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The compositional structure ofmultipartite quantum entanglement
Bob Coecke and Aleks Kissinger
Oxford University Computing Laboratory
Abstract. Multipartite quantum states constitute a (if not the) keyresource for quantum computations and protocols. However obtaining ageneric, structural understanding of entanglement in Nqubit systemsis a longstanding open problem in quantum computer science. Here weshow that multipartite quantum entanglement admits a compositionalstructure, and hence is subject to modern computer science methods.
We consider Nqubit states to be equivalent as computational re
sources if they can be interconverted by stochastic local (quantum)operations and classical communication (SLOCC). There are only twoSLOCCclasses of genuinely entangled 3qubit states, the GHZclass andthe Wclass, and we show that these exactly correspond with two kindsof internal commutative Frobenius algebras on C2 in the symmetricmonoidal category of Hilbert spaces and linear maps, namely specialones and antispecial ones. Within the graphical language of symmetricmonoidal categories, the distinction between special and antispecialis purely topological, in terms of connected vs. disconnected.
These GHZ and W Frobenius algebras form the primitives of a graphical calculus which is expressive enough to generate and reason about representatives of arbitrary Nqubit states. This calculus refines the graphical calculus of complementary observables due to Duncan and one of theauthors in [5, ICALP08], which has already shown itself to have many
applications and admit automation. Our result also induces a generalisedgraph state paradigm for measurementbased quantum computing.
1 Introduction
Spatially separated compound quantum systems exhibit correlations under measurement which cannot be explained by classical physics. Bipartite states areused in protocols such as quantum teleportation, quantum key distribution, superdense coding, entanglement swapping, and many other typically quantumphenomena. The tripartite GHZstate allows for a purely qualitative Belltypeargument demonstrating the nonlocality of quantum mechanics [16], a phenomenon which has recently been exploited to boost computational power [2]. In
oneway quantum computing, multipartite graph states which generalise GHZstates constitute a resource for universal quantum computing [18]. There are
This work is supported by EPSRC Advanced Research Fellowship EP/D072786/1,by a Clarendon Studentship, by US Office of Naval Research Grant N000140910248and by EU FP6 STREP QICS. We thank Jamie Vicary for useful feedback.
arXiv:1002.25
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also many other applications of GHZstates and graph states in the areas offaulttolerance and communication protocols [24]. The tripartite Wstate, whichis qualitatively very different from the GHZstate, supports distributed leaderelection [11]. However, very little is known about the structure and behavioursof general multipartite quantum states.
What is known is that the variety in different possible behaviours is huge. Forexample, there is an infinite subset of all 4qubit states of which no two states canbe interconverted by a combination ofstochastic local(quantum) operations andclassical communication (SLOCC) [28]. States that are not SLOCCequivalentcorrespond to incomparable forms of distributed quantumness, so each willexpose distinct behaviours which may translate in distinct applications.
For three qubits there are only two nondegenerate SLOCCclasses [15], onewhich contains the GHZstate and another one which contains the Wstate:
GHZ = 000 + 111 W = 100 + 010 + 001 .
The dominant operations in algebra have binary input arity i.e. 2 inputs(cf. (a, b)) and 1 output (cf. a + b). Quantum information has taught us notto care much about distinguishing inputs and outputs, as data can flow horizontally across entangled states, and hence the total arity 2 + 1 = 3 becomesthe cogent feature. Therefore, tripartite qubit states can be seen as algebraic operations on C2. We claim that in the same manner as + and generate allpolynomials in calculus, by composing GHZstates and Wstates one obtainsrepresentatives of arbitrary Nqubit SLOCCclasses.
The formal realm where our study takes place is that of categorical quantum mechanics, initiated by Abramsky and one of the authors in [1, LiCS04].An appealing feature of this framework is that since it is formulated withinsymmetric monoidal categories, it comes with a PenroseJoyalStreet graphicalcalculus [25,17,27]. More recently, Frobenius algebras, as presented by Carboni
and Walters [4], have become a key player in categorical quantum mechanics byaccounting for observables and corresponding classical data flows [8,9,7]. Complementarity of these observables was axiomatised in [5, ICALP08], resulting ina powerful graphical calculus, Z/Xcalculus, with applications both in quantumfoundations [6], in various models of quantum computation and for the analysisof quantum cryptographic protocols [5,13,14,10]. Meanwhile there exists software, named quantomatic [12], which semiautomates reasoning within it.
Section 2 reviews commutative Frobenius algebras in a categorical context.In section 3 we introduce a calculus which is similar to Z/Xcalculus, in thatit also has two commutative Frobenius algebras (CFAs) as its main ingredients.However, while in Z/Xcalculus the CFAs are both special [20], here we introduceantispecial CFAs which will account for the Wstate behaviour. We then showin section 4 that this calculus is a powerful tool for generating and reasoning
about multipartite states. In section 5 we show how the W/GHZcalculus actually refines Z/Xcalculus, by reconstructing the latters generators and relations.Since Z/Xcalculus subsumes reasoning with graph states [13], GHZ/Wcalculusprovides a generalised graph state paradigm, and could also lead to a generalised measurementbased quantum computational model, among many other
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potential applications. We conclude by noting how the quantomatic softwarecan easily be adjusted to semiautomate reasoning within the more powerfulGHZ/Wcalculus, giving a computational handle on this much richer language.
Appendix A contains all proofs. Appendix C.2 explains the relationship between graphical languages and the circuit model of quantum computing. Appendix C is a survey of elements in the theory of symmetric monoidal categories(SMCs), including graphical languages, compactness and internal (co)monoids.
For standard quantum computer science notation we refer to [29]. We formally dont distinguish states from vectors; it should be clear to the readerthat by state  we mean the ray spanned by the vector . By i we referto a vector of a chosen orthonormal basis, also referred to as the computationalbasis. It is in this basis that our GHZ and Wstates are expressed.
2 Background: commutative Frobenius algebras
Let C be a symmetric monoidal category. We work within the induced graphical calculus and take associativity and unit laws to be strict. A commutativeFrobenius algebra (CFA) in C is an object A and morphisms ,,, such that:
(A , : A A A , : I A) is an internal commutative monoid, (A , : A A A , : A I) is an internal cocommutative comonoid, and (1 ) ( 1) = .
Depicting , , , respectively as , , and the Frobenius identitybecomes:
=
Definition 1. For a CFA A = (A,,,,), an Agraph is a morphism obtained
from the following maps: 1A, A,A (the swap map), , , , and , combined withcomposition and the tensor product. An Agraph is said to be connectedpreciselywhen its graphical representation is connected.
Remark 1. The Frobenius identity guarantees that any connected Agraph isuniquely determined by its number of inputs, outputs, and its number of loops.1
This makes CFAs highly topological, in that Agraphs are invariant under deformations that respect the number of loops.
In the special case where there are 0 loops, we make the following notationalsimplification, called spidernotation.
Snm
=
...
...:= ... S0
m:= S1
m Sn0 :=
Sn1
1 The number of loops is the maximal number of edges one can remove without disconnecting the graph, i.e. the number of holes in the corresponding cobordism [ 20].
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For example, we can use this notation to construct the following maps:
S02
= cap= S20
= cup=
For caps and cups, we usually omit the dots when there is no ambiguity:
:= := :=
The notion of compactness [19] follows as a special case of Rem 1:
=
Definition 2. A CFA is special (a SCFA), resp. antispecial (an ACFA) iff:
= resp. =
While the notion of special is standard [20] that of antispecial seems new.For CFAs we assume that circles admit an inverse2 i.e. = = 1I.
3 GHZ and Wstates as commutative Frobenius algebras
Let be a distributed Nqubit system. A local operation on is an operationon a single subsystem of .3 Two states of are SLOCCequivalent if withsome nonzero probability they can be interconverted with only local physicaloperations and classical communication. Of importance here is that SLOCC
equivalence admits an easy characterisation in terms of linear maps:
Theorem 1 ([15]). Two states ,  of are SLOCCequivalent iff thereexist invertible linear maps Li : Hi Hi such that  = (L1 . . . LN).
By the induced tripartite state of a CFA we mean S03
= . We denote thesymmetric monoidal category of finitedimensional Hilbert spaces, linear maps,the tensor product and with C as the tensor unit by FHilb. In FHilb, anymorphism : C H uniquely defines a vector  := (1) H and vice versa.Theorem 2. For each SCFA (resp. ACFA) onC2 in FHilb its induced tripartite state is SLOCCequivalent to GHZ (resp. W). Conversely, any symmetric state inC2 C2 C2 that is SLOCCequivalent to GHZ (resp. W) is the
induced tripartite state for some SCFA (resp. ACFA) onC2
in FHilb.2 In FHilb, = D, the dimension of the underlying space. Therefore = 1/D.3 These may also be generalised measurements, i.e. those arising when measuring the
system together with an ancillary system by means of a projective measurement, aswell unitary operations applied to extended systems.
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Each SCFA on H in FHilb induces a state that is SLOCCequivalent toGHZ because the copied states i H, that is, those satisfying (i) =
i
i
, form a basis of
H[9]. The induced tripartite state is then i iiiwhich is evidently SLOCCequivalent to i iii.
For the GHZstate the induced SCFA is:
= 0 00 + 1 11 = + := 12
(0 + 1)
= 00 0 + 11 1 = + := 12
(0 + 1)(1)
and for the Wstate the induced ACFA is:
= 1 11 + 0 01 + 0 10 = 1= 00 0 + 01 1 + 10 1 = 0 (2)
We know from [15] that W and GHZ are the only tripartite qubit states, upto SLOCC. Thus, the result above offers an exhaustive classification of CFAson C2, up to local operations.
Corollary 1. Any CFA onC2 in FHilb is locally equivalent to a SCFA or anACFA. Concretely, there exists an invertible linear map L : C2 C2 such thatthe induced maps
L
and
L L
define a CFA that is either special or antispecial.
Corollary 2. Every (commutative) monoid onC2 in FHilb extends to a CFA.
We now show that an SCFA and an ACFA each admit a normal form. Bythe previous theorem these govern the graphical calculi associated to GHZ andW SLOCCclass states.
Theorem 3. Let C be any SMC. For an SCFA on C, any connected CFAmorphism is equal to a spider, for an ACFA, any connected CFAmorphism iseither equal to a spider or of the following form:
...
...scalar
where the scalar is some tensor product of , , and . The total number of
loops minus the number of is moreover preserved.
A key ingredient in the proof is that (resp. ) copies (resp. ).
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Proposition 1. For any ACFA we have: =
Thm 2 shows that the structure of either an SCFA or an ACFA alone generates relatively few morphisms, and the only nondegenerate multipartite stateswhich arise are the canonical nqubit analogous to the GHZ and Wstate. However, when combining these two a wealth of states emerges, as we show now.
4 Generating general multipartite states
For the specific cases of the GHZSCFA and the WACFA as in Eqs (1) and(2) there are many equations which connect ( , , , ) and ( , , , ). Asmall subset of these provide us with a calculus that helps us to realise themain goal of this section, to show that representatives of all known multipartiteSLOCCclasses arise from the interaction of a SCFA with an ACFA.
The cups and caps induced by each CFA in general do not coincide, e.g.
10 + 01 = = = 00 + 11
Therefore we reintroduce explicit dots in order to distinguish them.
Definition 3. A GHZ/Wpair consists of a SCFA ( , , , ) and an ACFA
( , , , ) which satisfy the following four equations.
(i.)  := = (ii.)
=

(iii.) = (iv.)  =
In FHilb these conditions have a clear interpretation. By compactness ofcups and caps, the first condition implies that a tick on a wire is selfinversewhich together with the second condition implies that it is a permutation of thecopiable points of the SCFA (see [7]3.4). The third condition asserts that isa copiable point. The fourth condition implies that is also a (scaled) copiablepoint since it is the result of applying a permutation to a scalar multiple of .Indeed:
=

=   = 
=
We have shown abstractly that the ACFA structure defines the copiable pointsof the SCFA structure, which uniquely determines the structure itself. We showthe converse in Appendix A.
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Theorem 4. The equations in Def 3 suffice to establish a bijective correspondence between SCFAs onC2 and ACFAs onC2 in FHilb. That is to say, fixingone Frobenius algebra uniquely determines the other.
There are necessarily infinite SLOCC classes when N 4 [15]. To obtainfinite classification results many authors have expanded to talk about SLOCCsuperclasses, or families of SLOCC classes parameterised by one or more continuous variables. The current state of the art of this approach is [21], wherethe authors introduced a classification scheme based upon the right singularsubspace of a pure state. They partition out one of the qubits, regarding anNpartite state as a map M from
N1C2 to C2. Performing a singular value
decomposition on such a matrix yields a 1 or 2dimensional right singular subspace, spanned by two vectors in
N1C2. The SLOCC superclass of M is
then labeled by the SLOCC superclasses of these vectors, thus performing theinductive step. The base case is then C2C2, where the only two SLOCC classesare represented by the product state and the Bell state. An alternative way oflooking at this scheme is to consider Npartite states as controlled (N 1)partite states. That is to say, the right singular space of a state is spanned by{, } iff there exists a SLOCCequivalent state of the form 0+ 1. Thisprovides an operational description of a SLOCC superclass. Namely, a state is in the SLOCC superclass {, } if there exists some (twodimensional,possibly nonorthogonal) basis B such that measuring the first qubit in B yieldsa state that is SLOCCequivalent to  for outcome 1 and  for outcome 2.
First we show that the language of GHZ/Wpairs realises the inductive step.
Theorem 5. InFHilb, when considering the GHZ/Wpair onC2 as in Eqs (1)and (2), the linear map
QMUX := 



(3)
takes states   C2 C2 to
10 + 11 .
From this it follows that more generally,
QMUX ...
...
... ...
QMUX QMUX
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takes the states   N1
C2
N1
C2
to
1 . . . 1 N1 0 + 1 . . . 1 N1 1 .
Proof. We show this by means of GHZ/Wcalculus, for 0 = and 1 = :




=
 

=
 

=



 =

 =




 =
 

 =
 

 =



=


 =


=
2
Appendix A contains many computations of this kind and Appendix B compares the QMUXgate to the classical multiplexer.
Scalars 1 . . . 1 and 1 . . . 1 can be assumed to be nonzero, since toachieve this it suffices to vary the representatives of SLOCCclasses. The GHZ/Wcalculus generates the whole SLOCCclass given a representative.
Theorem 6. InFHilb, when considering the GHZ/Wpair onC2 as in Eqs (1)and (2), a linear map L : C2
C2 can always realised either as:
L :=


 or as L :=


for some singlequbit states , and . Consequently, given a representative ofa SLOCCclass we can reproduce the whole SLOCCclass when we augment theGHZ/Wcalculus with variables, i.e. singlequbit states.
Since we can produce a witness for any SLOCCclass, as well as any othermultipartite state that is SLOCCequivalent to it, we can produce any multipartite state. Via mapstate duality, from arbitrary N + Mqubit states we obtainarbitrary linear maps L : NC2 MC2.Corollary 3. If we adjoin variables to the graphical language of GHZ/Wpairsthen any Nqubit entangled state and consequently also any linear map L :N
C2 MC2 can be written in graphical language.
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What is important here, since all variables are local, is that all genuine newkinds of entanglement arise from the GHZ/Wcalculus only.
Of course, the graphs that we obtain in the above prescribed manner are notat all the simplest representatives of a SLOCCsuperclass. For example, with 4qubits it is easy to construct much simpler representatives of distinct SLOCCsuperclasses. Here are the representatives of five distinct superclasses:
where:
 :=  = 
The first two are the 4 qubit GHZ and Wstate respectively for which we have:
GHZ4
=
0
000
+
1
111
W4 = 0W + 1000and one easily verifies that the other three are:
0 (000 + 101 + 010) SLOCC
W
+1(0 (01 + 10) SLOCC
Bell
)
0000 + 1(1 (01 + 10) SLOCC
Bell
)
0 (000 + 111) SLOCC
GHZ
+1010
respectively, from which we can read off the corresponding right singular vectors.We can also obtain examples of fully parametrized SLOCCsuperclasses. That
is, the values of the variables yield all SLOCCclasses that the superclass contains.
corresponds with the parametrized SLOCCsuperclass:
0((00 + 1) SLOCC
Bell
) + 10Bell
So instead of relying on a heavyhanded inductive method to generate ar
bitrary multipartite states, the graphical calculus provides an intuitive tool toproduce and reason about multipartite states. Rather than producing entangledstates at random, and then exploring their applications, one could constructentangled states following a particular behavioural specification or preparationtechnique.
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5 Induced Z/Xcalculus
Z/Xpairs (also called complementary classical structures) provide a graphicalmeans to reason about the behaviour of interacting, maximally noncommutingobservables in a quantum system. They also provide a handle on embedding classical data in a quantum state space [5]. Here, we show how (in two dimensions)the theory of GHZ/Wpairs subsumes the theory of Z/Wpairs.
Definition 4. A Z/Xpair consists of two SCFAs ( , , , ) and ( , , , )which satisfy the following equations:
(I.) = (II .) =
(III.) = (IV .) =
as well as the horizontal mirror image of these equations.
The key example of a Z/Xpair on C2 in FHilb are the SCFAs correspondingto the Z and the Xeigenstates, i.e. a pair of complementary observables. Bycomposition one for example obtains:
CNOT = := =
and the calculus then enables to reason about circuits, measurementbased quantum computing, QKD and many other things [5,13,6,14,10]. Meanwhile there isthe quantomatic software which semiautomates reasoning within Z/Xcalculus.
Theorem 7. Under the assumption of the plugging rule:4 f...
= g...
f
...
= g
...
f...
= g...
(4)
and for
: I I an isomorphism such that
= , and with inverse : I I, each GHZ/Wpair induces a Z/Xpair, in particular:
=





=

 =
 =
(5)
One particular application of Z/Xcalculus is that it enables to write downgraph states and reason about them. Graph states constitute the main resourcein measurementbased quantum computing. What is particular appealing aboutthem is that they relate physical properties to combinatorial properties of graphs.
4 In FHilb, this condition can be read as the vectors { ,  } span C2.
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Z/Xgraph graph state
The results in this section tell us that the GHZ/Wcalculus gives rise to a morepowerful generalised notion of graph state. Following [5], we can derive methodsto prepare multipartite states, given a supply of GHZ and W resource states,directly from the GHZ/Wcalculus.
6 Conclusion and outlook
In the light of the results in Sec 5, and given the power of the graphical calculus of complementary observables, we expect even more to come from ourGHZ/Wcalculus. As the Wstate and the GHZstate have very different kinds
of applications [16,11] we expect their blend to give rise to a wide range of newpossibilities. The graphical paradigm has been very successful in measurementbased quantum computing [18,3,14] and other areas [24], and our generalisedgraphical paradigm would substantially expand the scope of this practice, whichwith support of a forthcoming quantomatic mark II can be semiautomated.
We can now ask which fragment of equations that hold for Hilbert spacequantum mechanics can be reproduced with GHZ/Wcalculus, augmented withthe plugging rule Eq (4). Does there exist an extension which provides a completeness result? This question is less ambitious then it may sound at first, givenSelingers theorem that dagger compact closed categories are complete with respect to Hilbert spaces [26]. The obvious next step is to explore the states representable in this theory, and the types of (provably correct) protocols they canimplement.
References
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A Proofs
A.1 Graphical Lemmas
We begin by developing several lemmas based upon the following axioms for anGHZ/Wpair. First, we recall the definition of GHZ/Wpair from section 4.
Definition. A GHZ/Wpair consists of a SCFA ( , , , ) and an ACFA
( , , , ) which satisfy the following four equations.
(i.)  := = (ii.)
=

(iii.) = (iv.)  =
Definition 5. In a symmetric monoidal category C, we define an operationtranspose as follows for a CFA ( , , , ):
f......
T = f.........
...
Proposition 2. LetC be the subcategory ofC generated by tensor copies of A.() T extends to a contravariant, involutive functor fromC to itself.Proof. Using Frobenius identities, it is easy to verify that:
(1A) T = 1A , (f g) T = g T f T and (f T) T = f .2
From hence forth, assume the ACFA ( , , , ) and the SCFA ( , , , )form an GHZ/Wpair.
Lemma 1. = 
Proof. From axiom (i.) we conclude that the  is selfinverse. The result thenfollows from axiom (ii.).
=

= 
=  =  =
2
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Lemma 5.  =
Proof.
 =  = =  =  =
 =
2
A.2 Proof of Theorem 2.
The proof to follow is greatly assisted by this lemma.
Lemma 6 (Mathonet et al. [a5]). If  and  are symmetric Nqubitstates such that and are SLOCCequivalent, then there exists an invertiblelinear map L : C2 C2 such that  = LN.
(; SCFAGHZ) Let ( , , , ) be a SCFA on C2 in FHilb. Define anotherSCFA as follows:
:: 0 00, 1 11; := +; :=
; := ( )
It is easy to verify that this is indeed an SCFA. It can also be verified that:
= GHZ
Let u, v be the pair of linearly independent vectors copied by . Define aninvertible map L :: 0 u, 1 v. We define in terms ofL and :
=
L1
L L
This induces as follows.
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= =L1
L L
=
L
= L
It then follows that
=
L L L
= (L L L)GHZ
(; SCFAGHZ) In the other direction, we start with a symmetric state  thatis SLOCC with GHZ. By Lemma 6, we know  must be of the following form,for L invertible.
 := (L L L)GHZ (6)Let := + L1. Let cap := ( 1 1) . Now, pick a unique cup such thatcup and cap form a compact structure. We can then define a SCFA as follows:
:= (cup 1 1) (1 ):= + L1:= (1
cup)
(1
1
cup
1)
(
1
1)
:= (1 ) cap
Taking  to be of the form of Eq (6), we obtain the following values:
cap := (L L)(00 + 11)cup := (00 + 11)(L1 L1)
:= (L L)(00 0 + 11 1)L1:= + L1:= L(0 00 + 1 11)(L1 L1):= L+
This set of generators does indeed obey the axioms of an SCFA.
(; ACFAW) ( , , , ) be a ACFA on C2 in FHilb Since is left andright unital, it is not separable i.e. it cannot be expressed as one of these forms:
A b, a L, L a. Note that = trR( ), the result of tracing the input to
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the right leg. Assume without loss of generality that is normal, which can beachieved by rescaling with some scalar . To avoid confusion we denote these
recalled variants of and as and t. Let B := t, t be an orthonormalbasis for C2. By Prop 1 we have t = att. So, for some :
= att t +  tTake the right partial trace of both sides:
t = at + trR(
t)
So, trR(
t) = (1 a)t. We can express  in the basis B:
 = ut + vtand
(1 a)t = trR( t) = vNow, letting u = bt + ct,
 = btt + ctt+ (1 a)ttPlugging in to , letting d = (1 a):
= att t + btt + ctt+ dtt td = 0, otherwise is separable. Let s = 1
d(bt + ct):
= att t +
dst + d
tt
t
Note that s = kt, otherwise is separable. Also, by definition of Frobenius algebra, is nondegenerate (i.e. entangled). So, choose nonproportional s, tsuch that:
(t 1) = 1as (t 1) = 1
dt
Then, the state associated with (, ) is:
= ( 1) = (tts + stt + ttt)Now, define the following local maps:
L1 :: t 0, s 1L2 :: t 0,
t 1L3 :: t 0, s 1
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These are all invertible because they take bases to bases. Then
(L1
L2
L3)(
1) =W
(; ACFAW) For the converse, we mirror the construction from the proof ofthe SCFAGHZ case. The only difference is the choice of counit. Start with asymmetric state  that is SLOCC with W. By Lemma 6, we know  mustbe of the following form, for L invertible.
 := (L L L)W (7)
Let := 0 L1. Let cap := ( 1 1) . Now, pick a unique cup such thatcup and cap form a compact structure. We can then define an ACFA as follows:
:= (cup 1 1) (1 ):= 0 L1:= (1 cup) (1 1 cup 1) ( 1 1):= (1 ) cap
Taking  to be of the form of Eq (7), we obtain the following values:
cap := (L L)(10 + 01)cup := (10 + 01)(L1 L1)
:= (L L)(10 1 + 01 1 + 00 0)L1:=
0
L1
:= L(1 11 + 0 01 + 0 10)(L1 L1):= L1
This set of generators does indeed obey the axioms of an ACFA.
A.3 Proof of Corollary 1.
Each CFA on C2 in FHilb induces a tripartite qubit state S03 = , which, asa consequence of the CFA being unital, is nondegenerate. Hence it either mustbe SLOCCequivalent to GHZ or W, so we can apply the construction in theproof of Thm 2. This preserves , yet yields a new cup:
. Let L from the
corollary then be the composition of the old cap with the new cup.
L =
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A.4 Proof of Corollary 2.
We begin by recalling an equivalent definition of a Frobenius algebra (see e.g.
[a3]), spelled out for the specific case of FHilb.
Definition 6. A Frobenius algebra over C is a monoid (A,,) with a linearfunctional : A C such that is a nondegenerate pairing. That is, theinduced map: :: u (1 u)is invertible. In such a case, is called a Frobenius form.
Since fixes an isomorphism with the dual space, it uniquely determines acomultiplication , from which we can recover the definition of Frobenius algebragiven in section 2.
Let (A,,) be a (commutative) monoid. is unital, so in particular, mustbe connected. Therefore, using [a2], for M one of the following:
M = GHZ = 0 00 + 1 11 or M = W = 0 00 + 1(10 + 01)we have = (L1)M(L2 L3), for some invertible maps Li. In the case whereM = GHZ , let = + (L 11 ). Then = (00+11)(L2L3). For Li invertible, is a nondegenerate pairing. In the case where M = W, let = 1 (L 11 ).Then = (10 + 01)(L2 L3), which is also a nondegenerate pairing. Itthen follows that (A,,) extends to a Frobenius algebra. Furthermore, if iscommutative, is symmetric and the induced FA is also commutative.
Theorem 8 ([a3]). Any connected CFAmorphism is always of the form:
...
...
...
(8)
A.5 Proof of Proposition 1.
We can show this using Thm 8 and antispeciality:
= = =
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A.6 Proof of Theorem 3.
(SCFA) Substituting speciality in Eq (8) removes all loops, yielding a spider.
(ACFA) If Eq (8) has no loops then it is a spider. If it has two or more loops, itcan be reduced to a product of a graph with one loop and copies of . Supposesome graph G has N 2 loops. Then, we can find an equivalent graph with onefewer loop.
G =
H
= H
By induction, we can always rewrite a connected graph G to or anothergraph with at most one loop.
If the graph has zero inputs and zero outputs, the above result suffices to finda normal form. So, suppose it has zero inputs, at least one output, and exactlyone loop. Then it must be of this form:
...
By Prop 1 this can be written as:
... ...
The case of at least one input, zero outputs, and one loop is treated similarly.
It remains only to consider the case of one or more inputs and outputs, and oneloop:
...
...
=
...
...
=
...
...
This is then a product of previously considered cases.
A.7 Proof of Theorem 4.
First, we fix a SCFA ( , , , ) and show that the axioms of an GHZ/Wpair
given in Def 3 can uniquely determine the ACFA ( , , , ). By [9], we know
that the vectors ei such that ei = eiei form a basis for C2. By axiom
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(iii.), is such a vector, so let e1 = . Now, from axioms (ii.) and (iii.)  is alsosuch a vector. Suppose  = e1. Then by axiom (iv.) for some nonzero scalark, k  = k = , but this is impossible for an ACFA of dimension 2, for itwould imply that the identity is a rank 1 matrix: 1/k( ). So,  = e2. Byaxiom (i.) the tick is an involution, so it must be the permutation e1 e2.Therefore we have defined the black cap.
:= 
Furthermore, by axiom (iv.) and antispecialness, the following identity holds.
= = =  
Now, we have completely defined .
::
  The data ( , ) suffice to define W. Conversely, fix W. We have already
established that the points { , } span C2. We showed in section 4 that
=   =
So, the basis { , } completely determines .
::
By Lemmas 1 and 4, we can see that { , } totally determines .
::
1C
The data ( , ) suffice to define G.
A.8 Proof of Theorem 6.
For A defined as follows:
L :=



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any 1qubit linear map can be expressed as A or A (  ). To see why, recall thatan arbitrary 2 2 matrix admits a decomposition:
A = PLDU (9)
Where P is a permutation, L and U are unitdiagonal lowertriangular anduppertriangular matrices, and D is a diagonal matrix.
For a vectors , ,  C2, we can construct the following maps:
L :=



=
2 01 2
D :=
=
1 00 2
U :=
=
2 10 2
The only two permutations over C2 are the identity and NOT, so if we set2 = 2 = 1, we obtain the decomposition in Eq (9).
A.9 Proof of Theorem 7.
Let
: I I be an isomorphism such that = . We denote itsinverse as
and note that
= . Under the following encoding:
=





=

 =
 =
we need to show that ( , ) is an SCFA and furthermore satisfies these interaction properties:
(I.) = (II .) =
(III.) = (IV .) =
and their upsidedown counterparts. It shall suffice however to show just (IIV.),as transpose induces the suitable functor to turn any Z/Xpair graph upsidedown. T
= ( )T
= T
= ( )T
=
From Lem 4 we have =
. Thus condition (I.) follows immediately fromaxiom (iii.). We have already seen the following two identities in the proof fromthe previous section.
From antispecialness and Lem 5, we can derive the following two identities:
=
 =



=  (10)
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is cocommutative, because is. Thus we can conclude that is a left andright counit for . Furthermore,



=


= 
So, we conclude by plugging that

 =



from which these equations (including condition (III.)) follow:
=  =
= = (11)
Proposition 3. is coassociative.
Proof. Since
is an isomorphism, { , } is a valid set for plugging, as in Eq(4). We can show associativity using this fact and Eqs ( 10) and (11).
= =
=  =  =

2
So, ( , ) is a cocommutative comonoid. We can show similarly that ( , )is an associative monoid. The following proposition then suffices to show that( , , , ) is a CFA.
Proposition 4. =
Proof. = =  =



=

= 2
One final appeal to plugging suffices to show that this CFA is special.
Proposition 5. =
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Proof. We show this by plugging along { ,  }.
= =
 =
=  
= 
=



=
 =
=

=  =

= 
2
So, it only remains to show that conditions (II.) and (IV.) hold. These canboth be seen by straightforward plugging arguments on the basis { , }.
B Classical multiplexers versus QMUX
The classical multiplexer (MUX) is a device that acts as a twoway switch. Aninput of 0 on the control line connects the first input to the output, and an inputof 1 connects the second. It is often represented schematically as follows:
in 1
in 2
ctrl
out
The logic gate that does the critical work of the MUX is the 3state gate, ortristate, which behaves as an electronic switch, with the side input taking acontrol bit.
0
=
1
=
This looks very similar, topologically, to the behaviour of the multiplication
for an ACFA . Let and be the two possible inputs for the control qubit, then:
= =
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Using this intuition, let us pretend (roughly) that SCFA dots acts as junctions, ACFA dots act as tristates and the tick acts as a NOT gate. We get aquantumised diagram that is very similar to the classical one.
in 1
in 2
ctrl
out in1
in2
ctrl
out
For our purposes, we would like to have the control line on an output, sodoing a bit of manipulation, we get a more convenient form.
QMUX := 




This provides a quantum analogue to the classical multiplexor. However,rather than impeding (i.e. disconnecting) the unwanted line, linearity forcesus to project it out. Luckily, when considering inputs and/or outputs only up toSLOCC, this oddity can often be rectified.
C Symmetric monoidal categories and graphical calculus.
In categories where objects admit a tensor product, such as FHilb, the category
of finitedimensional complex Hilbert spaces and linear maps, is a monoid onobjects up to isomorphism, with the underlying field C as its unit. This monoidstructure on objects extends in a natural way to a bifunctor.
A monoidal category is a category C, with a bifunctor
( ) : C C C
that is (weakly) associative and unital. Weak associativity means there exists anatural isomorphism such that for all objects A , B, C C,
A,B,C : (A B) C = A (B C).
Weak unity means there exists an object I and two natural isomorphisms
and such that A : A = I A and A : A = A I. These isomorphisms aresubject to certain coherence properties (see e.g. [a4]).
A strict monoidal functorF : C D between monoidal categories is a functorthat preserves the monoidal product: F(A B) = F A F B. A strict monoidalnatural transformation is a natural transformation such that AB = AB .
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In many monoidal categories, the two objects AB and BA are isomorphic.If there is a natural isomorphism A,B : AB = BA that interacts in a sensibleway with
and , i.e. this equation holds:
(1B A,C) B,A,C (A,B 1C) = (B,C,A (B,CA) A,B,Cthen is called a braiding, and C a braided monoidal category. If furthermore,A,BA,B = 1AB, then is called a symmetry, and C is a symmetric monoidalcategory. A braided (resp. symmetric) monoidal functor is just a monoidal functor that preserves braidings (resp. symmetries).
There are many examples of symmetric monoidal categories. Our main example will be FHilb. For any two Hilbert spaces A and B, there is a symmetry
A,B : A B B A :: Since the tensor unit in FHilb is C, we note that a linear map s : C C canbe identified by the scalar s(1) C. We extend this to a general notion. Moregeneral, in a monoidal category (C, , I), any map f : I I is called a scalar.
Another important structure in FHilb we wish to capture is the property ofhaving a dual space. We say A has a dual if there exists an object A and mapsdA : I A A and eA : A A I such that
(1A dA) (eA 1A) = 1A (dA 1A) (1A eA) = 1A (12)A symmetric monoidal category is compact closed if all objects have du
als. When we fix an object A, its dual is only determined up to isomorphism.Therefore, when A is isomorphic with its dual, as is always the case in finitedimensional Hilbert spaces, we can choose the categorical dual to be A itself. Insuch a situation, we say A is self dual. For example, objects which come with acommutative Frobenius algebra (see Section 2) always are selfdual.
Again these maps have to interact coherently with the other elements ofthe symmetric monoidal structure. For an explicit statement of these coherenceproperties, see [a6]. We avoid the verbose technical definition here, but we willstill make an exact definition of coherence shortly, by introducing the graphicalnotation for compact closed categories.
C.1 Graphical representation
The theory of monoidal categories is in a strong sense a 2dimensional theory.One interpretation for this dimensionality is that the tensor product provides aspacial dimension, while the composition of arrows provides temporal, or causaldimension. The interplay of these two dimensions is represented by the bifunc
toriality of the tensor product.(f2 g2) (f1 g1) = (f2 f1) (g2 g1) (13)
One can interpret this equation by thinking of these four arrows occupyinga piece of 2dimensional space.
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f1
f2
g1
g2
From this point of view, the bracketing in Eq (13) is a piece of essentiallymeaningless syntax, which is required to make something that is 2dimensionalby nature expressible as a (1dimensional) term. To address this issue, we shallintroduce a graphical notation for symmetric monoidal categories, similar to thatof circuit diagrams. Edges represent objects and nodes represent morphisms.Normally, both edges and nodes are labeled, but here we shall consider onlygraphs where every edge represents the same object, so we shall omit edge labels.Tensoring is done by juxtaposition and composition is performed by plugging, or
gluing the inputs of one graph to the outputs of another. The identity arrow isrepresented by an empty edge and the tensor unit by an empty graph.
f g = f g g f =f
g
(14)
Since we can express A A as a pair of lines, we can, for instance, express amap h : A A A A and compose in various ways with other maps.
h (f 1) =f
h
(1 f) h =f
h
(15)
We can express the bifunctoriality of and the naturality of as follows:
f
g
=
f
g f g=
fg
(16)
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Edges, nodes, and edge crossings provide a graphical language for symmetricmonoidal categories. This language captures exactly the coherence propertiespresent in a symmetric monoidal category. Selinger states this precisely in [a6].
Theorem 9. (Coherence for symmetric monoidal categories). A wellformedequation between morphisms in the language of symmetric monoidal categories
follows from the axioms of symmetric monoidal categories if and only if it holds,up to isomorphism of diagrams, in the graphical language.
So far, we have introduced a graphical language for describing terms in asymmetric monoidal category. These terms are precisely the directed acyclicgraphs generated by the arrows in that category. If the line represents an objectA, and A has a dual, we can actually express terms as arbitrary graphs. Werepresent the type A as a line directed down, A as a line directed up, and themaps dA and eA from the previous section as caps and cups.
dA = eA = = (17)
Theorem 10. (Coherence for compact closed categories). A wellformed equation between morphisms in the language of compact closed categories follows fromthe axioms of compact closed categories if and only if it holds, up to isomorphismof diagrams, in the graphical language.
C.2 Relationship with the Circuit Model
Anyone with some familiarity of the quantum circuit model will have seen diagrams that look like (14) and (15). This is because quantum circuit diagrams are
a special case of the graphical representation of monoidal categories. Traditionally, circuit diagrams are just graphs that are planar and directed acyclic. Theseare exactly the kinds of graphs one can construct in a monoidal category if onerefrains from using the symmetric, compact structure of the category. Selingerssurvey [a6] offers a result (originally due to Joyal and Street) about these kindsof graphs as well, for the sake of completeness.
Theorem 11. (Coherence for planar monoidal categories). A wellformed equation between morphism terms in the language of monoidal categories follows fromthe axioms of monoidal categories if and only if it holds, up to planar isotopy,in the graphical language.
Here, planar isotopy means simply that one graph can be deformed into
another without crossing edges. Circuit diagrams also tend to run in tracks,parallel lines that clearly distinguish qubits. This rigidity comes from the factthat circuit components are unitary, so tend to have the same number of inputand output wires. However, this convention is purely pedagogical, as arbitraryplanar, directed acyclic graphs can be interpreted as circuits without ambiguity.
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What symmetry makes explicit is the act of swapping qubits, rather thanencoding it with quantum gates, such as 3 controllednot gates:
:=
(18)
Making this definition allows line crossings, and we can do away with theneedless restriction that diagrams be planar. Compact structures can be encodedin the circuit model as well, using Bellstates and Belleffects [a1].
C.3 Internal monoids, comonoids, and Frobenius algebras
A monoid internal to a monoidal category (C, , I), is an object A and a pairof maps : A A A defining multiplication and : I A picking out theunit. Multiplication is associative, so this diagram commutes.
(A A) A A A
A
A (A A) A A
A
A
Multiplication is left and right unital, so this diagram also commutes.
A A I
A A
I A
A A
A
A
We now have the ability to equip a large variety of objects with a multiplicative structure. In the case where C = Set, this recovers the usual notion ofmonoid. Monoids internal to Ab, the category of abelian groups, are rings, andmonoids internal to Vectk are associative kalgebras.
The dual of a monoid is a comonoid. For an object A in C, one can define ainternal comonoid as a triple (A, : A A A, : A I) where the followingdiagrams commute. Coassociativity:
(A A) AA AA
A (A A)A A
A
A
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Counit:
A A 1
A A
1 A
A A
A
A
In the graphical language, the axioms of a monoid (A, , ) become:
= = =
The axioms of a comonoid are just the previous ones, upsidedown:
= = =
As we have already noted in section 2, the following additional conditiondefines an internal Frobenius algebra.
(1 ) ( 1) = = ( 1) (1 )
Of course, in the commutative case, the rightmost equation is redundant.Graphically, this identity is:
= =
References
a1. S. Abramsky and B. Coecke (2004) A categorical semantics of quantum protocols.LiCS04. Revision: arXiv:quantph/0808.1023
a2. W. Dur, G. Vidal and J. I. Cirac (2000) Three qubits can be entangled in twoinequivalent ways. Phys. Rev. A 62, 062314. arXiv:quantph/0005115
a3. J. Kock (2003) Frobenius Algebras and 2D Topological Quantum Field Theories.Cambridge
a4. S. Mac Lane (1998  2nd edition) Categories for the Working Mathematician.SpringerVerlag.
a5. P. Mathonet, S. Krins, M. Godefroid, L. Lamata, E. Solano, T. Bastin (2009)
Entanglement Equivalence of nqubit Symmetric States. arXiv:0908.0886a6. P. Selinger (2009) A survey of graphical languages for monoidal categories. In: New
Structures for Physics, B. Coecke (ed), 275337, SpringerVerlag. arXiv:0908.3347
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