A categorical semantics for causal structure

0.29 MB·

by  Aleks Kissinger

A categorical semantics for causal structure
AleksKissinger and SanderUijlen
InstituteforComputingandInformationSciences,RadboudUniversity
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1 {aleks,suijlen}@cs.ru.nl
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2 January 26,2017

n
a
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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 di-
agrams 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 = B`A
∼ ∼
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