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A categorical semantics for causal structure AleksKissinger and SanderUijlen InstituteforComputingandInformationSciences,RadboudUniversity 7 1 {aleks,suijlen}@cs.ru.nl 0 2 January 26,2017 n a J Abstract 5 2 We presenta categorical construction for modellingboth definite and indefinite causal struc- tureswithinageneralclassofprocesstheoriesthatincludeclassicalprobabilitytheoryandquan- ] tumtheory. Unlikepriorconstructionswithincategoricalquantummechanics, theobjectsofthis h p theoryencode fine-grainedcausal relationshipsbetween subsystemsand givea newmethod for - expressingandderivingconsequencesforabroadclassofcausalstructures.Toillustratethispoint, t n we show that this framework admits processes with definite causal structures, namely one-way a signallingprocesses,non-signallingprocesses,andquantumn-combs,aswellasprocesseswithin- u definitecausalstructure,suchasthequantumswitchandtheprocessmatricesofOreshkov,Costa, q and Brukner. We furthermoregive derivationsof their operational behaviour using simple, dia- [ grammaticaxioms. 2 v 2 1 Introduction 3 7 Broadly,causalstructuresidentifywhicheventsorprocessestakingplaceacrossspaceandtimecan, 4 0 inprinciple,serveascausesoreffectsofoneanother. Forinstance,thecausalstructureofrelativistic . spacetimeisgivenbydemandingeventscanonlyhaveacausalinfluenceonothereventswhichcan 1 0 bereachedwithoutexceedingthespeedoflight. 7 Inthe context of quantum theory, causalrelationships between inputs and outputs to quantum 1 processes have been expressed in a variety of ways. Perhaps the simplest are in the form of non- : v signalling constraints, which guarantee that distant agents are not capable of sending information i X fasterthatthespeedoflight,e.g. toaffecteachother’smeasurementoutcomes[3]. Quantumstrate- gies [15] and more recently quantum combs [4] offer a means of expressing more intricate causal r a relationships,intheformofchainsofcausallyorderedinputsandoutputs. Furthermore,ithasbeen shown recently that one can formulate a theory that is locally consistent with quantum theory yet assumes no fixed background causal structure [22]. Interestingly, such a theory admits indefinite causalstructures.Namely,itallowsonetoexpressprocesseswhichinhabitaquantumsuperposition of causal orders. If physically realisable, such processes can outperform any theory that demands a fixed causalordering in certainnon-local games such as ‘guess your neighbour’s input’ [22] and quantum computational tasks such as single-shot channel discrimination [6]. Perhaps even more surprisingly,ithasbeenshownrecentlythat,inthepresenceofthreeormoreparties,causalbounds canbeviolatedevenwithinatheorythatbehaveslocallylikeclassicalprobabilitytheory[2]. The keyingredientin studying (andvarying) causalstructuresof processesis the development ofacoherenttheoryofhigherordercausalprocesses. Thatis,ifwetreatlocalagents(orevents,labora- tories,etc.) asfirstordercausalprocesses,thentheactofcomposingtheseprocessesinaparticular causal order should be treated as a second-order process. In this paper, we develop a categorical 1 framework for expressing and reasoning about such higher order processes. This starts from the generalcontextofaprecausalcategory,whichisacompactclosed categorythatsatisfiesafourextra axiomsthatprovideenough additionalstructuretoreasonaboutcausalrelationshipsbetweensys- temsandtoprovesimpleno-goresultssuchasnotime-travel,i.e.theimpossibilityofprocesssending informationintoitsowncausalpast. Most importantly, precausal categories have a special discarding process defined on each object, whichenablesonetostatethecausalitypostulateforaprocessΦ: A B: → B Φ = A A Thishasaclearoperationalintuition: iftheoutputofaprocessisdiscarded,itdoesn’tmatterwhichprocess occurred.Whileseeminglyobvious,thiscondition,originallygivenby[5]inthecontextofoperational probabilistic theories, is surprisingly powerful. For instance, it is equivalent to the non-signalling propertyforjointprocessesarisingfromsharedcorrelations[13]. Startingfromaprecausalcategory ,wegiveaconstructionofthe -autonomouscategoryCaus[ ] C ∗ C ofhigher-ordercausalprocesses,intowhichthecategoryof(first-order)causalprocessessatisfying the equation above embeds fully and faithfully. Our main examples start from the precausalcate- goriesofmatricesofpositiverealnumbersandcompletelypositivemaps,whichwillyieldcategories ofhigher-orderclassicalstochasticprocessesandhigher-orderquantumchannels,respectively. Whilecategoricalquantummechanics[1]hastypicallyfocussedoncompactclosedcategoriesof quantumprocesses,weshowthatthis -autonomousstructureyieldsamuchrichertypesystemfor ∗ describingcausalrelationships betweensystems. A simple, yetstriking example is in the use of ⊗ vs. `toformthetypesofprocessesonajointsystem: (A⊸A′) (B⊸B′) non-signallingprocesses ⊗ ← (A⊸A′)`(B⊸B′) allprocesses ← Usingthistypesystem,wegivelogicalcharacterisationsofnon-signallingandone-waynon-signalling processes and then prove that these are equivalent to the operational characterisations in terms of their interactions with the discardingprocess. We thengo on tocharacterisehigher-ordersystems, notablybipartite,andn-partitesecond-ordercausalprocessesandgiveseveralexampleswhichare knowntoexhibitindefinitecausalstructure,namelytheOCBprocessfrom[22],theclassicaltripar- titeprocessfrom[2],andanabstractversionofthequantumswitchdefinedin[6]. Finally,weprove using just the structure of Caus[ ] thatthe switchdoes notadmita causalorderingbyreducingto C notime-travel. Relatedwork. Thisworkwasinspiredby[23],whichaimsforauniformdescriptionofhigher- orderquantum operations in termsof generalised Choi operators. However, rather thanrelyingon the linearstructure of spacesof operators, we workpurelyintermsof the -autonomous structure ∗ andtheprecausalaxioms,whichconcernthecompositionalbehaviourofdiscardingprocesses. The constructionofCaus[ ]isavariantofthedoublegluingconstructionusedin[17]toconstructmodels C oflinearlogic. Inthelanguageofthatpaper,ourconstructionconsistsofbuildingthe‘tightorthog- onality category’induced bya focussed orthogonality on 1 (I,I), then restrictingtoobjects I { } ⊆ C satisfying the flatness condition in Definition 4.2. Since it is -autonomous, Caus[ ] indeed gives ∗ C a model of multiplicative linear logic, enabling us to enlist the aid of linear-logic based tools for provingtheoremsaboutcausaltypes. Wecommentbrieflyonthisintheconclusion. Acknowledgements.BothauthorsaregratefultoPauloPerinottiforhisinputandsharingadraft versionof[23].WewouldalsoliketothankCˇaslavBrukner,MattyHoban,andSamStatonforuseful discussions. This work is supported by the ERC under the EuropeanUnion’s Seventh Framework Programme(FP7/2007-2013)/ERCgrantno320571. 2 2 Preliminaries We work in the context of symmetric monoidal categories (SMCs). An SMC consists of a collection of objects ob( ), for every pair of objects, A,B ob( ) a set (A,B) of morphisms, associative C ∈ C C sequentialcomposition‘ ’withunits1 forall A ob( ),associative(uptoisomorphism)parallel A ◦ ∈ C composition ‘ ’ for objects and morphisms with unit I ob( ), and swap maps σ : A B A,B ⊗ ∈ C ⊗ → B A, satisfying the usual equations one would expect for composition and tensor product. For ⊗ simplicity,wefurthermoreassume isstrict,i.e. C (A B) C = A (B C) A I = A = I A ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ThisisnolossofgeneralitysinceeverySMCisequivalenttoastrictone.Fordetails,[21]isastandard reference. WewishtotreatSMCsastheoriesofphysicalprocesses,henceweoftenrefertoobjectsassystems and morphisms a processes. We will also extensively use string diagram notation for SMCs, where systemsaredepictedaswires,processesandboxes,and: g g f := f g:= g f ◦ ⊗ f B A 1A := A 1I := σA,B := A B Note that diagrams should be read from bottom-to-top. A process ρ : I A is called a state, a → processπ : A I iscalledaneffect,andλ : I I iscalledanumber. Depictedasstringdiagrams: → → state:= effect:= π number:=λ ρ NumbersinanSMCalwaysformacommutativemonoidwith‘multiplication’ andunittheidentity ◦ morphism1 . Wetypicallywrite1 simplyas1. I I We will begin with a category and construct a new category Caus[ ] of higher-order causal C C processes. In order to make this construction, we first need a mechanism for expressing higher- orderprocesses. Compactclosureprovidessuchamechanismthatisconvenientwithinthegraphical language andalreadyfamiliarwithin the literatureonquantum channels, in the guise of the Choi- Jamiołkowskiisomorphism. Definition2.1. AnSMC iscalledcompactclosedifeveryobject A hasadualobject A . Thatis,for ∗ C everyAthereexistsmorphismsη : I A Aandǫ : A A I,satisfying: A ∗ A ∗ → ⊗ ⊗ → (ǫ 1 ) (1 η ) =1 (1 ǫ ) (η 1 )=1 A A A A A A A A A A ⊗ ◦ ⊗ ∗⊗ ◦ ⊗ ∗ ∗ Werefertoη andǫ asacupandacap,denotedgraphicallyas and ,respectively. A A Inthisnotation,theequationsinDefinition2.1become: A A∗ = = A∗ A A A∗ A A∗ 3 Itisalwayspossibletochoosecupsandcapsinsuchawaythatthecanonicalisomorphisms I = I, ∗ ∼ (A B) = A B ,andA = A areallinfactequalities. Wewillassumethisisthecasethroughout ⊗ ∗ ∼ ∗⊗ ∗ ∼ ∗∗ thispaper. Crucially, two morphisms in a compact closed category are equal if and only if their string diagrams are the same. That is, if one diagram can be continuously deformed into the other while mainingtheconnectionsbetweenboxes. Hence,whenwedrawastringdiagram,wemeananycom- positionofboxesviacups,caps,andswapswhichyieldsthegivendiagram,uptodeformation. See [24]foranoverviewofstringdiagramlanguagesformonoidalcategories. Compactclosed categoriesexhibitprocess-stateduality,thatis, processes f : A B arein1-to-1 → correspondencewithstatesρ : I A B: f → ∗⊗ f f (1) 7→ ρf Hence, we treateverythingasa‘state’in and write f : X asshorthand for f : I X. Inthis C → notation,statesareoftheformρ : A,effectsπ : A ,andgeneralprocesses f : A B. Furthermore, ∗ ∗ ⊗ wewon’trequire‘output’wirestoexitupward,andweallowirregularly-shapedboxes.Forexample, wecanwriteaprocessw : A B C Dusing‘comb’notation: ∗ ∗ ⊗ ⊗ ⊗ D w C := A∗ B C∗ D (2) B w A Note that we adopt the convention that an A-labelled ‘input’ wire is of the same type as an A - ∗ labelled‘output’. While both the LHS and the RHS in equation (2) are notation for the same process w, the LHS is strongly suggestive of a second-order mapping from processes B C to processes A D. → → Composition in this notation simply meansapplying the appropriate‘cap’processesto plug wires together: D A∗ D C w Φ := B C∗ B∗ C w Φ B A Remark2.2. Sinceoddly-shapedboxesdon’tuniquelyfixanyorderingofsystemswithrespectto , ⊗ wewilloften‘name’eachsystembygivingitaunique typeandassumesystemsarepermutedvia σ-mapswhenevernecessary. Thisisacommonpracticee.g. inthequantuminformationliterature. Our key exampleswill be Mat(R+) and CPM, which contain stochastic matricesand quantum channels,respectively. Example 2.3. The category Mat(R+) has as objects natural numbers. Morphisms f : m n are → n mmatrices.Compositionisgivenbymatrixmultiplication: (g f)j :=∑ fkgj,m n :=mnand × ◦ i k i k ⊗ f gistheKroneckerproductofmatrices: ⊗ 4 (f g)kl := fkgl ⊗ ij i j Consequently, thetensor unit I = 1,sostatesarecolumnvectorsρ : 1 n, effectsarerowvectors → π : n 1,andnumbersareλ R+. Mat(R+)iscompactclosedwithn = n∗,wherecupsandcaps → ∈ aregivenbytheKroneckerdeltaδ : ij ηij :=δ =: ǫ ij ij Example 2.4. The category CPM has as objects (H), (K),... for Hilbert spaces H,K,... and as B B morphisms completely positive maps Φ : (H) (K). (H) (K) = (H K), hence I = B → B B ⊗B B ⊗ (C) = C. Statesarethereforepositive operatorsandeffectsare(un-normalised)quantum effects, B ∼ i.e. completely positive maps of the form: π(ρ) := Tr(ρP), for some positive operator P. As with Mat(R+), numbers are R+. CPM is also compact closed, with cups and caps given by the (un- normalised)Bellstate: η = Φ Φ ǫ(ρ)=Tr(ρ Φ Φ ) 0 0 0 0 | ih | | ih | where Φ =∑ i i .Consequentially, (H) = (H)andequation(1)givesthebasis-dependent | 0i i| i⊗| i B ∗ B version,i.e. ‘Choi-style’,oftheChoi-Jamiołkowskiisomorphism[19]. The biggest convenience of a compact closed structure is also its biggest drawback: all higher- order structure collapses to first-order structure! As we will soon see, there is a pronounced dif- ferencebetweenfirst-ordercausalprocesseswhichwe introduceinthe nextsection, andgenuinely higher-order causal processes. Thus, while it is natural to take to be compact closed, we expect C Caus[ ]tobeadifferentkindofcategory,whichallowsthisgenuinehigher-orderstructure. C Definition2.5. A -autonomouscategoryisasymmetricmonoidalcategoryequippedwithafulland ∗ faithfulfunctor( ) : op suchthat,byletting: ∗ − C → C A⊸B :=(A B∗)∗ (3) ⊗ thereexistsanaturalisomorphism: (A B,C)= (A,B⊸C) (4) C ⊗ ∼ C ⊸ Asthe notationandtheisomorphism (4)suggest, A B isthe systemwhose statescorrespond ⊸ toprocessesfromAtoB.Weadopttheprogrammers’conventionthat hasprecedenceover and ⊸ ⊗ associatestotheright: A B⊸C :=(A B)⊸C A⊸B⊸C := A⊸(B⊸C) ⊗ ⊗ Either expressionabove representsthe system whose states areprocesseswith two inputs. Indeed ⊸ ⊸ ⊸ (4)impliesthatA B C = A B C. Since issymmetric,wecanre-arrangetheinputsatwill, ⊗ ∼ C i.e. ⊸ ⊸ ⊸ ⊸ A B C = B A C (5) ∼ ⊸ Unlikein acompactclosed category, the object A B canbe differentfromsimply A B. In- ∗ ⊗ deedanycompactclosedcategoryis -autonomous,whereitadditionallyholdsthat: ∗ A⊗B ∼=(A∗⊗B∗)∗ (6) inwhichcase: A⊸B :=(A⊗B∗)∗ ∼= A∗⊗B∗∗ ∼= A∗⊗B However,ina -autonomouscategory,theRHSof(6)isnotA B,butsomethingnew,calledthe ∗ ⊗ ‘par’of AandB: 5 A`B:=(A∗ B∗)∗ (7) ⊗ Thisnewoperationinheritsitsgoodbehaviourfrom : ⊗ A`(B`C) =(A`B)`C A`B = BÀ ∼ ∼ Soacompactclosedcategoryisjusta -autonomouscategorywhere = `. However,thislittle ∗ ⊗ tweakyieldsamuchricherstructureofhigher-ordermaps. WethinkofA Basthejointstatespace ⊗ ofAandB,whereasA`BisliketakingthespaceofmapsfromA toB.For(firstorder)statespaces, ∗ thesearebasicallythesame,butaswegotohigherorderspaces,A`Btendstobemuchbiggerthan A B. ⊗ 3 Precausal categories Precausalcategoriesgiveauniverseofallprocesses,andprovideenoughstructureforustoidentify whichofthoseprocessessatisfyfirst-orderandhigher-ordercausalityconstraints. As noted in [10, 11, 12, 5], the crucial ingredient for defining causality is a preferred discarding process fromeverysystem Ainto I. Usingthiseffect,wecandefinecausalityasfollows: A Definition3.1. For systems A and B with discardingprocesses, a process Φ : A B issaid to be → causalif: B Φ = A (8) A Intuitively,causalitymeansthatifwedisregardtheoutputofaprocess,itdoesnotmatterwhich processoccurred.Wedefinediscardingon A Band I intheobviousway: ⊗ := := 1 (9) A B A B I ⊗ Hencecausalstatesproduce1whendiscarded: causal = 1 ρ ⇐⇒ ρ Sincediscardingthe‘output’ofaneffectπ : A I istheidentity,thereisauniquecausaleffectfor → anysystem,namelydiscardingitself: π causal π = ⇐⇒ Example3.2. ForMat(R+),thediscardingprocessisarowvectorconsistingentirelyof1’s;itsends astatetothesumoveritsvectorentries: = 1 1 1 = ∑ ρi ρ ··· i (cid:0) (cid:1) So, causality is precisely the statement that a vector of positive numbers sums to 1, i.e. forms a probabilitydistribution. Consequentially, thecausalityequation(8)foraprocessΦstatesthateach columnofΦmustsumto1. Thatis,Φisastochasticmap. Example 3.3. For CPM, discard is the trace. Hence causal states are density operators and causal processesaretrace-preservingcompletelypositivemaps,a.k.a.quantumchannels. 6 Discardingnotonlyallowsustoexpresswhenaprocessiscausal,italsoallowsustorepresent causalrelationshipsbetweenthesystemsinvolved. Forexample Definition3.4. AcausalprocessΦisone-waynon-signallingwith AbeforeB(written A B)ifthere (cid:22) existsaprocessΦ : A A suchthat ′ ′ → A′ B′ A′ Φ = Φ (10) ′ A B A B Itisone-waynonsignallingwithB AifthereexistsΦ suchthat ′′ (cid:22) A′ B′ B′ Φ = Φ (11) ′′ A B B Suchaprocessiscallednon-signallingifitisbothnon-signallingwith A BandB A. (cid:22) (cid:22) If A hasdiscarding,wecanalsoproduceastatefor A: ∗ A := A A∗ (12) Theseingredientsallowustomakethefollowingdefinition: Definition3.5. Aprecausalcategoryisacompactclosedcategory suchthat: C (C1) hasdiscardingprocessesforeverysystem,compatiblewiththemonoidalstructureasin(9). C (C2) Forevery(non-zero)system A,thedimensionof A: dA := A isaninvertiblescalar. (C3) hasenoughcausalstates: C  ρcausal. f = g  = f = g ∀ ⇒ ρ ρ       (C4) Second-ordercausalprocessesfactorise: ∀Φcausal. ∃Φ1,Φ2causal.    Φ  2 =  w Φ =  ⇒  w = Φ1      7 (C1)enablesonetotalkaboutcausalprocesseswithin . (C2)enablesustorenormalisecertain C processestoproducecausalones. Forexample,everynon-zerosystemhasatleastonecausalstate, calledtheuniformstate. Itisobtainedbynormalising(12): A := d−A1 A Then: A = d−A1 A = d−A1dA = 1 Notethatweallow tohaveazeroobject,inwhichcased =0isnotrequiredtobeinvertible. 0 C (C3)saysthatprocessesarecharacterisedbytheirbehaviouroncausalstates. Since iscompact C closed,(C3)impliesthatitsufficestolookonlyatproductstatestoidentifyaprocess. Proposition3.6. Foranycompactclosedcategory ,(C3)isequivalentto: C ∀ρ1,ρ2causal.  Φ = Φ  = Φ = Φ (13) ′ ⇒ ′    ρ1 ρ2 ρ1 ρ2      Proof. (C3) follows from (13) by taking one of the two systems involved to be trivial. Conversely, assumethepremiseof(13). Applying(C3)onetimeyields: Φ = Φ ′ ρ2 ρ2 forallcausalstatesρ . Bendingthewireyields: 2 Φ = Φ′ ρ2 ρ2 Hencewecanapply(C3)asecondtime. Bendingthewirebackdowngivestheresult. (C4) is perhaps the least transparent. It says that the only mappings from causal processes to causal processes are ‘circuits with holes’, i.e. those mappings which arise from plugging a causal processintoalargercircuitofcausalprocesses.Thiscanequivalentlybesplitintotwosmallerpieces, whichmaylookmorefamiliartosomereaders. Proposition3.7. Foracompactclosedcategory satisfying(C1),(C2),and(C3),condition(C4)is C equivalenttothefollowingtwoconditions: (C4′) Causalone-waysignallingprocessesfactorise: ∃Φ′causal.  ∃Φ1,Φ2causal.    = Φ2 Φ = Φ ⇒  Φ =   ′   Φ1        8 (C5′) Forallw : A B∗: ⊗ ∀Φcausal. ∃ρcausal.     = w Φ = 1 ⇒ w =            ρ          Proof. Assume(C4)holds,andletΦsatisfythepremiseof(C4′). Then,forcausalΨ,wehave: Φ Φ Ψ = ′ = = Φ Ψ Ψ ′ Hence,by(C4)thereexistcausalΦ ,Φ suchthat: 1′ 2 Φ Φ = 2 Φ 1 Deformingthengivesthefactorisationin(C4′). Togetto(C5′)from(C4),wetaketheinputandoutputsystemsofwtrivial,whichgives: Φ 2 w = = =: ρ Φ Φ 1 1 where the second ‘=’ follows from the factthat discardingis the unique causal effect. Conversely, suppose(C4′)and(C5′)hold. Letwbeasecondordercausalprocess. Then,foranycausalρandΦ: w Φ = 1 ρ So,by(C5′): Lemma3.8 w = = w = w ⇒ ρ′ ρ ρ ρ Then,by(C3): 9 w = w =: w′ Hence w satisfies the premise of (C4′), up to diagram deformation, where Φ := w and Φ′ := w′, whichimpliesthatitfactorsasin(C4). Lemma3.8. Foranyw : A B : ∗ ⊗  ρ. w =   w = w  ∃ ⇐⇒  ρ                Proof. ( )immediatelyfollowsfromdiagramdeformation. For( ),assumetheleftmostequation ⇐ ⇒ above. Then we canobtainanexpressionfor ρ byplugging the uniform state into w and applying causalityoftheuniformstate: w = = ρ ρ Substitutingthisexpressionbackinforρyields: B B B w = = = w B A A w ρ A A Example3.9. Mat(R+) and CPMareboth precausalcategories. Seeappendix for proofsof condi- tions(C1)-(C4). Notethatwecanreadilygeneraliseone-waynon-signallingfromDefinition(3.4)tonsystems. Definition3.10. AprocessΦisone-waynon-signalling(A ... A A )if 1 n 1 n (cid:22) (cid:22) − (cid:22) ... ... Φ = Φ ′ ... ... withΦ one-waynon-signallingwith A ... A . ′ 1 n 1 (cid:22) (cid:22) − Then(C4′)impliesangeneraln-foldversionofitself: Proposition3.11. LetΦbeone-waynon-signallingwith A ... A ,thenthereexistsΦ ,...,Φ 1 n 1 n (cid:22) (cid:22) suchthat 10