Categories and Quantum Informatics: Monoidal categories

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1 Cateories and Quantum Inormatics: Monoidal cateories Chris Heunen Sprin 2018 A monoidal cateory is a cateory equipped with extra data, describin how objects and morphisms can be combined in parallel. This chapter introduces the theory o monoidal cateories, and shows how our example cateories Hilb, Set and Rel can be iven a monoidal structure. We also introduce a visual notation called the raphical calculus, which provides an intuitive and powerul way to work with them. 1.1 Monoidal structure Throuhout this book, we interpret objects o cateories as systems, and morphisms as processes. A monoidal cateory has additional structure allowin us to consider processes occurrin in parallel, as well as sequentially. In terms o our example cateories rom the introduction, one could interpret this in the ollowin ways: lettin independent physical systems evolve simultaneously; runnin computer alorithms in parallel; takin products or sums o alebraic or eometric structures; usin separate proos o P and Q to construct a proo o the conjunction (P and Q). It is perhaps surprisin that a nontrivial theory can be developed at all rom such simple intuition. But in act, some interestin eneral issues quickly arise. For example, let A, B and C be processes, and write or the parallel composition. Then what relationship should there be between the processes (A B) C and A (B C)? You miht say they should be equal, as they are dierent ways o expressin the same arranement o systems. But or many applications this is simply too stron: or example, i A, B and C are Hilbert spaces and is the usual tensor product o Hilbert spaces, these two composite Hilbert spaces are not exactly equal; they are only isomorphic. But we then have a new problem: what equations should that isomorphism satisy? The theory o monoidal cateories is ormulated to deal with these issues. Deinition 1.1. A monoidal cateory is a cateory C equipped with the ollowin data: a tensor product unctor : C C C; a unit object I Ob(C); an associator natural isomorphism (A B) C α A,B,C A (B C); a let unitor natural isomorphism I A λ A A; a riht unitor natural isomorphism A I ρ A A. 1

2 This data must satisy the trianle and pentaon equations, or all objects A, B, C and D: α A,I,B (A I) B A (I B) ρ A id B id A λ B (1.1) A B α A,B C,D ( A (B C) ) D A ( (B C) D ) α A,B,C id D ( (A B) C ) D id A α B,C,D A ( B (C D) ) (1.2) α A B,C,D α A,B,C D (A B) (C D) The naturality conditions or α, λ and ρ correspond to the ollowin equations: α A,B,C (A B) C A (B C) I A λ A A ρ A A I ( ) h ( h) I I (1.3) α A,B,C (A B ) C A (B C ) I B λ B B ρ B B I The tensor unit object I represents the trivial or empty system. This interpretation comes rom the unitor isomorphisms λ A and ρ A, which witness the act that the object A is just as ood as, or isomorphic to, the objects A I and I A. The trianle and pentaon equations each say that two particular ways o reoranizin a system are equal. Surprisinly, this implies that any two reoranizations are equal; this is the content o the Coherence Theorem. Theorem 1.2 (Coherence or monoidal cateories). Given the data o a monoidal cateory, i the pentaon and trianle equations hold, then any well-typed equation built rom α, λ, ρ and their inverses holds. In particular, the trianle and pentaon equation toether imply ρ I = λ I. To appreciate the power o the coherence theorem, try to show this yoursel. Coherence is the undamental motivatin idea o a monoidal cateory, and ives an answer to question we posed earlier in the chapter: the isomorphisms should satisy all possible well-typed equations. So while these morphisms are not trivial or example, they are not necessarily identity morphisms it doesn t matter how we apply them in any particular case. Our irst example o a monoidal structure is on the cateory Hilb. Deinition 1.3. The monoidal structure on the cateory Hilb, and also by restriction on FHilb, is deined in the ollowin way: 2

3 the tensor product : Hilb Hilb Hilb is the tensor product o Hilbert space; the unit object I is the one-dimensional Hilbert space C; associators (H J) K α H,J,K H (J K) are the unique linear maps satisyin (u v) w u (v w) or all u H, v J and w K; let unitors C H λ H H are the unique linear maps satisyin 1 u u or all u H; riht unitors H C ρ H H are the unique linear maps satisyin u 1 u or all u H. Althouh we call the unctor o a monoidal cateory the tensor product, that does not mean that we have to choose the actual tensor product o Hilbert spaces or our monoidal structure. There are other monoidal structures on the cateory that we could choose; a ood example is the direct sum o Hilbert spaces. However, the tensor product we have deined above has a special status, since it correctly describes the state space o a composite system in quantum theory. While Hilb is relevant or quantum computation, the monoidal cateory Set is an important settin or classical computation. Deinition 1.4. The monoidal structure on the cateory Set, and also by restriction on FSet, is deined as ollows or all a A, b B and c C: the tensor product is Cartesian product o sets, written, actin on unctions A B and C D as ( )(a, c) = ( (a), (c) ) ; the unit object is a chosen sinleton set { }; associators (A B) C α A,B,C A (B C) are the unctions iven by ( (a, b), c ) ( a, (b, c) ) ; let unitors I A λ A A are the unctions (, a) a; riht unitors A I ρ A A are the unctions (a, ) a. The Cartesian product in Set is a cateorical product. This is an example o a eneral phenomenon: i a cateory has products, then these can be used to ive a monoidal structure on the cateory. The same is true or coproducts, which in Set are iven by disjoint union. This hihlihts an important dierence between the standard tensor products on Hilb and Set: while the tensor product on Set comes rom a cateorical product, the tensor product on Hilb does not. We will discover many more dierences between Hilb and Set, which provide insiht into the dierences between quantum and classical inormation. There is a canonical monoidal structure on the cateory Rel. Deinition 1.5. The monoidal structure on the cateory Rel is deined in the ollowin way, or all a A, b B, c C and d D: the tensor product is Cartesian product o sets, written, actin on relations A R B and C S D by settin (a, c)(r S)(b, d) i and only i arb and csd; the unit object is a chosen sinleton set = { }; associators (A B) C α A,B,C A (B C) are the relations deined by ( (a, b), c ) ( a, (b, c) ) ; let unitors I A λ A A are the relations deined by (, a) a; riht unitors A I ρ A A are the relations deined by (a, ) a. The Cartesian product is not a cateorical product in Rel, so althouh this monoidal structure looks like that o Set, it is in act more similar to the structure on Hilb. 3

4 Example 1.6. I C is a monoidal cateory, then so is its opposite C op. The tensor unit I in C op is the same as that in C, whereas the tensor product A B in C op is iven by B A in C, the associators in C op are the inverses o those morphisms in C, and the let and riht unitors o C swap roles in C op. Monoidal cateories have an important property called the interchane law, which overns the interaction between the cateorical composition and tensor product. Theorem 1.7 (Interchane). Any morphisms A B, B C, D h E and E j F in a monoidal cateory satisy the interchane law: ( ) (j h) = ( j) ( h) (1.4) Proo. This holds because o properties o the cateory C C, and rom the act that : C C unctor: C is a ( ) (j h) (, j h) = ( (, j) (, h) ) (composition in C C) = ( (, j) ) ( (, h) ) (unctoriality o ) = ( j) ( h) Recall that the unctoriality property or a unctor F says that F ( ) = F () F (). 1.2 Graphical calculus A monoidal structure allows us to interpret multiple processes in our cateory takin place at the same time. For morphisms A B and C D, it thereore seems reasonable, at least inormally, to draw their tensor product A C B D like this: B A The idea is that and represent processes takin place at the same time on distinct systems. Inputs are drawn at the bottom, and outputs are drawn at the top; in this sense, time runs upwards. This extends the one-dimensional notation or cateories. Whereas the raphical calculus or ordinary cateories was onedimensional, or linear, the raphical calculus or monoidal cateories is two-dimensional or planar. The two dimensions correspond to the two ways to combine morphisms: by cateorical composition (vertically) or by tensor product (horizontally). One could imaine this notation bein a useul short-hand when workin with monoidal cateories. This is true, but in act a lot more can be said: the raphical calculus ives a sound and complete lanuae or monoidal cateories. The (identity on the) monoidal unit object I is drawn as the empty diaram: D C (1.5) (1.6) The let unitor I A λ A A, the riht unitor A I ρ A A and the associator (A B) C α A,B,C A (B C) 4

5 are also simply not depicted: A A A B C (1.7) λ A ρ A α A,B,C The coherence o α, λ and ρ is thereore important or the raphical calculus to unction: since there can only be a sinle morphism built rom their components o any iven type (see Section??), it doesn t matter that their raphical calculus encodes no inormation. Now consider the raphical representation o the interchane law (1.4): C F C F j j B E = B E (1.8) h h A D A D We use brackets to indicate how we are ormin the diarams on each side. Droppin the brackets, we see the interchane law is very natural; what seemed to be a mysterious alebraic identity becomes clear rom the raphical perspective. The point o the raphical calculus is that all the supericially complex aspects o the alebraic deinition o monoidal cateories the unit law, the associativity law, associators, let unitors, riht unitors, the trianle equation, the pentaon equation, the interchane law melt away, allowin us to make use o the theory o monoidal cateories in a direct way. These alebraic eatures are still there, but they are absorbed into the eometry o the plane, o which our species has an excellent intuitive understandin. The ollowin theorem is the ormal statement that connects the raphical calculus to the theory o monoidal cateories. Theorem 1.8 (Correctness o the raphical calculus or monoidal cateories). A well-typed equation between morphisms in a monoidal cateory ollows rom the axioms i and only i it holds in the raphical lanuae up to planar isotopy. Two diarams are planar isotopic when one can be deormed continuously into the other within some rectanular reion o the plane, with the input and output wires terminatin at the lower and upper boundaries o the rectanle, without introducin any intersections o the components. For this purpose, we assume that wires have zero width, and morphism boxes have zero size. 5

6 Example 1.9. Here are examples o isotopic and non-isotopic diarams: h iso = h not iso h As we have done here, we will oten allow the heihts o the diarams to chane, and allow input and output wires to slide horizontally alon their respective boundaries, althouh they must never chane order. The third diaram here is not isotopic to the irst two, since or the h box to move to the riht-hand side, it would have to pass throuh one o the wires, which is not allowed. The box cannot pass over or under the wire, since the diarams are conined to the plane that is what is meant by planar isotopy. You should imaine that the components o the diaram are trapped between two pieces o lass. The correctness theorem is really sayin two distinct thins: that the raphical calculus is sound, and that it is complete. To understand these concepts, let and be morphisms such that the equation = is well-typed, and consider the ollowin statements: P (, ): under the axioms o a monoidal cateory, = ; Q(, ): the raphical representations o and are planar isotopic. Soundness is the assertion that or all such and, P (, ) Q(, ). Completeness is the reverse assertion, that Q(, ) P (, ) or all such and. Provin soundness is straihtorward: there are only a inite number o axioms, and one just has to check that they are all valid in terms o planar isotopy o diarams. Completeness is much harder, and beyond the scope o this book: one must analyze the deinition o planar isotopy, and show that any planar isotopy can be built rom a small set o moves, each o which independently leave the value o the morphism in the monoidal cateory unchaned. Let s take a closer look at the condition that the equation = must be well-typed. Firstly, and must have the same source and the same taret. For example, let = id A B, and = ρ A id B. Then their types are A B A B and (A I) B A B. These have dierent source objects, and so the equation is not well-typed, even thouh their raphical representations are planar isotopic. Also, suppose that our cateory happened to satisy A B = (A I) B; then althouh and would have the same type, the equation = would still not be well-typed, since it would be makin use o this accidental equality. For a careul examination o the well-typed property. The notation iso = to denote isotopic diarams, whose interpretations as morphisms in a monoidal cateory are thereore equal, will be used throuhout this book, to indicate an application o the correctness property o the raphical calculus. 1.3 States and eects I a mathematical structure lives as an object o a cateory, and we want to learn somethin about its internal structure, we must ind a way to do it usin the morphisms o the cateory only. For example, consider a set A Ob(Set) with a chosen element a A: we can represent this with the unction { } A deined by a. This inspires the ollowin deinition, which ives us a eneralized cateorical notion o state. 6

7 Deinition In a monoidal cateory, a state o an object A is a morphism I also called points. A. States are sometimes Since the monoidal unit object represents the trivial system, a state I a way or the system A to be brouht into bein. A o a system can be thouht o as Example We now examine what the states are in our three example cateories: in Hilb, states o a Hilbert space H are linear unctions C by considerin the imae o 1 C; H, which correspond to elements o H in Set, states o a set A are unctions { } imae o ; A, which correspond to elements o A by considerin the in Rel, states o a set A are relations { } R A, which correspond to subsets o A by considerin all elements related to. Deinition A monoidal cateory is well-pointed i or all parallel pairs o morphisms A, B, we have = when a = a or all states I a A. A monoidal cateory is monoidally well-pointed i or all, parallel pairs o morphisms A 1 A n B, we have = when (a1 a n ) = (a 1 a n ) or all states I a1 A 1,..., I an A n. The idea is that in a well-pointed cateory, we can tell whether or not morphisms are equal just by seein how they aect states o their domain objects. In a monoidally well-pointed cateory, it is even enouh to consider product states to veriy equality o morphisms out o a compound object. The cateories Set, Rel, Vect, and Hilb are all monoidally well-pointed. For the latter two, this comes down to the act that i {d i } is a basis or H and {e j } is a basis or K, then {d i e j } is a basis or H K. To emphasize that states I a A have the empty picture (1.6) as their domain, we will draw them as trianles instead o boxes. A a (1.9) 1.4 Product states and entanled states For objects A and B o a monoidal cateory, a morphism I c it raphically in the ollowin way. A B c A B is a joint state o A and B. We depict (1.10) Deinition A joint state I c or I a A and I b B: A B is a product state when it is o the orm I λ 1 I A B A B I I a b A B = (1.11) c a b Deinition A joint state is entanled when it is not a product state. 7

8 Entanled states represent preparations o A B which cannot be decomposed as a preparation o A alonside a preparation o B. In this case, there is some essential connection between A and B which means that they cannot have been prepared independently. Example Joint states, product states, and entanled states look as ollows in our example cateories: in Hilb: joint states o H and K are elements o H K; product states are actorizable states; entanled states are elements o H K which cannot be actorized; in Set: joint states o A and B are elements o A B; product states are elements (a, b) A B comin rom a A and b B; entanled states don t exist; in Rel: joint states o A and B are subsets o A B; product states are subsets U A B such that, or some V A and W B, (v, w) U i and only i v V and w W ; entanled states are subsets o A B that are not o this orm. This hints at why entanlement can be diicult to understand intuitively: classically, in the processes encoded by the cateory Set, it cannot occur. However, i we allow nondeterministic behaviour as encoded by Rel, then an analoue o entanlement does appear. 1.5 Eects An eect represents a process by which a system is destroyed, or consumed. Deinition In a monoidal cateory, an eect or costate or an object A is a morphism A I. Given a diaram constructed usin the raphical calculus, we can interpret it as a history o events that have taken place. I the diaram contains an eect, this is interpreted as the assertion that a measurement was perormed, with the iven eect as the result. For example, an interestin diaram would be this one: A x (1.12) a This describes a history in which a state a is prepared, and then a process is perormed producin two systems, the irst o which is measured ivin outcome x. This does not imply that the eect x was the only possible outcome or the measurement; just that by drawin this diaram, we are only interested in the cases when the outcome x does occur. An eect can be thouht o as a postselection: we run our entire experiment repeatedly, only acceptin the result when we ind that our measurement had the speciied outcome. Overall our history is a morphism o type I A, which is a state o A. The postselection interpretation tells us how to prepare this state, iven the ability to perorm its components. 8

9 Example These statements are at a very eneral level. To say more, we must take account o the particular theory o processes described by the monoidal cateory in which we are workin. In quantum theory, as encoded by Hilb, we require a, and x to be partial isometries. The rules o quantum mechanics then dictate that the probability or this history to take place is iven by the square norm o the resultin 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 x they may be arbitrary relations o the correct types. The overall composite relation then describes the possible ways in which A can be prepared as a result o this history. I the overall composite is empty, that means this particular sequence o a state preparation, a dynamics step, and a measurement result cannot occur. Thins are very dierent in Set. The monoidal unit object is terminal in that cateory, meanin Set(A, I) has only a sinle element or any object A. So every object has a unique eect, and there is no nontrivial notion o measurement. 1.6 Braidin and symmetry In many theories o processes, i A and B are systems, the systems A B and B A can be considered essentially equivalent. While we would not expect them to be equal, we miht at least expect there to be some special process o type A B B A that switches the systems, and does nothin more. Developin these ideas ives rise to braided and symmetric monoidal cateories, which we now investiate. We irst consider braided monoidal cateories. Deinition A braided monoidal cateory is a monoidal cateory equipped with a natural isomorphism A B σ A,B B A (1.13) satisyin the ollowin hexaon equations: α 1 A,B,C A (B C) σ A,B C (B C) A α 1 B,C,A (A B) C σ A,B id C B (C A) id B σ A,C (1.14) (B A) C B (A C) α B,A,C σ A B,C (A B) C C (A B) α A,B,C α C,A,B A (B C) id A σ B,C A (C B) α 1 A,C,B (A C) B (C A) B σ A,C id B (1.15) 9

10 We include the braidin in the raphical notation like this: (1.16) A B σ A,B B A B A σ 1 A,B A B Invertibility then takes the ollowin raphical orm: = = (1.17) This captures part o the eometric behaviour o strins. Naturality o the braidin and the inverse braidin have the ollowin raphical representations: = = (1.18) The hexaon equations have the ollowin raphical representations: = = (1.19) Each o these equations has two strands close to each other on the let-hand side, to indicate that we are treatin them as a sinle composite object or the purposes o the braidin. We see that the hexaon equations are sayin somethin quite straihtorward: to braid with a tensor product o two strands is the same as braidin separately with one then the other. Since the strands o a braidin cross over each other, they are not lyin on the plane; they live in threedimensional space. So while cateories have a one-dimensional or linear notation, and monoidal cateories have a two-dimensional or planar raphical notation, braided monoidal cateories have a three-dimensional notation. Because o this, braided monoidal cateories have an important connection to three-dimensional quantum ield theory. Braided monoidal cateories have a sound and complete raphical calculus, as established by the ollowin theorem. The notion o isotopy it uses is now three-dimensional; that is, the diarams are assumed to lie in a cube, with input wires terminatin at the lower ace and output wires terminatin at the upper ace. This is also called spatial isotopy. Theorem 1.19 (Correctness o raphical calculus or braided monoidal cateories). A well-typed equation between morphisms in a braided monoidal cateory ollows rom the axioms i and only i it holds in the raphical lanuae up to spatial isotopy. 10

11 Given two isotopic diarams, it can be quite nontrivial to show they are equal usin the axioms o braided monoidal cateories directly. So as with ordinary monoidal cateories, the coherence theorem is quite powerul. For example, try to show that the ollowin two equations hold directly usin the axioms o a braided monoidal cateory: = (1.20) = (1.21) Equation (1.21) is called the Yan Baxter equation, which plays an important role in the mathematical theory o knots. We now ive some examples o braided monoidal cateories. For each o our main example cateories there is a naive notion o a swap process, which in each case ives a braided monoidal structure. Deinition Our example cateories Hilb, Set and Rel can all be equipped with a canonical braidin: in Hilb, H K σ H,K in Set, A B σ A,B in Rel, A B σ A,B K H is the unique linear map extendin a b b a or all a H and b K; B A is deined by (a, b) (b, a) or all a A and b B; B A is deined by (a, b) (b, a) or all a A and b B. In act these are all symmetric monoidal structures, which we explore in Section Symmetric monoidal cateories In our example cateories Hilb, Rel and Set, the braidins satisy an extra property that makes them very easy to work with. Deinition A braided monoidal cateory is symmetric when or all objects A and B, in which case we call σ the symmetry. Graphically, condition (1.22) has the ollowin representation. σ B,A σ A,B = id A B (1.22) = (1.23) Intuitively: the strins can pass throuh each other, and nontrivial knots cannot be ormed. 11

12 Lemma In a symmetric monoidal cateory σ A,B = σ 1 B,A, with the ollowin raphical representation: = (1.24) Proo. Combine (1.17) and (1.23). A symmetric monoidal cateory thereore makes no distinction between over- and under-crossins, and so we simpliy our raphical notation, drawin (1.25) or the sinle type o crossin. The raphical calculus with the extension o braidin or symmetry is still sound: i the two diarams o morphisms can be deormed into one another, then the two morphisms are equal. Suppose we imaine our diarams as curves embedded in our-dimensional space. Then we can smoothly deorm one crossin into the other, in the manner o equation (1.24), by makin use o the extra dimension. In this sense, symmetric monoidal cateories have a our-dimensional raphical notation. The ollowin correctness theorem thereore uses the our-dimensional version o isotopy. Theorem 1.23 (Correctness o the raphical calculus or symmetric monoidal cateories). A well-typed equation between morphisms in a symmetric monoidal cateory ollows rom the axioms i and only i it holds in the raphical lanuae up to our-dimensional isotopy. 12