Table Of ContentPicturing Indefinite Causal Structure
Aleks Kissingerand Sander Uijlen
InstituteforComputingandInformationSciences,RadboudUniversity
{aleks,suijlen}@cs.ru.nl
Followingonfromthenotionof(first-order)causality,whichgeneralisesthenotionofbeingtrace-
preservingfromCP-mapstoabstractprocesses,wegiveacharacterizationforthemostgeneralkind
ofmapwhichsendscausalprocessestocausalprocesses. Thesenew,second-ordercausalprocesses
enableustotreattheinputprocessesas‘locallaboratories’whosecausalorderingneedsnotbefixed
inadvance.Usingthischaracterization,wegiveafully-diagrammaticproofofanon-trivialtheorem:
namelythatbeingcausality-preservingonseparableprocessesimpliesbeing‘completely’causality-
preserving. That is, causality is preserved even when the ‘local laboratories’ are allowed to have
ancilla systems. An immediate consequence is that preserving causality is separable processes is
equivalencetopreservingcausalityforstronglynon-signalling(a.k.a.localizable)processes.
1 Causality and non-signalling
Throughout this extended abstract, we will work in a self-dual compact closed category C, that is, a
symmetricmonoidalcategorywhichhasforeveryobjectapairofmorphisms:
h :I →A⊗A e :A⊗A→I
A A
whichwerefertoascupsandcapsrespectively, satisfying thefollowing‘yanking’ identities:
(e ⊗1 )◦(1 ⊗h )=1 g ◦h =h e ◦g =e
A A A A A A A A A A A
where g : A⊗A → A⊗A is the symmetry natural isomorphism. Furthermore, we will adopt string
A
diagramnotionfordepictingcompositions ofmorphisms(seee.g. [9]). Usingthisnotion, cupsandcaps
resembletheirnamesakes:
h := e :=
A A
andhencetheequations abovebecome:
= = =
Note that the monoidal unit I is depicted as empty space. Throughout the paper, we will think of mor-
phismsinthiscategoryasphysicalprocessesofsomekind,henceweadopt‘process-theoretic’ language.
Namely,werefertoobjectsassystemsandmorphismsasprocesses. Furthermore,wegivespecialnames
toprocesses fromandtothetrivialsystem:
p
states := y effects :=
R.DuncanandC.Heunen(Eds.):QuantumPhysicsandLogic(QPL)2016
EPTCS236,2017,pp.87–94,doi:10.4204/EPTCS.236.6
88 PicturingIndefiniteCausalStructure
In addition tothe compact closed structure, wealso assume C has adistinguished effect d :A→I
A
foreverysystemAcalleddiscarding. Thisispictured as:
d :=
A
andiscompatiblewith⊗andI asfollows:
d = = = d ⊗d d = = 1
A⊗B A B I I
Theutilityofthediscarding processisitenablesustodefinecausality, following[7,3]:
Definition1.1. AprocessY :A→Biscalledcausalifd ◦Y =d ,orpictorially:
B A
Y =
Themottoforcausalprocesses istherefore:
Ifwediscardtheoutputofaprocess, itdoesn’t
matterwhichprocesshappened.
InthecategorywhoseobjectsarequantumstatespacesandwhosemorphismsareCP-maps,causality
corresponds tobeingatrace-preserving CP-map,i.e. aquantum channel.
The utility of causality is that it enables us to use diagrams to represent the causal relationships
between processes [7]. For example, if we wish to express that Alice can signal to Bob (but not vice-
versa!), wecanrequire thatacausalprocess:
F :A ⊗A →B ⊗B
1 2 1 2
factorises as:
Y
B
F =
Y
A
whereY andY arealsocausal. Following[6],weseefromthis factorisation, thatitisindeed impos-
A B
sible for Bobtosignal Alice. Indeed, ifwediscard Bob’soutput (towhich Alicedoes not have access),
thewholeprocessdisconnects:
Y
B
F = = =: Y ′
Y Y
A A
We say such a process is non-signalling from B to A, and write A (cid:22) B. Similarly, a process is non-
signalling fromAtoBifitfactorises as:
Y
A
F =
Y
B
AleksKissinger&SanderUijlen 89
andwesayitissimplynon-signalling ifitadmitsbothfactorisations.
A typical example of a non-signalling process is a Bell-type scenario. That is, Alice and Bob share
somebipartite state,towhichtheyperformlocaloperations:
Y Y
A B
F = (1)
r
Thisclearlyadmitsthetwofactorisations forA(cid:22)BandB(cid:22)A:
Y Y
B A
Y Y
A B
r r
so one might ask if in fact all non-signalling processes arise this way. In quantum theory, surprisingly
theanswerisno.
Definition1.2. Amorphism iscalled strongly non-signalling ifitfactorises asinequation (1)forsome
causalmorphismsY ,Y ,andr .
A B
Itwasshownin[1]thatthereindeedexistquantumchannelswhicharenon-signallingbutnotstrongly
non-signalling (conditions referredtoas‘causal’ and‘localizable’ in[1],respectively).
2 Second-order causality
Recently, frameworks have been proposed to discuss quantum correlations which do not necessarily
have a fixed causal ordering [8, 4]. Both of these frameworks rely on the notion of a ‘higher-order
quantum channel’ [2],i.e. amapping whichsendschannels tochannels. Inthissection, wewillprovide
acharacterisation ofsuchamapinanycompactclosedcategorywithdiscarding.
Asis the usual trick ina compact-closed category, wecan obtain higher-order maps by firstturning
firstordermapsintostatesby‘bending up’theinputwire:
F 7→ F
Thisbendingissometimescalledprocess-state duality, whichinduces abijection between:
processes F :A→B ∼= statesF˜ :I→A⊗B (2)
(cid:8) (cid:9) (cid:8) (cid:9)
Hence, we can express a map which sends aprocess of type A →A to a process of type B →B
1 2 1 2
asamapoftheform:
B B
1 2
W :: F 7→ W
F
A A
1 2
90 PicturingIndefiniteCausalStructure
Definition2.1. Aprocessiscalledsecond-order causal (SOC)ifitsendscausalprocesses (encoded via
process-state duality(2))tocausalprocesses. Diagrammatically,W isSOCifforallF :
F = =⇒ W = (3)
F
Itisoftenmoreenlightening towriteSOCmapsusing‘comb’notation(cf. [5]):
B
2
B B
1 2
A
W W 2 (4)
A
1
A A
1 2
B
1
Thenprocesses arecomposedfromtheinside-out, ratherthanbottom-to-top. Hence,(3)becomes:
F = =⇒ W F =
However,processeswithjustone‘hole’arenotthatinteresting,sowewillconsideramoreinteresting
kindofsecond-order causalmap,whichhastwoholes:
Definition2.2. Aprocess:
W :(A ⊗A )⊗(B ⊗B )→C ⊗C
1 2 1 2 1 2
iscalledbipartite second-order causal (SOC )ifforallcausalF ,F :
2 A B
F W F
A B
iscausal.
Particularly simpleexamplesofSOC mapssimplywireF together withF insomeorder:
2 A B
(5)
AleksKissinger&SanderUijlen 91
However,interestingly, theorderthattheyarewiredtogether ishiddenifwetreatW asablackbox,and
it can be shown (see for example [8]) that wecan even define SOC maps which don’t admit any fixed
2
causalorder.
It is natural to ask whether separate notions of bipartite (or more generally, n-partite) second-order
causal maps is really necessary. We could, after all, define an SOC map as in (4) where A :=X ⊗Y
1 1 1
andA :=X ⊗Y ,i.e. oftheform:
2 2 2
B
2
X Y
2 2
W
X Y
1 1
B
1
butthisisveryrestrictivesinceitneedstosendanycausalmaptoacausalmap,ratherthanjustseparable
ones. In fact, the simple example of an SOC map which wires two processes together in some fixed
2
orderisalreadynotSOC.Supposeforinstancethatwepluga(non-separable)swapmapintotheleftmost
process in(5):
=
Thenweintroduce aloop, whichfor mostcategories C (including CP-maps)willimmediately kill nor-
malisation, andhencecausality.
Definition2.3. Wesayacategory C hasenough causalstatesif:
F F ′
∀r causal . = =⇒ F =F ′
r r
Since C is compact closed, we can prove that if C has enough states, it also has enough separable
causalstates:
··· ···
F F ′
∀r ,...,r causal . = =⇒ F =F ′
1 n
··· ···
r r r r
1 n 1 n
because wecansimplyapplyDefinition2.3oneinputatatime,via:
F F ′ F F ′
= ⇐⇒ =
r r r r
92 PicturingIndefiniteCausalStructure
3 Second-order causality and non-signalling
Nowwearereadytoprovethemaintheorem ofthisextended abstractandgiveasimplecorollary.
Theorem3.1. IfaprocessinacategorywithenoughcausalstatesisSOC ,thenitis‘completely’SOC
2 2
inthesensethat,foranycausalprocesses:
F :A′ ⊗A →A′ ⊗A F :B′ ⊗B →B′ ⊗B
A 1 1 2 2 B 1 1 2 2
theprocess:
F W F
A B
iscausal.
Proof. Foranycausalstatesr ,r ,thefollowingprocesses arecausal:
A B
F F
A B (6)
r r
A B
whichcanbeseenjustbydiscarding therespective outputs andapplying causality ofF ,F ,r andr
A B A B
individually. Then, ifW is SOC , plugging in the causal maps (6) yields a causal map. Hence, for any
2
r ,wehave:
W
F W F = =
A B r
W
r r
A B
r
W
Sincetheprocessaboveagreeswithdiscarding forallr ,r ,r :
A B W
=
r r r
A W B
we can conclude, using the fact that C has enough causal states (and hence enough separable causal
states)that:
F W F =
A B
AleksKissinger&SanderUijlen 93
Wecan nowshow thatSOC processes notonly preserve causality forseparable processes, butalso
2
forstrongly non-signalling processes:
Corollary 3.2. IfaprocessW isSOC ,thenitsendseverycausal, stronglynon-signalling process:
2
F :A ⊗B →A ⊗B
1 1 2 2
toacausalprocess.
Proof. IfF isstrongly non-signalling, itfactorsasin(1). Then:
W
W
W
Y Y
F = Y A Y B = A B
r
r
(3.1)
= =
r
In[4]itisshownthatpreserving causality forproduct channels isequivalent topreserving causality
for all non-signalling channels. This can be shown straight-forwardly in the concrete case of CP-maps
using the fact that non-signalling channels always arise as affine combinations of separable channels.
Onecouldthereforeextendtheproofabovetoworkforallnon-signalling processesifwereplacer with
a‘pseudo-state’ givenby,e.g.
(cid:229)
:= r |iihi|⊗|iihi|
rrr i
i
for(possibly negative)coefficientsr summingto1. Thenwestillhave:
i
=
rrr
andwecanfurthermore realise anyaffinecombination ofseparable CP-maps(hence anynon-signalling
channel) via:
Y Y
A B
F =
rrr
Thentheproof ofCorollary 3.2proceeds identically, replacing r withrrr. However, thishasthe undesir-
able property that we have to go outside of the category of ‘physically realisable’ processes to get this
(non-positive)pseudo-staterrr. Whetheronecangiveafullydiagrammaticproofwithoutresortingtosuch
tricksisanopenquestion.
94 PicturingIndefiniteCausalStructure
4 Acknowledgements
Wewouldliketothankeveryonefromthe2016JointMcGill-OxfordWorkshoponCausalityinQuantum
Foundations for fruitful discussions as well as the anonymous reviewers for their useful comments and
references. The research leading to these results has received funding from the European Research
Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant
agreement no. 320571
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