Categorical Probabilistic Theories

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by  Stefano Gogioso

Categorical Probabilistic Theories
StefanoGogioso CarloMariaScandolo
UniversityofOxford UniversityofOxford
[email protected] [email protected]
Wepresentasimplecategoricalframeworkforthetreatmentofprobabilistictheories,withtheaim
ofreconcilingthefieldsofCategoricalQuantumMechanics(CQM)andOperationalProbabilistic
7
Theories(OPTs). Inrecentyears,bothCQMandOPTshavefoundsuccessfulapplicationtoanumber
1
0 ofareasinquantumfoundationsandinformationtheory: theypresentmanysimilarities,bothinspirit
2 andinformalism,buttheyremainseparatedbyanumberofsubtleyetimportantdifferences. We
attempttobridgethisgap,byadoptingaminimalnumberofoperationallymotivatedaxiomswhich
n
a providecleancategoricalfoundations,inthestyleofCQM,forthetreatmentoftheproblemsthat
J OPTsareconcernedwith.

5
2
1 Introduction

]
h
p Categoricalmethodsarefindingmoreandmoreapplicationsinthefoundationsofquantumtheoryand
-
t quantuminformation,withtwomainframeworkscurrentlydominatingthescene: CategoricalQuantum
n
Mechanics(CQM)andOperationalProbabilisticTheories(OPTs). TheCQMframework[4,7,26,30,31]
a
u concernsitselfwiththecompositionalandalgebraicstructureofquantumsystemandprocesses,andhasa
q
heavyfocusondiagrammaticmethods[24,48,49]. TheOPTframework[17,18,19,21,43,44,45]was
[
developedwiththeaimofobtainingacharacterisationofquantumtheoryintermsofinformation-theoretic

1
axioms,insidethewidercommunityofgeneralisedprobabilistictheories[9,10,11,36,46,50].
v
5 Both frameworks are based on symmetric monoidal categories, and at first sight it looks like they
7 should be complementing each other. However, a number of subtle differences have resulted in a
0
significantdisconnectbetweenthetwocommunities,withduplicationofeffortsandlossofsynergy. The
8
0 OPTframeworkisfoundedonexplicitprobabilisticstructure,axiomsofconcretephysicalinspirationand
.
1 traditionalproofmethods, whichmakeithardtoworkwithinacategoricalanddiagrammaticsetting.
0 Conversely,theCQMframeworkhascleancategoricalfoundationsanddiagrammaticproofmethods,but
7
attheexpenseofdirectphysicalinterpretation.
1
: Wesetofftoreconcilethetwoframeworks,byproposingcategoricalfoundationsinthestyleofCQM
v
forthekindofsystemsandresultsthatareofinterestfortheOPTcommunity. Wedefineaprobabilistic
i
X
theory as a symmetric monoidal category satisfying three additional requirements: the existence of
r
a classicalsystems,theexistenceofprobabilisticstructure,andthepossibilityofdefininglocalstates. Our
requirementsareformulatedinstandardcategoricalterms,andcanbereadilyadoptedwithintheCQM
framework;atthesametime,theycorrespondtorelatablephysicalandoperationalrequirementsthatany
probabilistictheoryshouldpossess. Theendresultisaframeworkwhichissimpleandrigorousinits
foundations,showsdirectoperationalsignificance,andcomeswithmoreexpressivepowerthaneitherof
theCQMorOPTframeworksalone.
ItshouldbenotedthatsimilareffortstoconnecttheCQMandOPTcommunitiesarebeingundertaken
byRef.[53](andongoingworkbyitsauthor): theworkthereinishoweverclosertothelogicaltakeof
effectustheory,whileourworkhasamorephysicaltake,withclassicalsystemsatitsveryheart.
(cid:13)c S.Gogioso&C.M.Scandolo
Submittedto:
ThisworkislicensedundertheCreativeCommons
QPL2017
Attribution-Noncommercial-ShareAlikeLicense.
2 CategoricalProbabilisticTheories
2 Classical Theory
ThemainingredientdistinguishingOPTsfromthecategoriesusedinthecontextofCQMistheexplicit
presenceofprobabilisticmixtures,orequivalentlytheexistenceofclassicalsystemstakingpartinthe
processes. FromthepointofviewofOPTs,theseclassicalsystemsareabsorbedintotheformalism,using
indicesandsummations. FromthepointofviewofCQM,ontheotherhand,theonlydistinctionbetween
classicalandquantumsystemsisoperational,andbothshouldbemodelledexplicitlybyatheory.
2.1 ClassicaltheoryasaSMC
Classicalsystems1 andstochasticprocessesbetweenthemformasymmetricmonoidalcategory(SMC),
usuallyknownasStoch. TheobjectsofStocharelabelledbyfinitesets,andthemorphismsX →Y in
Stoch are theY-by-X stochastic matrices, with matrix composition as sequential composition and the
Kroneckerproductasparallelcomposition(akatensorproduct).
Fromacategoricalstandpoint,therestrictiontoconvexcombinationsrequiredbyStochisinconvenient:
if we wish to use a summation operation, as is usually done in the treatment of classical systems and
mixed-statequantumtheory,wehavetointroduceexternalrestrictionsonthekindofsumsthatweallow.
Alternatively, we could see the summation as defined on a larger category, with Stoch arising as the
sub-categoryofsuitablynormalised processes. Thislatterapproachisinlinewiththegeneralphilosophy
ofCQM:onestudiesabroadercollectionofabstractprocesses,easiertodescribeandtomanipulate,and
then assembles/restricts them appropriately to obtain the subcategory of concrete, physically relevant
processes. Asourmodelofclassicalsystemsand(potentiallynon-normalised)non-deterministicprocesses
betweenthem,wetaketheSMCR+-Mat: justlikeStoch,theobjectsarethefinitenon-emptysets,butthe
morphismsX →Y areALLtheR+-valuedmatricesin(cid:0)R+(cid:1)Y×X,notjustthestochasticones. Wereferto
thesemapsastheprobabilisticprocesses,andusenormalisedtodenotethestochasticones.
2.2 Linearstructure
WecanequipR+-Matwithawell-definednotionofsummationbysuitablyenrichingitincommutative
monoids: we endow the morphisms X →Y with the linear structure of (cid:0)R+(cid:1)Y×X, i.e. we take matrix
additionasoursummationoperationandthezeromatrixasitsneutralelement(modellingtheimpossible
process). Traditionally, the only requirement for a category enriched in commutative monoid is that
sequentialcompositionbelinear2,butinSMCsitmakessensetofurtherrequirethatparallelcomposition
belinearaswell3. Thisisindeedthecaseforourchoiceofenrichment.
2.3 Deterministicprocesses
AspecialplaceamongstthemanyprobabilisticprocessesbetweentwoclassicalsystemsX andY isheld
bythedeterministicprocesses4,thefunctionsX →Y. Thedeterministicprocessesformthesub-SMC
fSet of R+-Mat, and the Kronecker product restricts to the Cartesian product on them. The states in
1Thiswork,likemostworksinOPTsandCQM,isconcernedwithfiniteclassicalsystemsalone.
2Specifically,werequire f◦(g+h)=(f◦g)+(f◦h),(g+h)◦f =(g◦f)+(h◦f),0◦f =0and f◦0=0.
3Thingsareabitmorecomplicatedwiththetensorstructure:furthertothetensorproductbeinglinear,justlikecomposition,
butweneedtorequiretheassociatorandunitorstobelinearaswell.
4Inlinewithitscommonmeaningincomputerscience,weusetheworddeterministictodenoteclassicalprocesseswhich
mapeachdefiniteinputtoasingledefiniteoutput. Weusestochastic(ornormalisedprobabilistic,inthiswork)todenote
classicalprocesseswhichmoregenerallymapdefiniteinputstoprobabilitydistributionsoverdefiniteoutputs.
S.Gogioso&C.M.Scandolo 3
fSet for a classical system X are its deterministic states, the probability distributions δ concentrated
x
entirely on a single point x∈X (or, if you prefer, the linear extensions of the points of X themselves).
The singleton set 1:={1} is the terminal object in fSet, and we refer to the unique effect X →1 as
thediscardingmap5 ontheclassicalsystemX,whichisgraphicallydenotedbyagroundsymbol .
X
Thedeterministicprocessessitinsidealargersub-SMCfPFunofpartialdeterministicprocesses,with
partialfunctionsasmorphisms.
2.4 Resolutionoftheidentity
Inordertoclearlydistinguishclassicalsystemsfrommoregeneralsystemsinthetheory,wewilladopt
a graphical convention in which dashed wires are used to denote systems which are guaranteed to be
classical,sothattheidentitymorphismX →X foraclassicalsystemX willbedenotedasfollows:
X X (2.1)
Usingtheenrichment,weobtainthefollowingresolutionoftheidentity:
X X = ∑ X x x X (2.2)
x∈X
Thestatelabelledbyx∈X aboveisthedeterministicstatecorrespondingtox;theeffectlabelledbyx
aboveistheonesendingxto1∈R+ andallx(cid:48)(cid:54)=xto0∈R+ (or,ifyouprefer,thelinearextensionofthe
partialfunctionX (cid:42)1whichsendsxto1∈1andisundefinedonallotherx(cid:48)∈X).
2.5 Marginalisation
Because the resolution of the identity is in terms of deterministic states, we immediately obtain the
followingresolutionofthediscardingmap:
X = ∑ X x (2.3)
x∈X
ThismeansthatdiscardingasubsysteminR+-Matcorrespondstotheusualnotionofmarginalisation.
From the CQM perspective, the process of localisation of states and processes in arbitrary SMCs is
captured by the notion of environment structure [25,32]. Because classical states and processes are
localisedbymarginalising,itisnosurprisethatdiscardingmapsinR+-Matsatisfytherequirementsto
formanenvironmentstructure:
X
X⊗Y = 1 = (2.4)
Y
Theemptydiagramontherighthandsideofthesecondequationisthescalar1,whichasadiagrammatic
conventionisusuallyomitted.
5Inanot-at-all-unexpectedcoincidence,itmakestotalsensetocallthisthedeterministiceffect.
4 CategoricalProbabilisticTheories
2.6 Normalisedprocesses
Achoiceofenvironmentstructurealwayssinglesoutasub-SMCofnormalisedprocesses,namelythose
processes f satisfyingthefollowingcondition:
f = (2.5)
TheabstractnormalisationconditionaboveallowsustorecoverStochinacategoricallyfashionableway
as the sub-SMC of normalised processes in R+-Mat. As a bonus, we also get fSet as the sub-SMC of
normalisedprocessesinfPFun.
2.7 Probabilitiesasaresource
Incomputerscience,itiscommonforcommutativesemiringstomodelresourcesusedincomputation:
forexample,theprobabilitysemiring(R+,+,×)isusedtomodelprobabilisticcomputation,theboolean
semiring(B,∨,∧)isusedtomodelexistenceproblems,andthenaturalnumbers(N,+,×)areusedto
modelcountingproblems6. Asimilarapproachcouldbeadoptedinphysics: probabilities(corresponding
tothesemiringR+)couldbeseenasaresourceprovidingnon-determinisminclassicalsystems,andwe
mightwishtostudythetoytheoriesobtainedbyusingotherresources(correspondingtoothersemirings)
intheirplace. Thisis,forexample,theroadtakenbythesheaf-theoreticframeworkofRef.[2],wherea
numberofimportantresultsonnon-localityandcontextualityareobtainedbyconfrontingtheprobabilities
(thesemiringR+)withpossibilities[5](thesemiringB)andsignedprobabilities[6,39](thesemiringR,
whichisgivenanoperationalinterpretationinRef.[3]).
IfweuseagenericcommutativesemiringRtomodelnon-determinism,theSMCofclassicalsystems
andR-probabilisticprocessesbetweenthemisgivenbythecategoryR-MatofR-valuedmatrices: we
refertothenormalisedstatesasR-distributions,andtothenormalisedprocessesasR-stochastic. Asan
example,B-MatisthecategoryfReloffinitesetsandrelationsbetweenthem,awell-studiedtoymodel
inCQM[27,33,40]. AllthetheorywehavedevelopedaboveforR+-Matstraightforwardlyextendsto
thegeneralsemirings,aslongaswereplaceStochwithanappropriatenotionofnormalisedprocesses.
ThecategoriesfSetandfPFunoftotalandpartialdeterministicprocessesarealwayssub-SMCsofR-Mat,
showingthatourinterpretationofRasmodellinganotionofnon-determinismissound.
3 Probabilistic Theories
When talking about a theory, we broadly mean a categorical model (a SMC) that captures physical
systems and the compositional structure of processes between them. In the spirit of CQM, we avoid
restrictingourattentiontophysicalprocessesalone,butinsteadallowthepresenceofawholespectrum
ofidealised,abstractprocessesthatprovidebuildingblocksforphysicalprocesses,orotherwisehelpin
reasoningaboutthem. ContrarytotheOPTframework, wetaketheviewthatatheoryshouldspecify
objectsandprocessesinapurelycompositionalway,andneednotnecessarilymakeimmediatereference
to probabilistic outcomes of tests and measurement: as a consequence, we do not take any quotient
withrespecttoprobabilisticoutcomes,andweallowforthepossibilitythatatheoryspecifiesprocesses
whicharedifferentbutcannotbedistinguishedbymeansoftestsormeasurementsalone(thismeansthat
somedifferent“hiddenvariabletheories”withthesameoperationalpredictionscanbemodelledinthis
framework).
6Onecouldalsomentionthetropicalsemiring(R,min,+)usedintheFloyd-Warshallalgorithm,ortheViterbisemiring
([0,1],max,×)usedintheViterbialgorithm.
S.Gogioso&C.M.Scandolo 5
3.1 Probabilistictheories
Whentalkingaboutaprobabilistictheory,wemeanaSMCwhichincludesatleasttheclassicalsystems
amongstitsranks,andwhichisfurthermorecompatiblewithacoupleofbasicoperationalfeatures,namely
theirprobabilisticstructureandmarginalisation. Webrieflymotivateourrequirementsbelow.
1. Everyprobabilistictheoryhasclassicalprobabilisticsystemsunderthehood: becausetheseare
themselves physical systems, we model them explicitly. In particular, their interface with other
systems(e.g. measurementsandpreparations)canbetalkedaboutincompositionalterms.
2. Itmakesnosensetotalkaboutaprobabilistictheoryiftheprobabilisticstructuredoesnotextend
fromclassicalsystemstoarbitrarysystems. Ifthiswerenotthecase,onewouldnotnecessarilybe
abletoworkwithscenariosinwhichmultiplexedprocessesarecontrolledbyaclassicalrandom
variable,ortoconditionaprocessbasedonaclassicaloutput.
3. Sensible theories without a notion of local state do exist (e.g. Everettian quantum theory), but
theypresentanumberofoperationalchallenges,andtheyarenotprobabilisticinnature. Indeed,
the absence of a notion of local state compatible with marginalisation makes most multi-party
measurementscenariosandprotocolspecificationsmeaningless,afatetheysharewiththenotionof
no-signallingandwiththeprobabilisticfoundationsofthermodynamics.
Wenowproceedtoformalisetheserequirementsincategoricalterms.
Definition1(ProbabilisticTheory).
AprobabilistictheoryisasymmetricmonoidalcategoryC whichsatisfiesthefollowingrequirements.
1. Thereisafullsub-SMCofC,whichwedenotebyC ,whichisequivalenttoR+-Mat.
K
2. TheSMCC isenrichedincommutativemonoids7,andtheenrichmentonC coincideswiththe
K
onegivenbythelinearstructureofR+-Mat.
3. The SMC C comes with an environment structure, i.e. with a family ( H :H →1)H∈objC of
morphismswhichsatisfythefollowingrequirements:
H
H ⊗G = 1 = (3.1)
G
TheenvironmentstructureonC coincideswiththeonegivenbythediscardingmapsofR+-Mat.
K
TherequirementsabovecanbegeneralisedbyreplacingR+ withanarbitrarycommutativesemiringR,in
whichcasewewilltalkaboutanR-probabilistictheory.
InthecontextofaspecificR-probabilistictheory,werefertoC astheclassicaltheory,toitsobjectas
K
theclassicalsystemsandtoitsmorphismsastheclassicalprocesses. Inparticular,1isthetensorunit
forC,andthescalarsofC formthesemiringR. ForeachpairofobjectsH ,K ∈objC,theprocesses
H →K inC formacommutativemonoid,withasummationoperation+andazeroelement0which
we refer to as the impossible process. In line with the nomenclature adopted for classical systems,
werefertothe H mapsinvolvedintheenvironmentstructureasthediscardingmaps, andtothose
processes f satisfyingthefollowingequationasnormalised:
f = (3.2)
R-probabilistictheoriescomewithanumberofhandybuilt-infeatures,whichwenowpresent.
7Withlinearityofbothcompositionandtensorproduct.
6 CategoricalProbabilisticTheories
3.2 Tests
Consideraprocess f withaclassicaloutputvaluedinsomefinitesetY:
G
H f (3.3)
Y
Operationally, we can think of each output value y∈Y as corresponding to a different process being
performed8,andthatprocesscanbeobtainedbytestingagainsttheoutputvaluey:
G
H f G := H f (3.4)
y
y
Itshouldbenotedthat,evenwhen f isnormalised,theprocess f definedinEquation3.4isnotnecessarily
y
so,insteadbeingweightedbytheprobabilityofyoccurring. Inthissense,aprocessintheformof3.3
correspondstothenotionofatestinOPTs[18,21];iftheinputsystemH istrivialthenitisapreparation
test,whileiftheoutputsystemK istrivialthenitisanobservationtest. Notethatinthisframeworkthe
differencebetweenatestandamoregeneralprocessisentirelyintheeyeofthebeholder: whensaying
“test”insteadof“process”,wewillmeanthatwearenotinterestedinitsclassicalinputs,butwewillnot
excludethepresenceofclassicalsystemsamongsttheinputsystemsoftheprocess.
3.3 Outputprobability
Theprobability9 P (y)ofoutputvaluey∈Y occurringinapreparationtestcanbeobtainedasfollows:
ρ
P (y) := ρ (3.5)
ρ
y
Whentheprobabilityofoutcomeyforapreparationtestρ isinvertible10,wecanconditiononytoobtain
thestate 1 ρ ,whichisnormalisedwhenevertheoriginalstateρ is:
P (y) y
ρ
H
ρ (3.6)
y 1
P (y)
ρ
If ρ is normalised and the probabilities of all classical outcomes are either zero or invertible, then the
reducedstate ρ| canbewrittenasusualintheformofaconvexcombinationoftheconditionedstates:
H
H H
ρ = ∑ P (y) ρ (3.7)
ρ
ys.t.Pρ(y)(cid:54)=0 y P1(y)
ρ
8Avalueywhichisneveroutputcan,withoutlossofgenerality,bethoughttocorrespondtotheimpossibleprocess.
9WeusethewordlooselytointendthevalueinagenericcommutativesemiringR: asthelatterisfixedbythechoiceof
R-probabilistictheory,thisshouldcausenoconfusion.Inspecificapplications,onemightwanttousemorespecialisedterms,
suchaspossibilityforR=B,orcountforR=N.
10InthecaseofR+,thisboilsdowntotheusualrequirementthattheprobabilitybenon-vanishing.
S.Gogioso&C.M.Scandolo 7
3.4 Classicalcontrol
ProcessesinR-probabilistictheoriesarenotlimitedtotests: initsmostgeneralform,aprocess f includes
bothsomeclassicaloutputsetY andsomeclassicalinput/controlsetX:
H G
f (3.8)
X Y
Theprocess f(x) correspondingtoadefiniteinputvaluex∈X isimmediatelyobtainedbyapplying f to
thedeterministicstatecorrespondingtox:
G H G
H f(x) := f (3.9)
Y x Y
More in general, we can apply the process f above to an arbitrary state p:=(p ) on X to obtain a
x x∈X
linearcombinationoftheindividualprocesses f(x) correspondingtoeachdefiniteinputvaluex:
H G H G
f = ∑ px f(x) (3.10)
p Y x∈X Y
Whenthestateisnormalised,thisgivesaconvexcombination∑xpxf(x) oftheindividualprocesses.
3.5 Preparations
As an example of the difference between the constructions above, we look at preparations. In an R-
probabilistic theory, there are three possible notions of preparation: we have controlled preparations,
preparationtests,andconvexmixturesofthepreparedstates. Acontrolledpreparationtakesaclassical
inputx∈X andproducesastate f(x) inH ,whileapreparationtestisastateonthejointsystemH ⊗X:
H
X f H ρ (3.11)
X
controlledpreparation preparationtest
GivenanormalisedcontrolledpreparationandanormalisedstateqonX,i.e. anR-distribution(q ) ,
x x∈X
wecanalwaysobtainacorrespondingpreparationtestasfollows:
classicalcopymaponX
ρ H := f H (3.12)
X q X
Thenormalisationrequirementensuresthattheprobabilityofoutputx∈X inthepreparationtestisthe
samespecifiedbytheprobabilitydistribution(q ) :
x x∈X
f
ρ
= q = q = q X
X X X
(3.13)
GivenapreparationtestobtainedasinEquation3.12,onecanfurtherdiscardtheclassicaloutputsystem
X toobtainthecorrespondingconvexcombinationofpreparedstates,asshowninEquation3.7. There
isnocompositionalwayofobtainingthecontrolledpreparationbackfromthepreparationtest, orthe
preparationtestbackfromtheconvexcombinationofpreparedstates.
8 CategoricalProbabilisticTheories
3.6 Coarse-graining
Sometimes the classical outputs of a process f are not immediately interesting on their own, and one
mightwishtoapplysomeformofclassicalpost-processingtothem,bywhichwemeantheapplication
ofsomeadditionalclassical(probabilistic)processtotheoutputsystem. Mostcommonly,oneobtains
a new process g by applying a deterministic function q:X →Z to the relevant output system X of f,
mappingtheoriginaloutputstosome“moreinteresting”valuesinsomeothersetZ:
H G H G
g := f (3.14)
X Z X Y q Z
If we now focus on our new values in Z, we see that the process we obtain is exactly what the OPT
frameworkreferstoasacoarse-grainingofprocess f:
H H G H
gz G = f = ∑ fy G
X X Y q z ys.t.q(y)=z X
(3.15)
As a consequence, by a coarse-graining of a process f we will mean a process g taking the form of
Equation3.14forsomedeterministicfunctionq:X →Z. JustlikeintheOPTframework,wesaythat f
isarefinementofgwhenevergisacoarse-grainingof f.
3.7 Sharppreparations/observationpairs
A controlled preparation can be seen as a way of encoding the values of its classical input set X into
its output system H . However, not all preparations admit a corresponding decoding process which
deterministicallyreturnstheoriginalinputvalues11: werefertothosepreparationsadmittingonesuch
decodingassharppreparations,andtheycorrespondtoperfectlydistinguishablestates[18]. Wesay
thatapair(p,m)ofacontrolledpreparation p:X →H andanobservationtestm:H →X isasharp
preparation/observationpair(orSPOpair,forshort)iftheobservationtestmwitnessesthesharpness
ofthecontrolledpreparation p:
X p H
preparation X p m X = X X (3.16)
H m X
observation
WesaythatanSPOpairisnormalised12 whenboth pandmarenormalisedprocesses. Infact,itsuffices
toaskformtobenormalised,asnormalisationof palwaysfollows13:
p = p m = (3.17)
11Thesimplestexamplebeinggivenbynon-injectivedeterministicfunctions.
12NormalisedSPOscorrespondtoperfectlydistinguishablestatesinOPTs.
13Theconverseisnotingeneraltrue.Asasimplecounterexample,takeadeterministicfunctionp:X→Y whichisinjective
butnotsurjective,andletm:Y (cid:42)X beitsleftinverse,whichisnevertotal:thenpisnormalised,asitisatotalfunction,butm
isnottotal,andhencenotnormalised.
S.Gogioso&C.M.Scandolo 9
3.8 Decoherencemaps
If(p,m)isanormalisedSPOpair,werefertothefollowingprocessastheassociateddecoherencemap:
H dec H := H m p H (3.18)
p,m
decoherence
InordertokeeptrackoftheclassicalsystemX interveninginthedecoherencemap,wewilluseK to
p,m
denoteit. Thereisgoodreasonforourchoiceofname. Firstly,theprocessdec isnormalised(because
p,m
compositionofnormalisedprocesses)andidempotent:
decp,m decp,m = m p m p = m p = decp,m
(3.19)
Secondly,decoherencemapscanbeusedtoreconstructclassicalsystemsfromarbitraryones: tounder-
standhowthisworks,weuseamildvariationonacommonconstructionfromcategorytheory,known
asKaroubienvelope(oridempotentcompletion). IfC isaSMCwithanenvironmentstructure,thenwe
definethenormalisedKaroubienvelopeofC,denotedbySplit [C],tobetheSMCdefinedasfollows:
• thesystemsaregivenbypairsintheform(H ,h),whereH isasystemofC andh:H →H is
anormalisedidempotentprocess;
• theprocesses f :(H ,h)→(G,g)areexactlythoseprocesses f :H →G inC whichareinvariant
underthespecifiedidempotents,i.e. thosewhichsatisfy f =g◦ f ◦h;
• the tensor product on systems is given by (H ,h)⊗(G,g):=(H ⊗G,h⊗g), while the tensor
productonprocessesistheoneinheritedfromC.
Bylookingatobjectsintheform(H ,idH),itisimmediatetoseethatC isafullsub-SMCofSplit [C];
in particular, the two categories have the same scalars. In fact, we can prove that Split [C] is an
R-probabilistictheorywheneverC is.
Lemma2. IfC isanR-probabilistictheory, thensoisSplit [C], withclassicalsystems, enrichment
anddiscardingmapsdirectlyinheritedfromC.
AmongstthemanyobjectsofthenormalisedKaroubienvelopeforanR-probabilistictheorysitallobjects
intheform(H ,dec ),with(p,m)anormalisedSPOpaironH : werefertosystemsinthisformas
p,m
decoheredsystems,andwenowshowthattheycanbeusedtorecovertheclassicalsystemsofthetheory.
Theorem3(Decoheredsystems).
LetC beanR-probabilistictheory,andletDecoh[C]bethefullsubcategoryofSplit [C]spannedby
thedecoheredsystems. ThenDecoh[C]isasub-SMC,anditisequivalenttothesub-SMCC ofclassical
K
systemsinC. Theequivalencesendsadecoheredsystem(H ,dec )totheclassicalsystemK ,anda
p,m p,m
process f :(H ,dec )→(G,dec )tothefollowingclassicalprocess:
p,m q,n
Kp,m p f n Kq,n (3.20)
Inthecaseofquantumtheory,thedecoherencemapsassociatedwith(possiblydegenerate)observables
arisefromthenormalisedSPOpairgivenbyconsideringthedemolitionmeasurementintheobservable
together with its adjoint. However, there are more decoherence maps in quantum theory than those
associatedwithobservables. Indeed,wehaveimposednorequirementfortheobservationtobeadjoint
tothepreparation: hencedecoherencemapsarealwaysnormalisedandidempotent,butnotnecessarily
self-adjoint. As such, our notion of decoherence maps in quantum theory is more general than the
self-adjointnormalisedidempotentnotionusedinRef.[52]andrelatedworks.
10 CategoricalProbabilisticTheories
4 Quantum Theory
InCQM,daggercompactcategoriesaretakentobeabstractmodelsofpure-statequantumtheory,and
thecorrespondingmodelsofmixed-statequantumtheoryareobtainedviatheCPMconstruction[51].
The CPM category CPM[fdHilb] contains all positive states and completely positive maps of finite-
dimensional quantum theory, it comes with an environment structure given by the trace, and it can be
enrichedincommutativemonoids,withR+ assemiringofscalars(theusualR+-linearstructureofCP
maps). Unfortunately, CPM[fdHilb]doesnotcomewithawayoftalkingaboutclassicalsystems, and
inordertodosooneusuallyappealstotheCP*construction[29,35], whichisrelatedtoC*algebras
andquantumlogic. Inthiswork,wewillnotappealtotheCP*construction: instead,wewillrelyonthe
normalisedKaroubienvelopetogetaprobabilistictheoryoutofCPM[fdHilb],followingthestepsof[52].
Aswehaveseenbefore,thenormalisedKaroubienvelopeSplit [CPM[fdHilb]]containsCPM[fdHilb]
asafullsub-SMC,andwewillrefertotheobjectsinthatsub-SMCasquantumsystems. Italsocon-
tains objects in the form (H ,dec ), where is a special commutative †-Frobenius algebra on the
finite-dimensional Hilbert space H : we will refer to objects in this form as classical systems. The
decoherencemapdec for isthenormalisedidempotentprocessdefinedasfollows:
dec := = ∑ x x (4.1)
x∈K( )
where(|x(cid:105)) istheorthonormalbasisformedbytheclassicalstatesof (seeRef.[34]).
x∈K( )
Theorem4(Classicalsystems[52]).
LetD bethefullsubcategoryofSplit [CPM[fdHilb]]spannedbytheclassicalsystems. ThenD isafull
sub-SMCequivalenttoR+-Mat. Theequivalencesendsaclassicalsystem(H ,dec )tothesetK( )of
classicalstatesfor ,andaprocess f :(H ,dec )→(G,dec )tothefollowingclassicalprocess:
(cid:16) (cid:17)
(cid:104)y|f|x(cid:105) :K( )→K( ) (4.2)
(y,x)∈K( )×K( )
When talking about quantum theory in the context of probabilistic theories, we will refer to the full
sub-SMCofSplit [CPM[fdHilb]]jointlyspannedbythequantumsystemsandtheclassicalsystems.
5 Causality and no-signalling
5.1 Causality
CausalityisperhapsthetrickiestpointofcontactbetweentheOPTframeworkandprobabilistictheories
aswehavedefinedthem. Ontheonehand,causalityispartoftheverydefinitionofprobabilistictheories:
adistinguishedfamilyofdiscardingmapsissingledoutbythebasicoperationalrequirementthatstates
belocalisable. ThisishowcausalityisusuallyunderstoodintheCQMcommunity[25,32]. Ontheother
hand,however,provingcausalityinapurelycompositionalwayisimpossiblewithoutreferencetothat
very environment structure. Indeed, one encounters the following issue when trying to formulate the
causality axiom as understood by the OPT community: the observation tests in OPTs have additional
requirementsthatmakethem“normalised”inasuitablesense,whiletheonesdefinedinthisworkhave
nosuchrequirementplaceduponthem. Inordertosingleoutasetofobservationtestswhichwouldbe
suitabletoformulatethecausalityaxiom(asunderstoodinOPTs),wewouldhavetosomehowreferto
theenvironmentstructuretoimposeanappropriatenormalisationcondition14.
14Forexample,notethatCPM[fdHilb]admitsenvironmentstructuresdifferentfromthecanonicalone.