 Chapter 0 Background material The ideal oundations or reading these notes are a amiliarity with basic elements o both category theory and quantum computer science. This sel-contained chapter ixes our notation and conventions, while briely recalling the basic notions rom both subjects that we will be using in these notes: categories, unctors, natural transormations, vector spaces, Hilbert spaces, and tensor products. 0.1 Category theory Category theory is quite dierent rom other areas o mathematics. While a category is itsel just an algebraic structure much like a group, or a ring, or a ield we can use categories in a powerul way to organize and understand other mathematical objects. This happens in a surprising way: by neglecting all inormation about the structure o the objects, and ocusing entirely on relationships between the objects. Category theory is the study o the patterns which are ormed by these relationships. In this sense, category theory is more like a social science which studies how individuals behave within a society, than a physical science in which objects are reduced to their internal components. The categorical perspective is that we cannot know the internal structure o the systems we are studying, and we may only learn about them by observing their behaviour rom the outside. While at irst this seems limiting, in act it is enormously powerul, as it becomes a very general language or the 1

8 2 CHPTER 0. BCKGROUND MTERIL description o many diverse structures. Categories The view o a category as a collection o objects and relationships between them is axiomatized in the ollowing deinition. The crucial point is that relationships should compose. Deinition 0.1 (Category). category C consists o: a collection Ob(C) o objects; or every two objects, B a collection C(, B) o morphisms; or every two morphisms C(, B) and g C(B, C), a morphism g C(, C); or every object a morphism id C(, ). These must satisy the ollowing properties, or all objects, B, C, D, and all morphisms C(, B), g C(B, C), h C(C, D): associativity: h (g ) (h g) ; identity: id B id. The prototypical example is the category Set, with sets or objects and unctions or morphisms. 1 nother example is the category Vect whose objects are complex vector spaces and whose morphisms are linear transormations, which we will discuss later on. We will meet more examples in Section 1.3. We write B instead o C(, B) when no conusion can arise. Sometimes we will not even bother to name the objects and just talk about the morphism. Then dom() is its domain, and B cod() is its codomain. 1 Our deinition o a category reers to collections o objects and morphisms, rather than sets, because sets are too small in general. The category Set illustrates this very well, since Russell s paradox prevents a set o all sets. However, such set-theoretical issues will not play a role in these notes, and we may use set theory naively.

9 0.1. CTEGORY THEORY 3 In category theory we oten draw diagrams o morphisms, which indicate the way they can be composed. Here is an example. g B C h i j D k E We say a diagram commutes when every possible path rom one object in it to another is the same. In the above example, this means i k h and g j i. It then ollows that g j k h, where we do not need to write parentheses thanks to the associativity property o Deinition 0.1. Thus we have two ways to speak about equality o composite morphisms: by algebraic equations, or by commuting diagrams. O central importance in these notes will be a third way, called the graphical calculus, which we introduce in Section 1.2. morphism B is an isomorphism when there exists a morphism B 1 satisying 1 id B and 1 id. We say in this case that and B are isomorphic. category in which every morphism is an isomorphism is called a groupoid. I the isomorphic objects are all actually equal, then we say the category is skeletal. ny category C has an opposite category C op, with the same objects, but with C op (, B) given by C(B, ). That is, the morphisms B in C op are morphisms B in C. I C and D are categories, there is a product category C D, whose objects are pairs (, B) o object rom C and B rom D, and whose morphisms are pairs (, g) o morphisms in C and g in D. Functors Remember the motto that morphisms are more important than objects. Categories are interesting mathematical objects in their own right. Category theory, the study o categories, takes its own medicine here: there are interesting notions o morphisms between categories, as in the ollowing deinition.

10 4 CHPTER 0. BCKGROUND MTERIL Deinition 0.2 (Functor). Given categories C and D, a unctor F : C D is deined by the ollowing data: an object F () in D or each object in C; a morphism F () F () F (B) in D or each morphism B in C. This data must satisy the ollowing properties: F (g ) F (g) F () or all morphisms B and B g C in C; F (id ) id F () or every object in C. These are also called covariant unctors. There are also contravariant unctors that reverse the direction o morphisms: a contravariant unctor C D is a covariant unctor C op D. Natural transormations Going urther, there is also an interesting notion o morphisms between unctors. Deinition 0.3 (Natural transormation). Given unctors F : C D and G: C D, a natural transormation α: F G is an assignment or every object in C o a morphism F () α G() in D, such that the ollowing diagram commutes or every morphism B in C. F () F () F (B) α α B G() G(B) G() I every component α is an isomorphism then α is called a natural isomorphism, and F and G are said to be naturally isomorphic.

11 0.2. HILBERT SPCES Hilbert spaces In the traditional approach to quantum theory, the state space o a quantum system is ormalized as a Hilbert space. The linear structure accounts or superposition o states, and the inner product gives the amplitudes o observing one state given that the system is in another. mplitudes are complex numbers in general, and we convert them to probabilities by taking the square o their absolute value. The state space o a compound system is given by the tensor product o the state spaces o the component systems. We will now briely recall each o these notions. Inner product spaces vector space is a collection o elements that can be added to one another, and scaled. Deinition 0.4 (Complex vector space). complex vector space is a set V with a chosen element 0 V, an addition operation +: V V V, and a scalar multiplication operation : C V V, satisying the ollowing properties or all u, v, w V and a, b C: additive associativity: u + (v + w) (u + v) + w; additive commutativity: u + v v + u; additive unit: v + 0 v; additive inverses: there is a v V such that v + ( v) 0; additive distributivity: a (u + v) (a u) + (a v) scalar unit: 1 v v; scalar distributivity: (a + b) v (a v) + (b v); scalar compatibility: a (b v) (ab) v. The prototypical example o a vector space is C n, the cartesian product o n copies o the complex numbers..

12 6 CHPTER 0. BCKGROUND MTERIL unction : V W between vector spaces is called linear when (u + v) (u) + (v) or u, v V, and (a v) a (v) or v V and a C. Vector spaces and linear unctions orm a category Vect. We will use some more structure on vector spaces. n inner product on a vector space lets us measure amplitudes between two vectors, and lengths o vectors. Deinition 0.5 (Inner product). n inner product on a vector space V is a unction : V V C satisying the ollowing properties, or u, v, w V and a C: conjugate-symmetric: v w w v or v, w V ; linear in the second argument: v a w a v w, positive deinite: v v 0, u v + w u v + u w ; v v 0 i and only i v 0. n inner product gives rise to a norm v : v v. In turn, we can speak about the distance between two vectors u and v as u v. Hilbert space is a vector space with an inner product in which it makes sense to sum up certain ininite sequences o vectors. The ollowing deinition makes this precise. Deinition 0.6 (Hilbert space). Hilbert space is a vector space H with an inner product that is complete in the ollowing sense: i a sequence v 1, v 2,... o vectors satisies i1 v i <, then there is a vector v such that v n i1 v i tends to zero. ny vector space with an inner product can be completed to a Hilbert space. linear map : H K between Hilbert spaces is bounded when there exists a number b R such that (v) b v or all v H. set {e i } o vectors is called orthonormal when e i e i 1 or all i and e i e j 0 or all i j. It is called an orthonormal basis when every vector can be written as an ininite linear combination o e i, i.e. when any vector v allows coordinates v i C or which v i v i e i tends to zero. It is always possible to choose an orthonormal basis, but remember

13 0.2. HILBERT SPCES 7 that Hilbert spaces allow many dierent orthonormal bases. When there can be no conusion about the chosen orthonormal basis e i, we sometimes write i. The dimension o a Hilbert space is the size, or cardinality, o an orthonormal basis; this is independent o the orthonormal basis used. We will mostly be concerned with inite-dimensional Hilbert spaces. initedimensional vector space with an inner product is automatically a Hilbert space, and any linear map between inite-dimensional Hilbert spaces is automatically bounded. I H is a Hilbert space, then so is Hilb(H, C), the set o bounded linear unctions H C. ny Hilbert space H has a dual Hilbert space H, with the same set o vectors as Hilb(H, C) and the same addition and inner product, but where scalar multiplication is conjugated: z v in H equals z v in Hilb(H, C). Hilbert space is always isomorphic to its dual: the map H H that sends v H to the unction w v w is an invertible bounded linear unction. djoints We use the inner product to deine the adjoint to a linear map. Deinition 0.7 (djoint o a bounded linear map). For a bounded linear map : H K, its adjoint : K H is the unique linear map with the ollowing property, or all φ H and ψ K: (φ) ψ φ (ψ). (1) It ollows immediately rom (1) by uniqueness o adjoints that ( ), (g ) g, and id H id H. partial isometry is a bounded linear map : H K satisying. This means that H ker() H and that preserves norm on the Hilbert space H. This class o maps includes projections (p: H H such that p p p p ) and isometries ( : H K such that id H ). Tensor products I V and W are vector spaces, then so is V W ; this is called the direct sum and is also denoted by V W. This way o grouping two vector

14 8 CHPTER 0. BCKGROUND MTERIL spaces is classical, in the sense that states o the direct sum are completely determined by states o the constituent vector spaces. The tensor product is another way to make a new vector space out o two given ones, that allows or entangled states. With some work it can be constructed explicitly, but it is only important or us that it exists, and is deined up to isomorphism by a universal property. I U, V and W are vector spaces, a unction : U V W is called bilinear when it is linear in each variable: when the unction u (u, v) is linear or each v V, and the unction v (u, v) is linear or each u U. Deinition 0.8 (Tensor product). The tensor product o vector spaces U and V is a vector space U V together with a linear unction : U V U V such that or every bilinear unction g : U V W there exists a unique linear unction h: U V such that g h. U V (linear) U V (bilinear) g h The unction usually stays anonymous and is written as (u, v) u v. Thus arbitrary elements o U V take the orm n i1 a iu i v i or a i C, u i U, and v i V. The tensor product also extends to linear maps. I 1 : U 1 V 1 and 2 : U 2 V 2 are linear maps, there is a unique linear map 1 2 : U 1 U 2 V 1 V 2 that satisies ( 1 2 )(u 1 u 2 ) 1 (u 1 ) 2 (u 2 ) or u 1 U 1 and u 2 U 2. In this way, the tensor product becomes a unctor : Vect Vect Vect. I V and W carry inner products, we can urnish their direct sum with an inner product by (v 1, w 1 ) (v 2, w 2 ) v 1 v 2 + w 1 w 2. We can also urnish their tensor product as vector spaces with an inner product by v 1 w 1 v 2 w 2 v 1 w 1 v 2 w 2. I H and K are Hilbert spaces, their direct sum is again a Hilbert space H K, now called their orthogonal direct sum; the completion o their tensor product as vector spaces is again a Hilbert space, that we denote by H K. In this way, the tensor product is a unctor : Hilb Hilb Hilb. I {e i } is an orthonormal W

15 0.2. HILBERT SPCES 9 basis or H, and { j } is an orthonormal basis or K, then {e i j } is an orthonormal basis or H K. So when H and K are inite-dimensional, there is no dierence between their tensor products as vector spaces and as Hilbert spaces. Notes and urther reading Categories arose in algebraic topology and homological algebra in the 1940s. They were irst deined by Eilenberg and Mac Lane in Early uses o categories were mostly as a convenient language. With applications by Grothendieck in algebraic geometry in the 1950s, and by Lawvere in logic in the 1960s, category theory became an autonomous ield o research. It has developed rapidly since then, with applications in computer science, linguistics, cognitive science, philosophy, and many other areas, including physics. s a good irst textbook, we recommend [7], but the more mathematically inclined might preer the gold standard [56]. bstract vector spaces as generalizations o Euclidean space had been gaining traction or a while by Two parallel developments in mathematics in the 1900s led to the introduction o Hilbert spaces: the work o Hilbert and Schmidt on integral equations, and the development o the Lebesgue integral. The ollowing two decades saw the realization that Hilbert spaces oer one o the best mathematical ormulations o quantum mechanics. The irst axiomatic treatment was given by von Neumann in 1929, who also coined the name Hilbert space. lthough they have many deep uses in mathematics, Hilbert spaces have always had close ties to physics. For a rigorous textbook with a physical motivation, we reer to [66].

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17 Chapter 1 Monoidal categories monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined in parallel. This chapter introduces the theory o monoidal categories. They orm the core o these notes, as they provide the basic language with which the rest o the material will be developed. We introduce a visual notation called the graphical calculus, which provides an intuitive and powerul way to work with them. We also introduce our main examples o monoidal categories Hilb, Set and Rel which will be used as running examples throughout these notes. 1.1 Monoidal categories Scope We will soon give the precise mathematical deinition o a monoidal category. To appreciate it, it is good to realize irst what sort o situation it aims to represent. In general, one can think o objects, B, C,... o a category as systems, and o morphisms B as processes turning the system into the system B. This can be applied to a vast range o structures: physical systems, and physical processes governing them; data types in computer science, and algorithms manipulating them; 11

18 12 CHPTER 1. MONOIDL CTEGORIES algebraic or geometric structures in mathematics, and structurepreserving unctions; logical propositions, and implications between them; or even ingredients in stages o cooking, and recipes to process them rom one state to another. The extra structure o monoidal categories then simply says that we can consider processes occurring in parallel, as well as one ater the other. In the examples above, one could interpret this as: letting separate physical systems evolve simultaneously; running computer algorithms in parallel; taking products or sums o algebraic or geometric structures; proving conjunctions o logical implications by proving both implications; chopping carrots while boiling rice. Monoidal categories provide a general ormalism or describing these general sorts o composition. It is perhaps surprising that a nontrivial theory can be developed at all rom such simple intuition. But in act, the theory o monoidal categories is remarkably rich, and provides a potent and elegant language or many developments in modern mathematics, physics and computer science. Deinition and coherence Deinition 1.1 (Monoidal category). monoidal category is a category C equipped with the ollowing data, satisying a property called coherence: a unctor : C C C, called the tensor product; an object I C, called the unit object; a natural isomorphism whose components ( B) C α,b,c (B C) are called the associators;

19 1.1. MONOIDL CTEGORIES 13 a natural isomorphism whose components I λ are called the let unitors; a natural isomorphism whose components I ρ are called the right unitors. The coherence property is that every well-ormed equation built rom,, id, α, α 1, λ, λ 1, ρ and ρ 1 is satisied. Interesting examples o such equations are the ollowing triangle and pentagon identities. α,i,b ( I) B (I B) ρ id B id λ B (1.1) B α,b C,D ( (B C) ) D ( (B C) D ) α,b,c id D ( ( B) C ) D id α B,C,D ( B (C D) ) α B,C,D α,b,c D ( B) (C D) (1.2) By the coherence property, these diagrams must commute in any monoidal category. Conversely, and perhaps surprisingly, it turns out that these identities (1.1) and (1.2) are suicient to ensure coherence. The ollowing very important and beautiul theorem, which is too deep or us to prove here, records this. Theorem 1.2 (Coherence or monoidal categories). The data or a monoidal category are coherent i and only i identities (1.1) and (1.2) hold.

20 14 CHPTER 1. MONOIDL CTEGORIES This theorem implies the nontrivial but useul equation ρ I λ I (see Exercise 1.6.2). Strictness Some types o monoidal category are particularly simple. Deinition 1.3 (Strict monoidal category). monoidal category is strict i all components o the natural isomorphisms α, λ, and ρ, are identities. In act, every monoidal category can be made into a strict one. The ollowing deep theorem, which we state without proo, is tightly related to the Coherence Theorem 1.2. Theorem 1.4 (Strictiication). Every monoidal category is monoidally equivalent to a strict monoidal category. These notes will not give a deinition o monoidal equivalence, which determines when two monoidal categories encode the same systems and processes. This theorem means that, i you preer, you can always strictiy your monoidal category to obtain an equivalent one or which α, λ and ρ are all identities. However, this is sometimes not very useul. For example, you oten have some idea o what you want the objects o your category to be but this might have to be abandoned to construct a strict version o your category. In particular, it s oten useul or categories to be skeletal, meaning that i any pair and B o objects are isomorphic, then they are equal. Every monoidal category is equivalent to a skeletal monoidal category, and skeletal categories are oten particularly easy to work with. However, not every monoidal category is monoidally equivalent to a strict, skeletal category. I you have to choose, it oten turns out that skeletality is the more useul property to have. The interchange law Monoidal categories enjoy an important property, called the interchange law, which governs the interaction between the categorical composition and tensor product.

21 1.2. GRPHICL CLCULUS 15 Theorem 1.5 (Interchange). ny morphisms B, B g C, D h E and E j F in a monoidal category satisy the interchange law (g ) (j h) (g j) ( h) (1.3) Proo. This holds because o properties o the category C C, and rom the act that : C C C is a unctor. (g ) (j h) (g, j h) ( (g, j) (, h) ) (deinition o C C) ( (g, j) ) ( (, h) ) (unctoriality o ) (g j) ( h) Recall that the unctoriality property or a unctor F says that F ( g) F () F (g). 1.2 Graphical calculus We now describe a graphical way to denote the basic protagonists o monoidal categories: objects, morphisms, composition, and tensor product. This graphical calculus aithully captures the essence o working with monoidal categories. nd in act, in most cases, it makes them a lot easier to work with. Graphical calculus or ordinary categories We begin by describing a graphical notation or ordinary categories without any monoidal structure. We draw an object as ollows. (1.4) It s just a line. In act, really, you shouldn t think o this as a picture o the object ; you should think o it as a picture o the identity morphism id. Remember, in category theory, the morphisms are more important than the objects.

22 16 CHPTER 1. MONOIDL CTEGORIES We draw a general morphism B as ollows, as a box with one input at the bottom, and one output at the top. B (1.5) Composition o B and B g C is then drawn by connecting the output o the irst box to the input o the second box. C g B (1.6) Let s use this to see what the identity law id id B looks like. B B B (1.7) It s completely trivial we just have to remember that what is important is the connectivity o the diagram, not whether our lines are perectly

23 1.2. GRPHICL CLCULUS 17 straight, or wobble slightly. Categories also have an associativity axiom: given C h D, we must have (h g) h (g ). Graphically, this becomes the ollowing. D h C g B D h C g B (1.8) The brackets are here to show how we have built up each picture; they are not part o the notation. This associativity condition is trivial in the graphical representation; again, we just have to remember that only the connectivity o our diagram is important or identiying the morphism that the diagram deines. So even or ordinary categories without any monoidal structure, the graphical calculus is already useul: it somehow absorbs our axioms, making them a consequence o the notation. This is because the axioms o a category are about stringing things together in sequence. t a undamental level, this connects to the geometry o the line, which is also one-dimensional. O course, this graphical representation isn t so unamiliar we usually draw it horizontally, and call it algebra. Graphical calculus or monoidal categories Now let s bring tensor products into the graphical notation. n object B or rather, the morphism id B is drawn as two lines side-by-

24 18 CHPTER 1. MONOIDL CTEGORIES side. (1.9) B Morphisms and composition are drawn in the same way as or ordinary categories. Given morphisms B and C g D, we draw C g B D in the ollowing way. B D g (1.10) C The idea is that and g represent separate processes, taking place at the same time. Whereas the graphical calculus or ordinary categories was one-dimensional or linear, the graphical calculus or monoidal categories is two-dimensional or planar. The two dimensions correspond to the two ways to combine morphisms: by categorical composition (vertically) or by tensor product (horizontally). The monoidal unit object I is drawn as the empty diagram. (1.11) The let unitor λ : I, the right unitor ρ : I and the associator α,b,c : ( B) C (B C) are drawn as ollows. B C (1.12) λ ρ α,b,c

25 1.2. GRPHICL CLCULUS 19 They are completely trivial. The coherence o α, λ and ρ is thereore important or the graphical calculus to unction: since there can only be a single morphism ormed rom these natural isomorphisms between any two given objects, it doesn t matter that their graphical calculus encodes no inormation. We now consider the graphical representation o the interchange law (1.3). C g B F j E h D C B g F E h D j (1.13) We use brackets to indicate how we are orming the diagrams on each side. Dropping the brackets, we see that the interchange law is in act very natural what seemed to be a mysterious algebraic identity becomes very clear rom the graphical perspective. The point o the graphical calculus is that all o the supericially complex aspects o the algebraic deinition o monoidal categories the unit law, the associativity law, associators, let unitors, right unitors, the triangle equation, the pentagon equation, the interchange law simply melt away, allowing us to use the ormalism much more directly. These eatures are still there, but they are absorbed into the geometry o the plane, o which our species has an excellent automatic understanding. It can be ormally proven that the morphisms represented by two given diagrams are equal under the axioms o a monoidal category i and only i one diagram can be deormed into the other respecting the geometry o the plane. That is, you can continuously move boxes around in the plane, as long as you don t introduce crossings or allow wires to be detached rom the upper and lower boundaries.

26 20 CHPTER 1. MONOIDL CTEGORIES 1.3 Examples It is now high time to have some examples. The ollowing three monoidal categories will be our running examples throughout these notes. Hilbert spaces Our irst example is Hilb, the monoidal category o Hilbert spaces, which will play a central role in these notes. We also discuss the closely related categories FHilb and FHilb ss. See Section 0.2 or a brie irst introduction to the theory o Hilbert spaces. Deinition 1.6. The monoidal category Hilb is deined in the ollowing way: Objects are Hilbert spaces H, J, K,...; Morphisms are bounded linear maps, g, h,...; Composition is composition o linear maps; Identity maps are given by the identity linear maps; Tensor product : Hilb Hilb Hilb is the tensor product o Hilbert spaces; The unit object I is the one-dimensional Hilbert space C; ssociators α H,J,K : (H J) K H (J K) are the unique linear maps satisying φ ( χ ψ ) ( φ χ ) ψ or all φ H, χ J and ψ K; Let unitors λ H : C H H are the unique linear maps satisying 1 φ φ or all φ H; Right unitors ρ H : H C H are the unique linear maps satisying φ 1 φ or all φ H. You might have noticed that this deinition o Hilb makes no mention o the inner products on the Hilbert spaces. This structure is crucial or

27 1.3. EXMPLES 21 quantum mechanics, so it s perhaps surprising it hasn t made an appearance here. In act, in the development o this subject, it took quite a while or people to understand the correct way to deal with it categorically. We will encounter the inner product in Section??. We also deine a inite-dimensional variant o Hilb. Deinition 1.7. The monoidal category FHilb has inite-dimensional Hilbert spaces as objects; the rest o the structure is the same as or Hilb, in particular morphisms and tensor products. This is particularly appropriate or the purposes o quantum inormation theory, where the main results are oten in inite dimensions. Neither o the monoidal categories Hilb or FHilb are strict, and neither o them are skeletal. However, or FHilb, there is a monoidally equivalent monoidal category which is strict and skeletal, which we call FHilb ss. Deinition 1.8. The strict, skeletal monoidal category FHilb ss is deined as ollows: Objects are natural numbers 0, 1, 2,...; Morphisms n m are matrices o complex numbers with m rows and n columns; Composition is given by matrix multiplication; Tensor product : FHilb ss FHilb ss FHilb ss is given by n m : nm on objects, and on morphisms by Kronecker product o matrices: ( 11 g ) ( 21 g ) ( 1n g ) ( 12 g ) ( 22 g ) (... 2n g ) ( g) :..... (. m1 g ) ( m2 g ) (... mn g ) ; The tensor unit is the natural number 1; ssociators, let unitors and right unitors are the identity matrices.

28 22 CHPTER 1. MONOIDL CTEGORIES Objects n in Hilb ss can be thought o as the Hilbert space C n, which are equipped with a privileged basis. Linear maps between such Hilbert spaces can be canonically represented as matrices. In practice, this monoidal category FHilb ss is the most convenient place to work when doing calculations involving inite-dimensional Hilbert spaces. I you have done calculations with inite-dimensional Hilbert spaces, or example as part o an exercise in quantum computing, you have really been working in this category. We do not give a ull treatment o the notion o monoidal equivalence in these notes, but it seems intuitively possible that FHilb ss somehow captures everything that is important about FHilb as a monoidal category. Sets and unctions While Hilb is relevant or quantum physics, the monoidal category Set is an important setting or classical physics. Deinition 1.9. The monoidal category Set is deined in the ollowing way: Objects are sets; Morphisms are unctions; Composition is unction composition; Identity morphisms are given by the identity unctions; Tensor product is Cartesian product o sets, written ; The unit object is a chosen 1-element set { }; ssociators α,b,c : ( B) C (B C) are the unctions given by ( (a, b), c ) ( a, (b, c) ) or a, b B, and c C; Let unitors λ : I are the unctions given by (, a) a or a ; Right unitors ρ : I are the unctions given by (a, ) a or a.

29 1.3. EXMPLES 23 Deinition The monoidal category FSet has inite sets or objects, and the rest o the structure is the same as in Set. I you have studied some category theory, you might know that the Cartesian product in Set is a (categorical) product. We have an example here o a general phenomenon: i a category has products, then these can be used to give a monoidal structure. The same is true or coproducts, which in Set are given by disjoint union. This gives us an important dierence between Hilb and Set: while the tensor product on Set comes rom a categorical product, the tensor product on Hilb does not. (See also Chapter 6 and Exercise ) We will discover many more dierences between Hilb and Set, which oten tells us about the dierences between quantum and classical inormation. Sets and relations While Hilb is a setting or quantum physics and Set is a setting or classical physics, Rel, the category o sets and relations, is somewhere in the middle. It seems like it should be a lot like Set, but in act, its properties are much more like those o Hilb. This makes it an excellent test-bed or investigating dierent aspects o quantum mechanics rom a categorical perspective. Deinition Given sets and B, a relation R B is a subset R B. I elements a and b B are such that (a, b) R, then we oten indicate this by writing a R b, or even a b when R is clear. Since a subset can be deined by giving its elements, we can deine our relations by listing the related elements, in the orm a 1 R b 1, a 2 R b 2, a 3 R b 3, and so on. We can think o a relation R B in a dynamical way, as indicating the possible ways or elements o to evolve into elements o B. This

30 24 CHPTER 1. MONOIDL CTEGORIES suggests the ollowing sort o picture. R B (1.14) This suggests we interpret a relation as a sort o nondeterministic classical process: each element o can evolve into any element o B to which it is related. Nondeterminism enters here because an element o can be related to more than one element o B, so under this interpretation, we cannot predict how it will evolve. n element o could also be related to no elements o B: we interpret this to mean that, or these elements o, the dynamical process halts. Because o this interpretation, the category o relations is important in the study o nondeterministic classical computing. Suppose we have a pair o relations, with the target o the irst equal to the source o the second. R S B B C Our interpretation o relations as dynamical processes then suggests a natural notion o composition: an element a is related to c C i there is some b B with a R b and b S c. For our example above, this gives

31 1.3. EXMPLES 25 rise to the ollowing composite relation. S R C This deinition o relational composition has the ollowing algebraic orm. S R : {(a, c) b B : arb and bsc} C (1.15) We can write this dierently as a (S R) c b (bsc arb), (1.16) where represents logical OR, and represents logical ND. Comparing this with the deinition o matrix multiplication, we see a strong similarity: (g ) ij g ik kj (1.17) k This suggests another way to interpret a relation: as a matrix o truth values. For the example relation (1.14), this gives the ollowing matrix, where we write 0 or alse and 1 or true: R B (1.18) Composition o relations is then just given by ordinary matrix composition, with OR and ND replacing + and.

32 26 CHPTER 1. MONOIDL CTEGORIES This gives an interesting analogy between quantum mechanics and the theory o relations. Firstly, a relation R B tells us, or each a and b B, whether it is possible or a to produce b, whereas a complexvalued matrix H L J gives us an amplitude or a to evolve to b. Secondly, relational composition tells us the possibility o evolving via an intermediate point, whereas matrix composition tells us the amplitude or this to happen. Deinition The monoidal category Rel is deined in the ollowing way: Objects are sets; Morphisms R B are relations; Composition o two relations R B and B in (1.15) above; S C is given as Identity morphisms id are the relations {(a, a) a } ; Tensor product is Cartesian product o sets, written ; The unit object is a chosen 1-element set { }; ssociators α,b,c : ( B) C (B C) are the relations deined by ( (a, b), c ) ( a, (b, c) ) or all a, b B and c C; Let unitors λ : I are the relations deined by (, a) a or all a ; Right unitors ρ : I are the relations deined by (a, ) a or all a. The monoidal category FRel is the restriction o the monoidal category Rel to inite sets.

33 1.4. STTES States States o general objects Morphisms out o the tensor unit I play a special role in a monoidal category. In many cases we can think o such morphisms as generalized states or points, even though the objects might not be sets at all and thus have no recognizable elements, points, or states. Deinition 1.13 (State). In a monoidal category, a state o an object is a morphism I. We now examine what the states are in our three example categories. In Hilb, points o a Hilbert space H are linear unctions C H, which correspond to elements o H by considering the image o 1 C; In Set, points o a set are unctions { }, which correspond to elements o by considering the image o ; In Rel, points o a set are relations { } R, which correspond to subsets o by considering all elements related to. Deinition 1.14 (Well-pointed). monoidal category is well-pointed i or all parallel pairs o morphisms,g B, we have g when a g a or all points I a. The idea is that in a well-pointed category, we can tell whether or not morphisms are equal just by seeing how they aect points o their domain objects. The categories Set, Hilb, and Rel are all well-pointed. However, using well-pointedness somewhat goes against the philosophy o category theory that you should not try to use internal structure o objects.

34 28 CHPTER 1. MONOIDL CTEGORIES Graphical representation To emphasize that states I a have the empty picture (1.11) as their domain, we will draw them as triangles instead o boxes. a (1.19) We can think o this dynamically as a method o preparing : we begin with the empty system at the bottom o the diagram, and then ater the process occurs, we have an instance o system. The emphasis here is on the process that takes place, rather than the coniguration o which results. Entanglement and product states For objects and B o a monoidal category, a morphism I a B is a joint state o and B. We depict it graphically in the ollowing way. B a (1.20) joint state is a product state, or separable, when it is o the orm I λ 1 I I I a b B or I a and I b B. a B b (1.21) n entangled state is a joint state which is not a product state. Entangled states represent preparations o B which cannot be decomposed as a preparation o alongside a preparation o B. In this case, there is some

35 1.4. STTES 29 essential connection between and B which means that they cannot have been prepared independently. Let s see what these look like in our example categories. In Hilb: In Set: In Rel: Joint states o H and J are elements o H J; Product states are actorizable states; Entangled states are elements o H J which cannot be actorized. Joint states o and B are elements o B; Product states are elements (a, b) B coming rom a and b B; Entangled states don t exist! Joint states o and B are subsets o B; Product states are subsets P B such that, or some R and S B, (a, b) P i and only i a R and b S; Entangled states are subsets o B that are not o this orm. This gives us an insight into why entanglement can be diicult or us to understand intuitively: classically, in the worldview encoded by the category Set, it simply does not occur. I we allow possibilistic behaviour as encoded by Rel, then an analogue o entanglement can be described in a classical way. Eects n eect represents a process by which a system is destroyed, or consumed. Deinition In a monoidal category, an eect or costate or an object is a morphism I.

36 30 CHPTER 1. MONOIDL CTEGORIES Eects are opposite to states, in the sense that states are morphisms o type I. Given a diagram constructed using the graphical calculus, we can interpret it as a history o events that have taken place. I the diagram contains an eect, this is interpreted as the assertion that a measurement was perormed, with the given eect as the result. For example, an interesting diagram would be this one: b (1.22) a This describes a history in which a state a is prepared, and then a process is perormed producing two systems, the irst o which is measured giving outcome b. This does not imply that the eect b was the only possible outcome or the measurement; just that by drawing this diagram, we are only interested in the cases when it is. n eect in a string diagrams can be thought o as a postselection: we run our entire experiment, only choosing whether to keep the resulting state ater checking that our measurement had the correct outcome. Overall our history is a morphism o type I, which is a state o. The postselection interpretation tells us how to prepare this state, given the ability to perorm its components. These statements are at a very general level. To say more, we must take account o the particular theory o processes described by the monoidal category in which we are working. In quantum theory, as encoded by Hilb, we require a, and b to be partial isometries. The rules o quantum mechanics then dictate that the probability or this history to take place is given by the square norm o the resulting state. So in particular, the history described by this composite is impossible exactly when the overall state is zero. In nondeterministic classical physics, as described by Rel, we need put no particular requirements on a, and b they may be arbitrary relations

37 1.5. BRIDING ND SYMMETRY 31 o the correct types. The overall composite relation then describes the possible ways in which can be prepared as a result o this history. I the overall composite is zero, that means this particular sequence o a state preparation, a dynamics step, and measurement result cannot occur. Things are very dierent in Set. The monoidal unit object is terminal in that category, meaning Hom(, I) has only a single element or any object. So every object has a unique eect, and there is no nontrivial notion o measurement. Indeed, the deterministic classical physics encoded by this category is very dierent rom our other example categories, as we will see repeatedly throughout these notes. 1.5 Braiding and Symmetry We have seen that the graphical calculus or monoidal categories allows us to move around boxes, as long as we don t cut wires or introduce crossings. We now discuss the kinds o monoidal categories or which crossings are allowed. Braided monoidal categories We irst consider braided monoidal categories. Deinition braided monoidal category is a monoidal category C equipped with a natural isomorphism whose components satisy the ollowing hexagon identities. σ,b : B B (1.23) (B C) σ,b C (B C) α,b,c ( B) C σ,b id C α B,C, B (C ) (1.24) id B σ,c (B ) C B ( C) α B,,C

38 32 CHPTER 1. MONOIDL CTEGORIES (B C) σ 1 B C, (B C) α,b,c α B,C, ( B) C σ 1 B, id C B (C ) id B σ 1 C, (1.25) (B ) C B ( C) α B,,C We can include the braiding in our graphical notation by drawing them as: σ,b : B B σ 1,B : B B (1.26) Invertibility then takes the ollowing graphical orm: (1.27) This captures part o the geometric behaviour o strings. Since they cross over each other, they are not lying on the plane they live in three-dimensional space. So while categories have a one-dimensional or linear notation, and monoidal categories have a two-dimensional or planar graphical notation, we see that braided monoidal categories have a three-dimensional or spatial notation. Because o this, braided monoidal categories have an important connection to certain three-dimensional quantum ield theories.

39 1.5. BRIDING ND SYMMETRY 33 Symmetric monoidal categories Deinition braided monoidal category is symmetric when σ B, σ,b id B (1.28) or all objects and B. Graphically, this has the ollowing representation. (1.29) Intuitively, this means the strings can pass through each other, and there can be no nontrivial linkages. Lemma In a symmetric monoidal category we have σ,b σ 1 B,, with the ollowing graphical representation: (1.30) Proo. Combine (1.27) and (1.29). symmetric monoidal category thereore makes no distinction between over- and under-crossings, and so we simpliy our graphical notation, drawing (1.31) or both. The graphical calculus with the extension o braiding or symmetry is still sound: i the two diagrams o morphisms can be deormed into one another, then the two morphisms are equal. This relies on an extension o the Coherence Theorem with symmetries. The statement is more involved than that o Theorem 1.2 because id σ, ; basically it says that every diagram built rom associators, unitors, and braidings

40 34 CHPTER 1. MONOIDL CTEGORIES or symmetries, commutes, as long as all paths have the same underlying permutation. Suppose we imagine our pictorial diagrams as curves embedded in ourdimensional space. Then we can smoothly deorm one crossing into the other, by making use o the extra dimension. In this sense, symmetric monoidal categories have a our-dimensional graphical notation. Since all our example categories are symmetric monoidal, we will not consider braided monoidal categories explicitly in the rest o these notes. However, many o the theorems that we prove or symmetric monoidal categories also hold or braided monoidal categories. Examples Our example categories Hilb, Set and Rel can all be equipped with a symmetry: In Hilb, σ H,K : H K K H is the unique linear map extending φ ψ ψ φ or all φ H and ψ K; In Set, σ S,T : S T T S is deined by (s, t) (t, s) or all s S and t T ; In Rel, σ S,T : S T T S is the deined by (s, t) (t, s) or all s S and t T. 1.6 Exercises Exercise Let, B, C, D be objects in a monoidal category. Construct a morphism ((( I) B) C) D (B (C (I D))). Can you ind another? Exercise Suppose given the data o a monoidal category satisying (1.1) and (1.2).

41 1.6. EXERCISES 35 (a) Prove that the marked triangle in the diagram below commutes. (Hint: consider the rest o the diagram irst.) ( B) (I D) α B,I,D id B λ D α,b,i D ρ B id D (( B) I) D ( B) D (B (I D)) α,b,i id D (id ρ B ) id D (id α B,I,D ) α,b I,D ( (B I)) D (b) Prove that the ollowing triangle commutes. α,b,i ( B) I (B I) ρ B B id ρ B (c) Prove that the ollowing square commutes. ρ I I (I I) I I I ρ I id I ρ I I I ρ I I

42 36 CHPTER 1. MONOIDL CTEGORIES (d) Use your answers to (a) (c) to conclude that ρ I λ I. Exercise Convert the ollowing algebraic equations into graphical language. Which would you expect to be true in any symmetric monoidal category? (a) (g id) σ ( id) ( id) σ (g id) or,g. (b) ( (g h)) k (id ) ((g h) k), or k B C, C h B and B,g B. (c) (id h) g ( id) (id ) g (h id), or,h and g. (d) h (id λ) (id ( id)) (id λ 1 ) g h g λ ( id) λ 1, or g B C, I I and B C h D. Exercise Recall Deinition 1.8. (a) Show explicitly that the Kronecker product o three 2-by-2 matrices is strictly associative. (b) What might go wrong i you try to include ininite-dimensional Hilbert spaces in a strict, skeletal category as in Deinition 1.8? Exercise Recall that an entangled state o objects and B is a state o B that is not a product state. (a) Which o these states o C 2 C 2 in Hilb are entangled? ( ) ( ) ( ) ( )

43 1.6. EXERCISES 37 (b) Which o these states o {0, 1} {0, 1} in Rel are entangled? {(0, 0), (0, 1)} {(0, 0), (0, 1), (1, 0)} {(0, 1), (1, 0)} {(0, 0), (0, 1), (1, 0), (1, 1)} Exercise We say that two joint states I u,v B are locally equivalent, written u v, i there exist invertible maps, B g B such that u g v (a) Show that is an equivalence relation. (b) Find all isomorphisms {0, 1} {0, 1} in Rel. (c) Write out all 16 states o the object {0, 1} {0, 1} in Rel. (d) Use your answer to (b) to group the states o (c) into locally equivalent amilies. How many amilies are there? Which o these are entangled? Exercise Recall equation (1.22) and its interpretation. (a) In FHilb, take I. Let be the Hadamard gate 1 ( 1 1 ) 2 1 1, let a be the 0 state ( 1 0 ), let b be the 0 eect ( 1 0 ), and let c be the 1 eect ( 0 1 ). Can the history b a occur? How about c a? (b) In Rel, take I. Let be the relation {0, 1} {0, 1} given by {(0, 0), (0, 1), (1, 0)}, let a be the state {0}, let b be the eect {1}, and let c be the eect {1}. Can the history b a occur? How about c a?

44 38 CHPTER 1. MONOIDL CTEGORIES Notes and urther reading (Symmetric) monoidal categories were introduced independently by Bénabou and Mac Lane in 1963 [11, 55]. Early developments centred around the problem o coherence, and were resolved by Mac Lane s Coherence Theorem 1.2. For a comprehensive treatment, see the textbooks [56, 14]. The graphical language dates back to 1971, when Penrose used it to abbreviate tensor contraction calculations [62]. It was ormalized or monoidal categories by Joyal and Street in 1991 [41], who later also introduced and generalized to braided monoidal categories [43]. For a modern survey, see [72]. The relevance o monoidal categories or quantum theory was emphasized originally by bramsky and Coecke [4, 16], and was also popularized by Baez [9] in the context o quantum ield theory and quantum gravity. It has since become a popular ormalism or work in quantum oundations. Our remarks about the dimensionality o the graphical calculus are a shadow o higher category theory, as irst hinted at by Grothendieck [34]. For a modern overview, see [51]. Monoidal categories are 2-categories with one object, braided monoidal categories are 3-categories with one object and one 1-cell, and symmetric monoidal categories are 4-categories with one object, one 1-cell and one 2-cell and n-categories have an n-dimensional graphical calculus; see [8].