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B. Coecke, J. Selby & S. Tull
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Two Roads to Classicality
Bob Coecke∗
University of Oxford
bob.coecke@cs.ox.ac.uk
John Selby†
University of Oxford & Imperial College London
john.selby08@imperial.ac.uk
Sean Tull ‡
University of Oxford
sean.tull@cs.ox.ac.uk
Mixing and decoherence are both manifestations of classicality within quantum theory, each of
which admit a very general category-theoretic construction. We show under which conditions these
two ‘roads to classicality’ coincide. This is indeed the case for (finite-dimensional) quantum the-
ory, where each construction yields the category of C*-algebras and completely positive maps. We
present counterexamples where the property fails which includes relational and modal theories. Fi-
nally, we provide a new interpretation for our category-theoretic generalisation of decoherence in
terms of ‘leaking information’.
Physical, computational, and many other theories can very generally be described by (monoidal)
categories. Examples include categorical logic [21], categorical programming language semantics [2],
and more recently, categorical quantum mechanics [1]. More specifically, we think of any (monoidal)
category as a candidate theory of physical systems (objects) and processes (morphisms). When view-
ing morphisms as quantum processes, two universal constructions provide roads to classical physics,
allowing one to build new systems to describe classical data, respectively embodying a generalisation of
mixing and of decoherence.
Firstly, one may represent mixing in a category Cby means of sum-enrichment. This is justified
by the fact that, when Cis monoidal, endomorphisms of the tensor unit (i.e. scalars) allow one to then
form weighted mixtures of morphisms, generalising the probabilistic mixtures appearing in information
theory. When Cis sum-enriched, we may apply a universal construction, the biproduct completion
C⊕[22] to generate classical set-like systems, with the biproduct playing the role of the set-union.
The second road to classicality, decoherence, is given in the quantum formalism by an idempotent,
causal operation that sets all off-diagonal entries of a density matrix to 0. Causality may be discussed
in any category Ccoming with suitable ‘discarding’ morphisms, and we generalise decoherence to any
causal idempotent in such a category. Applying our next universal construction, (a variant of) the Karoubi
Envelope Split (C)(splitting of idempotents or Cauchy completion) [5] generates systems equipped with
such a ‘decoherence’ map which ‘classicise’ their processes.
In this article, we investigate when there is an embedding C⊕→Split (C)(Theorem 2.13) and when
it is an equivalence (Corollary 2.15), showing that this is the case for quantum theory (Corollary 3.4). In
this case one recovers not only classical theory, but also intermediate systems described by C*-algebras,
and remarkably C*-algebras only. In its most concrete form, our main result may be stated as follows:
in the category of finite-dimensional C*-algebras, all causal idempotents split.
Correspondences between these different manners of encoding classicality within categorical quan-
tum mechanics were already studied by Heunen, Kissinger and Selinger [20]. Our result strengthens and
greatly generalises theirs by removing the assumption that the idempotents are self-adjoint. Abstractly,
this allows our approach to apply beyond the ‘dagger compact categories’ considered there to arbitrary
∗Supported by AFOSR grant Algorithmic and Logical Aspects when Composing Meanings.
†Supported by the EPSRC through the Controlled Quantum Dynamics Centre for Doctoral Training.
‡Supported by EPSRC Studentship OUCL/2014/SET.
B. Coecke, J. Selby & S. Tull
105
categories without a dagger, or in fact even monoidal structure. Nonetheless, it is a straightforward corol-
lary that our result respects the monoidal structure when present. While the passage from self-adjoint
idempotents to general idempotents might seem minor, it is precisely this relaxation that allows for a
clear interpretation.
There was a third road in [20] sandwiched between the two others, corresponding to the use of ‘dag-
ger Frobenius algebras’ as a generalisation of C*-algebras. This leads to another interesting interpreta-
tion of our result: as already hinted at above, a structure much weaker than the full-blown axiomatization
suffices to capture all finite dimensional C*-algebras. Moreover this weaker structure has a very clear
interpretation as resulting from leakage of information into the environment, see Section 5. Two ex-
tremes are the fully quantum C*-algebras with minimal leakage, and the fully classical C*-algebras with
maximal leakage. Hence, from a physical perspective, the additional structure of C*-algebras is merely
an artefact of the Hilbert space representation.
This work is related to and draws on several earlier works. In particular, Corollary 2.15 draws on a
result of Blume-Kohout et al. [4], and our approach can be seen as a generalisation of that of Selinger
and collaborators [28, 20] as discussed in depth in Section 6.
1 Setup
Physical theories can be described as categories where we think of the objects in a category as (physical)
systems, of the morphisms as processes, and ◦as sequential composition of these processes1. We often
call a process an event when we think of it as forming a part of a probabilistic process (e.g. the occurrence
of a particular outcome in a quantum measurement).
The categories we consider here will typically come with a chosen object to represent ‘nothing’,
denoted I. We call morphisms a:I→A states,e:A→I effects, and s:I→I scalars. Admittedly, this
terminology is slightly abusive unless we take Ito be the tensor unit in a monoidal category (C,⊗,I).
In such a case, where we think of the tensor product ⊗as parallel composition of processes, we can
introduce or remove the ‘nothing’-object at will via the natural isomorphisms
λ
A:I⊗A→Aand
ρ
A:A⊗
I→A. While most of our example categories are monoidal, our results do not require any monoidal
structure, though importantly they are compatible with any which is present (see Remark 2.17.
Example 1.1 (Pure quantum events). These can be described by a category Quantpure where the objects
are finite-dimensional Hilbert spaces Hand morphisms are linear maps identified up to a global phase
(i.e. f∼g⇐⇒ f=ei
θ
gfor some
θ
). The monoidal product is the standard Hilbert space tensor product
and the monoidal unit is C. States are therefore vectors in Hup to a phase, and effects dual-vectors
again up to a phase. Scalars are complex numbers up to a phase, i.e. positive real numbers.
Two important structures are lacking in this theory of pure quantum events, namely our two roads to
classicality: discarding and mixing.
Definition 1.2. Acategory with discarding (C,)is a category coming with a chosen object Iand
family of morphisms A:A→I, with I=idI. A morphism f:A→Bis causal when B◦f=A.
The reason why the term causality is justified is that when restricting to causal processes, a (monoidal)
theory is non-signalling [11]. Hence compatibility with relativity theory boils down to the requirement
that all effects are causal and hence equal to the discarding effect; this notion of causality was first in-
troduced in [7]. However, in this form causality has a very lucid interpretation: whether we discard an
object before or after applying some process is irrelevant, either way the result is the same — the object
1Indeed we will use the terms category/morphism/object and theory/process/system interchangeably throughout.
106
Two Roads to Classicality
ends up discarded [12, 13]. This guarantees that processes outside the direct surroundings of an exper-
iment can be ignored, and hence is vital to even be able to perform any kind of scientific experiment
without, for example, intervention from another galaxy. We now move on to generalised mixing.
Definition 1.3. A category Cis semi-additive when it is enriched in the category CMon of commutative
monoids. That is, each homset forms a commutative monoid (C(A,B),+,0), with +and 0 preserved
by composition. In particular, Chas a family of zero morphisms 0=0A,B:A→B, for all A,B∈C,
satisfying 0 ◦f=0=0◦gfor all morphisms f,g. We often write ∑n
i=1fifor f1+···+fn.
Remark 1.4. We use the term mixing here as in quantum theory it is precisely this semi-additive structure
that allows one to discuss probabilistic mixtures of processes. More generally, in any monoidal category
where the scalars s:I→Ican be interpreted as probabilities, we may discuss probabilistic weightings
of processes by setting s·f=
λ
B◦(s⊗f)◦
λ
A−1:A→B, and hence probabilistic mixtures ∑isi·fi. If
∑isi=1 then {si}is a interpreted as a normalised probability distribution, and any such mixing of causal
processes will again be causal.
We now consider the role that these two structures play in quantum theory.
Example 1.5 (Quantum events). These can be modelled as a category Quant which has the same
objects and monoidal product as Quantpure. Note that the bounded operators on a Hilbert space Hdefine
an ordered real vector space B(H)with associated positive cone B(H)+. Morphisms between these
are then defined as linear order-preserving maps f:B(H)→B(H′)which moreover are completely
positive, meaning f⊗idKpreserves positivity of elements, for all objects K. The semi-additive
structure is defined as the usual addition of linear maps. States therefore correspond to
ρ
∈B(H)+(i.e.
are density matrices), effects can be written in terms of states via the trace inner product h
ρ
,i=tr(
ρ
)
(i.e. are POVM elements) and scalars are positive real numbers. The discarding effect is given by tr(I)
and so causal states have trace 1 and general causal morphisms are trace preserving (i.e. are CPTP maps).
This category can also be defined in terms of Selinger’s CPM construction [27], which we will return to
in Section 6. There is an embedding of Quantpure in Quant sending a process f:H→H′to the map
f◦ − ◦ f†:B(H)→B(H′).
Remark 1.6. The relationship between Quantpure and Quant as described in the above example can
be viewed in two different ways. Firstly note that Quantpure is a subcategory of Quant and then it is a
standard result in quantum information that there are two equivalent ways to write any general quantum
transformation f:A→Bin terms of processes in Quantpure , firstly, via the Kraus decomposition, as
a sum of pure transformations, f=∑kak, and secondly, via Stinespring dilation, as a pure process
with an extra output which is discarded, f=
ρ
B◦(idB⊗C)◦gwhere
ρ
Bis the monoidal coherence
isomorphism. Therefore all of the processes of Quant can be obtained from those in Quantpure either by
means of the semi-additive or discarding structure.
This is an important feature of quantum theory, for example, in the form of the purification postulate
[7] which has been used as an axiom in reconstructing quantum theory [8]. More generally, the CPM
construction provides a recipe for producing categories with discarding and a form of purification, see
Section 6. Conceptually, this provides both an ‘internal’ and ‘external’ view on the origins of general
quantum transformations, which this paper develops with the two constructions.
Here are some other example theories which serve as useful points of comparison for quantum theory.
Example 1.7 (Probabilistic classical events). These can be modelled in the category Class, objects are
natural numbers n, and morphisms f:n→mare n×mmatrices with positive real entries. The monoidal
unit is 1 and the monoidal product is n⊗m=nm. Semi-additive structure is provided by the matrix sum.
States are therefore column vectors with positive real elements, effects are row vectors and scalars are
B. Coecke, J. Selby & S. Tull
107
positive real numbers. The discarding effects are those of the form (1,1, ...,1)such that causal states
are normalised probability distributions over a finite set and general causal morphisms are stochastic
matrices.
Example 1.8 (Possibilistic classical events). These can be modelled in the symmetric monoidal cate-
gory Rel of sets and relations. Here objects are sets A,B,..., morphisms R:A→Bare relations from
Ato B, i.e. subsets R⊆A×B, sequential composition of relations R:A→Band S:B→Cis de-
fined by: S◦R={(a,c)∈A×C|R(a,b)∧S(b,c)}and parallel composition is the set-theoretic product
A⊗B=A×B, so that the monoidal unit is given by the singleton set I={⋆}. In particular, the scalars
s:I→Iare the Booleans {0,1}.Rel has discarding, with A:A→Igiven by the unique relation with
a7→ ⋆for all a∈A. Hence, causal relations R:A→Bare those satisfying ∀a∃b R(a,b). This theory is
semi-additive under the union of relations R+S:=R∨S.
Example 1.9 (Modal quantum events). The events in modal quantum theories [24, 25] can be modelled
in the symmetric monoidal category where the objects are lattices of subspaces of finite dimensional
vector spaces over a particular finite field Zp, where the choice of field defines a particular modal theory
Modalp. Morphisms are ∨and ⊥preserving maps between these lattices. The monoidal product is
inherited from the tensor product of the underlying vector spaces. The monoidal unit is the lattice of
subspaces of a 1D vector space which gives two scalars 0 and 1 interpreted as impossible and possible
respectively. This allows us to define zero-morphisms by 0A,B(a) = 0 for all a. The discarding effect
A:A→Ican then be defined by: A◦a=0⇐⇒ a=0IA where a:I→A. This is again a semi-
additive category with (f∨g)(a) = f(a)∨g(a)for a∈A. In fact, Rel can be viewed as Modalpfor the
case ‘p=1’ in a certain precise sense [15].
Example 1.10. Any semi-ring (R,+,0,1)forms a one object, semi-additive discard category, with =1.
2 The Two Roads
We now introduce the two constructions which adjoin classicality to a theory. The first follows an external
perspective, describing how one may build mixtures of the existing objects. The second follows the
internal perspective, showing how classicality can emerge due to restrictions arising from decoherence.
The biproduct completion The first of these approaches is captured by a standard notion from cat-
egory theory. Recall that, in any semi-additive category, a biproduct of a finite collection {Ai}n
i=1of
objects consists of an object A=Ln
i=1Aiand morphisms AiLn
i=1AiAj
κ
i
π
jsatisfying:
π
i◦
κ
j=0 for i6=j
π
i◦
κ
i=idAi(1)
n
∑
i=1
π
i◦
κ
i=idA(2)
This makes Aboth a product and coproduct of the objects {Ai}n
i=1. An empty biproduct is the same
as a zero object — an object 0 which is both initial and terminal.
Definition 2.1. In a category with discarding a biproduct is causal when its morphisms
κ
iare causal.
In contrast, the structural morphisms
π
iwill usually not be causal. Note that a semi-additive category
has finite (causal) biproducts whenever it has a zero object and (causal) biproducts of pairs of objects.
108
Two Roads to Classicality
Examples 2.2. Class and Rel each have causal biproducts, given by the direct sum of vector spaces and
disjoint union of sets, respectively. Any grounded biproduct category in the sense of Cho, Jacobs and
Westerbaan (×2) [9] is a semi-additive category with discarding and causal biproducts.
Definition 2.3 (Biproduct completion). Any semi-additive category Cmay be embedded universally
into one with biproducts C⊕, its free biproduct completion (see [22, Exercise VIII.2.6]). The objects of
C⊕are finite lists hA1,...,Aniof objects from C, with the empty list forming a zero object. Morphisms
M:hA1,...,Ani → hB1,...,Bmiare matrices of morphisms hMi,j:Ai→Bjiand composition is the usual
one of matrices. The identity on hA1,...,Aniis the matrix with identities on its diagonal entries and
zeroes elsewhere.
There is a canonical full and faithful embedding of semi-additive categories C֒→C⊕given by
A7→ hAi. Moreover, this is universal in that any semi-additive functor F:C→Efrom Cto a semi-
additive category Ewith biproducts lifts to one ˆ
F:C⊕→E, unique up to natural isomorphism.
Example 2.4. Generalising Example 1.7, the biproduct completion of a semi-ring R, seen as a one-object
category, is the category MatRof R-valued matrices. In particular Class =MatR+, while the category
MatBof Boolean valued matrices is equivalent to FRel, the full subcategory of Rel on finite sets.
When Cis a category with discarding, C⊕is also with I=hIiand hA1,...,Ani=hAi:Ai→Iin
i=1.
To state its universal property, we will need the following notions.
Definition 2.5. A functor F:(C,)→(D,)between categories with discarding is causal when the
morphism F(I):F(I)→Iis an isomorphism and F(A) = F(I)◦F(A)for all objects A. A natural
transformation is causal when all of its components are.
In particular, the embedding C֒→C⊕is causal, with the same universal property as before when
restricted to causal biproducts, functors and natural transformations.
Splitting idempotents We now turn to decoherence, capturing it with the following categorical con-
cept. Recall that, in any category, an idempotent on an object Ais a morphism p:A→Asatisfying
p◦p=p. An idempotent splits when it decomposes as p=m◦efor a pair of morphisms e:A→B,
m:B→A, with e◦m=idB. We then denote the splitting pair by (m,e). Conversely, for any such pair of
morphisms, m◦eis always an idempotent on A.
Definition 2.6. In a category with discarding, an idempotent splits causally when it has a splitting (m,e)
with mcausal.
Definition 2.7 (Karoubi Envelope). Splittings for idempotents can always be added freely for any cat-
egory C. The Karoubi Envelope Split(C)of Cis the following category ([5], see also [28]). Objects are
pairs (A,p)where p:A→Ais an idempotent. Morphisms f:(A,p)→(B,q)are morphisms f:A→B
in Csatisfying f=q◦f◦pIn particular, the identity on (A,p)is p.
Split(C)has discarding whenever Cdoes, given by (A,p)=◦p:(A,p)→(I,id). In this case we
write Split (C)for the full subcategory on objects (A,p)with pcausal. If Cis semi-additive then so is
Split()(C), with addition lifted from C. The key property of the construction is as follows: there is a full
(causal) embedding C֒→Split()(C)given by A7→ (A,id), which gives every (causal) idempotent in C
a (causal) splitting in Split()(C). As before, Split( )(C)is universal with this property.
Remark 2.8. We may interpret the Split(C)construction as introducing new objects by imposing a
fundamental restriction on the allowed morphisms, (i.e. those satisfying f=q◦f◦p) in Section 5 we
discuss how such a restriction can arise due to ‘leaking’ information.
B. Coecke, J. Selby & S. Tull
109
Comparing the constructions It is now natural to ask what the relationship is between the C⊕and
Split (C)constructions, and when they coincide. We now answer this question for both Split (C)and
the more generally definable Split(C)simultaneously. We will require the following weakening of the
notion of a biproduct:
Definition 2.9. Let Cbe a category with zero morphisms. A disjoint embedding of a finite collection
of objects {Ai}n
i=1in Cis given by an object Aand morphisms
κ
i:Ai→A,
π
j:A→Ajsatisfying the
first collection of biproduct equations (1). When Cis semi-additive, the morphism p=∑n
i=1
κ
i◦
π
iis
then an idempotent on A. When Calso has discarding, a causal disjoint embedding is one for which the
morphisms pand
κ
iare all causal.
In particular, an empty (causal) disjoint embedding is just an object of the form (A,0)in Split()(C).
Note that a disjoint embedding is not necessarily a biproduct in C, since it may fail to satisfy (2).
Example 2.10. Any (causal) biproduct is in particular a (causal) disjoint embedding. Hence they are
present in our examples Class,Rel and MatR.
Example 2.11. A motivating example is Quant, which has disjoint embeddings which are not biprod-
ucts. Given a collection of (finite-dimensional) Hilbert spaces {Hi}i, we may form their Hilbert space
direct sum Ln
i=1Hi, which is their biproduct in the category FHilb of (finite-dimensional) Hilbert spaces
and continuous linear maps. However this is no longer a biproduct of the {Hi}iin Quant, where lin-
ear maps from FHilb are identified up to global phase, but only a disjoint embedding. Physically, the
distinction is that the addition in FHilb is given by superposition, while that of Quant refers to mixing.
This example generalises to categories of the form CPM(C), see Proposition 6.2 later.
Remark 2.12. Disjoint embeddings can be understood as a property of our theory allowing for the
encoding of classical data. Concretely, they provide the ability to store any collection of systems in a
disjoint way in some larger system. In particular, by forming a disjoint embedding Cof ncopies of I
we may store an n-level classical system in C. However, the choice of Cis non-canonical: there can be
many which need not be isomorphic.
Abstractly, the significance of disjoint embeddings is the following.
Theorem 2.13. Let Cbe semi-additive (with discarding). Then Split()(C)has finite (causal) biproducts
iff Chas (causal) disjoint embeddings.
Proof. We have already seen that Split()(C)is semi-additive, so the statement makes sense. Expanding
the definitions shows that a (causal) disjoint embedding is precisely a (causal) biproduct of the form
(A,p) = Ln
i=1(Ai,id)in Split()(C). Hence the conditions are clearly necessary. It’s easy to see that
an empty (causal) disjoint embedding (A,0)is a zero object in Split()(C), so it suffices to check this
category has binary (causal) biproducts.
Now for any pair of objects (A1,p1),(A2,p2)in Split( )(C), by assumption the objects A1A2have
a (causal) disjoint embedding (A,p), with morphisms
κ
i:Ai→Aand
π
j:A→Aj. Then q=∑2
i=1
κ
i◦
pi◦
π
iis a (causal) idempotent on A. Further, it is easy to check that
κ
i◦pi:(Ai,pi)→(A,q)and
pi◦
π
i:(A,q)→(Ai,pi)are well-defined morphisms in Split()(C)making (A,q)a (causal) biproduct
(A1,p1)⊕(A2,p2).
Observe that the idempotents arising from disjoint embeddings are those with the following property.
Definition 2.14. An idempotent p:A→Ahas a finite decomposition when it may be written as a finite
sum p=∑n
i=1mi◦eiof split idempotents pi=mi◦eifor which pi◦pj=0 for i6=j. Such a decomposition
is causal when all of the morphisms miare causal.
110
Two Roads to Classicality
We can now determine when our two constructions coincide (cf. [28, Corollary 4.7]).
Corollary 2.15. When Chas (causal) disjoint embeddings, there is a canonical full, semi-additive
(causal) embedding F:C⊕→Split()(C). Further, the following are equivalent:
1. Fis a (causal) equivalence of categories;
2. every (causal) idempotent splits (causally) in C⊕;
3. every (causal) idempotent in Chas a (causal) finite decomposition.
When the causal form of these hold we say Chas the finite decomposition property.
Proof. To see that Fexists, apply the universal property of C⊕to the (causal) embedding C→Split()(C),
using Theorem 2.13. Concretely, Facts on objects by (A1,··· ,An)7→ Ln
i=1(Ai,id).
(1)⇒(2): Since Fis a (causal) equivalence and (causal) idempotents split (causally) in Split()(C),
they do in C⊕. (2) ⇒(3): It’s easy to see that a (causal) idempotent p:A→Asplits (causally) over
(Ai)n
i=1in C⊕iff it has a (causal) finite decomposition p=∑n
i=1piwith each pisplitting over Ai.
(3) ⇐⇒ (1): By construction, Fis (causally) essentially surjective on objects iff every object (A,p)
forms a (causal) biproduct Ln
i=1(Ai,idAi)in Split()(C). By definition, this holds iff every phas a (causal)
finite decomposition.
Example 2.16. When Calready has biproducts the embedding C֒→C⊕is an equivalence, and Corol-
lary 2.15 amounts to the fact that any finitely decomposable idempotent p=p1+···+pnin Calready
splits over Ln
i=1Ai, where pisplits over Ai, for each i.
Remark 2.17 (Monoidal Structure). These results are compatible with monoidal structure whenever it
is present, in a straightforward way:
•a(symmetric) monoidal category with discarding (C,⊗,)is a (symmetric) monoidal category
(C,⊗,I)which is also a category with discarding (C,,I)with Ibeing the monoidal unit, for
which all coherence isomorphisms are causal and A⊗B=
λ
I◦(A⊗B)for all objects A,B, where
λ
I:I⊗I→Iis the coherence isomorphism.
•acausal (symmetric) monoidal functor F:(C,⊗,)→(D,⊗,)is a causal functor which is strong
(symmetric) monoidal, with causal structure isomorphisms I→F(I)and F(A)⊗F(B)→F(A⊗B).
•Asemi-additive monoidal category is a monoidal category which is monoidally enriched in CMon.
Explicitly, it is semi-additive with f⊗(g+h) = f⊗g+f⊗h,(f+g)⊗h=f⊗h+g⊗hand f⊗0=
0=0⊗gfor all morphisms f,g,h. for all morphisms f,g,h.
When Cis a semi-additive (symmetric) monoidal category (with discarding) so are each of C⊕and
Split()(C), and they satisfy the same universal properties with respect to such categories and (causal,
symmetric) monoidal semi-additive functors and (causal) monoidal natural transformations between
them. In particular, the functors and equivalences of Corollary 2.15 are now monoidal ones.
Results on Idempotent Splittings Before turning to quantum theory, we briefly consider some abstract
results of use later. Firstly, recall that a category with a distinguished object Iis well-pointed when for
all f,g:A→Bwith f◦a=g◦afor all a:I→A, we have f=g. For any morphism f:A→Bwe set
Im(f):={f◦a|a:I→A}.
Example 2.18. All of the categories with discarding Quant,Class,Rel,MatRare well-pointed over their
usual object I.Real quantum theory provides a physically interesting non-well-pointed category [19].
Lemma 2.19. Let p,q:A→Abe (causal) idempotents in a well-pointed category (C,I), with Im(p) =
Im(q). Then psplits (causally) iff qdoes, and phas a (causal) finite decomposition iff qdoes.
B. Coecke, J. Selby & S. Tull
111
Proof. For all a:I→Awe have q◦a=p◦bfor some band so p◦q◦a=p◦p◦b=p◦b=q◦a. Hence
by well-pointedness p◦q=q, and dually q◦p=palso. This states precisely that (A,p)and (A,q)are
(causally) isomorphic in Split()(C). Then phas a (causal) finite decomposition iff it forms a (causal)
biproduct Ln
i=1(Ai,id), iff qdoes. Taking i=1 shows that psplits (causally) precisely when qdoes.
Next we observe that for many theories, including quantum theory and our other probabilistic exam-
ples, splittings for causal idempotents in fact suffice to provide splittings of a broader class.
Definition 2.20. In a semi-additive category with discarding, a morphism f:A→Bis sub-causal when
there is some x:A→Iwith (◦f) + x=.
Examples 2.21. Any causal process is in particular sub-causal. In Quant, a process is sub-causal when it
is trace non-increasing. A sub-causal process in Class is a sub-stochastic matrix, i.e. one whose columns
have sum bounded by 1. In Rel and Modalpevery process is sub-causal.
Proposition 2.22. Let Cbe a semi-additive category with discarding satisfying:
•Cancellativity: f+g=f+h=⇒g=h;
•For every non-zero f:A→Bthere exists a:I→Awith ◦f◦a=idI.
If causal idempotents causally split in C, so do (non-zero) sub-causal idempotents.
Proof. Let p:A→Abe a non-zero sub-causal idempotent, with (◦p) + x=. Note that we have
◦p= ( ◦p+x)◦p= ( ◦p) + (x◦p)and so by cancellativity x◦p=0. By assumption there exists
a:I→Awith ◦p◦a=1. One may then check that q=p+(p◦a◦x)is a causal idempotent satisfying
p◦q=qand q◦p=p. These ensure that if qhas splitting (m,e)then phas splitting (m,e◦p).
3 Quantum theory
We now turn to the main result of this paper, that for the example of quantum theory our two constructions
coincide, both leading to the (symmetric monoidal) category of finite dimensional C*-algebras.
Example 3.1 (C*-algebras). In the category CStar objects are finite dimensional C*-algebras and mor-
phisms completely positive linear maps (in the same sense as defined for Quant). The monoidal product
is the standard tensor product of finite-dimensional C*-algebras and the monoidal unit is B(C). Semi-
additive structure is provided by the standard sum of linear maps. Discarding effects are provided by the
trace, and biproducts by the direct sum of C*-algebras.
Note that Quant and Class are each equivalent to full subcategories of CStar, corresponding to C*-
algebras of the form B(H), and the commutative C*-algebras, respectively. These subcategories can
also be characterised in terms of leaks, see Section 5.
There is a well known classification result [6] stating that any finite dimensional C*-algebra is iso-
morphic to a direct sum of complex matrix algebras. It is therefore unsurprising that our first road, the
biproduct completion, leads to this category: see [20, Example 3.4.] for the details.
Example 3.2. There is a monoidal, causal equivalence of categories CStar ≃Quant⊕.
The second road, however, requires some more work.
Proposition 3.3. Quant has the finite decomposition property.
112
Two Roads to Classicality
Proof. We saw in Example 2.11 that Quant has causal disjoint embeddings. Now let p∈Quant(H,H)
be a causal idempotent. Then pis an idempotent, trace-preserving, completely positive linear map on
B(H). By [4, Theorem 5], there is a decomposition H≃LkAk⊗Bkand set of positive semidefinite
matrices
τ
Kon Bkwith:
p(B(H)) = {∑kMAk⊗
τ
k:MAk∈B(Ak)}
Without loss of generality, assume
τ
k6=0 and so Tr(
τ
k)6=0, for all k. Then p(B(H)) = q(B(H)) for
the causally finitely decomposable idempotent q=∑k∈Kmk◦ek, where mk:B(Ak)→B(H)is given
by M7→ (1/Tr(
τ
k))M⊗
τ
k, and ek:B(H)→B(Ak)is given by M7→ TrBk(TrAj⊗Bj,j6=k(M)).
Since states
ρ
∈Quant(C,H)may be viewed as elements of B(H), on which pis an idempotent
map, this gives that Im(p) = Im(q)in the well-pointed category (Quant,C). Hence by Lemma 2.19,
since qhas a causal finite decomposition, so does p.
By combining Example 3.2, Proposition 3.3 and Corollary 2.15, we reach our main result:
Corollary 3.4. There is a monoidal, causal equivalence:
Split (Quant)≃CStar ≃Quant⊕
Hence all causal (trace-preserving) idempotents causally split in CStar. By Proposition 2.22, the
same in fact holds for all idempotents which are sub-causal, i.e. trace-non increasing. However, it
remains an open question whether all idempotents split.
4 Further Examples
Classical probability theory There are (monoidal, causal) equivalences:
Class⊕≃Class ≃Split(Class)≃Split (Class)
The left hand equivalence holds since Class already has biproducts. Conversely, it follow from a The-
orem of Flor ([18, Theorem 2], see also [17, Theorem 4]) that any idempotent pin Class has a finite
decomposition p=z1+···+zkwhere the zisatisfy zi◦zj=
δ
i,jziand each are of rank one, hence split-
ting over I. In particular, every idempotent in Class has a finite decomposition and so splits, yielding the
other equivalences. Hence, as in the quantum case, these constructions coincide.
Possibilistic theories In contrast, the constructions will generally fail to coincide in theories of a possi-
bilistic nature, such as Modalpor Rel. By a possibilistic theory we mean one in which the addition +is
idempotent, the scalars s:I→Iunder (◦,+) are the Booleans {0,1}, and we have ◦f=0=⇒f=0
for all morphisms f. We will consider possibilistic theories with a particular physically motivated prop-
erty. Call a pair of states a0,a1:I→Aon A perfectly distinguishable when there exists a pair of effects
¯a0,¯a1on Awith ¯a0+¯a1=and ¯ai◦aj=
δ
i,j. We say that a theory satisfies perfect distinguishability
when every system not isomorphic to Ior a zero object 0 has a pair of perfectly distinguishable states,
and there exists at least one such system. Then we have the following (see App. A for proof):
Proposition 4.1. Any possibilistic theory (C,)with perfect distinguishability lacks the finite decom-
position property.
Example 4.2. Both Modalpand Rel are possibilistic theories with discarding and perfect distinguisha-
bility, and so for these theories the constructions do not coincide.
For example, in Rel, any set Anot isomorphic to I={⋆}or 0 =/0 has at least two distinct elements,
forming a pair of perfectly distinguishable states. Hence Rel lacks the finite decomposition property.
Concretely, the causal idempotent (5) is the relation on {0,1}given by 0 7→ 0, 1 7→ 0,1, does not split.
B. Coecke, J. Selby & S. Tull
113
Information units A somewhat contrived example can be motivated by considering the idea of an
information unit for a theory: a particular object Uin a theory Csuch that any process in the theory
can be ‘simulated’ on an n-fold monoidal product of U. The existence of an information unit has been
used as a postulate in reconstructing quantum theory [23] where Uis a qubit, and, moreover, underlies
the circuit model of quantum computation. We define the information unit subtheory, ChUi, as a full
subcategory restricting to objects of the form U⊗n.
Example 4.3. For the quantum case with U=C2we find that Quant⊕
hUi(CStar ≃Split (QuanthUi).
To see the equivalence note that we can obtain an arbitrary n-level quantum system by splitting a causal
idempotent, i.e., consider msuch that n≤M:=2mand consider a sub-causal projector onto an ndi-
mensional subspace then Proposition 2.22 shows there is a causal idempotent which splits over the same
system. On the other hand is not possible to construct (for example) a qutrit with direct sums or tensor
products of qubits and hence the biproduct completion does not give the entirety of CStar. In fact, this
result can be seen as a demonstration that the qubit is indeed an information unit for quantum theory as
this provides a way to simulate any other quantum system on some composite of qubits.
5 Idempotents from leaks
Idempotents naturally arise, in symmetric monoidal theories with discarding, from information leakage.
Definition 5.1. Aleak on an object Ais a morphism l:A→A⊗Lwhich has discarding as a right counit,
that is:
ρ
A◦(idA⊗L)◦l=idA. It follows that all leaks are causal processes.
Example 5.2. In Quant all leaks are constant: i.e. of the form (idA⊗
σ
)◦
ρ
−1
Afor a causal state
σ
on L.
Example 5.3. Abroadcasting map [3, 12] on an object Ais a leak l:A→A⊗A,i.e. with L=A, for
which discarding is also a left counit. Both of our classical theories Class and Rel have such a map.
Remark 5.4. In fact, minimal and maximal leakage characterises quantum and classical theory respec-
tively [26]. Specifically, Quant and Class are equivalent to the full subcategories of CStar on objects
with only constant leaks, and on objects with maximal leaks, i.e. broadcasting maps, respectively.
It is natural, given any theory (C,), and for each system Aa chosen process lA:A→A⊗LAto con-
struct a new theory CLin which lArepresents the ongoing ‘leakage’ of that system into the environment.
The result of this leakage on Awould be the process:
ι
A:=
ρ
A◦(idA⊗LA)◦lA(3)
Processes A→Bin the new theory should then consist of applying the leakage to the inputs and outputs
of every process f:A→Bin the original theory, i.e.
ι
B◦f◦
ι
A. Demanding the
ι
Abe idempotent
ensures that CLis a category —the full subcategory of Split (C)given by the objects (A,
ι
A). It is again
symmetric monoidal whenever our choice of leaks lAensures that
ι
A⊗B=
ι
A⊗
ι
Bfor all A,B. In this case,
ι
Abecomes the identity on A, guaranteeing that lAprovides a leak in the new theory, [(
ι
A⊗
ι
LA)◦lA◦
ι
A].
Alternately, starting again from C, if we want to obtain a theory which can describe all possible
systems that could arise from some leakage then we must instead take systems to be all possible pairs
(A,lA)with the above property, i.e. such that
ι
Ais idempotent. But this is none other than Split (C).
Indeed, any causal idempotent p:A→Ais of the form (3) for some lA, for example by taking lA:=
ρ
−1
A◦p:A→A⊗I. A more insightful manner is by taking lAto be any ‘purification’ of
ι
A, that is,
any pure process satisfying (3). The existence of such a process is guaranteed in quantum theory by the
Stinespring dilation theorem, and more generally for theories arising from the CPM-construction [14],
114
Two Roads to Classicality
or, those satisfying the purification postulate of [7]. Hence we have generalised the idea that decoherence
results from information-leaking, from the specific case of quantum theory to a much broader class of
theories. There is another important conclusion we can draw from idempotents resulting from leaks.
Example 5.5. A result of Vicary [29] states that finite dimensional C*-algebras are precisely (dagger
special) Frobenius structures in FHilb: comonoids
δ
:A→A⊗Awhich additionally satisfy (together
with their ‘daggers’, see Section 6 below) two equations called the dagger Frobenius law: and speciality.
However, we now know that the much weaker concept of a process l:A→A⊗Lin the directly
physically interpretable category Quant, for which
ι
is a causal idempotent as above, is already sufficient
to guarantee the rest of this comonoid structure. Hence, one is tempted to deduce that the essential
physical structure of C*-algebras is captured by that of a leak.
6 Comparison with dagger categorical approaches
The use of biproducts and idempotent splittings to model hybrid quantum-classical systems has in fact
already been studied by Selinger [28], and Heunen, Kissinger and Selinger [20]. Crucially, however,
these previous works have relied on a feature of quantum theory not present in general physical theories –
the existence of a dagger. Recall that a dagger category (C,†)is a category Ccoming with an involutive,
identity on objects functor †: Cop →C. A dagger compact category is additionally symmetric monoidal,
in a way compatible with the dagger, with every object having a dagger dual – see [1, 27] for details.
Examples 6.1. Rel,Quantpure ,Quant,FHilb,Class and CStar are all dagger compact categories. In
Rel the dagger is given by relational converse, and in the other cases it extends the adjoint of linear maps.
The results of Section 2 can easily be adapted to include daggers, as follows. A dagger idempotent
is one pwith p=p†, a dagger splitting p=m◦eis one with e=m†, and a dagger disjoint embedding
or biproduct is one with
π
i=
κ
†
ifor all i. Then D⊕and Split†(D), the full subcategory of Split (D)
on the causal dagger idempotents, satisfy the same universal properties as before with respect to dagger-
respecting functors, and Corollary 2.15 becomes an equivalence of dagger categories D⊕≃Split†(D).
The CPM construction Given a dagger compact category Cof ‘pure’ processes, we may construct
a new one, CPM(C), with the same objects but interpreted as consisting of mixed processes [27]. An
axiomatization of the construction closely resembling our treatment is provided by the following no-
tion [10, 14]. An environment structure (D,Dpure ,)consists of a dagger compact category Dwith
discarding respecting the dagger compact structure, along with a chosen dagger compact subcategory
Dpure satisfying an axiom relating and †. It satisfies purification whenever every morphism in Dis
of the form
λ
◦(⊗id)◦ffor some morphism fin Dpure. In this case there is a (dagger monoidal)
isomorphism of categories D≃CPM(Dpure). Conversely, every D=CPM(C)arises in this way.
The key example is that Quant and Quantpure form an environment structure with purification, with
Quant ≃CPM(Quantpure)≃CPM(FHilb). We have seen that, while FHilb has biproducts, Quant
merely has disjoint embeddings. In fact, these suffice to deduce properties of the CPM construction
previously shown using biproducts (cf. [28, Theorem 4.5]) - see App. A for a proof.
Proposition 6.2. If Cis a dagger compact category with zero morphisms and dagger (causal) disjoint
embeddings, so is CPM(C). If Cis also dagger semi-additive so is CPM(C), and then there is a full
embedding CPM(C)⊕֒→Split†(CPM(C)).
B. Coecke, J. Selby & S. Tull
115
The CP∗construction There is another treatment of hybrid classical-quantum systems applicable to
any dagger compact category C. Dagger special Frobenius structures in C(see Example 5.5) form the
objects of a category CP∗(C), with the motivating example being that CP∗(FHilb)≃CStar [29].
An axiomatisation of the CP∗construction, extending that for CPM, has been given by Cunningham
and Heunen [16]. Given an environment structure, observe that for each Frobenius structure (A,
δ
)in
Dpure, the morphism
λ
A◦(⊗id)◦
δ
is a causal dagger idempotent. A decoherence structure is a
choice of dagger splitting for each such idempotent, in a way compatible with the monoidal structure.
Immediately, we have the following.
Proposition 6.3. If Dhas an environment structure, Split†(D)has a canonical decoherence structure.
Under a mild extra assumption known as positive dimensionality, any decoherence structure induces
a faithful (causal, dagger) functor [16]: CP∗(Dpure )→Split†(D). Moreover, results of Heunen, Kissinger
and Selinger [20] tell us that, whenever purification is satisfied, so that Dis of the form CPM(C)for
some C, this is a full causal embedding, and that when Chas dagger biproducts the causal embeddings
from each construction factor as:
CPM(C)⊕֒→CP∗(C)֒→Split†(CPM(C)) ֒→Split (CPM(C)) (4)
Then using Corollary 2.15 we have the following:
Corollary 6.4. Let Chave dagger biproducts, and suppose CPM(C)has the finite decomposition prop-
erty. Then each of the inclusions (4) are (causal, monoidal) equivalences of categories.
In particular, the case C=FHilb gives equivalences:
Quant⊕≃CStar ≃Split†(Quant)≃Split (Quant)
extending the result from [20].
7 Conclusion
We explored two seemingly unrelated categorical representations of classicality, through the Split (C)
and C⊕constructions, finding clear properties ensuring that they coincide – the existence of disjoint
embeddings and the finite decomposition property. In Corollary 3.4 we showed that these hold for
quantum theory, strengthening the result of Heunen-Kissinger-Selinger [20] by removing all mention of
daggers. This strengthening allows for a clear physical interpretation. In particular, we saw that we may
obtain all of the usual C*-algebraic structure simply from the concept of leaking information (Section 5).
The generality of the approach here leaves many interesting open questions. Firstly, note that the
results of Section 2 do not crucially rely on the discarding structure.Though the restriction to sub-causal
idempotents is well motivated physically, as it is only these that have an interpretation as probabilistic
outcomes, an obvious open question is whether it is needed. Do all idempotents in CStar split? Also,
since these results do not rely on dagger, compact closed or even monoidal structure, they are applica-
ble to the infinite-dimensional case. Do all (sub-causal) idempotents split (causally) in the category of
arbitrary C*-algebras (or von Neumann algebras) and completely positive maps?
Finally, while the existence of disjoint embeddings has a reasonable physical interpretation (see
Remark 2.12), our main result also relied on the finite decomposition property, which is less well un-
derstood. Are there any clear physical principles which allow one to deduce the finite decomposition
property? A positive answer to this would yield a more insightful proof of Corollary 3.4. Conversely,
what other interesting physical consequences does this property have?
116
Two Roads to Classicality
A Proof of Lemma 4.1
In this appendix we prove that any possibilistic theory (C,)with perfect distinguishability lacks the
finite decomposition property.
Proof. It is easy to see that the biproduct completion C⊕is again a possibilistic theory satisfying perfect
distinguishability, and hence it suffices to show that causal idempotents do not split in C. Pick a system
Apossessing a pair of states {a0,a1}perfectly distinguishable by some effects {¯a0,¯a1}. Then since
¯a1+=, the following defines a causal idempotent on A:
p=a0◦+a1◦¯a1(5)
Suppose p:A→Ahas a splitting (m,e)over an object B. Next suppose Bhas two perfectly distin-
guishable states b0,b1, via the effects {¯
b0,¯
b1}. We define new states and effects on Aby ci=m◦biand
¯ci=¯
bi◦e, respectively, for i=1,2. Then:
0=¯c0◦c1=¯c0◦p◦c1= ( ¯c0◦a0)◦(◦c1) + ( ¯c0◦a1)◦(¯a1◦c1)
Since the scalars are the Booleans, and the biand hence ciare non-zero, in particular we must have
¯c0◦a0=0. Dually, ¯c1◦a0=0 holds also. But then ◦e◦a0= ( ¯
b0+¯
b1)◦e◦a0= ( ¯c0+¯c1)◦a0=0.
By positivity, e◦a0=0, and hence a0=p◦a0=m◦e◦a0=0, contradicting ¯a0◦a0=1.
We conclude that no such pair of states exists on B. Since p6=0, Bcannot be a zero object. Hence
we must have B≃Iand so p=x◦yfor some state xand effect yon A, respectively. But then, for
any effect zon A, since the scalars are the Booleans we must have either z◦p=0 or z◦p=y. Hence
¯a1=¯a1◦p=y=◦p=, and so ¯a1◦a0=1, a contradiction.
B Proof of Proposition 6.2
Suppose that Cis dagger compact with zero arrows and disjoint embeddings. There is always a dagger
functor C→CPM(C)which preserves zeroes and is surjective on objects, and hence CPM(C)is also.
When Chas disjoint embeddings, CPM(C)is closed under addition in C– this is proven just as for
biproducts in [27, Lemma 4.7(c)]. Finally, (the dagger version of) Theorem 2.13 gives an embedding
CPM(C)⊕֒→Split†(CPM(C)). .
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