Categorical. Operational Physics

Transcription

1 ategorical arxiv: v1 [quant-ph] 1 Feb 2019 Operational Physics Sean Tull St atherine s ollege University o Oxord thesis submitted or the degree o Doctor o Philosophy Trinity 2018

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3 For Lion.

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5 cknowledgements Firstly, I wish to thank my supervisors ob oecke and hris Heunen. To ob I am grateul or irst suggesting the goal o a categorical quantum reconstruction, which motivated this entire project, or his guidance in the academic world, and or ostering the kind o lively atmosphere which leads to invitations to arbados with two days notice. hris deserves equal thanks as my main teacher o the practice o research, and or all the time and excellent advice he has given me, remaining a constantly available source o eedback and collaboration even ater moving to the distant town o Edinburgh. Next I wish to thank my examiners Paolo Perinotti and Dan Marsden or an enjoyable viva and or helpul eedback which has improved the presentation o this thesis. I thank art Jacobs and leks Kissinger or immediately making me eel at home during two visits to their group in Nijmegen which were very inluential on this work, and everyone there or enlightening discussions, notably as and ram Westerbaan and John van de Wetering. Extra thanks goes to art or suggesting the copowers in hapter 2, and to Kenta ho or the collaborative work underpinning hapter 3. Next I extend my gratitude to Paolo, once again, and Mauro D riano, who warmly welcomed me on an interesting visit to Pavia University and whose work with Giulio hiribella motivates much o this thesis. I thank Marino Gran or hosting several enjoyable and productive visits to ULouvain discussing purer categorical topics. I also thank my other collaborators on work not in this thesis but which has enriched my studies; ob, hris, John Selby, leks, as, and Pau Enrique Moliner. In Oxord I thank all the colleagues who ve made this time so enjoyable and inluenced my thinking; including Steano Gogioso, Sam Staton or suggesting connections with multicategories, Robin Lorenz and visitor Johannes Kleiner or distracting me with talks about consciousness, and hristoph Dorn or encouraging me to see a bit o college lie. I am grateul also to the EPSR or all o their inancial support. Lastly, I wish to thank my riends and amily or all o their help through the years; particularly my parents, whose support made this possible, and Jon or his help in my inal year. Special thanks goes to Jon and arol Field or so oten putting up with me in their home, where some o the main results o this thesis were reached. Most importantly, I thank Lottie or being with me every step o the way, and bringing so much joy to these years.

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7 bstract Many insights into the quantum world can be ound by studying it rom amongst more general operational theories o physics. In this thesis, we develop an approach to the study o such theories purely in terms o the behaviour o their processes, as described mathematically through the language o category theory. This extends a ramework or quantum processes known as categorical quantum mechanics (QM) due to bramsky and oecke. We irst consider categorical rameworks or operational theories. We introduce a notion o such theory, based on those o hiribella, D riano and Perinotti (DP), but more general than the probabilistic ones typically considered. We establish a correspondence between these and what we call operational categories, using eatures introduced by Jacobs et al. in eectus theory, an area o categorical logic to which we provide an operational interpretation. We then see how to pass to a broader category o super-causal processes, allowing or the powerul diagrammatic eatures o QM. Next we study operational theories themselves. We survey numerous principles that a theory may satisy, treating them in a basic diagrammatic setting, and relating notions rom probabilistic theories, QM and eectus theory. Particular ocus is paid to the quantum-like eatures o puriications and superpositions. We provide a new description o superpositions in the category o pure quantum processes, using this to give an abstract construction o the more well-behaved category o Hilbert spaces and linear maps. Finally, we reconstruct inite-dimensional quantum theory itsel. More broadly, we give a recipe or recovering a class o generalised quantum theories, beore instantiating it with operational principles inspired by an earlier reconstruction due to DP. This reconstruction is ully categorical, not requiring the usual technical assumptions o probabilistic theories. Specialising to such theories recovers both standard quantum theory and that over real Hilbert spaces.

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9 ontents Introduction 1 1 ategories o Processes Examples Operational Theories and ategories Operational Theories Operational ategories Further xioms or Theories ategories o Tests Eectuses From Sub-causal to Super-causal Processes ddition and iproducts Totalisation ompact and Dagger ategories Principles or Operational Theories Minimal Dilations Kernels ombining Minimal Dilations and Kernels Perect Distinguishability and Ideal ompressions Puriication Pure Exclusion Puriication and Daggers Superpositions and Phases Phased oproducts From Phased oproducts to oproducts Phases in Monoidal ategories Phased iproducts Phases in ompact ategories Phases in Dagger ategories Reconstructing Quantum Theories Recipe or Reconstructions The Operational Principles Deriving Superpositions i

10 ii ontents 6.4 Properties o Pure Morphisms Reconstruction Further Reconstructions Outlook 153 Index o ategories 158 Index o Notation 161 Index o Subjects 162 ibliography 162 ii

11 Introduction The state o contemporary physics is one o contradiction. Our deepest insights into nature come rom quantum mechanics, yet even a century ater its conception the underlying reality this theory describes remains deeply mysterious, with debates over its proper interpretation continuing to this day. t the same time, quantum theory provides us with experimental predictions o unprecedented accuracy, and in more recent years it has emerged that quantum systems can be incredibly useul, allowing one to quickly perorm computations that may take vastly longer using classical computers. Together, these acts have encouraged many to take an operationalist perspective on physical theories. In this approach, one studies a theory in terms o the operations it allows one to perorm through physical experiments, rather than any underlying reality that it may describe. Though this could be seen as a denial that any such reality exists, the operational approach may simply be taken as a practical one, allowing physics to progress in the absence o any such clear underlying picture o the world. entral to the operational perspective is the notion o a process between two physical systems. Examples include the preparation o a system into a particular state, the evolution o a system over time, and the perorming o measurements. The mathematical language o such composable processes is category theory, a powerul and very general one which can also be used to study connections between dierent ields and ideas, and even as a oundation or mathematics [ML78]. Over the past decade and a hal, the categorical perspective has led to a new approach to the study o physical theories purely in terms o their process-theoretic properties. ategories provide an intuitive calculus or reasoning about these processes using diagrams [Sel11], and lie at the heart o new connections emerging between the oundations o physics, quantum inormation, mathematics and computer science [D95, 04, S10, T11, P11]. The greatest successes o the categorical method in physics so ar have been in the study o quantum theory itsel, and particularly its pure processes as captured by the now well-understood category o Hilbert spaces [Heu09], including the development o a high-level diagrammatic ormalisation o quantum computation [D11, K14]. However, in more recent years, categorical methods relevant to the study o more general theories, including classical physics, have begun to emerge [Jac15, JWW16]. The goal o this thesis is to develop such a categorical approach to the study o operational theories o physics. 1

12 2 Introduction ategories o processes Let us now be a bit more precise about the kinds o theories we will be considering. The basic ingredients are physical systems and processes between them. We depict a process which takes us rom a system o type to one o type as a box Operationally, we might wish to think o a process as a piece o experimental apparatus in our laboratory. Like these, processes can be plugged together and placed alongside each other, allowing us to orm circuit diagrams like: g e ρ It is well-known that such a speciication o processes corresponds simply to a symmetric monoidal category, whose objects are systems and morphisms are the processes. The use o these diagrammatic methods in physics was pioneered by bramsky and oecke [04] in a ield o research now known as categorical quantum mechanics (QM). Since categories are very general, more we will be required in order or us to view a given category as being o an operational nature. particular characteristic o the operational perspective is that, given any system, we should always have some process which simply discards it, which we may depict as Such symmetric monoidal categories with discarding provide a very general ramework or reasoning about operational procedures, and will be the basic setting throughout this work. Examples include quantum theory, in which morphisms are given by so-called completely positive maps between Hilbert spaces, as well as classical probabilistic or possibilistic physics, and even more exotic theories such as Spekkens toy model [Spe07, E12]. In hapter 1 we introduce this categorical ramework more ormally and provide numerous such examples. Tests and operational theories long with the structure o processes, there are urther eatures which are typically included in notions o operational theories. t a basic level, the only way in which we may actually interact with systems in such a theory is through experimental tests or measurements. Such a procedure takes a given system and returns one o a range o possible outcomes, which the experimenter then records, perhaps by 2

13 Introduction 3 reading the value o a pointer on some device: i 0 n Each possible outcome corresponds to the occurrence o a particular physical process or event, so that a test such as the above is given simply by an indexed collection o events rom to. Imagining that an experimenter should be ree to choose which test to perorm next based on outcomes o earlier experiments, however, quickly leads one to realise that tests should more generally take the orm i i allowing or varying output systems (though this is not always standard, see e.g. [DP10, p.12-13]). Tests should satisy some basic rules relecting our interpretation; or example that like processes we should be allowed to place them side-by-side to orm new ones. Moreover, given any test, we may also imagine an experimenter choosing to not care which out o two (or more) o its events, say and g, occur, thus merging them into a new coarse-grained event denoted n i1 Ŕ g One may then deine an operational theory to be a collection o events, given by a symmetric monoidal category with discarding, along with a speciication o tests and such a partially deined addition, satisying suitable axioms. Examples include quantum theory, in which tests are given by so-called quantum instruments [N10], as well as classical and possibilistic theories. Now, the typical approach in physics is to only consider probabilistic such theories, which come with extra structure explicitly relating tests to probabilistic experiments, along with technical assumptions ensuring that the processes o any given type generate a inite-dimensional real vector space [DP10, ar07]. In this thesis we will not use these assumptions, showing that operational theories may in act be studied in a ully categorical manner, much in the spirit o QM. s a irst step, it is useul to know that the ull structure o an operational theory may in act be studied in terms o the properties o a single category. This may be done by considering its partial tests, i.e. subsets o tests, which orm a category with discarding in a straightorward manner. In doing so we gain the ability to represent the eatures o tests, their outcomes and coarse-graining all using categorical eatures called coproducts +. In particular, any (partial) test may now be represented as a single morphism o the orm n 3

14 4 Introduction onversely, any suitable category with coproducts in act deines a whole operational theory in this way. The use o these eatures comes rom a categorical ormalism or classical, probabilistic and quantum computation known as eectus theory [JWW16], which gains a new operational interpretation rom this perspective. The two categorical ormalisms we have mentioned can be compared in terms o their main eatures as ollows. Main Feature Description Formalism ategorical Logical Operational QM nd Parallel Processes Eectus Theory + Or Tests In hapter 2 we properly deine operational theories and study their correspondence with certain categories with coproducts which we call operational categories, along with connections to eectus theory. eyond sub-causal processes From irst principles we have seen how a physical theory may be described by a category coming with a partial addition on its morphisms. The act that is typically only partially deined relates to the assumption that every morphism belongs to a test, and so is sub-causal meaning that Ŕ e or some process e. For example in quantum theory the only maps with a direct interpretation, satisying the above, are those which are trace non-increasing. However, it is oten much easier to instead work with a total addition operation + g on morphisms. To do so, we must consider more general super-causal processes. In hapter 3 we present a general construction, which given any category with a suitable partial addition operation, constructs a new one T() with a total addition, its totalisation, within which sits as the sub-category o sub-causal morphisms. This construction can be seen to connect the eectus and QM ormalisms, which typically study sub-causal and super-causal processes respectively. Working with super-causal processes also allows us to consider powerul extra diagrammatic eatures which are central to the QM approach; most notably that our category is dagger-compact [04, Sel07]. In diagrams, this means that we may lip pictures upside-down, made visible through the use o pointed boxes, and also bend wires to exchange inputs and outputs o our morphisms, and so produce diagrams like k g h In hapter 3 we introduce and study the T() construction, beore recalling these extra diagrammatic eatures. 4

15 Introduction 5 Principles or operational theories major beneit o the study o generalised physical theories is the ability they provide to isolate particular physical principles, and examine their consequences. Several surprising aspects o the quantum world, such as the amous no-cloning theorem, have been ound to in act hold in all non-classical probabilistic theories [LW07], while others such as quantum teleportation have been ound to be more special [LW12]. For example, a principle which has been shown to lead to many quantum-like eatures in the setting o probabilistic theories is the ability to write every process in terms o those which are maximally inormative in the ollowing sense[dp10]. We call a morphism pure when any dilation o it is trivial: g g ρ or some ρ with ρ and we say that puriication holds when every morphism has a dilation which is pure. Quantum theory has particularly well-behaved puriications given by the Stinespring dilation o any completely positive map. In contrast, the ollowing principle is much more general, holding in both the quantum and classical settings. Firstly, many categories come with zero morphisms, special morphisms 0: with which every morphism composes to give 0. Such a category then has kernels when every morphism comes with another ker(), satisying g 0 (!h) g ker() h The existence o certain such kernels in act captures the essential structure o subspaces ound in classical and quantum theory, as historically treated in the ield o quantum logic [HJ10]. Many principles, such as puriication, have typically only been studied in the context o probabilistic theories, while others such as kernels only appear in speciic categorical settings. In hapter 4 we study a range o principles or operational theories, seeing that they may in act be treated in the very general setting o symmetric monoidal categories with discarding. In doing so we ind close relations between eatures that have arisen in the rameworks o probabilistic theories, categorical quantum mechanics and eectus theory. Superpositions and phases In order to move our attention away rom general theories and towards quantum theory itsel, we will require an account o arguably its most characteristic eature; the ability to orm superpositions o pure processes. The most amous example is o course Schrödinger s cat, which exists in a superposition o the pure states live + Dead 5

16 6 Introduction In act there is already a well-known categorical description o superpositions; abstractly, they are given by an addition operation on morphisms in the category o Hilbert spaces and linear maps. In turn this arises rom the existence o biproducts H K in this category, which are given concretely by the direct sum o Hilbert spaces [Sel07]. Indeed states o such a direct sum are precisely superpositions o states o H with those o K. However, there is a problem. Pure quantum processes are not simply given by linear maps between Hilbert spaces, since physically we must identiy any two such maps whenever they are equal up to some global phase e iθ, or real-valued θ. In act, in the category o pure quantum processes H K is no longer a biproduct. Nonetheless, it has similar properties which we are able to capture using the new notion o a phased biproduct, or more general phased coproduct + in a category. These resemble coproducts, but come with extra isomorphisms called phases. In quantum theory their presence relects the act that we may equally have replaced the state o Schrödinger s cat with any one o the orm live + e i θ Dead In hapter 5 we introduce and study phased coproducts, showing that rom any suitable category with them we may construct a new one GP() with coproducts rom which it arises by quotienting out some global phases as above. In particular this lets us recover the category o Hilbert spaces and linear maps rom that o pure quantum processes. Reconstructing quantum theory The primary motivation or the study o operational theories has always been to ind new understandings o the quantum world. Just a short time ater giving the irst precise ormulation o quantum theory in the language o Hilbert spaces [vn55], von Neumann himsel expressed his dissatisaction with this ormalism [Réd96], and since then there have been many attempts to reconstruct the ull apparatus o the theory rom instead more basic operational statements about experimental procedures. Early results were given in terms o quantum logic [vn75, Pir76, Sol95], and various versions o the convex probabilities ramework pursued by Mackey, Ludwig and many others [Mac63, Lud85, Gud99, FR81, DL70]. Unortunately, each o these results relied on some technicalities which could not be said to be ully operational. The birth o quantum inormation led to a renewed interest in these questions and, ater a proposal by Fuchs [Fuc02], a goal to understand quantum theory in terms o inormation-theoretic principles. The irst orm o such a reconstruction o inite-dimensional quantum theory was provided by Hardy [Har01], and the irst entirely operational reconstruction by hiribella, D riano and Perinotti [DP11], using puriication as its primary principle. long with these other such reconstructions have been presented in various rameworks [H03, Wil09, D + 10, Har11, FS11, MM11, Wil17b, Höh17, SS18, vdw18]. However, these reconstructions all typically rely on the standard technical assumptions o probabilistic theories. We may wonder whether these eatures are 6

17 Introduction 7 integral to the process o recovering quantum theory, or whether instead a purely process-theoretic reconstruction is possible. In hapter 6 we provide such a categorical reconstruction o quantum theory. We show that any suitable category with discarding which is non-trivial and: is dagger-compact; has essentially unique puriications; has kernels; and whose scalars satisy a basic boundedness property is in act equivalent to that o a generalised quantum theory Quant S over a certain ring S. When our scalars have an extra eature - the presence o square roots - we ind that S resembles either the real or complex numbers. Specialising to probabilistic theories we then immediately obtain either standard quantum theory or more unusually that over real Hilbert spaces. Recovering quantum theory in this manner provides us with a new elementary axiomatization o the theory which will hopeully be o use in the ormalisation o quantum computation, thanks to the many established uses o categories rom across computer science[t11]. More speculatively, it suggests that uture theories o physics may be ormulated in a manner which takes processes as their most undamental ingredients. Prerequisites Throughout we will assume a very basic knowledge o category theory, though we aim to introduce all key deinitions or our purposes, including simple notions such as coproducts. For later reerence, some standard ones we will use are as ollows. In any category a morphism : is monic when g h g h, epic when g h g h, and an isomorphism when there exists a morphism 1 with 1 id and 1 id. The appropriate notion o mapping F: D between categories is that o a unctor, and between these is that o a natural transormation. pair o unctors F : D and G: D orm an equivalence o categories D when there are natural isomorphisms G F id and F G id D, and an isomorphism when these are strict equalities. ssuming choice, an equivalence may also be given simply by a unctor F : D which is is ull (every g: F() F() has g F() or some : ), aithul (F() F(g) g), and has that every object o D is isomorphic to one o the orm F(). y an embedding we will simply mean a aithul unctor. Occasionally we will also mention the concept o an adjunction between categories. The standard text on category theory is [ML78], while riendlier introductions are given by [T11, Lei14] and the physicist-targeted [P11]. Statement o originality ll work here is my own, unless otherwise stated. The results o Section 3.2 are in collaboration with Kenta ho. This thesis is based on the papers [Tul16], [Tul18b], [Tul18a] and new material. During my DPhil I also co-authored the articles [HT15, KTW17, Tul17, EMHT18, ST18, GHT18]. 7

18 8 Introduction 8

19 hapter 1 ategories o Processes In the process-theoretic approach to physics, we imagine a physical theory simply as a speciication o certain systems and processes that may occur between them. general process may be depicted and thought o as a physical occurrence which transorms a system o type into one o type. Given another process taking as input the system we should be able to compose them to orm a new process g which we typically interpret as occurs, and then g occurs.the ormal structure capturing this notion o composable processes is the ollowing. Recall that a category consists o: a collection o objects,,...; or each pair o objects, a collection (,) o morphisms : ; along with a rule or composing any pair o morphisms :, g: to give a morphism g :. Some basic axioms are also satisied; composition is associative, with (h g) h (g ), and every object comes with an identity morphism id : satisying id id or all :. long with the notation :, morphisms may be drawn just like our processes above, with identities and composition depicted id g g 9

20 10 hapter 1. ategories o Processes so that the identity and associativity rules become trivial diagrammatically, e.g. or associativity we have D D D h g h g h g When interpreting a category physically, it is natural to assume we also have a spatial composition,,,g g allowing us to place objects (systems) and morphisms (processes) side-by-side in diagrams: D g D g We also oten wish to consider processes with no input. This is expressed by having some object I interpreted as nothing, and depicted by the empty diagram: I id I I s is well-known, these eatures are captured by the ollowing extra structure on a category. Recall that a monoidal category (, ) is a category together with a unctor : ; a distinguished object I called the unit object; natural coherence isomorphisms ( ) α,, ( ) I I λ ρ satisying some equations [P11]. The diagrammatic notation above in act orms a precise graphical calculus or reasoning about monoidal categories [Sel11], allowing one in practice to avoid the technicalities o the coherence isomorphisms, and making many acts about monoidal categories immediately apparent. In any monoidal category, we call morphismsρ: I, e: I and s: I I states, eects and scalars respectively. Since (the identity on) I is given by an empty picture, these are respectively depicted as: ρ e s 10

21 11 The scalars s: I I in any monoidal category orm a commutative monoid under composition. This is surprising rom the ormal deinition o a monoidal category, but immediate rom the graphical calculus since we have: s r s r r s They also allow us to deine a scalar multiplication s on morphisms by s : s We may have alternatively chosen to multiply by scalars on the other side. However, in categories arising rom physical theories the order in which we compose via is typically unimportant, thanks to the ollowing extra structure. Recall that a symmetric monoidal category is one coming with a natural swap isomorphism σ, : satisying σ, σ, id, along with some coherence equations. We depict σ by crossing wires, so that naturality and this equation become: D D g g ategories with discarding In this work our ocus will be on categories with an interpretation as operational processes one may perorm within some domain o physics; such categories have also been called process theories [K14, Sel17]. distinguishing eature o this operational setting is the ability that any agent should have to simply discard or ignore a sub-system which is no longer o interest. This leads to the ollowing central notion o this thesis. Deinition 1.1. category with discarding is a category with a distinguished object I and a chosen morphism : I or each object, with I id I. monoidal category with discarding isoneorwhichismonoidal, with I being the monoidal unit, and such that or all objects,. The presence o discarding relects the perspective o an experimenter who may choose to only examine a smaller part o a larger process or system, as opposed to that o the underlying physics o the world which is typically taken to be reversible 11

22 12 hapter 1. ategories o Processes and so lack any such notion o discarding a system. We capture this idea o restricting to smaller parts o processesby saying that amorphism is a marginal o another morphism g when g and in this case we reer to g as a dilation o. The existence o a unique way to discard a system has also been ound to be closely related to notions o causality in a physical theory [DP10, p. 10] [L13, oe14], leading to the ollowing deinition. Deinition 1.2. [K15] In any category with discarding, a morphism : is called causal when it satisies Intuitively, i is a causal process it should have no inluence on earlier processesandsomakenodierencewhetherweirstdiscardoursystemorirstperorm and then discard its output. Lemma 1.3. Let be a (symmetric) monoidal category with discarding. Then all coherence isomorphisms α,λ,ρ, σ are causal, and the collection o causal morphisms orms a monoidal subcategory caus. Proo. learly all identities arecausal, andi,g arethensois g. Thecoherence isomorphisms ρ are all causal by naturality since ρ ρ I I Simple naturality argument show that the α,, and λ are all causal also. Finally, whenever : and g: D are causal then so is g, since: D g D g 1.1 Examples Let s now meet our main examples o symmetric monoidal categories both with and without discarding. 12

23 1.1. Examples 13 Deterministic classical physics 1. There is a category Set whose objects are sets,,... and morphisms are unctions :. This orms the causal subcategory o the symmetric monoidal category with discarding PFun whose morphisms are now partial unctions : between sets. The monoidal structure is given by the artesian product o sets and (partial) unctions, with the unit object being the singleton set I 1 { }. In this category the scalars may be seen as simply 0 and 1. Eects on an object are ound to correspond to subsets, while a state o is either empty or corresponds to a unique element a. Discarding is given by the unique unction : { }, so that a morphism is causal precisely when it is total, i.e. belongs to Set. lgebraic examples 2. ny commutative monoid (M, ) orms a symmetric monoidal category with one object in which morphisms are elements m M, with and being multiplication in M. Here every morphism is a scalar. 3. Let S be a semi-ring(a ring without subtraction ) which is commutative. There is a symmetric monoidal category Mat S whose objects are natural numbers n N and morphisms M: n m are m n matrices M i,j with elements in S. Such a matrix composes with another N: m k by standard matrix multiplication m (M N) i,k N j,k M i,j using multiplication and addition in the semi-ring S. The identity morphism on n is the n n matrix with 1 as each diagonal entry and 0 elsewhere. The monoidal product is given on objects by n m n m and on morphisms by the usual Kronecker product o matrices M N j1 a 11 N... a 1m N..... a n1 N... a nm N with I 1. The scalars in Mat S correspond to elements s S, while states and eects on n are n-tuples o elements o S, seen as column and row vectors respectively. Mat S has a choice o discarding given by n (1,...1): n 1, so that a matrix M is causal whenever each o its columns sum to 1. lassical probability theory 4. In the category lass the objects are sets and morphisms : are unctions sending each element a to a inite distribution over elements o with values in the positive real numbers R + : {r R r 0}. That is, they are unctions : R + or which (a,b) is non-zero only or only initely many values o, or each a. 13

24 14 hapter 1. ategories o Processes lternatively, we may view such morphisms as matrices, in which each column has initely many non-zero entries. The composition o : and g: is then that o matrices (g )(a)(c) b (a)(b) g(b)(c) This category is symmetric monoidal with I { }, and g deined as or the Kronecker product o matrices. The scalars here are given by the unnormalised probabilities R +. lass has discarding given by the unique map : { } with (a)( ) 1 or all a. Then a morphism is causal precisely when it sends each element a to a probability distribution, i.e. or all a we have (a)(b) 1 b In particular, causal states o an object are simply inite probability distributions over. More broadly, at an operational level we are oten interested in the sub-category lass p o morphisms : which send each element to a inite sub-distribution, i.e. or all a (a)(b) 1 b In lass p the scalars are then probabilities p [0,1], and an eect on an object simply assigns a probability e(a) to each element a. bstractly we may describe lass and lass p as Kleisli categories, o the R + -multiset and sub-distribution monad respectively [JWW16]. More generally, or continuous probability we can consider the Kleisli category Kl(G) o the Giry monad G on measure spaces [Jac13, Jac15]. 5. Restricting the above example to inite sets is equivalent to considering the category Flass : Mat R +, a special case o Example 3. The scalars here are given by R +, and causal morphisms are precisely (transposed) Stochastic matrices. Quantum theory 6. In the symmetric monoidal category Hilb objects are complex Hilbert spaces H,K... and morphisms are bounded linear maps : H K. The monoidal structure is given by the usual tensor product H K o Hilbert spaces, with unit object I. Then states ω o an object H correspond to elements ψ H by taking ψ ω(1), and so by taking adjoints so do eects. In particular the scalars are given by. We write FHilb or the ull subcategory given by restricting to inite-dimensional Hilbert spaces. oth categories can be seen to describe pure quantum theory, which thanks to the no-deleting theorem [P00] comes with no canonical choice o discarding. We may extend this example to include discarding and so describe more general quantum operations as ollows. 14

25 1.1. Examples In the symmetric monoidal category Quant, objects are inite-dimensional complex Hilbert spaces and morphisms H K are completely positive linear maps : (H) (K) between their spaces o operators. The monoidal structure is the usual one or such maps, inherited rom that o Hilbert spaces, again with I. Scalars now correspond to elements r R +. y Gleason s Theorem, states and eects on an object H now correspond to unnormalised density matrices ρ (H). This category has a canonical choice o discarding with H being the map sending each a (H) to its trace Tr(a). Then a morphism is causal whenever it is trace-preserving as a completely positive map, and causal states are simply density matrices in the usual sense. From an operational perspective we are oten interested in the subcategory Quant sub otracenon-increasing completely positivemaps, inwhichthescalars are probabilities p [0,1]. There is a unctor FHilb Quant which sends each linear map : H K to the induced Kraus map : ( ) : (H) (K) ny two linear maps,g induce the same such map whenever they are equal up to global phase, i.e. when e iθ g or some θ [0,2π). Hence the subcategory o all such Kraus maps is equivalent to the category FHilb o equivalence classes [] o morphisms in FHilb under equality up to global phase. More broadly we deine Hilb to be the category o equivalence classes [] o maps in Hilb up to global phase, in the same way. 8. Extending our previous example to ininite dimensions, and uniying it with our classical examples, we may consider the category Star op o unital complex *-algebras, where morphisms are completely positive linear maps :. Note that we work in the opposite category, with maps going the other way to morphisms. There are several dierent tensors available or (ininite-dimensional) operator algebras; we will take as the so-called minimal tensor product o *-algebras. Here I, so that scalars are given by elements o R +. States on an object correspond to those ω: on the algebra in the usual sense, while eects are positive elements e. Discarding is given by the unique completely positive map sending 1 to 1. Then a morphism is causal whenever its corresponding completely positive map : is unital, with (1 ) 1. More generally the maps with a direct operational interpretation are those which are sub-unital, with (1 ) 1, orming the subcategory Star op su. When working in inite dimensions one oten simply takes morphisms to go in the same direction as maps; we write FStar or the category o initedimensional *-algebras with morphisms being completely positive maps :. This is symmetric monoidal just as or Star op. Every inite-dimensional *-algebra comes with a trace, so that :. here is given by a Tr(a). There is an embedding FStar Star op sending trace non-increasing maps to sub-unital ones. 15

26 16 hapter 1. ategories o Processes Star op contains a version o classical probabilistic theory given by restricting to the ull subcategory o all commutative *-algebras, with Flass equivalent to the respective subcategory o FStar. To model quantum theory we can alternatively restrict to those algebras given by the bounded operators (H) o some Hilbert space H. In particular this gives an embedding Quant FStar. 9. particularly well-behaved class o *-algebras are those which are von Neumann algebras. We write vn op or the (opposite o) the subcategory o Star op given by all von Neumann algebras and normal completely positive maps between them, as studied in depth in [JWW16]. We are also oten interested in its subcategory vn op su o sub-unital morphisms. Our main examples o categories with discarding so ar are either deterministic, with scalars {0,1}, or more generally probabilistic, with scalars belonging to R +. It is common in the oundations o physics to work only with such general probabilistic theories, and to make some extra assumptions. The irst, tomography, ensures that morphisms are determined entirely by the probabilities they produce: e ω e g ω ω,e g This in turn ensures that maps o any given type generate a real vector space (up to some size issues) [hi14a]. Secondly, tomography is assumed to be inite, meaning that this space is inite-dimensional. In this thesis we will not make any o these assumptions, aiming to work in a purely process-theoretic manner. In particular this allows us to consider more general theories whose scalars are not given by probabilities, such as the ollowing. Possibilistic examples 10. Thereis acategory Rel whoseobjects aresets andwhosemorphismsr: are relations R. omposition o R: and S: is given by S R (a,c) b such that (a,b) R and (b,c) S Here is given by the artesian product, with I being the singleton set { }. The scalars are the ooleans : {, }, with states and eects on an object each corresponding to subsets o. There is a canonical choice o discarding given by the relation : { } relating every a with. Then a relation R: is causal when it relates every element o to some element o. 11. The previous example can be greatly generalised. For any category which is regular [G04] we may similarly deine a symmetric monoidal category with discarding Rel() o internal relations in in the same way. For some examples, Rel is the special case where Set. Taking to be the category Vec k o vector spaces over a ield k gives the category Rel(Vec k ) o 16

27 1.1. Examples 17 linear relations over k, i.e. subspaces R V W. Setting instead to be the category Grp o groups leads to relations which are subgroups R G H. The author explored Rel() with hris Heunen in [HT15], and with Marino Gran also in [GHT18], applying its diagrammatic eatures to topics in categorical algebra. More generally still, any such category Rel() is a special case o a bicategory o relations in the sense o arboni and Walters [W87]. 12. physically interesting possibilistic example somewhere in-between Rel and quantum theory is provided by Spekkens toy model [Spe07]. Spekkens originally presented the theory in terms o its states, which are subsets o sets o the orm IV n, where IV {1,2,3,4}, obeying the so-called knowledge balance principle. The theory was then given an inductive categorical deinition in [E12, Edw09]. We write Spek or the smallest symmetric monoidal subcategory o Rel closed under,, identities, swap maps and relational converse, and containing the objects I { } and IV {1,2,3,4}, all permutations IV IV, and the relations IV :: 1,3 IV IV IV :: 1 (1,1),(2,2) 2 (1,2),(2,1) 3 (3,3),(4,4) 4 (3,4),(4,3) Spek contains many similar eatures to FHilb, closely resembling stabilizer quantum mechanics [Pus12, D16]. In the original paper [Spe07] (which uses only unctional relations as morphisms) quantum eatures such as steering and teleportation are studied in the theory. It may be extended to a category with discarding MSpek [E12], deined to be the smallest monoidal subcategory o Rel closed under relational converse and containing Spek as well as the discarding morphisms rom Rel. Morphisms o categories with discarding t times we will also consider mappings between categories. y a morphism F: (, ) (D, ) o categories with discarding we mean a unctor F : D which preserves discarding in that F(I) is an isomorphism and F( ) is causal or all objects. When and D are (symmetric) monoidal with discarding we moreover require F to be a strong(symmetric) monoidal unctor and that its structure isomorphism I F(I) is causal; rom this it ollows that those isomorphisms F() F() F( ) will be causal also, similarly to Lemma 1.3. In either case a morphism F is an equivalence D when it is ull and aithul, and every object o D is causally isomorphic to one o the orm F(). 17

28 18 hapter 1. ategories o Processes 18

29 hapter 2 Operational Theories and ategories side rom the categorical structure o processes, there are other eatures which are typically included as basic components o an operational theory o physics. Most notably, such a theory should also describe multiple-outcome experimental procedures or tests which we may perorm on our systems, along with the outcome data obtained rom these experiments. ramework combining these eatures with the categorical approach is ound in the notion o an operational-probabilistic theory due to hiribella, D riano and Perinotti [DP10]. Such a theory is given by a (strict) symmetric monoidal category o processes, along with additional structure speciying which processes orm admissible tests, modelling the use o experimental outcome data, and allowing one to assign probabilities to these outcomes. In this chapter, we introduce a similar general notion o such an operational theory o physics. We then see how such theories may in act be presented entirely categorically, simply through the properties a single category which we call an operational category. This provides categorical descriptions o all o the main eatures o operational-probabilistic theories, such as the ability to orm convex combinations o physical events, and allows us to extend these notions beyond the probabilistic setting. In act the categorical eatures we will use are not themselves new, being based on eectus theory, an area o categorical logic developed by Jacobs and collaborators or the study o classical, probabilistic and quantum computation [Jac15, JWW16]. We will see a correspondence between basic properties o a theory and its associated category, in particular providing eectus theory with an operational interpretation. 2.1 Operational Theories asic operational theories Let us begin by introducing a basic ramework or what may be described as an operational theory o physics. s outlined in hapter 1, we will start with a symmetric monoidal category, whose objects here we call systems and morphisms 19

30 20 hapter 2. Operational Theories and ategories : we call events. s we have seen this means that events may be composed to orm circuit diagrams like ρ h g Tests On top o this category, an operational theory concerns experimental procedures which we call tests. Formally, a test is given by a inite non-empty collection ( i ) (2.1) i X o events o the same type. Such a test is to be thought o as an operation we may perorm on a system o type, leaving us with a system o type, with initely many possible outcomes indexed by the non-empty set X. On any run o the test precisely one event i will occur, with the outcome i then recorded. Ourtheory will speciy which initecollections ( i : ) i X orm admissible tests. More generally we call a inite non-empty collection ( i ) i X a partial test when it orms a sub-collection o a test ( j ) j Y, with X Y. We require some basic properties o tests. xiom 1. Tests satisy the ollowing: every event belongs to some test; tests are closed under relabellings o outcomes; whenever ( i ) i X and (g j ) j Y are tests, so is ( i g j ) i X,j Y The latter assumption states that, like events, we may place tests side-by-side to orm new ones. nother way we may expect to orm new tests is by using outcome data rom earlier ones as input, which we capture as ollows. xiom 2 (asic ontrol). Let ( i : ) i X be a test and, or each o its outcomes i, let (g(i,j): ) j Yi be a test. Then the ollowing is a test: ( i g(i,j) ) i X,j Y i We reer to the above as a controlled test, interpreting it as perorming the test ( i ) i X and then depending on the outcome i X choosing which test g(i, ) to perorm next. This axiom appears as an optional assumption in the ramework o [DP11], which allows or theories without any simple causal structure and hence any such straightorward notion o conditioning. 20

31 2.1. Operational Theories 21 oarse-graining second way in which an agent should be able to make use o the outcome data rom a test is simply to discard it, thus merging several o its events. all a collection o events o the same type ( i : ) i X compatible when they orm a partial test. n operational theory should come with a rule or merging any compatible pair o events,g: into a coarse-grained event g:, which we interpret as either or g occurs. The partial operation should ulill some basic rules to match this interpretation. xiom 3. The operation satisies the ollowing. i (,g,h 1,...h n ) is a test, g is deined and ( g,h 1,...h n ) is a test; g g or all compatible (,g); ( g) h (g h) or all compatible (,g,h); or all compatible (g,h) and events,k we have (g h) ( g) ( h) (g h) k (g k) (h k) (g h) ( g) ( h) Each o the above requirements has a straightorward operational interpretation. For example, the irst o the inal three equations above states that the events either g or h, then and either g then, or h then coincide. Note that both sides o the equations above are indeed well-deined thanks to our assumptions about tests. These properties allows us to deine the coarse-graining o any non-empty compatible collection o events by nï i : 1 ( 2 (... n )) i1 Itwillalsobehelpultoassumetheexistenceounits0: orcoarse-graining, which we think o as the unique impossible event between any two systems. Recall that a category has zero morphisms when it has a (necessarily unique) amily o morphisms0 0, : satisying 0 0 g 0 or all morphisms,g, and in the monoidal setting we also similarly require g. xiom 4. The category o events has zero morphisms. Moreover atuple ( 1,... n ) orms a test i ( 1,..., n,0) does also, and we have 0 or all events. Finally we will require the operational ability to discard systems as well as outcome data. The presence o such discarding maps will also allow us to speciy tests in terms o partial tests. xiom 5 (ausality). The category o events has discarding, and a partial test ( i ) i X is a test precisely when it satisies Ï i (2.2) i X 21

32 22 hapter 2. Operational Theories and ategories Intuitively, a test should be a partial test which always returns some outcome, as a whole being causal in our earlier sense. Note that in particular the above tells us that is the unique eect on any system which orms a test on its own. s remarked in hapter 1, this is indeed closely related to notions o causality in probabilistic theories [DP11]. Deinition 2.1. basic operational theory Θ consists o a symmetric monoidal category Event Θ with discarding, a choice o tests, and coarse-graining operations satisying xioms 1-5. Remark 2.2. lternatively, one may instead deine such a theory in terms o partial tests and coarse-graining, then deining tests as those satisying (2.2). However we view tests as a more primitive notion so have used them as our starting point. Many o our motivating examples o operational theories will be probabilistic, here meaning that their scalars are given by probabilities p [0,1], with p q : p + q being deined whenever this value is 1. This is assumed in rameworks such as [DP10]. More generally scalars in a theory behave much like probabilities, orming a commutative monoid with a similar partial addition. For example, we may call a test consisting o scalars (p i : I I) n i1 a distribution, by analogy with inite probability distributions. Given any collection o n events i : we may then consider their convex combination nï i1 pi which is well-deined thanks to the control axiom. One may go on to deine many typical notions rom the study o probabilistic theories such as completely mixed states, reasoning much like in [DP10] Extending the notion o test So ar we have taken the common approach o deiningtests as collections o events o the same type ( i : ) i X, as in e.g. [DP10, GS18]. However, there are standard operational procedures which cannot immediately be described in this manner (typically requiring extra structure to do so [DP10, Remark, p.12-13]). For example consider an agent who irst perorms such a test and then, depending on the outcome i, chooses between perorming one o several tests having dierent output systems i. simple case would be, conditioned on the outcome o a coin lip, preparing some state ω o a system or ρ o another system : ( heads ω I I i tails ρ, I I To account or such procedures, we must allow tests to have the general orm or inite sets X, now with varying output systems. ( i ) i (2.3) i X ) 22

33 2.1. Operational Theories 23 Operational theories o this new sort may be deined just as previously. s beore, such a theory speciies a category o events, certain collections o which orm tests or partial tests. We now include the empty collection as a partial test o any given type. oarse-graining g should still only be deined on events o the same type,g: which belong to some test, whose other events may have dierent types. More generally a collection o events o the same type ( i : ) n i1 are again called compatible when they orm a partial test, and their coarse-graining will be deinable as beore, with that o the empty partial test now set to 0. To include the procedures discussed above we now require a stronger control axiom. xiom 6 (ontrol). Let ( i : i ) i X be a test and, or each o its outcomes i, let (g(i,j): i,j ) j Yi be a test. Then the ollowing is a test: ( i g(i,j) ) i i,j i X,j Y i The rest o our earlier axioms were careully worded to apply immediately to theories o this new orm, which we reer to simply as ollows. Deinition 2.3. n operational theory Θ is given by a symmetric monoidal category with discarding Event Θ along with a speciication o tests o the orm o (2.3), and operations satisying xioms 1, 3, 4, 5 and 6. To distinguish these rom basic theories, we sometimes call such theories proper operational theories. ecause o the common practice o taking tests the orm 2.1, in this chapter we will consider both kinds o theories. Despite their name, the axioms o proper operational theories are in some sense weaker than those o basic ones, by the ollowing. Lemma 2.4. Let : be an event in a theory o either kind. 1. In an operational theory belongs to a test (,e) or some e: I. 2. In a basic operational theory belongs to a test o the orm (,g: ), and every object has a causal state. Proo. 1. ny belongs to some test (:,g 1 : 1,...,g n : n ). Then using control (,e) is a test where e Ŕ n i1 g i. 2. Here by assumption belongs to some test (,g 1,...,g n ) with each g i :. Then g Ŕ n i1 g i is well-deined and (,g) is a test. For the second statement take to be the zero state Examples Many o our examples o categories rom hapter 1 extend to orm operational theories. In each case these also orm basic operational theories by restricting to tests o theorm (2.1) and excludingobjects such as or 0which lack causal states. 1. The theory lassdet o deterministic classical physics has category o events PFun. Here a collection o partial unctions ( i : i ) i X orm a test when their domains are disjoint and partition, with being disjoint union. 23