Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras

0.17 MB·

by  Mathys Rennela

Infinite-Dimensionality in Quantum Foundations:
W*-algebras as Presheaves over Matrix Algebras
MathysRennela SamStaton RobertFurber
RadboudUniversity OxfordUniversity AalborgUniversity
Nijmegen,TheNetherlands Oxford,UnitedKingdom Aalborg,Denmark
In this paper, W*-algebras are presented as canonical colimits of diagrams of matrix algebras and
completelypositivemaps. Inotherwords,matrixalgebrasaredenseinW*-algebras.
Introduction
In the foundations of quantum mechanics and quantum computing, there is often a split between re-
searchusinginfinitedimensionalstructuresandresearchusingfinitedimensionalstructures. Ontheone
hand, in axiomatic quantum foundations there is often a focus on finite dimensional spaces and matrix
mechanics(e.g.[1,45,23,46,5,3,8,9,10,13,27]),andthesameistrueforcircuitbasedquantumcom-
puting (e.g. [16, 31]). On the other hand, infinite dimensional spaces arise naturally in subjects such as
quantumfieldtheory[47],andmoreovertheregisterspaceinascalablequantumcomputerarguablyhas
aninfinitedimensionalaspect(seee.g.[36]),whichhasledsomeresearcherstouseinfinitedimensional
spaces in the semantics of quantum programming languages [6, 37, 38, 21]. The ‘spaces’ in quantum
theoryarereallynon-commutative,soweunderstandthemasW*-algebras,byanalogytoGelfanddual-
ity[11,1.4].
A natural question, then, is whether foundational research frameworks that focus on finite dimen-
sional structures can approximate their infinite-dimensional counterparts. In brief, the answer to this
questionispositivewhenonedealswithW*-algebras. Indetail, whenwefocusoncompletelypositive
maps, as is usual in quantum foundations and quantum computation, one can show that every infinite
dimensional W*-algebra is a canonical colimit of matrix algebras. This characteristic is expressed in
the following theorem, which constitutes our main result. Note that it is about colimits in the opposite
category of W*-algebras and completely positive maps, and therefore about limits in the category of
W*-algebrasandcompletelypositivemaps.
Theorem. Let W∗-Alg be the category of W*-algebras together with completely positive maps. Let
CP
N be the category whose objects are natural numbers, with n considered as the algebra of n×n
CP
complex matrices, and completely positive maps between them. Let Set be the category of sets and
functions.
Thehom-setfunctorW∗-Alg (−,=):W∗-Algop →[N ,Set]isfullandfaithful.
CP CP CP
Recallthatthecategoryofpresheaves[N ,Set]isafreecolimitcompletionofNop (e.g.[26,III.7]).
CP CP
Recalltoothatafull-and-faithfulfunctoristhesamethingasafullsubcategory,uptocategoricalequiv-
alence. Thus we can say that every W*-algebra is a canonical colimit of matrix algebras. A category
theoristwouldsaythatthematrixalgebrasaredenseintheW*-algebras(following[25,Ch.5]),making
ananalogywithtopology(e.g. therationalnumbersaredenseamongthereals). Wephrasetheresultin
termsofthedualcategoryW∗-Algop insteadofW∗-Alg withtheideathatW∗-Algop isacategoryof
CP CP CP
non-commutativespaces.
(cid:13)c M.Rennela,S.Staton&R.Furber
R.DuncanandC.Heunen(Eds.):QuantumPhysicsandLogic(QPL)2016
Thisworkislicensedunderthe
EPTCS236,2017,pp.161–173,doi:10.4204/EPTCS.236.11
CreativeCommonsAttributionLicense.
162 Infinite-dimensionalityinQuantumFoundations
Relatedideas. Ourtheoremisnovel(asfaraswecantell)butthethemeisrelatedtovariousresearch
directions.
• Densitytheoremsoccurthroughoutcategorytheory. Perhapsthemostfamoussituationissimpli-
cial sets, which are functors [∆op,Set], where ∆ is a category whose objects are natural numbers,
with n considered as the n-simplex. There is a restricted hom-functor Top→[∆op,Set]; it is full
andfaithfulup-tohomotopywhenonerestrictsToptoCW-complexes.
• There is a long tradition of studying limits and colimits of *-homomorphisms, rather than com-
pletelypositivemaps. Notably,AFC*-algebrasarelimitsofdirecteddiagramsoffinite-dimensional
C*-algebrasand*-homomorphisms[4].
• In a dual direction, C*-algebras and *-homomorphisms form a locally presentable category [34]
andsothereexistsasmalldensesetofC*-algebraswithrespectto*-homomorphisms. Thisdense
setofC*-algebrashasnotbeencharacterizedexplicitly,toourknowledge,butitislikelytoalready
containinfinitedimensionalC*-algebras.
• Operator spaces and operator systems are generalizations of C*-algebras that still permit matrix
constructions. Thesearealsorelatedtothepresheafconstruction,asexplainedinSection4.
• Inprogramminglanguagetheory,asidefromquantumcomputation,theideaofdefiningcomputa-
tionalconstructsondensesubcategoriesisincreasinglycommon(e.g.[29]).
• Density also appears in quantum contextuality. For example, the boolean algebras are dense in
the effect algebras [42], and compact Hausdorff spaces are dense in piecewise C*-algebras [17,
Thm. 4.5]. This is a compelling way to study contextuality: the base category offers a classical
perspective on the quantum situation. However, it is unclear how to study tensor products of
C*/W*-algebrasinthisway.
• The Karoubi envelope of a category is a very simple colimit completion. The Karoubi envelope
of N contains the category of finite-dimensional C*-algebras and completely positive maps, as
CP
discussedin[41,24].
• Pagani, SelingerandValiron[32]usedafreebiproductcompletionofN tomodelhigher-order
CP
quantumcomputation. Itremainstobeseenwhethereveryobjectoftheircategorycanbethought
of as a W*-algebra, and whether their type constructions correspond to known constructions of
W*-algebras.
• Malherbe, Scott and Selinger [28] proposed to study quantum computation using presheaf cate-
gories [Qop,Set], where Q is a category related to N . Thus our result links their proposal for
CP
higher-orderquantumcomputationwithworkbasedonoperatoralgebra.
1 Preliminaries on operator algebras
In this section we briefly recall some key concepts from operator algebra. See [39, 44] for a complete
introduction. We recall C*-algebras, which are, informally, non-commutative topological spaces, as a
step towards W*-algebras, which are, informally, a non-commutative measure spaces [11, 1.4]. The
positive elements of the algebras are thought of as observables, and so we focus on linear maps that
preserve positive elements. Completely positive maps are, roughly, positive maps that remain positive
whenquantumsystemsarecombined.
M.Rennela,S.Staton&R.Furber 163
C*-algebras Recall that a (unital) C*-algebra is a vector space over the field of complex numbers
that also has multiplication, a unit and an involution, satisfying associativity laws for multiplication,
involution laws (e.g. x∗∗ =x, (xy)∗ =y∗x∗, (αx)∗ =α¯(x∗)) and such that the norm of an element x is
givenbythesquarerootofthespectralradiusofx∗x,whichgivesaBanachspace.
Finite dimensional examples: qubits and bits A key source of examples of finite dimensional C*-
algebras are the algebras M of k×k complex matrices, with matrix addition and multiplication, and
k
where involution is conjugate transpose. In particular the set M =C of complex numbers has a C*-
1
algebrastructure,andthe2×2matrices,M ,containtheobservablesofqubits.
2
Anotherexampleisthealgebrasofpairsofcomplexnumbers,C2,withcomponentwiseadditionand
multiplication. Thiscontainstheobservablesofclassicalbits.
Positive elements and positive maps An element x ∈ A is positive if it can be written in the form
x = y∗y for y ∈ A. We denote by A+ the set of positive elements of a C*-algebra A and define the
following.
Let f :A→B be a linear map between the underlying vector spaces. The map f is positive if it
preserves positive elements and therefore restricts to a function A+ →B+. A positive map A→C will
becalledastateonA.
W*-algebras and normal maps In what follows, we will focus on W*-algebras, which are C*-
algebras A that have a predual, that is, such that there is a Banach space A whose dual is isomorphic
∗
to A [39]. The positive elements of a C*-algebra always form a partial order, with x≤y if and only if
(y−x)∈A+. Moreover,inaW*-algebra,ifadirectedsubsetofAhasanupperbound,thenithasaleast
upperbound. Itisnaturaltorequirethat(completely)positivemapsaremoreovernormal,whichmeans
thattheypreservesuchleastupperbounds.
W*-algebras encompass all finite dimensional C*-algebras, and also the algebras of bounded oper-
ators on any Hilbert space, the function space L∞(X) for any standard measure space X, and the space
(cid:96)∞(N)ofboundedsequences.
Matrix algebras and completely positive maps If A is a C*-algebra then the k×k matrices valued
in A also form a C*-algebra, M (A), which is a W*-algebra if A is. For instance M (C) = M , and
k k k
M (M)∼=M . Informally,wecanthinkoftheW*-algebraM (A)asrepresentingkpossibly-entangled
k l k×l k
copies of A. This can be thought of as a kind of tensor product: as a vector space M (A) is a tensor
k
productM (C)⊗A.
k
Let f : A → B be a linear map between the underlying vector spaces. The map f is completely
positiveifitisn-positiveforeveryn∈N,i.e. themapM (f):M (A)→M (B)definedforeverymatrix
n n n
[x ] ∈M (A)byM (f)([x ] )=[f(x )] ispositiveforeveryn∈N.
i,j i,j≤n n n i,j i,j≤n i,j i,j≤n
Completely positive maps and positive maps are related as follows: a positive map f : A → B of
C*-algebras, for which A or B is commutative, is completely positive. Hence, every (sub-)state on a
C*-algebraAiscompletelypositive.
WewriteW∗-Alg forthecategoryofW*-algebrasandnormalpositivemaps,andW∗-Alg forthe
P CP
categoryofW*-algebrasandnormalcompletelypositivemaps.
164 Infinite-dimensionalityinQuantumFoundations
2 Naturality and representations of complete positivity
Inthissection,werecallthecategoricalcharacterizationofcompletelypositivemapsasnaturalfamilies
of positive maps. This gives a technique for building representations of completely positive maps (see
[38]formoreexamples).
In the previous section we explained that for every W*-algebra A the matrices valued in A form a
W*-algebra again. This construction(A,m)(cid:55)→M (A)is functorial. To make thisprecise, we introduce
m
thecategoryN ofcomplexmatrices: theobjectsarenon-zeronaturalnumbersseenasdimensions,and
Mat
the morphisms m→n are m×n complex matrices. Composition is matrix multiplication. (We remark
thatthecategoryNMatisequivalenttothecategoryFdVectCoffinite-dimensionalcomplexvectorspaces
andlinearmaps,sinceeveryfinite-dimensionalvectorspaceisisomorphictoCn. Itisalsoequivalentto
the category FdHilb of finite-dimensional Hilbert spaces and linear maps, since every such space has a
canonicalinnerproduct.)
The construction of matrices of elements of a W*-algebra can be made into a functor W∗-Alg ×
CP
N →W∗-Alg . Ittakesapair(A,m)toM (A)andapairofmorphisms(f,F):(A,m)→(B,n)tothe
Mat P m
positivemapF∗(f )F :M (A)→M (B).
m n
We will consider this functor in curried form, M : W∗-Alg → [N ,W∗-Alg ]. It takes a W*-
CP Mat P
algebraAtoafunctor,i.e. anindexedfamilyofW*-algebras,M(A)={M (A)} . Acompletelypositive
n n
map f :A→BistakentothecorrespondingfamilyofpositivemapsM(f)={M (f):M (A)→M (B)} .
n n n n
Thisgivesthemainresultof[38]: thefunctorM isfullandfaithful,i.e. completelypositivemapsarein
naturalbijectionwithfamiliesofpositivemaps.
Theorem1([38]). ThefunctorM:W∗-Alg →[N ,W∗-Alg ]isfullandfaithful.
CP Mat P
Faithfulness is obvious, since for any CP-map f :A→B we have M(f) = f. Proving fullness is
1
moreinvolvedandrequiresthefollowinglemma.
Lemma1. Considertwopositivemaps f :M (B)→M (A)and f :B→AofC*-algebras. Thefollow-
n n n 1
ingconditionsareequivalent:
1. ∀y∈M (B),v∈Cn. v∗(f (y))v= f (v∗yv)
n n 1
2. f =M (f ).
n n 1
Theproofofthelemma[38]makesuseofstabilizerstatesinCn.
3 Main result: W*-algebras are colimits of CP-maps
This section gives our main contribution: we show that infinite dimensional W*-algebras are canonical
colimitsofmatrixalgebras.
Our first result is based on the representation of W*-algebras by their cones of positive linear
functionals. We say that an (abstract) cone is a module for the semiring of positive reals. Thus it
M.Rennela,S.Staton&R.Furber 165
is a set X that is equipped with both the structure of a commutative monoid (X,+,0) and a function
(−·−):R ×X →X that is a group homomorphism in each argument and such that rs·x=r·(s·x),
>0
1·x=x. Most examples of cones arise as subsets of a larger vector space that are not subspaces per se
butmerelyclosedunderadditionandmultiplicationbypositivescalars. Forexample,thesetofpositive
realsitselfformsacone. ThepositiveelementsofaC*-algebraalsoformacone.
LetConebethecategoryofconesandstructurepreservingfunctionsbetweenthem.
For any W*-algebras A and B, the set of normal positive maps A → B forms a cone: it is closed
under addition, zero, and multiplication by positive scalars. (Formally, we can say that W∗-Alg is
CP
enrichedinthecategoryCone,equippedwiththeusualsymmetricmonoidalstructure: compositionisa
cone-homomorphism in each argument. This also plays a role in [32].) In particular we have a functor
W∗-Alg (−,C):W∗-Algop→Cone.
P P
Proposition1. ThenormalpositivelinearfunctionalfunctorW∗-Alg (−,C):W∗-Algop→Coneisfull
P P
andfaithful.
Proof. Fullnessessentiallycomesfromthefactthattheclosednessandcompletenessofthepositivecone
imply that every positive linear map A →C is bounded (see e.g. [30] or [40, Th. V.5.5(ii)]). We refer
∗
theinterestedreadertoAppendixAforadetailedproof.
We define a category N as a full subcategory of W∗-Alg whose objects are W*-algebras of the
CP CP
form M . We consider the functor W∗-Algop →[N ,Set] which takes a W*-algebra A to the functor
n CP CP
M (cid:55)→W∗-Alg (A,M ).
n CP n
Theorem2. ThehomfunctorW∗-Alg (−,=):W∗-Algop →[N ,Set]isfullandfaithful.
CP CP CP
Proof. BycombiningProp.1withTheorem1,wehavethatthecomposite
W∗-Alg (−⊗=,C):W∗-Algop →[Nop ,W∗-Algop]→[Nop ,Cone]
CP CP Mat P Mat
isfullandfaithful. Ourfirststepistoshowthatthehom-conefunctor
W∗-Alg (−,=):W∗-Algop →[N ,Cone]
CP CP CP
isfullandfaithful. Indeed,byelementarycategorytheory,foranyfunctor
H :W∗-Algop ×D→Cone
CP
ifthereisanidentity-on-objectsfunctorF :Nop →Dandandafamilyofisomorphisms
Mat
W∗-Alg (M (A),C)∼=H(A,F(n)) naturalinn∈Nop andA∈W∗-Alg (1)
CP n Mat CP
thenthetransposeofH
W∗-Algop →[D,Cone]
CP
isfullandfaithful.
Inparticular,letH betherestrictedhom-functorH:W∗-Algop ×N →Cone. ForF :Nop →N ,
CP CP Mat CP
we first note that for any matrixV :m→n in N we have a completely positive mapV∗(−)V, with
Mat
reference to Choi’s theorem [7]. To turn this into a contravariant functor F : Nop → N , we note
Mat CP
that N is self-dual with the isomorphism Nop →N taking a matrix V to its transpose V(cid:62). We
Mat Mat Mat
166 Infinite-dimensionalityinQuantumFoundations
let F(V) = (V(cid:62))∗(−)V(cid:62). The natural isomorphism (1) is now the standard bijection between states
M (A)→CandcompletelypositivemapsA→M (seee.g.[43]). Thuswecanconcludethatthehom-
n n
conefunctorW∗-Algop →[N ,Cone]isfullandfaithful.
CP CP
Itremainstoshowthatthehom-setfunctor
W∗-Algop →[N ,Set]
CP CP
isfullandfaithful. Wemustshowthatifafamilyoffunctionsφ :W∗-Alg (B,M )→W∗-Alg (A,M )
n P n P n
between hom-sets is natural in n ∈ N then each φ is necessarily a cone homomorphism, i.e. that
CP n
φ (λ.f)=λ.φ (f)andφ (f+g)=φ (f)+φ (g). Thefirstfact,φ (λ.f)=λ.φ (f),comesimmediately
n n n n n n n
from naturality with respect to the CP-map M →M given by scalar multiplication with the scalar λ.
n n
Forthesecondfact,φ (f +g)=φ (f)+φ (g),weuseacharacterizationofpairsofmapsA→M . Let
n n n n
j:M →M betheidempotentcompletelypositivemap j(a b)=(a 0). Wehaveabijection
2n 2n c d 0 d
W∗-Alg (A,M )×W∗-Alg (A,M ) ∼= {h∈W∗-Alg (A,M )|h= j·h}. (2)
CP n CP n CP 2n
Thisbijectiontakesapairofmaps f,g:A→M tothemaph:A→M withh(a)=(f(a) 0 ). Under
n 2n 0 g(a)
the bijection (2), we can understand addition in the cone W∗-Alg (A,M ) as composition with the CP
CP n
map t : M → M given by t(a b) = a+d, and so, since φ is natural with respect to t, the addition
2n n c d
structureoftheconeispreservedbyeachφ .
n
Here is a higher level account of the previous paragraph. Let FdC∗-Alg be the category of
CP
all finite dimensional C*-algebras and completely positive maps between them. We have an equiva-
lence of categories [N ,Set](cid:39)[FdC∗-Alg ,Set], in other words, the Karoubi envelope of N con-
CP CP CP
tains FdC∗-Alg (e.g. [41, 24]). Now FdC∗-Alg has a full subcategory FdCC∗-Alg , the com-
CP CP CP
mutative finite dimensional C*-algebras and completely positive maps between them. In fact, this
category FdCC∗-Alg of commutative C*-algebras is equivalent to the Lawvere theory for abstract
CP
cones (c.f. [20, Prop. 4.3]), so the category Cone of cones is a full subcategory of the functor category
[FdCC∗-Alg ,Set]. Sonaturalmapsin[N ,Set]are,inparticular,conehomomorphisms.
CP CP
As discussed in the introduction, this theorem means that a W*-algebra can be understood as the
canonical colimit of a diagram of matrix algebras and completely positive maps. At this point, it is
important to stress that our result is about colimits in the opposite category W∗-Algop of W*-algebras
CP
andcompletelypositivemaps,andthereforeaboutlimitsinthecategoryW∗-Alg ofW*-algebrasand
CP
completelypositivemaps.
4 Some remarks on topological vector spaces
ItisnaturaltowonderwhetherwecanabstractfromthesettingofthetheoryofW*-algebrasandevolve
tothelargerscopeofthetheoryoftopologicalvectorspaces.
Recall that a topological vector space X over a topological field K is a vector space whose addition
X2 → X and scalar multiplication K×X → X are continuous with respect to the topology of K. All
HilbertspacesandBanachspacesareexamplesofsuchtopologicalvectorspaces. And,foreverynatural
number k and every topological vector space X, the k×k matrices whose entries are in X also form a
topologicalvectorspace,M (X),wherethetopologyistheproducttopologyonXk2.
k
M.Rennela,S.Staton&R.Furber 167
A continuous K-linear map f :X →Y between topological vector spaces over the field K therefore
alwaysextendstoacontinuousmapM (f):M (X)→M (Y)foreverynaturalnumbern∈N,essentially
n n n
byviewingM (f)as f ×···× f,withn2 factors.
n
The category of topological vector spaces over a given topological field K is commonly denoted by
TVS or TVect , taking topological vector spaces over K as objects and continuous K-linear maps as
K K
arrows. We will additionally restrict to topological vector spaces over C whose bornology (i.e. ideal of
boundedsets[40,§I.5])isdefinablebyanorm,apropertywhichistrueforC*-algebrasandW*-algebras.
ConsiderasubcategoryVofTVectC closedundermatrixalgebras,i.e. satisfying
C∈V and A∈V =⇒ M (A)∈V. (3)
n
We will call V the closure of the category V under matrices of morphisms. Then, one obtains the fol-
C
lowingtheoremanalogoustoTheorem1,inlinewith[38].
Theorem3. ConsiderasubcategoryVofTVectC satisfying(3)andsuchthatthematricesfunctor
VC×NMat→TVectC
factorsthroughV. ItinducesafullandfaithfulfunctorV →[N ,V].
C Mat
Fromtherewecanbuildrepresentationsforsomeofthecategoriesoftopologicalvectorspacesintro-
ducedintheliterature[2,19,40]. First,wewillrecallsomedefinitionsandthenstatetherepresentation
theoremassociatedtothem.
ABanachspaceisacompletenormedvectorspace. A(concrete)operatorspaceisaclosedsubspace
of a C*-algebra, or alternatively a Banach space given together with an isometric embedding into the
space of all bounded operators on some Hilbert space H [35]. We define Banach to be the category of
Banachspacesandboundedmaps,i.e. linearmapsϕ suchthat∃k.(cid:107)ϕ(x)(cid:107)≤k·(cid:107)x(cid:107),andOpSpacetobe
thecategoryofoperatorspacesandcompletelyboundedmaps[33],i.e. linearmapsϕ suchthatthenorm
(cid:107)ϕ(cid:107) :=sup (cid:107)M (ϕ)(cid:107) is finite, i.e. (cid:107)M (ϕ)(cid:107) is bounded by a constant which does not depend on n.
cb n n n
Alternatively,onecouldconsidercontractivemaps,i.e. linearmapsϕ suchthat(cid:107)ϕ(cid:107)≤1,andcompletely
contractivemaps,i.e. linearmapsϕ suchthat(cid:107)M (ϕ)(cid:107)≤1foreverynaturalnumbern. Then,thecate-
n
goryOpSpaceofoperatorspacesisafullsubcategoryofthecategoryBanach ofBanachspacesand
CP
completelyboundedmaps.
With those definitions in mind, we can simply explain why we needed to make an additional re-
quirement on the structure of the topological vector spaces that we are considering. In short, complete
boundedness is not formalizable in terms of topology. In detail, an abstract operator space is a com-
patible choice of norms on E, M (E), M (E), ··· for a Banach space E. No matter which norm we
2 3
choose,thetopologyisalwaysthesame,soitisalwaysthecasethatM (f)iscontinuous,andtherefore
n
bounded. Then,anoperatorspacestructureisachoiceofnormgeneratingthebornologyforeachn,and
completelyboundedmapsarethoseforwhichthenormofthemapsM (f)isboundedbysomeconstant
n
notdependingonn. InthecasewhereE isequippedwithaC∗-algebrastructure,wetakethenormstobe
theC∗-normsofeachM (E),andthisdefineswhatthenormsareuniquelyinthecaseofaconcreteop-
n
eratorspace. Inthecaseofcompletelycontractivemapsthenormboundisautomaticallyindependentof
168 Infinite-dimensionalityinQuantumFoundations
n(itis1)butonestillneedsachoiceofnormoneachM (E)andthereforecannotsticktotopologyalone.
n
An order-unit space (E,E ,u) is an ordered vector space (E,E ) equipped with a strong Archi-
+ +
median unit u∈E [2, Def. 1.12]. An operator system is an involutive vector space V such that the
+
vector space M (V) of n-by-n matrices whose entries are inV is an order-unit space, or alternatively a
n
closedsubspaceofaunitalC*-algebrawhichcontains1. WedefineOUStobethecategoryoforder-unit
spaces and unit-preserving positive maps between them, and OpSystem to be the category of operator
systems and unit-preserving completely positive maps between them. Then the category OpSystem of
operatorsystemsisafullsubcategoryofthecategoryOUS oforder-unitspacesandunit-preserving
UCP
completelypositivemaps.
Theorem 4. The following matrix functors M, taking a topological vector space X to a functor M(X):
n(cid:55)→M (X),arefullandfaithful
n
M:OpSpace→[N ,Banach] M:OpSystem→[N ,OUS]
Mat Mat
5 Further work
The present work only covers one specific aspect of the approximation of infinite-dimensional operator
algebrasbyfinite-dimensionalones. Determininghowourconstructionrelatestootherapproaches,like
TobiasFritz’perspectiveoninfinite-dimensionalstatespaces[18],isatopicworthinvestigating.
Moreover, Day’s construction [12] provides a canonical way to extend the tensor product of a base
category, like N , to a tensor product of presheaves (see also [28]). Thus one can define a tensor
CP
product A(cid:63)B of two W*-algebras A and B as the unique extension of the standard tensor product of
matrix algebras that preserves colimits of CP-maps in each argument. This seems related to the way
that Grothendieck defined tensor products on the category of Banach spaces [22], by starting with the
categoryoffinitedimensionalnormedspacesandatensordefinedonthere.
Finally, although ithas beenestablishedthat everyvector spaceisa filteredcolimit offinitedimen-
sionalspaces,itisunclearwhetherthereisananalogouscharacterizationforW*-algebras.
Acknowledgements The authors would like to thank Ross Duncan, Tobias Fritz, Shane Mansfield,
IsarStubbe,BramWesterbaanandFabioZanasiforhelpfuldiscussions,andtheanonymousrefereesfor
their comments and suggestions. 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 n. 320571, and a Royal Society University Research Fellowship. Robert
FurberwassupportedbyProject4181–00360fromtheDanishCouncilforIndependentResearch.
References
[1] Samson Abramsky & Bob Coecke (2004): A categorical semantics of quantum protocols. In: Logic
in Computer Science, 2004. Proceedings of the 19th Annual IEEE Symposium on, IEEE, pp. 415–425,
doi:10.1109/LICS.2004.1319636.
M.Rennela,S.Staton&R.Furber 169
[2] ErikMAlfsen&FrederikWShultz(2012): Statespacesofoperatoralgebras: basictheory, orientations,
andC*-products. SpringerScience&BusinessMedia,doi:10.1007/978-1-4612-0147-2.
[3] JohnBaez&MikeStay(2011):Physics,Topology,LogicandComputation:ARosettaStone.InBobCoecke,
editor: NewStructuresforPhysics, LectureNotesinPhysics813, SpringerBerlinHeidelberg, pp.95–172,
doi:10.1007/978-3-642-12821-9 2.
[4] OlaBratteli(1972): InductivelimitsoffinitedimensionalC*-algebras. TransactionsoftheAmericanMath-
ematicalSociety171,pp.195–234,doi:10.2307/1996380.
[5] GiulioChiribella,GiacomoMauroD’Ariano&PaoloPerinotti(2010): Probabilistictheorieswithpurifica-
tion. PhysicalReviewA81(6),p.062348,doi:10.1103/PhysRevA.81.062348.
[6] KentaCho(2016): SemanticsforaQuantumProgrammingLanguagebyOperatorAlgebras. NewGenera-
tionComputing34(1),pp.25–68,doi:10.1007/s00354-016-0204-3.
[7] Man-DuenChoi(1975): Completelypositivelinearmapsoncomplexmatrices. Linearalgebraanditsappli-
cations10(3),pp.285–290,doi:10.1016/0024-3795(75)90075-0.
[8] Bob Coecke (2008): Axiomatic description of mixed states from Selinger’s CPM-construction. Electronic
NotesinTheoreticalComputerScience210,pp.3–13,doi:10.1016/j.entcs.2008.04.014.
[9] Bob Coecke & Keye Martin (2011): A partial order on classical and quantum states. Springer,
doi:10.1007/978-3-642-12821-9 10.
[10] BobCoecke&DuskoPavlovic(2006):Quantummeasurementswithoutsums. doi:10.1201/9781584889007.
[11] AlainConnes(1992): Noncommutativegeometry. Springer,doi:10.1007/BFb0089204.
[12] Brian Day (1970): On closed categories of functors. In: Reports of the Midwest Category Seminar IV,
Springer,pp.1–38,doi:10.1007/BFb0060438.
[13] EllieD’Hondt&PrakashPanangaden(2006): Quantumweakestpreconditions. MathematicalStructuresin
ComputerScience16(03),pp.429–451,doi:10.1017/S0960129506005251.
[14] JacquesDixmier(1977): C*-Algebras. North-HollandMathematicalLibrary15,North-HollandPublishing
Company.
[15] JacquesDixmier(1981): VonNeumannAlgebras. North-HollandMathematicalLibrary27, North-Holland
PublishingCompany.
[16] Richard P Feynman (1986): Quantum mechanical computers. Foundations of physics 16(6), pp. 507–531,
doi:10.1007/BF01886518.
[17] Cecilia Flori & Tobias Fritz (2016): (Almost) C∗-algebras as sheaves with self-action.
arxiv:1512.01669v2.
[18] TobiasFritz(2013):Oninfinite-dimensionalstatespaces. JournalofMathematicalPhysics54(5),p.052107,
doi:10.1063/1.4807079.
[19] RobertFurber(2016): CategoricalDualityinProbabilityandQuantumFoundations. Ph.D.thesis,Radboud
University,Nijmegen.
[20] Robert Furber & Bart Jacobs (2013): From Kleisli categories to commutative C*-algebras: probabilistic
Gelfandduality. In: AlgebraandCoalgebrainComputerScience,Springer,pp.141–157,doi:10.1007/978-
3-642-40206-7 12.
[21] RomanGielerak&MarekSawerwain(2010): Generalisedquantumweakestpreconditions. QuantumInfor-
mationProcessing9(4),pp.441–449,doi:10.1007/s11128-009-0151-8.
[22] AlexanderGrothendieck(1996):Re´sume´delathe´orieme´triquedesproduitstensorielstopologiques. Resen-
hasdoInstitutodeMatema´ticaeEstat´ısticadaUniversidadedeSa˜oPaulo2(4),pp.401–481.
[23] LucienHardy(2001): Quantumtheoryfromfivereasonableaxioms. arXiv:quant-ph/0101012.
[24] Chris Heunen, Aleks Kissinger & Peter Selinger (2013): Completely positive projections and biproducts.
arXiv:quant-ph/1308.4557.
[25] MaxKelly(1982): Basicconceptsofenrichedcategorytheory. 64,CUPArchive.
170 Infinite-dimensionalityinQuantumFoundations
[26] Saunders Mac Lane (1971): Categories for the Working Mathematician. Graduate Texts in Mathematics,
SpringerVerlag,doi:10.1007/978-1-4612-9839-7.
[27] Ciara´nMLee&JonathanBarrett(2015): Computationingeneralisedprobabilisitictheories. NewJournal
ofPhysics17(8),p.083001,doi:10.1088/1367-2630/17/8/083001.
[28] Octavio Malherbe, Philip Scott & Peter Selinger (2013): Presheaf models of quantum computation: an
outline.In:Computation,Logic,Games,andQuantumFoundations.TheManyFacetsofSamsonAbramsky,
Springer,pp.178–194,doi:10.1007/978-3-642-38164-5 13.
[29] Paul-Andre´ Mellie`s (2010): Segal Condition Meets ComputationalEffects. In: Proc. LICS 2010, pp. 150–
159,doi:10.1109/LICS.2010.46.
[30] IsaacNamioka(1957): Partiallyorderedlineartopologicalspaces. MemoirsoftheAmericanMathematical
Society24,AmericanMathematicalSoc.
[31] MichaelANielsen&IsaacLChuang(2010): Quantumcomputationandquantuminformation. Cambridge
universitypress,doi:10.1017/CBO9780511976667.
[32] MichelePagani,PeterSelinger&BenoˆıtValiron(2014): ApplyingQuantitativeSemanticstoHigher-order
Quantum Computing. In: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of
ProgrammingLanguages,POPL’14,ACM,pp.647–658,doi:10.1145/2535838.2535879.
[33] VernPaulsen(2002): Completelyboundedmapsandoperatoralgebras. 78,CambridgeUniversityPress.
[34] JW Pelletier & J Rosicky´ (1993): On the equational theory of C*-algebras. Algebra Universalis 30, pp.
275–284,doi:10.1007/BF01196099.
[35] Gilles Pisier (2003): Introduction to Operator Space Theory. London Mathematical Society Lecture Note
Series,CambridgeUniversityPress,doi:10.1017/CBO9781107360235.
[36] TCRalph,AlexeiGilchrist,GerardJMilburn,WilliamJMunro&ScottGlancy(2003): Quantumcomputa-
tionwithopticalcoherentstates. PhysicalReviewA68(4),p.042319,doi:10.1103/PhysRevA.68.042319.
[37] Mathys Rennela (2014): Towards a Quantum Domain Theory: Order-enrichment and Fixpoints in W*-
algebras. In: Proceedings of the 30th Conference on the Mathematical Foundations of Program-
ming Semantics (MFPS XXX), 308, Electronic Notes in Theoretical Computer Science, pp. 289 – 307,
doi:10.1016/j.entcs.2014.10.016.
[38] Mathys Rennela & Sam Staton (2015): Complete positivity and natural representation of quantum com-
putations. In: Proceedings of the 31st Conference on the Mathematical Foundations of Program-
ming Semantics (MFPS XXXI), 319, Electronic Notes in Theoretical Computer Science, pp. 369–385,
doi:10.1016/j.entcs.2015.12.022.
[39] Shoˆichiroˆ Sakai (2012): C*-algebras and W*-algebras. Springer Science & Business Media,
doi:10.1007/978-3-642-61993-9.
[40] HelmutH.Schaefer(1966): TopologicalVectorSpaces. GraduateTextsinMathematics3,SpringerVerlag,
doi:10.1007/978-1-4612-1468-7.
[41] PeterSelinger(2008): Idempotentsindaggercategories. ElectronicNotesinTheoreticalComputerScience
210,pp.107–122,doi:10.1016/j.entcs.2008.04.021.
[42] Sam Staton & Sander Uijlen (2015): Effect algebras, presheaves, non-locality and contextuality. In:
Proc.ICALP2015,doi:10.1007/978-3-662-47666-6 32.
[43] ErlingStørmer(2012): PositiveLinearMapsofOperatorAlgebras. SpringerMonographsinMathematics,
SpringerBerlinHeidelberg,doi:10.1007/978-3-642-34369-8.
[44] MasamichiTakesaki(2002): Theoryofoperatoralgebras.I. Springer-Verlag,Berlin.
[45] SeanTull(2016): OperationalTheoriesofPhysicsasCategories. arXiv:quant-ph/1602.06284.
[46] JamieVicary(2011): Categoricalformulationoffinite-dimensionalquantumalgebras. Communicationsin
MathematicalPhysics304(3),pp.765–796,doi:10.1007/s00220-010-1138-0.