Spekkenss toy theory as a category of processes

Sp ekk ens’s to y theory as a category of pro cesses Bob Co ec k e and Bill Edwa rds Abstract. W e prov ide t wo mathematical descriptions of Sp ekk ens’s to y qubit theory , an inductiv ely one in terms of a small set of generat ors, as we ll as an explicit closed form description. It is a sub category MSpek of the categ ory of ﬁnite sets, r elations and the cartesian pro duct. States of maximal kno wl- edge form a sub category Spek . This establishes the consistency of the to y theory , which has pr eviously only been constructed f or at most four systems. Our mo del also shows that the theory is closed under b oth parallel and sequen- tial comp osition of op erations (= symmetric monoidal structure), that it ob eys map-state dualit y (= compact closure), and that states and eﬀects are i n bi jec- tiv e corresp ondence (= dagger structure). F r om the persp ectiv e of ca tegorical quan tum mecha nics, this provides an inte resting alternativ e mo del which en- ables us to descri b e many quant um phenomena in a discrete manner, and to which mathematical conce pts such as basis structures, and complementarit y thereof, still apply . Hence , the framework of categorical quan tum mec hanics has delivered on its promi s e to encompass theories other than quant um th eory . 1. In tro duction In 2007 Rob Spek kens prop osed a toy theory [ 20 ] with the aim o f showing that many of the c haracter istic features of quan tum mechanics could result from a restriction o n our knowledge of the state of a n essentially cla ssical system. The theory describ es a s imple type of system which mimics many of the features of a quantum qubit. The success of the toy theo ry in replicating ch ar acteristic quan tum behaviour is, in one s e nse, quite puzzling, since the ma thematical str uc tur es em- ploy ed b y the tw o theor ies ar e quite diﬀeren t. Quan tum mechanics represents s tates of systems b y vectors in a Hilber t space, while pro cesse s undergone by sy s tems ar e represented by linear maps. In co n trast, the toy theory represents states by s ubsets and proces ses b y relations. This ‘incomparability’ means that the mathematical origins of the similarities (and diﬀerences) betw een the t wo theories ar e not ea sy to pinpo in t. In this pap er we consider the toy theory from a new p ersp ective - by lo oking a t the ‘algebra’ of how the pro cesses of the theory combine. The mathematica l struc- ture formed b y the compos ition of pro ces ses in a n y physical theory is a symmetric 2000 Mathematics Subje ct Classiﬁc ation. 81P10, 18B10, 18D35. This work was supp orted b y EPSR C Adv anced Researc h F ellowship EP/D072786/1 and b y EU-FP6-FET STREP QICS. This work was suppor ted by EPSRC PhD Plus funding at Oxfor d Universit y Computing Laboratory and an FQXi Large Grant . 1 2 BOB COECKE AND BILL EDW ARDS monoidal c ate gory [ 14 ], with the ob jects of the catego r y repr e sent ing systems, and the morphisms representing pro ces ses undergo ne by these s ystems - we term this the pr o c ess c ate gory o f the theory . W ork initiated by Abra msky and Coecke [ 2 ], and contin ued by man y other author s [ 17, 5, 9 ] ha s inv estigated the structur e of the pro ces s category of quantum mechanics. Mathematical structures o f this c at- egory hav e b een iden tiﬁed whic h cor resp ond directly to key ph ysical fea tur es of the theo ry . In principle, any theo ry whos e pr o cesses form a category with these mathematical features should exhibit these qua ntum-like ph ysica l features. It will be gratifying then to note that the pr o cess category of the toy theor y shares many of these fea tures, thus ‘acco un ting for’ the similarities which this theory has with quantum mechanics. The main aim of this pap er is to character is e the pro cess c a tegory of the toy theory . This turns out to b e less s traightforw ard than one might hav e hop ed! In the case o f quantum mechanics it is straig h tforward to see that the states of a system corres p ond exactly with the vectors of the Hilbert space describing that system, and that proce s ses cor resp ond exactly with linea r maps; thus w e can immediately conclude tha t the pr o cess catego ry of quantum mec hanics is Hil b , whose ob jects are Hilb ert space s and whose morphisms a re linea r ma ps. Such a quick and easy statement of the pro cess category fo r the to y theor y is not po ssible. This is due largely to the w ay that the v alid states and pro cesses of the theo ry are deﬁned: this is via an inductive pro cedure w her e the v alid states and trans formations for a collection of n − 1 s ystems mu st b e kno wn b efore those fo r a collection o f n systems can b e deduced. There is also a ce rtain deg ree of ambiguit y in the way in which the theory was or iginally stated, and, since the toy theory was nev er fully constructed (un til now), its consistency w as no t even guar a nt eed. Another fea tur e whic h was not addressed in the theo ry is c omp ositio nal closur e , that is: If we c omp ose two valid pr o c esses, either in p ar al lel or se quential ly (when typ es match), do we again obtain a valid pr o c ess? Obviously , this is a natura l op er ational requirement, and as indi- cated a bove, this no tion o f comp ositio na l closure is the op erationa l corners tone to mo delling physical pro cesses in a sy mmetr ic mono idal ca tegory . In this pap er, it is exactly the comp ositiona l clo sure assumption which allows us to formulate the toy theory in terms of no more tha n a few generator s, with clea r op erational meaning s. The structure o f the pap er is as follo ws. W e beg in with a very brief summary of the toy bit theor y , and a discussio n of the am biguities in its deﬁnition. W e then deﬁne a ca tegories Sp ek a nd MSp e k which we claim is the pro cess ca tegory fo r the toy theor y . This is a sub-category o f Rel , whose o b j ects ar e sets a nd who se morphisms are rela tions. The next section is devoted to showing the genera l form of the relations which constitute the morphisms of MSp ek . W e then go on to argue that these r elations are exactly tho s e which describ e the pr o cesses o f the toy theory , thus demonstrating that MSp ek is the pro cess category of the toy theory . Finally we note some o f the key catego rical features of MSpe k and link these to the characteristica lly qua n tum b ehaviour exhibited b y the to y bit theory . The category Sp ek was ﬁr st pr op osed by us in [ 6 ], a nd we provide some e x plicit po in ters to useful infor mation therein throughout this tex t. SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 3 2. The to y theory F or full details of Sp ekkens’s toy bit theory , the re a der is r eferred to the o riginal pap er [ 20 ]. Here we provide a very brief summary of the key p oints. There is just one type of elementary system in the theory , which can e xist in one o f four states. W e will denote the se t of these four sta tes, the ontic st ate sp ac e by I V = { 1 , 2 , 3 , 4 } . Alternatively w e ca n depict it graphically a s: (2.1) The on tic states are to be distinguis hed from the epistemic states, which describ e the exten t of our knowledge abo ut whic h ontic s ta te the system o ccupies. F or example we might know that the system is either in ontic state 1 or 2. W e w ould depict such an epistemic state in the following fashion: (2.2) The e pis temic state is intended to be the analogue of the q uantu m state. Note that an epistemic state is simply a s ubset of the ontic state spa ce. Given a co mp os ite system o f n elementary systems, its ontic state spa ce is simply the Ca rtesian pro duct of the ontic state spa c es of the c ompo site systems, I V n . Thus such a system has 4 n ontic s tates. The key pr emise of the theory is that our kno wledge of the on tic state is re- stricted in a s peciﬁc w ay so tha t only certain epistemic states are allowed. W e refer the reader to the o riginal pap er fo r the full statemen t of this epistemic restriction, which Sp ekkens refers to as the kn ow le dge b alanc e principl e . It s uﬃces here to say that, given an n -compo nent system, an epistemic s tate is a 2 n , 2 n +1 , . . . or 2 2 n element subset of the o n tic state space, the 2 n case b eing the situation o f maximal knowledge a nd the 2 2 n case b eing the situation of total igno r ance. F or a single s ystem this means that there ar e six epistemic states o f maximal knowledge: (2.3) and one o f non-ma ximal knowledge: (2.4) A c onsequence of the knowledge balance principle is that an y measurement on a system inev itably results in a probabilistic disturbance of that system’s ontic state. A measurement in the toy theory essentially corresp onds to asking which of a collectio n o f subsets o f the ontic s tate space con tains the o n tic sta te. An exa mple would be a measurement on a single system cor resp onding to the questio n ”is the ontic s tate in the subset { 1 , 2 } o r the subset { 3 , 4 } ?”. If the initial epistemic state of the system w as the subse t { 1 , 3 } then answering this question w ould allo w us to pin down the o n tic state precise ly , to 1 if we get the ﬁrst answer or 3 if we get the seco nd, in vio lation of the knowledge balance principle. Thus we hypothesise that if, for exa mple, we get the ﬁrst answer to our question, then the ontic state undergo es a random disturba nce, either rema ining in s ta te 1 o r moving into state 2, with equal probability . The epistemic state following the measur ement then would be { 1 , 2 } . The eﬀect of such a probabilistic disturba nce on the epistemic state is best mo delled b y viewing it as a r elation on the ontic state space, deﬁned by 4 BOB COECKE AND BILL EDW ARDS 1 ∼ { 1 , 2 } , 2 ∼ { 1 , 2 } , 3 ∼ ∅ , 4 ∼ ∅ , w he r e the notation denotes that the elemen t 1 in the domain rela tes to b oth the elements 1 and 2 in the co domain, the element 2 in the doma in, while the elements 3 and 4 in the domain relate to no elements in the co domain. Belo w we will o mit sp eciﬁcation of those elemen ts in the domain that r elate to no element s in the co domain. Th us in gener al the transformations of the theor y are descr ibed b y r elations . In his o riginal pap er Sp ekkens derives the states a nd transformatio ns allow ed in the theo ry , appar ent ly a ppea ling only to the following three principles: (1) The epistemic state of any sys tem must satisfy the kno wledge ba lance principle globally i.e. it should be a 2 n , 2 n +1 , . . . or 2 2 n element subset of the ontic state space. (2) When considering a comp o site system, the ‘marginal’ epistemic state of any subsystem should a ls o sa tisfy the knowledge ba lance principle. By the marginal epistemic state we mean the following. Supp ose the whole system has n comp onents, and we are interested in an m -comp onent subsystem. The epistemic state o f the whole system will b e so me set of n -tuples: to get the marginal state on the m - comp o nent system we simply delete the n − m entries from each tuple whic h corr esp ond to the subsystems which are no t of interest to us. (3) Applying a v alid tr a nsformation to a v alid sta te should result in a v alid state. The second principle is used, for example, to rule this out as an epistemic state for a t wo-component system: (2.5) The third principle is used extensively . It is in vok ed to show that the transfor- mations on a single element system constitute permutations o n the set of ontic states, since any other function on I V would lead to some v alid epistemic state being transformed into an inv a lid state. Another illuminating example o f the third principle in action is the elimination of this state as a v alid epistemic sta te: (2.6) This state would be allow ed b y the ﬁr st and seco nd principles, but up on making a certain mea surement on one of the systems, the resulting mea surement disturbance would tr ansform it in to this state: (2.7) which clea rly fails to satisfy the ﬁrst princ iple . SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 5 The mo de of deﬁnition o f the theory , v ia the three principles sta ted earlier, raises some interesting is sues. Firstly , this approach seems to b e necessar ily itera- tive. Compar e it, for exa mple, to ho w we would des crib e the form of v alid states in quantum mechanics - in one line we can say that they are the norma lised v ectors of the system’s state spa ce. Secondly , do these rules actually uniquely deﬁne the toy theory ? There does seem to b e a problem with the third r ule. When w e use it to r ule out the state in diagr am 2.6, our argument is that we already k now what the v alid mea surement disturbance tr a nsformations on a single elementary system are, and w hen we a pply one of them to this s tate we obtain a state which clearly violates the knowledge ba lance principle. How ever, it is not cle ar that we could not hav e made the a lternative choice - that this state sho uld b e v a lid, and therefore that the transformation which we had previo usly thought was v alid co uld no longer be consider e d a s s uc h. It seems that considera tions other tha n the three rules ab ov e come into deciding whic h should b e the v alid s ta tes o f the theor y , but it is nowhere clearly stated exactly what they are. The second p o int touched slig h tly on our ﬁnal issue: is the theory as Spekkens presents it co ns isten t? He der ives v alid states a nd tra nsformations for s ystems with up to three elemen tary co mpo ne nts. Ho wev er, can w e b e sur e that these states and transformatio ns, when co m bined in more complex situations in volving four or more elementary systems, won’t yield a sta te whic h clear ly violates the knowledge bala nc e principle? Currently there see ms to b e no such pro of of consistency . In fact, in the pro cess of r e - expressing the theory in categorical terms, we will develop such a pro of. 3. Pro cess categories o f quantum-lik e theories W e brieﬂy r eview the key structura l features of the pro ce ss catego r ies of quantum- like theories . A ph ysicist-frie ndly tutor ial of the categor y theoretic preliminaries is in [ 8 ]. A survey on the recent applications of these to quantum theory and quantum information is in [ 4 ]. Deﬁnition 3.1. A symmetric monoidal c ate gory ( C , I , − ⊗− ) is a category equipp ed with the fo llowing extra structure: a bifunctor −⊗ − : C × C → C ; a unit obje ct I ; and four natur al iso mo rphisms, left and right unit: λ A : A ∼ = I ⊗ A , ρ A : A ∼ = A ⊗ I , a sso- ciative: α A,B ,C : ( A ⊗ B ) ⊗ C ∼ = A ⊗ ( B ⊗ C ) a nd commut ative: σ A,B : A ⊗ B ∼ = B ⊗ A . F urthermore these ob jects and natural isomor phisms ob ey a series of c oher en c e c on- ditions [ 14 ]. As arg ued in [ 8 ], the pro c e ss ca tegory of a ny physical theory is a SMC. The bifunctor − ⊗ − is int erpr eted as a djoining tw o sy s tems to make a larger comp ound system. Hi l b b ecomes a SMC with the tenso r product as the bifunctor, a nd C a s the unit ob ject. Rel is a SMC with the Cartesian pro duct as the bifunctor and the single element set a s the unit ob ject. There is a very useful g raphical language for descr ibing SMCs, due to J oy al and Stree t [ 1 2 ], whic h we will ma ke extens ive use o f. It traces back to Penrose’s earlier diag rammatic notation fo r abstra ct tensors [ 16 ]. In this language we represent a morphism f : A → B by a b ox: (3.1) f A B 6 BOB COECKE AND BILL EDW ARDS g ◦ f , the c o mpo sition of mor phisms f : A → B and g : B → C is depicted as: (3.2) f A B g C The identit y morphism 1 A is a ctually just w r itten as a straight line — this mak es sense if you imag ine co mpo s ing it with another mo r phism. T urning to the symmet- ric monoida l structure, a morphism f : A ⊗ B → C ⊗ D is depicted: (3.3) B A D C f = f A ⊗ B C ⊗ D and if f : A → B and g : C → D then f ⊗ g is depicted as: (3.4) C g D A f B f ⊗ g = B ⊗ D A ⊗ C The ident ity o b ject I is not actually depicted in the gr aphical la nguage. Morphisms ψ : I → A and π : A → I are wr itten as: (3.5) ψ A π A The a sso ciativity and left and right unit natur al is o morphisms a re also implicit in the lang uage. The symmetry natural isomor phism is depicted as: (3.6) A B B A In fact the graphical language is more than just a useful to o l; it enables one to derive a ll equational statements that fo llow fr om the a xioms of a SMC: Theorem 3. 2 ([ 12 , 18 ]) . Two morphisms in a s ymm et ric monoidal c ate gory c an b e shown to b e e qual using t he axioms of a SMC iﬀ the diagr ams c orr esp onding to these morphisms in the gr aphi c al language ar e isomorphic, wher e by diagr am isomorphism we me an t hat the b oxes and wir es of the ﬁrst ar e in bije ctive c orr esp ondenc e with the se c ond, pr eserving the c onne ctions b etwe en b oxes and wir es. The categorie s describ ed by the graphica l la nguage are in fact strict SMCs, t hat is, those for which λ A , ρ A and α A,B ,C ar e e qualities. Mac L ane’s Strictiﬁcation Theorem [ 14 , p.257], which e stablishes an equiv alence of an y SMC with a strict one, enables one to apply the diag rammatic notation to any SMC. W e will often r efer to mo rphisms of t yp e ψ : I → A a s states , since in a pro cess category they represent the prepa ration of a system A in a given state. In Hilb such mor phisms are in bijection w ith the v ectors of the Hilber t space A ; in Rel they corr esp ond to subse ts of the set A . The pro ces s categor ie s of quant u m-like theories p ossess a range of a dditio na l structures. Deﬁnition 3.3. A dag ger c ate gory is a ca tegory equippe d with a contrav ariant inv olutiv e identit y-on-ob jects functor ( − ) † . A dagger symm et ric monoidal c ate gory ( † -SMC) is a symmetric monoida l catego r y with a dagger functor such that: ( A ⊗ B ) † = A † ⊗ B † , λ − 1 A = λ † A , ρ − 1 A = ρ † A , σ − 1 A,B = σ † A,B , and α − 1 A,B ,C = α † A,B ,C . SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 7 Hilb is a † -SMC with the adj oint playing the role of the da gger functor. Rel is a † - SMC with r elatio nal c onverse as the dag ger functor. Deﬁnition 3. 4 ([ 13 ]) . In a SMC C a c omp act structur e on an ob ject A is a tuple { A, A ∗ , η A : I → A ∗ ⊗ A, ǫ A : A ⊗ A ∗ → I } , where A ∗ is a dual ob ject to A whic h may or may not be equal to A , and η A and ǫ A satisfy the conditions: (3.7) A ρ A ✲ A ⊗ I 1 A ⊗ η A ✲ A ⊗ ( A ∗ ⊗ A ) A 1 A ❄ ✛ λ − 1 A I ⊗ A ✛ ǫ A ⊗ 1 A ( A ⊗ A ∗ ) ⊗ A α A,A ∗ ,A ❄ and the dual diagram for A ∗ . A c omp act close d category C is a SMC in whic h all A ∈ Ob( C ) have compact structures. Hilb is a compact closed categor y , wher e for each ob ject A the Bel l-st ate ket P i | i i ⊗ | i i (with {| i i} a basis for the Hilbert spa ce A ) familiar from qua n tum me- chanics provides the morphism η A , and the corr esp onding bra acts a s ǫ A . Perhaps unsurprisingly , given this exa mple, Abra msky and Co eck e in [ 2 ] show ed that this structure underlies the capacity of a n y quantum-lik e theory to exhibit information pro cessing proto co ls such as telep ortation and entanglement s wapping . F urther- more Abra msky has sho wn [ 1 ] that an y theo ry whose pro cess category is compact closed will obey a gener alised version of the no - cloning theor em. F or the compact s tr uctures of in terest in this pap er A ∗ is always equal to A – see prop osition 3.9 b elow. 1 Hence, from here on we won’t dis ting uish these tw o ob jects. This simpliﬁes the gr a phical language, w hich can b e extended b y intro ducing sp ecial elements to r epresent the morphisms of the compac t structure η A and ǫ A : (3.8) A A A A η A ǫ A Equation 3.7 and its dua l ar e then depicted as: (3.9) A A = A A = This extension of the g raphical languag e now r enders it co mpletely equiv alen t to the axioms of a compact closed category ; for a detailed discussion we refer to [ 18 ]. Theorem 3.5. Two morphi sms in a c omp act close d c ate gory c an b e shown to b e e qual u sing the axioms of c omp act closur e iﬀ the diagr ams c orr esp onding to these morphisms in the gr aphic al language ar e isotopic. 1 An analysis of the coherence conditions f or these self-dual compact structures is i n [ 1 9 ]. 8 BOB COECKE AND BILL EDW ARDS Compact clos ure is of par ticular imp ortance to us since in a n y compact closed category there will b e map-state duality : a bijection b etw een the hom-sets C ( I , A ⊗ B ) and C ( A, B ) (in fact both these will further b e in bijection with C ( A ⊗ B , I )): (3.10) M Ψ M H B H B H A H A = and (3.11) H A M H A Ψ M = H B H B That this is a bijection follows fro m equation 3.9. If we have a mor phis m with large r comp osite domain a nd co domain the num b er of hom-sets in bijection increa ses dramatically . F or example the morphisms of the ho m-sets C ( A 1 ⊗ · · · ⊗ A m ⊗ X, B 1 ⊗ · · · ⊗ B n ) and C ( A 1 ⊗ · · · ⊗ A m , B 1 ⊗ · · · ⊗ B n ⊗ X ) are in bijection: explicitly the conv ersion be tw een mo rphisms from the tw o sets can b e depicted as: (3.12) A 1 A 2 A m B 1 B 2 B n X Clearly mano euvres like this can conv ert any ‘input’ line into an ‘output’ line, by using the unit and co-unit morphisms to ‘bend lines a round’. Deﬁnition 3. 6. A diagr am e quivale nc e class 2 (DEC) is a set of mor phisms in a co mpact closed categor y which can b e inter-conv erted by comp osition with the units and co-units of the factors of their domains and co domains . The ﬁnal s tructure of interest is the b asis structu r e , whic h can b e seen as the ‘dagger- v arian t’ of Car bo ni and W alters’s F r ob enius algebr as [ 3 ]. This structure is discussed at length in [ 5 ]. Deﬁnition 3. 7. In a † -SMC a b asis structur e ∆ on an ob ject A is a commutativ e isometric dagger F ro benius comonoid ( A, δ : A → A ⊗ A, ǫ : A → I ). F or more details on this deﬁnition see section 4 o f [ 5 ] where basis structures are referr ed to as ‘obse rv able s tructures’. W e represent the morphisms δ and ǫ graphically a s: (3.13) δ ǫ Basis structures a re so named b ecause in Hilb they a re in bijection with or- thonormal bases , via the corresp ondence: (3.14) δ : H → H ⊗ H :: | i i 7→ | i i ⊗ | i i ǫ : H → C :: | i i 7→ 1 2 The termi nology is inspired by the fact that the diagrams of all members of a class are essen tially the same, only diﬀering in the orien tations of their input and output arr o ws. SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 9 where {| i i} i =1 ,...,n is an o rthonormal bas is for H . This is proved in [ 10 ]. Prop ositio n 3.8 (See e.g. [ 5 ]) . Two b asis structur es ( A, δ A , ǫ A ) and ( B , δ B , ǫ B ) induc e a thir d b asis stru ctur e ( A ⊗ B , δ A ⊗ B , ǫ A ⊗ B ) , with: (3.15) δ A ⊗ B = (1 A ⊗ σ A,B ⊗ 1 B ) ◦ ( δ A ⊗ δ B ) ǫ A ⊗ B = ǫ A ⊗ ǫ B , Prop ositio n 3.9 (See e.g. [ 9 ]) . A ny b asis st ructur e induc es a self-dual dagger c omp act stru ctur e, with A = A ∗ and η A = δ ◦ ǫ † . Basis structures are identiﬁed in [ 5 ] as a key structur e underly ing a v ariety of quantum-lik e features, for example the existence of incompa tible observ ables, and information pr o cessing tasks such as the quantum fourier tra ns form. 4. The categories Sp ek and MSp ek In the s ection we g ive the deﬁnitions of tw o k ey catego ries. The ﬁrst deﬁnition is a c o nv enien t stepping stone to the second: Deﬁnition 4.1. The category Sp ek is a subcateg o ry of Rel , deﬁned inductively , as follows: • The o b j ects o f Spe k ar e the single- e lemen t set I = { ∗} , the four element set I V := { 1 , 2 , 3 , 4 } , and its n -fold Cartesian products I V n . • The morphisms of Sp ek ar e all those r elations genera ted by comp osition, Cartesian pro duct and rela tional conv erse from the following gener ating relations: (1) All per m utations { σ i : I V → I V } o f the four element set, repr esented diagramma tica lly b y: (4.1) σ i There ar e 24 suc h p ermutations and they form a group, S 4 . (2) A relation δ Sp ek : I V → I V × I V deﬁned by: 1 ∼ { (1 , 1 ) , (2 , 2) } 2 ∼ { (1 , 2 ) , (2 , 1) } 3 ∼ { (3 , 3 ) , (4 , 4) } 4 ∼ { (3 , 4 ) , (4 , 3) } ; represented diagramma tically by: (4.2) (3) a relatio n ǫ Sp ek : I V → I deﬁned by { 1 , 3 } ∼ ∗ and r epresented diagramma tica lly b y: (4.3) (4) the relev a n t unit, asso ciativity and symmetry natural iso mo rphisms. Prop ositio n 4.2. [ 6 ] (I V , δ Sp ek , ǫ Sp ek ) is a b asis s t ructur e in FR el and henc e also in Sp ek . 10 BOB COECKE AND BILL EDW ARDS This catego ry turns out to be the pro cess categ ory for the fragment of the toy theory cont aining only epistemic states of ma ximal kno wledge. In particular, as discussed in detail in [ 6 ], the interaction of the basis structure (I V , δ Sp ek , ǫ Sp ek ) and the p ermutations of S 4 results in three basis structur es, analog ous to the Z -, the X - and the Y -bases of a qubit. F urthermore, as also discussed in [ 6 ] § 4, Sp ek a lso contains op era tions which in quantum theory corr esp ond with pro jection o per ators, while these a re not included in the toy theo ry . Indeed, compa c t closure of Sp ek implies map-state duality : op erations and bipartite states are in bijective cor r esp ondence. 3 The pr o cess categor y for the full toy theory has a similar deﬁnition: Deﬁnition 4.3. The categor y MSpe k is a sub- c a tegory o f Rel , with the same ob jects as Sp ek . Its morphisms ar e all those rela tions generated by compositio n, Cartesian pro duct and r elational conv erse fr o m the generators of Sp ek plus an additional g e ne r ator: ⊥ MSp ek : I → I V :: {∗} ∼ { 1 , 2 , 3 , 4 } By constructio n, b oth Sp ek and MSp ek inher it dagg er symmetric monoidal structure from Rel , with Cartesian pro duct be ing the monoidal pro duct, and rela- tional converse acting as the da gger functor. Prop ositio n 4.4. Sp ek and MSp ek ar e b oth c omp act close d. Proof. By pr op o sitions 4.2 and 3.8 it follows that each ob ject in these ca te- gories has a basis str ucture, and hence b y pro po s ition 3.9 they b oth ar e compact closed.  The primar y task o f the remainder of this pap er is to demonstra te that MSp ek is indeed the pro cess ca tegory of Spekkens’s toy theory . 5. The general form of the morphisms of Sp ek and MSp ek W e ﬁrs t mak e so me preliminary observ ations. Deﬁnition 5.1. A Sp ek diagr am is any v alid diagr am in the gr aphical languag e int ro duced in section 3 whic h can b e formed by linking toge ther the dia grams of the Sp e k generato r s, a s describ ed in deﬁnition 4.1. There is clearly a bijection betw een the p ossible comp ositions of Sp ek gen- erators , and Sp ek diagr ams. A Sp ek diag ram with m inputs and n outputs represents a morphism of t yp e I V m → I V n : a r elation b e t ween sets I V m and I V n . The n umber of rela tions b etw een t wo ﬁnite sets A and B is clearly ﬁnite itself: it is the p ow er set of A × B . Thus the ho m-set FRel (A,B) is ﬁnite. Since Sp ek (I V m , I V n ) ⊆ FRel (I V m , I V n ) w e can b e sure that the hom- sets o f Spe k ar e ﬁnite. On the other hand, there is clearly an inﬁnite n um b er of Sp ek dia grams which have m inputs and n inputs - we can add more and more internal lo ops to the dia grams. Thus many diag r ams repres ent the s a me morphism. How ever the morphisms of Sp ek are, by deﬁnition, a ll those rela tions resulting from arbitra ry comp ositions of the generating relations, i.e. a ny relatio n that corr esp onds to one 3 W e see this as an impro vemen t, and in [ 21 ] also Sp ekk ens expressed the desire f or theories to hav e this prop er t y , as well as having a dagger-like structure, in the sense that states and eﬀects should b e in bijective corresp ondence. SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 11 of the inﬁnity of Spe k diagra ms. Hence any pro of ab out the form of the mo rphisms in Sp ek is go ing to hav e to b e a result a bo ut the relations corr esp onding to each po ssible Sp ek dia gram, ev en though in general many dia grams corr esp ond to a single mor phism. If we know the r elation corr esp onding to one diagr am in one of Sp ek ’s diag ram equiv alence classes (reca ll deﬁnition 3 .6), then it is stra ightf or ward to determine the relations cor resp onding to all of the other diagr a ms. Lemma 5. 2. Given a Sp ek diagr am and c orr esp onding r elation: (5.1) R X 1 X 2 X m Y 1 Y 2 Y n R : X 1 × X 2 × · · · × X m → Y 1 × Y 2 × · · · × Y n then the r elatio n c orr esp onding to the fol lowing diagr am: (5.2) R Y 1 Y 2 Y n X m X 1 X 2 is given by, for al l x i ∈ X i : (5.3) ( x 1 , . . . , x m − 1 ) ∼    ( y 1 , . . . , y n , x m )      y i ∈ Y i x m ∈ X m ( y 1 , . . . , y n ) ∈ R ( x 1 , . . . , x m )    wher e R ( x 1 , . . . , x m ) is the subset of Y 1 ×· · ·× Y n which is re late d by R to ( x 1 , . . . , x m ) . Every diagra m equiv alence class in Sp ek has at least one diagram of t yp e I → I V n , r e presenting a sta te, where w e mak e ev ery external line an output. Relations of this type ca n b e viewed as subsets o f the se t I V n and it will b e conv enient for us to concen trate on characterising these mor phisms. Via lemma 5.2 an y res ults on the ge neral form of states will tr anslate in to results on the g eneral for m of all morphisms. In what follows w e will therefore mak e no distinction betw een the inputs and outputs of a Sp ek -diagram: a diagr am with m inputs a nd n outputs will simply be referred to as a ( m + n )-leg g ed diagram. Our pro of will inv olve building up Sp ek diag rams by connecting tog ether the generating morphisms. Here we show what v ario us diag r am manipula tions mean in concr ete ter ms for the corres po nding rela tions. Henceforth, r emembering that we o nly have to consider states, we will ass ume that the relatio n corr e spo nding to any n -leg ged diagr am is of t yp e I → I V n . First we introduce some terminology . The c omp osite of an m -tuple ( x 1 , . . . , x m ) and an n -tuple ( y 1 , . . . , y n ) is the ( m + n )-tuple ( x 1 , . . . , x n , y 1 , . . . , y n ) fro m I I m + n . By the i th -r emnant o f a n n -tuple we mea n the ( n − 1)-tuple obtained by deleting its i th comp onent. By the i, j th -r emnant of a n n -tuple (where i > j ) w e mean the ( n − 2)-tuple o bta ined by deleting the j th comp onent of its i th -remnant (or equiv alen tly , deleting the ( i − 1) th comp onent of its j th -remnant). 12 BOB COECKE AND BILL EDW ARDS Example 5.3. Consider linking two diagr ams, the ﬁrst r epr esenting the r elation R : I → X 1 × · · · × X m the se c ond r epr esenting t he r elation S : I → Y 1 × · · · × Y n via a p ermutation P , t o form a new diagr am as shown: (5.4) R S X 1 X m Y n Y 1 Y j X i P The r ela tion c orr esp onding to this diagr am is given by (5.5) ∗ ∼    ( x 1 , . . . , x i − 1 , x i +1 , . . . , x m , y 1 , . . . , y j − 1 , y j +1 , . . . , y n )      ( x 1 , . . . , x m ) ∈ R ( ∗ ) ( y 1 , . . . , y n ) ∈ S ( ∗ ) x i = P ( y j )    Or, in less fo rmal language, for every p ai r of a tuple fr om R and a tu ple fr om S ob eyi ng the c ondition x i = P ( y j ) , we form c omp osite of t he i th r emnant of the tuple fr om R , and the j th r emnant of t he tuple fr om S . Example 5.4 . Given a diagr am r epr esenting the re lation R : I → X 1 × · · · × X m , c onsider forming a new diagr am by linking the i th and j th le gs of t he original diagr am via a p ermutation P . (5.6) R P X 1 X m X i X j The r ela tion c orr esp onding to this diagr am is given by: (5.7) ∗ ∼  ( x 1 , . . . , x i − 1 , x i +1 , . . . , x j − 1 , x j +1 , . . . , x m )    ( x 1 , . . . , x m ) ∈ R ( ∗ ) x i = P ( x j )  Or, in less formal language, we take the i, j th -r emnant of every tu ple for whi ch x i = P ( x j ) . Example 5.5. Consider linking two diagr ams, the ﬁrst r epr esenting the r elation R : I → X 1 × · · · × X n the se c ond r epr esenting the r elation S : I → X i via a SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 13 p ermutation P , t o form a new diagr am as shown: (5.8) R S X 1 X n X i P The r ela tion c orr esp onding to this diagr am is given by: (5.9) ∗ ∼  ( x 1 , . . . , x i − 1 , x i +1 , . . . , x n )    ( x 1 , . . . , x n ) ∈ R ( ∗ ) x i ∈ P ( S ( ∗ ))  Or, in less formal language, we take the i th r emnant of every tuple for which x i ∈ P ( S ( ∗ )) . 5.1. Structure of the construction. It is conv enien t to single out a partic- ular sub-gr oup of the S 4 per mu tation sub-gro up. This consists of the four p ermuta- tions which don’t mix b et ween the sets { 1 , 2 } and { 3 , 4 } : (1)(2)(3)(4) (the identit y), (12)(3)(4), (1)(2)(34) and (1 2)(34). W e term these the phase d p ermutations. All other p ermutations ar e ter med unphase d . W e sing le out this sub-g roup b ecause the relations corresp onding to Sp ek diagrams generated fr om δ Sp ek , ǫ Sp ek , and the four phased permutations hav e a signiﬁcantly simpler genera l for m. W e term such diagrams phase d diagr ams . All o ther diag rams are ter med unphase d diagr ams . A morphism which cor resp onds to a pha s ed diagram is termed a phase d morphi sm , all other morphisms b eing un phase d morphisms . It is straightforw ard to see that a n y unphased Sp e k diagram ca n b e viewed as a co lle ction of phased sub- dia grams linked together via unphased p ermutations. W e refer to these sub-diag rams as zones . F urthermo r e, note that any p er mutation in S 4 can b e wr itten as a pr o duct of phased p ermutations and the unphas ed p ermutation (1)(3)(24) (which we denote by Σ). Since a s ingle phased p ermutation co nstitutes a phased zone with t wo legs, w e can in fa c t view an unphased dia gram a s a collectio n of phased z ones linked together by the per m utation Σ: (5.10) D 1 D 2 D 3 D 4 D 5 Here D i represents a phas ed sub- diagram and a squa re b ox re pr esents a n unphased per mu tations. Note that such a dia gram is not necessar ily planar, i.e. it may in volve crossing wires . W e distinguish b etw een external zones whic h hav e external legs (e.g. D 1 , D 2 and D 5 in dia gram 5 .10), a nd intern al zones all of whose legs ar e connected to other legs within the diagr am (e.g. D 3 and D 4 ). T o the exter na l legs we asso ciate an enumeration, such that legs from the s ame externa l zone app ear cons e cutiv ely . In the ﬁrst s tage of the proo f we determine the gener al for m of the rela tion corres p onding to a phase d diagr am . This stage its e lf splits into tw o phases: ﬁrst we determine the genera l for m of the morphisms of a new categor y HalfSp ek , and 14 BOB COECKE AND BILL EDW ARDS then we show how to use this result to prov e our main result in Sp ek . In the second stage we draw on the res ults o f the ﬁr st to determine the ge neral form of the re la tion corresp onding to any Sp ek diagra m. 5.2. The general form of the morphisms of HalfSp ek. W e build up to the full theorem via a simpliﬁed ca s e. F or this we need a new category . Deﬁnition 5.6. The categor y HalfSp ek is a sub categor y o f FRel . It is deﬁned inductively , as follows: • The ob jects of HalfSpek are the single-element set I = {∗} , the tw o element set I I := { 0 , 1 } , and its n -fold Car tesian pro ducts I I n . • The morphisms of H al fSp ek are all those relations generated b y co m- po sition, Ca rtesian pro duct a nd relational converse from the following generating re lations: (1) All p ermutations { σ i : I I → I I } of the tw o element set. There ar e 2 such p ermutations, the identit y and the op era tion σ whic h swaps the elements of I I. T ogether they form the g roup Z 2 . (2) A relation δ Half : I I → I I × I I deﬁned by: 0 ∼ { (0 , 0) , (1 , 1 ) } 1 ∼ { (0 , 1) , (1 , 0) } ; (3) a relation ǫ Half : I I → I :: 0 ∼ ∗ Prop ositio n 5.7. [ 6 ] (I I , δ Half , ǫ Half ) is a b asis structur e in FR el and henc e also in HalfSp ek . Remark 5.8. The existenc e of this b asis structu r e c ame as a surprise to the au- thors. Naively one might think (as many working in the ar e a of c ate goric al quantum me chanics initial ly did) that on a set X in FR el ther e is a single b asis structu re with δ given by x ∼ ( x, x ) for al l x ∈ X . The ‘b asis ve ctors’ (or c opy able p oints in the language of [ 5 ] ) ar e then the elements of this set. But this is not the c ase. Ther e ar e many ‘non-wel l-p ointe d’ b asis structur es such as (I I , δ Half , ǫ Half ) for which the numb er of c opyable p oints is less t han the nu mb er of elements of the s et . In r ela te d work, Pavlovic has classiﬁe d al l b asis structur es in FR el a nd Evans et al. have identiﬁe d the p airs of c omplementary b asis structur es (in the sense of [ 5 ] ) among these [ 15, 11 ] . Next we determine the ge neral form of the relatio ns which constitute the mor - phisms of HalfSp ek , to which the consider ations made a t the beginning of Section 5 also a pply . W e sa y that an elemen t of I I n has o dd parity if it has an odd n umber of ‘1’ elements, a nd that it has even pa rity if it has an even num ber of ‘1’ elemen ts. W e will use P to r epresent a particula r parity , o dd or even, and P ′ will represent the opp osite parity . Whether an o dd-pa rity n -tuple has an o dd or even num ber of ‘0 ’ elements clear ly dep ends o n whether n itself is o dd or even. W e could ha ve chosen either 0 o r 1 to play the role of la belling the parity; w e ha ve chosen 1 since it will turn out to be more c onv enien t later on. Theorem 5. 9. The relation in HalfSp ek co rresp onding to an n -leg ged HalfSp ek - diagram is a subset of I I n , consisting of all 2 n − 1 n -tuples of a c ertain parity , whic h depe nding on the diagr am ma y either b e even or o dd: if the pro duct of a ll the per mu tations app earing in the dia g ram is the identit y , then parity is even, and if it is σ , then the parity is o dd. SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 15 Proof. W e prove this result by induction on the num ber of g enerators k re- quired to co ns truct the diag ram. Remember that we need only co nsider those diagrams whose corresp onding relatio ns are states. There is just one possible base case ( k = 1), a diagra m comp osed purely of the generator ǫ † Half for which n = 1: the corr esp onding sta te consists o f the single 1 -tuple (0), which is indeed the uniq ue 1-tuple of even parit y . No w consider a diagra m D built from k ge nerators with a corres p onding state ψ co nsisting all 2 n − 1 n -tuples of parity P . It is easily seen that comp osing D with either ǫ † Half , δ Half or δ † Half resp ectively yie lds a diagra m whose corres p onding state consists of all 2 n − 2 n − 1-tuples, all 2 n n + 1-tuples, and a ll 2 n − 2 n − 1-tuples of parity P ; and that comp osing with σ yields a ll 2 n − 1 n -tuples of parity P ′ . Finally consider pro ducing a disconnected diagra m b y laying the ǫ † Half diagram along side D : it is easily seen that the corr esp onding state consists o f 2 n n + 1-tuples of parity P .  5.3. The general form of phased m orphisms in Sp ek. W e wan t to apply our r e sults o n HalfSp ek to the c a tegory of rea l interest, Sp ek . T o do this w e ﬁrst need to digress to discuss s ome structural features of relations. The categor y Rel has a nother symmetr ic monoidal structure, namely the disjoint union o r dir e ct sum , denoted by ⊔ . Concretely , if we ca n partition a set A into m subsets A i , then we hav e A = ⊔ i A i , and reca lling that a rela tion R : A → B is a subset of A × B , we can decomp ose R in to mn co mponents of the for m R i,j : A i → B j , such that R = F i,j R i,j . The relations R i,j : A i → B j are termed the c omp onents of R with r esp e ct to p artitions A = ⊔ i A i , B = ⊔ j B j . In categor y theoretic terms, this is bipr o duct , and there is a distributive law with res pect to the Cartesian pro duct: (5.11) A × ( ⊔ i B i ) = ⊔ i ( A × B i ) R × ( ⊔ i T i ) = ⊔ i ( R × T i ) for sets A, B i and relations R, T i . F or A = ⊔ i A i , B = ⊔ j B j , C = ⊔ k C k and D = ⊔ l D l , and relations R : A → B , S : B → C and T : C → D , w e then ha ve: (5.12) ( S ◦ R ) i,k = G j S j,k ◦ R i,j ( R × T ) i,j,k,l = T k,l × R i,j R c j,i = ( R i,j ) c F or A = A 1 ⊔ A 2 by distributivity w e have: (5.13) A m 1 ⊔ ( A m − 1 1 × A 2 ) ⊔ ( A m − 2 1 × A 2 × A 1 ) ⊔ · · · ⊔ ( A m − 2 1 × A 2 2 ) ⊔ · · · ⊔ A m 2 If for A = A 1 ⊔ A 2 and R : A m → A n the only non-empt y comp onents ar e R 1 : A m 1 → A n 1 and R 2 : A m 2 → A n 2 we call it p ar al lel . Giv en pa rallel relatio ns R : A m → A n , S : A n → A p and T : A p → A q with respec t to the partition A = A 1 ⊔ A 2 , the relations: (5.14) S ◦ R : A m → A p T × R : A m + p → A n + q R c : A n → A m are all also easily seen to b e parallel with respect to the same partition o f A . W e ca n use these insights to make a connection b etw een HalfSp ek a nd Sp ek . Prop ositio n 5.10. The gener ators of the pha se d morphi sms of Sp ek , i. e. δ Sp ek , ǫ Sp ek and the phase d p ermutations on I V , ar e al l p ar allel with r esp e ct to the fol lo wing p artition of I V = { 1 , 2 } ⊔ { 3 , 4 } . We c onclude that al l phase d morph isms of Sp ek ar e also p ar al lel with r esp e ct to this p artition. We r efer t o the two c omp onent s of a phase d Sp ek morphism as its { 1 , 2 } - c omp onent and { 3 , 4 } -c omp onent. 16 BOB COECKE AND BILL EDW ARDS Prop ositio n 5.11 . The { 1 , 2 } - c omp onents of t he gener ators of the ph ase d mor- phisms of Sp ek ar e simply the gener ators of HalfSp ek with the elements of I I = { 0 , 1 } r e-lab el le d ac c or ding t o 0 7→ 1 , 1 7→ 2 . Similarly the { 3 , 4 } -c omp onents of the gener ators of the phase d morphisms of Sp ek ar e simply the gener ators of HalfSp ek with the elements of I I = { 0 , 1 } r e-lab el le d ac c or ding to 0 7→ 3 , 1 7→ 4 . Prop ositio n 5.12. A state ψ ⊂ I V n c orr esp onding t o a phase d Sp ek diagr am D is e qual to the union of two st ates ψ 12 ⊂ { 1 , 2 } and ψ 34 ⊂ { 3 , 4 } . ψ 12 and ψ 34 ar e obtaine d by the fol lowing pr o c e dur e. F orm a HalfSp ek diag r am D 12 by r eplacing every o c cur enc e of δ Sp ek and ǫ Sp ek in D with δ Half and ǫ Half , and r epla cing every o c cure nc e of a p ermu tation with its { 1 , 2 } c omp onent, r e-lab el le d as a HalfSp ek p ermutation as describ e d in pr op ositio n 5.1 1 . F orm a se c ond HalfSp ek dia gr am D 34 in the obvious analo gous fashion using { 3 , 4 } c omp onents of p ermutations. ψ 12 and ψ 34 ar e t he st ates c orr esp onding to D 12 and D 34 , onc e aga in under the re - lab el ling describ e d in pr op osition 5.11. Note that D 12 and D 34 will appe a r identical as graphs, b oth to each o ther and to D , but the la bels on so me of their p er m utations will diﬀer. F rom pro po s ition 5.12 a nd theorem 5.9 now fo llows: Theorem 5. 13. A phase d morphism in Sp ek of typ e I → I V n is a su bset of I V n , c onsisting of 2 n n -tuples, divid e d int o two classes of e qual numb er: • The ﬁrst class c onsists of t u ples of 1s and 2s, al l of either o dd or even p arity. • The se c ond class c onsists of t uples of 3s and 4s, again al l of either o dd or even p arity. Note tha t we are adopting the conv en tion that tuples of the ﬁrs t cla s s hav e o dd parity if they ha ve an o dd n umber of 2 s, even parity if they ha ve an even num ber of 2s. T uples o f the ﬁr st clas s hav e o dd parity if they hav e an o dd n umber o f 4s, even parit y if they hav e an ev en num be r of 4s. 5.4. The general form of arbitrary morphism s i n Sp ek. Recall fr om section 5.1 that an y unphased diagram can be viewed as a collection of phased zones linked together by the pe rmut ation Σ, (1)(3)(24). W e also enumerated the external legs. Theorem 5.14 . The r elation in Sp ek c orr esp onding to an n -le gge d Sp ek -diagr am with m zones, none of which is internal, is a subset ψ of I V n with the fol lowing form: (1) It c ontains 2 n n -tuples; e ach entry c orr esp onding to an external le g. (2) Al l those ent ries c orr esp onding to a given zone of the diagr am ar e terme d a zone of the tuple (whether we ar e r eferring to a zone of a diagr am, or zone of a t uple should b e cle ar fr om the c ontext). Each zone of the tu ple has a wel l -deﬁne d type ( c omp onents either al l 1 or 2, or al l 3 or 4) and parity (as deﬁne d for phase d r elations). (3) A blo ck B is a subset of ψ su ch t hat t he i th zone of al l n -tuples in B has the same p arity and typ e. The se quenc e of typ es and p arities of e ach zone is c al le d the sig nature of the blo ck. The 2 n tuples of ψ ar e p artitione d into 2 m e qual ly size d blo cks e ac h with a u n ique signatur e. (4) Each of t he 2 m blo cks has a diﬀer ent typ e signatu r e - these exhaust al l p ossibl e typ e signatur es. SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 17 (5) The p arity signatur e of a blo ck is the fol lowing simple function of its typ e signatur e: (5.15) P i = Ψ i ( T i ) + X j ∈ adj ( i ) ( T i + T j ) Her e P i and T i ar e Bo ole an variable s r epr esenting the p arity and typ e of the i th zone. We adopt the c onvention that an o dd p arity is r epr esent e d by 0 and an even p arity by 1, whilst the typ e { 1 , 2 } is r epr esente d by 0 and the typ e { 3 , 4 } by 1. The set adj( i ) c onsists of the z ones dir e ctly adjac ent to the i th zone. Ψ i ( T i ) denotes the p arity of the typ e T i tuples in the r elation c orr esp onding t o t he i th zone se en as an indep endent phase d diagr am (r e c al l the or em 5.13). Theorem 5.15 . The r elation in Sp ek c orr esp onding to an n -le gge d Sp ek -diagr am with m zones, of which m ′ ar e external is either: • A subset, ψ of I V n which satisﬁes c onditio ns (1) and (2) of the or em 5.14 and which is p artitio ne d into 2 m ′ blo cks. The signatur es of these blo cks ar e determine d as fol lows: (1) Be gin with t he state c orr esp onding to t he diagr am obtaine d by addi ng an external le g to every internal zone (i.e. we wil l ha ve 2 m blo cks, e ach with m zones, exhausting al l p ossible typ e signatur es). (2) Eliminate al l blo cks whose typ e signatur es do not satisfy the fol lowing c onstr aints, one for e ach internal zone: (5.16) Ψ i ( T i ) + X j ∈ adj ( i ) ( T i + T j ) = 0 wher e i is t he lab el of the internal zone, and j lab els its adjac ent zones. (3) Final ly fr om e ach blo ck delete the zones c orr esp onding to internal zones. • or it is e qual to the empty set, ∅ . This se c ond p ossibility o c curs iﬀ the c onstr aints in e quation 5.16 ar e inc onsistent. A simple counting argument shows that within each blo ck, every tuple with the correct type and parity signature occur s. Theorems 5.14 and 5.15 th us completely characterise the state corresp onding to any Spe k -diagram. The input data is the shap e of the diagram, which deter mines adj(i) , and the ‘intrinsic parities’ Ψ i ( T i ) of each zone . F uthermor e, note from theorem 5.13 that, as we would exp ect, the general for m o f phas ed morphisms is a sp ecial case of the for m des c rib ed a bove, with m = m ′ = 1. Example 5.16. We now give an example of the or em 5.14 in action. Consid er the fol lowing schematic Sp ek -diagr am (cir cles simply denote zones), which has thr e e zones, al l ex ternal ( m = 3 ) and ﬁve extern al le gs ( n = 5 ): (5.17) ❥ ❥ ❥ 1 2 3 Odd Even Odd Odd Odd Even 18 BOB COECKE AND BILL EDW ARDS wher e the lab els by e ach zone denote the intrinsic p arities of that zone (i.e. the p arity of the typ e-12 tuples and the p arity of the typ e-34 tu ples, r e c al l the or em 5.13). We c onclude fr om the or em 5.14 that the state c orr esp onding to this diagr am wil l c onsist of 2 5 = 32 5-tu ples e ach with 3 zones, and that these wil l b e p artitione d into 2 3 = 8 e qual ly size d blo cks. Every c ombination of typ es app e ars exactly onc e amongst these blo cks, and the p arity signatur es ar e e asily determine d fr om e quation 5.1 5 as fol lows. First we note, fr om the int rinsic p arities that Ψ 1 ( T 1 ) = 0 , Ψ 2 ( T 2 ) = T 2 and Ψ 3 ( T 3 ) = 1 + T 3 . We then se e t hat e qu ation 5.15 her e is essential ly thr e e e quations: (5.18)    P 1 = T 2 + T 3 P 2 = T 1 + T 2 + T 3 P 3 = 1 + T 1 + T 2 + T 3 F r om this we c an est ablish the signatur es of the eight blo cks: (5.19) ( Odd , 12; Odd , 12; Even , 12) ( Even , 1 2; Even , 12; O dd , 34 ) ( Even , 12; Even , 34; Odd , 1 2) ( Odd , 12 ; O dd , 34 ; Even , 34) ( Odd , 34; Even , 12; Odd , 12) ( Even , 3 4; Odd , 1 2; Even , 34 ) ( Even , 34; Odd , 34; Even , 1 2) ( Odd , 34 ; Even , 3 4 ; Odd , 34) The ﬁ rst zone wil l have two elements, the se c ond zone one element and the t hir d zone two elements. Within e ach blo ck every p ossible c ol le ction of tu ples c onsistent with the signatur e wil l app e ar me aning that e ach blo ck wil l c onsist of four tuples. Example 5.17. We go on t o give an example of the or em 5.15. Consider the fol lowing diagr am: it is identic al t o the diagr am in the pr evious example exc ept that the se c ond zone has b e en internalise d. (5.20) ❥ ❥ ❥ 1 2 3 This diagr am has t wo external zones m ′ = 2 and four external le gs n = 4 ; thus we exp e ct the c orr esp onding state to c onsist of 2 4 = 16 4-tuples, e ach with 2 zones, p artitione d into 2 2 = 4 blo cks. We now determine t he signatur es of the blo cks applyi ng the or em 5.15. A c c or ding to step (1) we b e gin with the blo cks fr om the pr evio us example. Step (2) r e quir es that we eliminate al l blo cks whose t yp es do not satisfy the c onstr ai nt T 1 + T 2 + T 3 = 0 . This le aves: (5.21) ( Odd , 12; Odd , 12; Even , 12) ( Odd , 12 ; Odd , 3 4; Even , 34 ) ( Even , 3 4; Odd , 1 2; Even , 34 ) ( Even , 34; Odd , 34; Even , 1 2) Final ly we delete t he se c ond zone fr om e ach blo ck, le aving: (5.22) ( Odd , 12; Even , 12) ( Odd , 12; Even , 34) ( Even , 34; Even , 34) ( Even , 34; Even , 12) The pro o fs o f theorems 5 .14 and 5.1 5 pro ceed as inductio ns ov er the proces s of building up a diagra m b y linking toge ther phased z o nes via the p ermutation Σ. In the cours e of this pro ces s it is c le a rly po ssible for an ex ter nal zone to become an in ternal z o ne, as its la st external leg is linked to some other zone - we r e fer to SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 19 this as internalising a zone. It turns out that internalisation of a zo ne complicates the inductiv e pro of. T o get around this we do the induction in tw o stages. In the ﬁrst stage we build up a diagram identical to the one w e are aiming for , exc ept that every zone that should be internal is giv en a single exter nal leg in the following fashion. Supp ose we need to link together tw o zones via a p ermutation, and this will result in the in ternalisatio n of the left hand zone: (5.23) W e instead link the left hand zone to the p ermutation via a δ Sp ek morphism. The δ Sp ek morphism b ecomes part of the origina l zo ne, and provides it with an external leg: (5.24) In the se c o nd s tage we ca p o ﬀ a ll these e x tra exter nal legs with the ǫ Sp ek morphism. Since δ Sp ek and ǫ Sp ek constitute a basis structure the result is the diagram which we a re aiming for: (5.25) = 5.4.1. Diagr ams without internal z ones. W e now prov e theore m 5.14. The pro of uses an induction ov er the proc e s s of building up a diag ram. This building up ca n be split into tw o phases: ﬁrstly we connect all of the zo nes w hich will appea r in the ﬁnal diagram into a ‘tree-like’ structure with no clo sed lo ops, secondly we c lo se up any lo ops necessar y to yield the desired diag ram. Henceforth we will represent the signature of a tuple with m zones as ( P 1 , T 1 ; . . . ; P m , T m ) where P i is the parit y o f the i th zone, and T i is its type. Ag ain, if P is a parity , P ′ indicates the o ppo site parity , and likewise if T is a t yp e, T ′ represents the other t yp e . Lemma 5.18. Consider an n -le gge d n on- phase d diagr am D 1 with m zones. Sup- p ose the c orr esp onding state ψ 1 ⊂ I V n satisﬁes the c onditions in the or em 5.14. Now c onsider linking the i th le g of D 1 to the j th le g of an n ′ -le gge d ph ase d diagr am D 2 (with c orr esp ondi ng state ψ 2 ), via Σ , to cr e ate an ( n + n ′ − 2) - le gge d diagr am D 3 with m + 1 ex t ernal zones. We wil l assume that the i th le g of D 1 lies within its k th external zone. The state ψ 3 ⊂ I V n + n ′ − 2 c orr esp onding to D 3 also satisﬁes the c onditio ns in t he or em 5.14. Proof. By the 1- i th -remnants of ψ 1 we mea n the i th -remnants of those tuples in ψ 1 with a 1 in the i th po sition. W e deﬁne the 2 -, 3 -, and 4- i th -remnants similar ly . By prop osition 5.3 the elemen ts of ψ 3 comprise all the p ossible comp osites of the x - i th -remnants of ψ 1 and the Σ( x )- j th -remnants of ψ 2 , where x = 1 , . . . , 4. It is cle ar that the zo ne s tructure of the tuples of ψ 1 is inherited by these comp osites, a nd that the Σ( x )- j th -remnants of ψ 2 constitute an additiona l zone within the co mp osites 20 BOB COECKE AND BILL EDW ARDS - w e conven tionally c o nsider this to b e the ( m + 1) th zone. Thus the tuples of ψ 3 satisfy conditio n (2). Consider a block B ⊂ ψ 1 , with sig nature ( P 1 , T 1 ; . . . ; P k , T k ; . . . ; P m , T m ), in which for deﬁniteness we a ssume that T k = 0 (the ar gument runs entirely analo - gously if T k = 1). The comp osites of the 1 - i th -remnants of B and the 1 - j th -remnants of ψ 2 all hav e the same s ignature, ( P 1 , T 1 ; . . . ; P k , T k ; . . . ; P m , T m ; Ψ , T k ), and c o n- stitute a blo ck B 1 ⊂ ψ 3 . Lik ewise the comp osites o f the 2- i th -remnants of B and the 4- j th -remnants o f ψ 2 constitute a blo ck B 2 ⊂ ψ 3 of signa ture ( P 1 , T 1 ; . . . ; P ′ k , T k ; . . . ; P m , T m ; Ψ ′ , T ′ k ). Th us each ‘parent’ block in ψ 1 gives rise to tw o ‘prog eny’ blo cks in ψ 3 . By h yp othesis, each blo ck B ⊂ ψ 1 has a unique t yp e signature, th us the progeny blo cks derived from diﬀerent pa rent blocks are distinct. Th us ψ 3 is par- titioned into 2 m +1 blo cks, thus satisfying co ndition (3). It is also clear that if all po ssible type signatures are represe nted by the 2 m blo cks o f ψ 1 then this is also true for the 2 m +1 blo cks of ψ 3 , and th us tha t ψ 3 satisﬁes co ndition (1). Note that B consists o f 2 n − m n -tuples, a nd will hav e 2 n − m − 1 1- i th -remnants and a similar num ber of 2- i th -remnants, all of whic h will b e distinct. Similar ly ψ 2 will ha ve 2 n ′ − 2 1- i th -remnants and a simila r num ber of 4- i th -remnants, again all distinct. Thus, b oth B 1 and B 2 will consist of 2 n − m − 1 . 2 n ′ − 2 = 2 ( n + n ′ − 2) − ( m +1) tuples. This holds for all blocks B ⊂ ψ 1 , of which there are 2 m . Thus in total ψ 3 consists of 2 n + n ′ − 2 tuples, a nd so satisﬁes condition (4). Finally we turn to condition (5 ). Recall from ab ov e that a parent block of signature ( P 1 , T 1 ; . . . ; P k , T k ; . . . ; P m , T m ) yields tw o progeny blo cks, of s ig natures ( P 1 , T 1 ; . . . ; P k , T k ; . . . ; P m , T m ; Ψ , T k ) a nd ( P 1 , T 1 ; . . . ; P ′ k , T k ; . . . ; P m , T m ; Ψ ′ , T ′ k ). Note that those pro geny blocks for which the k th zone a nd its new a dja c e n t zone hav e diﬀeren t type s exhibit a parit y ﬂip on the k th zone, relative to the pa rent blo ck, and on the new adjacent zone, rela tiv e to its ‘intrinsic pa rity’. No suc h ﬂip o ccurs if the zones hav e the same t yp e. No te that the term T i + T j is equal to 0 if T i = T j and 1 if T i 6 = T j . If condition (2) ho lds for ψ 1 then the correct par it y signature for the blo cks of ψ 3 can b e obtained simply by adding the term T k + T m +1 to the P k and P m +1 equations (equation 5.15). Thu s we conclude that co ndition (5) will hold for ψ 3 as well.  Lemma 5. 19. Consider an n -le gge d diagr am D with m external zones. Supp ose the c orr esp onding state ψ satisﬁes the c onditions in the or em 5.14. Now c onsider forming a new ( n − 2) -le gge d diagr am D ′ , with c orr esp onding state ψ ′ , by linking the i th le g of D (in t he k th zone of D ), to the j th le g ( in the l th zone), via Σ . ψ ′ also satisﬁes the c onditio ns in the or em 5.1 4. Proof. By the x, y - i, j th -r emnants of a set of tuples we mean the i , j th -remnants of a ll those tuples with x in the i th po sition and y in the j th po sition. F rom prop o- sition 5.4, ψ ′ consists of the x, Σ( x )- i, j th -remnants of ψ . The zone structure of ψ is clear ly inherited by these remnan ts, thus the tuples of ψ ′ satisfy conditio n (2). Consider a blo ck B ⊂ ψ with signature ( P 1 , T 1 ; . . . ; P k , T k ; . . . ; P l , T l ; . . . ; P m , T m ). It is straig ht forward to se e that only one quarter of the tuples in this blo ck hav e i, j th -remnants whic h are x, Σ( x )- i, j th -remnants. All the x, Σ( x )- i, j th -remnants hav e the same signa ture: ( P 1 , T 1 ; . . . ; P k , T k ; . . . ; P l , T l ; . . . ; P m , T m ) if T k = T l and ( P 1 , T 1 ; . . . ; P ′ k , T k ; . . . ; P ′ l , T l ; . . . ; P m , T m ) if T k 6 = T l . Thus each ‘par en t’ block in ψ gives ris e to one ‘pr ogeny’ blo ck in ψ ′ , with o ne quarter as many tuples. The type sig natures of the pr ogeny blocks are identical to those o f the parent SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 21 blo cks; b y hypo thesis each parent blo ck had a diﬀerent t yp e signature and so the progeny blo cks deriv ing from diﬀeren t parent blo cks ar e all dis tinct. Thus ψ ′ is partitioned in to 2 m blo cks, each containing 2 n − m / 4 = 2 ( n − 2) − m tuples, a nd s o satisﬁes conditions (1) and (3). Since the type sig natures of progeny and pa rent blo cks ar e identical, if ψ satisﬁes condition (4), so will ψ ′ . W e now turn to condition (5 ). Closing a lo op betw een the k th and l th zones means that they no w become adjacent to eac h other. In the previous paragraph we sa w that a pro geny blo ck for which the k th and l th zones have diﬀerent types exhibits a parit y ﬂip on b oth these zo nes. Using similar reasoning a s in the previous lemma we co nclude that, if condition (2) holds for ψ , the cor rect par it y sig nature for the blo cks of ψ ′ can be obtained simply by adding the term T k + T l to the P k and P l equations (equatio n 5.15). Thus we conclude that condition (5) will ho ld for ψ ′ as well.  W e can no w prove Theorem 5.14 : Proof. By induction. The base cas e is a diagram cons is ting of a single zone, and it is clea r from theor em 5.13 that this satisﬁes all the conditions . An y other diagram is built up via tw o inductive steps: linking a new phased z one onto the existing diagram and clos ing up internal lo ops within the diagr am. Lemmas 5.1 8 and 5 .19 r espe ctively show that if a diag ram sa tis ﬁe d the conditions prior to either of these steps, the r esulting new diagram will also satisfy the conditions.  5.4.2. Diagr ams with internal zones. W e now addr ess the issue of in ternalising zones. Rec all that this step inv olves capping o ﬀ ex ternal legs with the ǫ Sp ek relation. Throughout this sectio n D will denote a diagra m with no internal zones, and D ′ will denote the diagr am obtained by internalising so me o f D ’s zones. The co rresp onding states will be ψ and ψ ′ . Prop ositio n 5.20. Supp ose we obtain D ′ by c apping t he k th external le g of D with the ǫ Sp ek morphism. F r om lemma 5.5 we c onclude that ψ ′ c onsists of t he 1- k th - and 3- k th -r emnants of ψ . Suppo se that in going from D to D ′ we internalised the i th zone o f D . F rom the pro po sition ab ov e we deduce that each blo ck B ⊂ ψ for which P i = 0 will give rise to one progeny blo ck B ′ ⊂ ψ ′ with the same num ber of tuples as B , while all those blocks fo r which P i = 1 will give rise to no progeny blo cks. F rom this w e can conclude that those blocks which do give rise to prog en y blo cks satisfy a constra int on their type signa tures, de r ived fr om setting P i = 0 in equation 5.15. Dep ending on the for m of Ψ i ( T i ), and whether the num b er of z o nes adjacent to the i th is o dd or even, this cons tr aint takes one of four forms: (5.26) X j ∈ adj( i ) T j =  0 1 (5.27) T i + X j ∈ adj( i ) T j =  0 1 W e descr ib e the constra in ts in (5.26) a s typ e-0 c onstr aints , and tho s e in (5.2 7) as typ e-1 c onst r aints . 22 BOB COECKE AND BILL EDW ARDS Suppo se D has n external leg s a nd m externa l zo nes . Suppose that in going to D ′ we internalise p of its zones . Each internalisation gives rise to a c orr esp onding c onstr aint. There ar e now tw o p ossibilities: (1) Not all of the c o nstraints are co nsistent. In this case none o f the blo cks in ψ satisfy a ll of the constraints, and none of them will give ris e to proge ny blo cks. Th us ψ ′ = ∅ . (2) All p c onstraints a re consistent, and of these p ′ are linear ly indep endent (this essen tially means that p − p ′ of the constraints can b e derived fro m the remaining p ′ ). E ach indep enden t constr aint reduces the num ber of blo cks which ca n g ive rise to pr ogeny by one half. Th us only 2 m − p ′ of the blocks in ψ give r ise to prog e n y blo cks in ψ ′ , and ψ ′ can ha ve a t most 2 m − p ′ blo cks - this maximum is attained if all of the progeny blocks are distinct. Lemma 5. 21. The fol lo wing ar e e quivale nt: (1) The c onstr aints ar e c onsistent and ther e ar e p ′ line arly indep endent c on- str aints. (2) The 2 m − p ′ blo cks in ψ which c an give rise to pr o geny blo cks in ψ ′ ar e p artitione d into 2 m − p sets, e ach c onsisting of 2 p − p ′ blo cks which al l yield identic al pr o geny blo cks. Thus in total ther e ar e 2 m − p distinct pr o geny blo cks. F or br evity we wil l descri b e this as ( p − p ′ ) - fold duplication of pr o geny blo cks. The pr o of of this lemma re q uires a num b er of preliminary deﬁnitions . Deﬁnition 5.22. The IZ-set is the set of zones which are in ternalised in go ing from D to D ′ . The non-int ern alise d adjac ent zones (nIAZs) o f an in ternalised zone are a ll the zones a djacent to it whic h are not themselves members o f the IZ -set. An adjac ency closur e set (ACS) is a subset of the IZ-s et with the minimal n umber of elements suc h that the disjoint union of the nIAZs of each element contains each nIAZ an ev en num b er of times. Example 5.23. Consider t he fol lowing seven zone diagr am (external le gs ar e sup- pr esse d for clarity). The ﬁl le d-in zones ar e those which we internalise. (5.28) ① ① ❤ ❤ ① ❤ ① 1 2 3 4 5 6 7 The IZ-set is { 1 , 3 , 6 , 7 } . The nIAZs for 1 ar e { 2 } for 3 ar e { 4 , 5 } , for 6 ar e { 4 } and for 7 ar e { 5 } . Zones 3, 6 and 7 t o gether c onstitute an ACS. Zone 1 is not p art of any A CS. Deﬁnition 5.24. Given a set S of zones in D , and a blo ck B ⊂ ψ , the S -mirr or of B , B S is the blo ck with the same t yp e signature a s B except o n the zones in S , where the t yp es ar e opp osite. Prop ositio n 5.25. Supp ose D has an ACS R . Now, so long as the blo cks B , B R ⊂ ψ (i.e. a blo ck and its R -mirr or) b oth yield pr o geny blo cks in ψ ′ , these pr o geny blo cks wil l b e identic al. Conversely, if any two blo cks B , B ′ ⊂ ψ yield identic al pr o geny blo cks in ψ ′ , they must b e mirr or e d with r esp e ct t o some A CS in D . SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 23 Proof. F o r t wo blo cks in ψ to give identical progeny blo cks in ψ ′ they must hav e identical t yp e and parity o n every zone which is no t in ternalis ed. Note from equation 5.15 that tw o blo cks in ψ which a re type mir rored o n a sing le zone will otherwise diﬀer only in parities on all the zo nes adjace nt to this zone . Now consider t wo blocks which are type-mirr ored on a s et of zones R : in the case wher e R constitutes an ACS all the pa rity ﬂips predicted by equation 5.15 ca nc e l one another out on the zones which will still b e visible in ψ ′ .  Prop ositio n 5. 26. Su pp ose D has an ACS R whose memb er zones have c orr e- sp onding c onstr aints which satisfy the fol lowing c onditio n: those zones with an o dd numb er of adjac ent zones within the IZ-set have t yp e- 1 c onstr aints, while those zones with an even n umb er of adjac ent zones within the IZ-set ha ve typ e-0 c onst r aints. Then if a blo ck B ⊂ ψ satisﬁes the c onstr aints and gives rise to pr o geny blo cks in ψ ′ , so do es its R -m irr or. Conversely, if a blo ck B ⊂ ψ and its mirr or with r esp e ct to some AC S R in D b oth give rise to pr o geny blo cks, the c onstr aints c orr esp onding to the zones of R must satisfy the c ondition ab ov e. Proof. If a co nstraint contains an even n umber of terms rela ting to zones from a s et R then given a block B ⊂ ψ either (i) b oth B a nd B R satisfy the constraint (ii) neither B nor B R satisfy the constr a int . Conv ersely , if b oth B and B R satisfy a constraint, it mu st cont ain a n even n umber o f terms relating to zo nes from R .  Prop ositio n 5. 27. Su pp ose D has an ACS R whose memb er zones have c orr e- sp onding c onstr aints which satisfy the c ondition in t he pr evious pr op osition. Then the c onstr aints to gether form a line arly dep endent set. The c onverse is also true. Proof. If R is an A CS a nd the condition on constraints is satisﬁed then each term a ppea rs in the constra in ts a n ev en n um b er of times a lto gether. Summing all the co nstraints tog e ther then r e sults in all the terms ca ncelling out, y ie lding the single equation 0=0. This is a necessa ry and suﬃcient condition for the constraints to b e line a rly dep endent.  W e can no w prove lemma 5.21. Proof. F r om pr op ositions 5 .2 5 and 5.26 w e conclude that for n -fold duplica- tion to take pla c e the IZ-set must contain n ACSs and that the constr aints must satisfy the condition in prop os ition 5.26. The conv erse is also clearly true. Ev ery linearly dependent set a mongst the c onstraints corre s po nding to the in ternalis ed zones reduces the total num ber of linearly indep endent constraints b y one. F rom prop osition 5 .27 we conclude that for there to be n linearly dependent sets the IZ-set must contain n A CSs and that the constraints m ust satisfy the co ndition in prop osition 5.2 6. The conv erse is also clear ly true. Thus we conclude that b o th statements in the lemma are equiv alent to a third statement: the IZ-set contains p − p ′ A CSs, and the constraints corr esp onding to the in ternalise d zones s atisfy the condition of prop osition 5.26.  Corollary 5. 28. Given a di agr am D without int ernal zones which satisﬁes the c onditio ns in the or em 5.14, a diagr am D ′ with internal zones forme d by c apping oﬀ external le gs of D with ǫ Sp ek morphisms wil l satisfy the c onditions in t he or em 5.15. Theorem 5 . 15 follows as a straightforward co rollary . 24 BOB COECKE AND BILL EDW ARDS 5.5. The g eneral form of the m orphisms of MSp ek. Theorem 5.29. All MSp ek mor phisms of type I → I V n are subsets of I V n con- taining 2 n , 2 n +1 , . . . , 2 2 n − 1 or 2 2 n n -tuples. Proof. An y MSp ek diagr am D ′ can be obtained fr om a Sp ek diagram D simply b y capping one or more legs of D with the morphism ⊥ MSp ek . Supp ose we obtain D ′ by ca pping a single externa l leg of D (the i th , say) with ⊥ MSp ek . The state ψ ′ corres p onding to D ′ consists of the i th -remnants of ψ , the state co r resp ond- ing to D . Suppo se D has n external legs: s ince D is a Sp ek -diagram ψ consists of 2 n tuples. Then, unless some o f the i th -remnants of ψ are iden tical, ψ ′ will also consist of 2 n tuples, despite only having n ′ = n − 1 external leg s. F urthermo re it is clea r that either all the i th -remnants of ψ are distinct, o r ψ is par titioned in to pairs, the elements o f which yield identical i th -remnants, meaning that the addition of each ⊥ MSp ek cap either halves the num ber o f tuples or leaves it unc hanged.  6. MSp ek is the pro cess category for the to y theory W e know that the epistemic states of the to y theory are subsets of the sets I V n , and that the transforma tions o n these states are relations b etw een these sets. Thu s we can see immediately that the toy theo ry’s pro cess ca teg ory m ust b e so me sub-categor y o f FRel , restric ted to the ob jects I V n . F ur thermore we know that it cannot b e the full sub-ca tegory r estricted to these o b j ects, since so me s ubsets of I V n clearly vio late the k nowledge bala nce principle. W e will show no w that (a strong candidate fo r) 4 the pro cess categor y for the toy theo r y in its en tirety is MSp ek , while if we r estrict the toy theo r y to states of maximal knowledge (consistent with the knowledge balance principle), the proce s s category is Sp ek . Prop ositio n 6.1. The morphisms of the pr o c ess c ate gory of the toy the ory ar e close d u nder c omp osition, Cartesian pr o duct and r elational c onverse. Proof. There is no feature of the toy theory which would put any re strictions on which op erations co uld b e comp osed, so we exp ect the states and tr ansformations to be closed under compos ition. Since the Cartesian product is used b y the toy theory to represe nt comp osite sy stems we a lso exp e ct the states and transfor mations to b e c losed under Cartesian pro duct. Every epistemic state cor resp onds to an outcome for at least o ne measurement (measurements co rresp ond to asking as ma ny questions as pos sible fro m canonical sets, epistemic states cor resp ond to the answ ers). Recalling the discus sion of mea - surement in se c tion 2, w e see that given a state ψ ⊂ I V n the disturba nce r esulting from the corresp onding measurement outcome ca n b e deco mpo sed a s ψ ◦ ψ † , wher e ψ † is the r elational conv erse of ψ . Thus w e exp ect the relational conv erse of each state a lso to feature in the ph ysical c ategory of the theory . The toy theory s ta te corres p onding to the subset Ψ Sp ek = { (1 , 1 ) , (2 , 2 ) , (3 , 3) , (4 , 4) } ⊂ I V × I V a long with its relationa l con verse a re then easily seen to co nstitute a compact structure on I V. W e th us ha ve map-state duality , a nd it is straightforw ard then to show that if states are closed under relational conv erse, so is any morphism in the physical category .  4 Giv en that the toy theory is in f act not unambiguously deﬁned f or more than three systems, there m a y b e other extensions to o. Ours is the mi ni mal extension given comp ositional closure. SPEKKEN S’S TO Y THEOR Y AS A CA TEGOR Y 25 Note that this p oint sharp ens our discussion ab out the consistency of the toy theory , in section 2. If the states and trans fo rmations which Sp ekkens has der ived for up to three systems, under the oper ations o f co mpos ition, Cartesian product and r elational c onv erse, yield states which violate the knowledge ba lance principle, then the theory as pre sent ed is inconsistent. Prop ositio n 6. 2 . Al l of the gener ating morphisms of MSp ek ar e states or tr ans- formations of the t oy the ory, or c an b e derive d fr om them by c omp osition, Cartesian pr o duct or re lational c onverse. Proof. The only gener ator for whic h this is less than obvious is δ Sp ek . This is formed by co mpo sing Spe k kens’s GHZ-lik e state (se e se ction V of [ 20 ]) with the relational converse of the state Ψ Sp ek deﬁned in the pro o f ab ov e.  Prop ositio n 6.3. Al l of the states and tr ansformatio ns derive d by Sp ekkens in his original p ap er [ 20 ] ar e morphisms of MSp ek . When we re strict to states of maximal know le dge al l of t he states and tr ansforma tions ar e morphi sms of Sp ek . Proof. By inspec tio n of [ 20 ].  Corollary 6.4. MSp ek is the minimal closur e u nder c omp ositio n, Cartesian pr o d- uct and r elational c onverse of the s t ates and tr ansformations describ e d in [ 20 ] . Sp ek is the minimal closur e under t hese op er ations of the states of [ 20 ] c orr esp onding to maximal know le dge and the tr ansformations which pr eserve them. Prop ositio n 6.5. Al l states ψ : I → I V n of MSp ek and Sp ek satisfy the know le dge b alanc e principle on the system c orr esp onding to I V n viewe d as one c omplete system. Al l those of Sp ek s atisfy the princip le maximal ly. Proof. Recall that the knowledge balance pr inciple requir es that we can know the answer to at most ha lf of a cano nical question set. A system with n elementary comp onents has 2 2 n ontic states. A canonical set for suc h a system consists of 2 n questions, each answer to a question halv ing the n umber o f p ossibilities for the ontic state. Th us, we kno w the answer to m s uc h questio ns ( m = 0 , . . . , n ), iﬀ our epistemic state is a subset of I V n with 2 2 n − m elements. W e conclude from theorem 5.2 9 that all states of MSp ek satisfy the knowledge balance principle on the system as a whole. W e conclude that a ll states of Sp ek corresp ond to the maximum knowledge ab out the system as a whole consistent with the knowledge balance principle.  Prop ositio n 6.6. Al l states ψ : I → I V n of MSp ek and Sp ek satisfy the know le dge b alanc e principle on every subsyst em of the system c orr esp onding to I V n . Proof. Giv en an epistemic state ψ ⊂ I V n of a comp osite system with n e le - men tary comp onents, the ‘marginal’ s tate on some subsystem is o btained from ψ by deleting from the tuples of ψ the comp onents cor resp onding to the elementary systems which a re no t part o f the subsystem of interest. Suppos e this epistemic state corr esp o nds to a Sp ek or MSp ek diagra m, D . The element ar y systems w hich are not part of the subsystem co rresp ond to a certain c ollection of external leg s of D , and, b y lemma 5.5, if we cap these with the MSp e k genera tor ⊥ MSp ek , the eﬀect o n the state ψ is exa ctly as just describ ed. Comp osing a Sp ek o r M Sp ek morphism with ⊥ MSp ek yields some morphism of MSp ek , which by prop ositio n 6.5 satisﬁes the knowledge balance principle.  26 BOB COECKE AND BILL EDW ARDS F rom cor ollary 6.4 a nd prop ositions 6.5 and 6 .6 we reach tw o key conclusio ns: • The states and transfor ma tions derived by Sp ekkens in [ 2 0 ] for systems of up to three comp onents are all consistent with the knowledge ba la nce principle. • The pro ce ss catego r y of the toy theory m ust, a t least, co nt ain a ll of the morphisms o f MSp ek . The seco nd co nclusion b eg s the question, could MSp ek b e a str ict sub-categ ory of the pro cess catego ry of the toy theo r y i.e. could the toy theory con tain op er- ations not con tained in MSp ek ? It is diﬃcult to answer this questio n, since, as discussed at the end of s ection 2 it is not clear what the rigoro us deﬁnition of the toy theory is, o r whether there is an una m biguous wa y to extend it b eyond three systems. Cer tainly , MSp ek is the pro cess catego ry o f a theory which c oincides with Sp ekkens’s theory up to the case of three qubits, and whose states and tr ans- formations are b o und to satisfy the thr e e rules of section 2 (the ﬁrst t wo rules by prop ositions 6.5 and 6.6, and the third simply by its deﬁnition as the c lo sure under comp osition of a set of generators). It is in this sense tha t we ear lier remarked that MSp ek is a strong candidate for the pro cess ca tegory of the toy theory . 7. Conclusion and outlo o k W e achiev ed our goal stated in the abstra ct, that is, to pr ovide a rig orous mathematical description of Sp ekkens’ to y theory , which pro ves its c o nsistency . This was established bo th in terms of generato rs for a dagger symmetric mono ida l sub c ategory of FRel , cons isting of s y mmetries for the elemen tary system and a basis structur e (and nothing more !), as well as in terms of an ex plicit descriptio n of these relations as in Theorems 5.1 5, 5.14 and 5.29. This des cription mean while a lr eady has prov ed to be of great use, for example, in pinpo int ing wha t the essen tial structura l diﬀerence is b etw een the toy theory and the r elev an t fragment of quantum theory . In joint w ork with Sp ekkens in [ 7 ] we s how ed that the key diﬀerence betw een the toy theory and relev an t fragment of quantum theo ry is the phase gr oup , a group tha t b y pure abstract nonsense can b e attributed to each basis structure. In the ca s e of the toy theory this phase g roup is Z 2 × Z 2 while in the case of the rele v a n t fragment of qua n tum theor y it is Z 4 . One can then show that it is this diﬀere nc e that causes the toy theory to be lo cal, while the relev a n t fra g ment of quantum theor y is no n-lo cal. In this context, one may wonder wether ther e is a genera l ca tegorica l co nstruc- tion which w ould turn a ‘local theo ry’ like Spek into a non-lo cal o ne. W e a lso exp ect that the constructio n in this pa p er can be fair ly str aightforw ardly extended beyond qubit theories, for example to qutrits [ 22 ]. References 1. S. Abramsky (2009 ) No-cloning in c ate goric al quantum me chanics . In: Seman tic T ec hniques for Quan tum Computation, I. Mac kie and S. Ga y (eds), pages 1–28, Camb ri dge Univ ersity Press. 2. S. Abramsky and B. Co eck e (2004) A c ate goric al semantics of quantum pr oto c ols . In: Pro- ceedings of 19th IEEE conference on Logic in Computer Science, pages 415–425 . IEEE Press. arXiv:quant-ph/04 02130 . Revised version (2009): Cate goric al quantum me chanics . 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Selinger (2010) Autonomous c ate gories in which A ≃ A ∗ . Pro ceedings of QPL 2010. 20. R. W. Sp ekk ens (2007) Evidenc e for the epistemic view of quantum states: A toy t he ory . Ph ysical R eview A 75 , 032110. arXi v:quan t-ph/0401052 21. R. W. Sp ekk ens (2007) Axiomatization thr ough foil the ories . T al k, July 5, Uni versity of Cam- bridge. 22. R. W. Sp ekk ens (2009) The p ower of epistemic r estrictions i n r e c onstructing quantum the ory . T alk, August 10, Perimeter Institute. PRISA:09080009 Oxf ord Un iversity, Dep ar tment of Computer Science E-mail addr ess : coecke@cs.o x.ac.uk Oxf ord Un iversity, Dep ar tment of Computer Science Curr ent addr ess : Perimeter Institute f or Theoretical Physics E-mail addr ess : wae28@hotma il.com

Spekkenss toy theory as a category of processes

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