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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 N-qubit systemsis a long-standing 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 N-qubit states to be equivalent as computational re-

sources if they can be inter-converted by stochastic local (quantum)operations and classical communication (SLOCC). There are only twoSLOCC-classes of genuinely entangled 3-qubit states, the GHZ-class andthe W-class, 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 anti-special ones. Within the graphical language of symmetricmonoidal categories, the distinction between special and anti-specialis purely topological, in terms of connected vs. disconnected.

These GHZ and W Frobenius algebras form the primitives of a graphi-cal calculus which is expressive enough to generate and reason about rep-resentatives of arbitrary N-qubit states. This calculus refines the graphi-cal 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 measurement-based quantum computing.

1 Introduction

Spatially separated compound quantum systems exhibit correlations under mea-surement which cannot be explained by classical physics. Bipartite states areused in protocols such as quantum teleportation, quantum key distribution, su-perdense coding, entanglement swapping, and many other typically quantumphenomena. The tripartite GHZ-state allows for a purely qualitative Bell-typeargument demonstrating the non-locality of quantum mechanics [16], a phe-nomenon which has recently been exploited to boost computational power [2]. In

one-way quantum computing, multipartite graph states which generalise GHZ-states 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 N00014-09-1-0248and by EU FP6 STREP QICS. We thank Jamie Vicary for useful feedback.

arXiv:1002.25

40v1

[quant-ph]1

2Feb2010

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also many other applications of GHZ-states and graph states in the areas offault-tolerance and communication protocols [24]. The tripartite W-state, whichis qualitatively very different from the GHZ-state, supports distributed leader-election [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 4-qubit states of which no two states canbe inter-converted by a combination ofstochastic local(quantum) operations andclassical communication (SLOCC) [28]. States that are not SLOCC-equivalentcorrespond to incomparable forms of distributed quantumness, so each willexpose distinct behaviours which may translate in distinct applications.

For three qubits there are only two non-degenerate SLOCC-classes [15], onewhich contains the GHZ-state and another one which contains the W-state:

|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 hori-zontally across entangled states, and hence the total arity 2 + 1 = 3 becomesthe cogent feature. Therefore, tripartite qubit states can be seen as algebraic op-erations on C2. We claim that in the same manner as + and generate allpolynomials in calculus, by composing GHZ-states and W-states one obtainsrepresentatives of arbitrary N-qubit SLOCC-classes.

The formal realm where our study takes place is that of categorical quan-tum 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 Penrose-Joyal-Street 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]. Com-plementarity of these observables was axiomatised in [5, ICALP08], resulting ina powerful graphical calculus, Z/X-calculus, 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 soft-ware, named quantomatic [12], which semi-automates 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/X-calculus, in thatit also has two commutative Frobenius algebras (CFAs) as its main ingredients.However, while in Z/X-calculus the CFAs are both special [20], here we introduceanti-special CFAs which will account for the W-state 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/GHZ-calculus actu-ally refines Z/X-calculus, by reconstructing the latters generators and relations.Since Z/X-calculus subsumes reasoning with graph states [13], GHZ/W-calculusprovides a generalised graph state paradigm, and could also lead to a gener-alised measurement-based quantum computational model, among many other

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potential applications. We conclude by noting how the quantomatic softwarecan easily be adjusted to semi-automate reasoning within the more powerfulGHZ/W-calculus, giving a computational handle on this much richer language.

Appendix A contains all proofs. Appendix C.2 explains the relationship be-tween graphical languages and the circuit model of quantum computing. Ap-pendix 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 for-mally 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 W-states are expressed.

2 Background: commutative Frobenius algebras

Let C be a symmetric monoidal category. We work within the induced graph-ical 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 A-graph is a morphism obtained

from the following maps: 1A, A,A (the swap map), , , , and , combined withcomposition and the tensor product. An A-graph is said to be connectedpreciselywhen its graphical representation is connected.

Remark 1. The Frobenius identity guarantees that any connected A-graph isuniquely determined by its number of inputs, outputs, and its number of loops.1

This makes CFAs highly topological, in that A-graphs are invariant under de-formations that respect the number of loops.

In the special case where there are 0 loops, we make the following notationalsimplification, called spider-notation.

Snm

=

...

...:= ... S0

m:= S1

m Sn0 :=

Sn1

1 The number of loops is the maximal number of edges one can remove without dis-connecting 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. anti-special (an ACFA) iff:

= resp. =

While the notion of special is standard [20] that of anti-special seems new.For CFAs we assume that circles admit an inverse2 i.e. = = 1I.

3 GHZ- and W-states as commutative Frobenius algebras

Let be a distributed N-qubit system. A local operation on is an operationon a single subsystem of .3 Two states of are SLOCC-equivalent if withsome non-zero probability they can be inter-converted 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 SLOCC-equivalent 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 finite-dimensional 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 tripar-tite state is SLOCC-equivalent to |GHZ (resp. |W). Conversely, any symmet-ric state inC2 C2 C2 that is SLOCC-equivalent 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 SLOCC-equivalent to|GHZ 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 SLOCC-equivalent to i |iii.

For the GHZ-state the induced SCFA is:

= |0 00| + |1 11| = |+ := 12

(|0 + |1)

= |00 0| + |11 1| = +| := 12

(0| + 1|)(1)

and for the W-state 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 anti-special.

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 SLOCC-class states.

Theorem 3. Let C be any SMC. For an SCFA on C, any connected CFA-morphism is equal to a spider, for an ACFA, any connected CFA-morphism 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 gener-ates relatively few morphisms, and the only non-degenerate multipartite stateswhich arise are the canonical n-qubit analogous to the GHZ- and W-state. How-ever, when combining these two a wealth of states emerges, as we show now.

4 Generating general multipartite states

For the specific cases of the GHZ-SCFA and the W-ACFA 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 multipartiteSLOCC-classes 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/W-pair 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 self-inversewhich 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 correspon-dence 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 SLOCCsuper-classes, or families of SLOCC classes parameterised by one or more con-tinuous 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 anN-partite state as a map M from

N1C2 to C2. Performing a singular value

decomposition on such a matrix yields a 1- or 2-dimensional right singular sub-space, spanned by two vectors in

N1C2. The SLOCC super-class of M is

then labeled by the SLOCC super-classes 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 N-partite 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 SLOCC-equivalent 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 (two-dimensional,possibly non-orthogonal) basis B such that measuring the first qubit in B yieldsa state that is SLOCC-equivalent to | for outcome 1 and | for outcome 2.

First we show that the language of GHZ/W-pairs realises the inductive step.

Theorem 5. InFHilb, when considering the GHZ/W-pair onC2 as in Eqs (1)and (2), the linear map

QMUX := -

-

-

-

-(3)

takes states | | C2 C2 to

1||0 + 1||1 .

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/W-calculus, for 0| = and 1| = :

-

-

-

-

-=

- -

-

-=

- -

-

-=

-

-

-

- =

-

- =

-

-

-

-

- =

- -

-

- =

- -

-

- =

-

-

-

=

-

-

- =

-

-

=

2

Appendix A contains many computations of this kind and Appendix B com-pares the QMUX-gate to the classical multiplexer.

Scalars 1 . . . 1| and 1 . . . 1| can be assumed to be non-zero, since toachieve this it suffices to vary the representatives of SLOCC-classes. The GHZ/W-calculus generates the whole SLOCC-class given a representative.

Theorem 6. InFHilb, when considering the GHZ/W-pair onC2 as in Eqs (1)and (2), a linear map L : C2

C2 can always realised either as:

L :=

-

-

- or as L :=

-

-

for some single-qubit states , and . Consequently, given a representative ofa SLOCC-class we can reproduce the whole SLOCC-class when we augment theGHZ/W-calculus with variables, i.e. single-qubit states.

Since we can produce a witness for any SLOCC-class, as well as any othermultipartite state that is SLOCC-equivalent to it, we can produce any multipar-tite state. Via map-state duality, from arbitrary N + M-qubit states we obtainarbitrary linear maps L : NC2 MC2.Corollary 3. If we adjoin variables to the graphical language of GHZ/W-pairsthen any N-qubit 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/W-calculus only.

Of course, the graphs that we obtain in the above prescribed manner are notat all the simplest representatives of a SLOCC-superclass. For example, with 4qubits it is easy to construct much simpler representatives of distinct SLOCC-superclasses. Here are the representatives of five distinct superclasses:

where:

- := - = -

The first two are the 4 qubit GHZ- and W-state respectively for which we have:

|GHZ4

=

|0

|000

+

|1

|111

|W4 = |0|W + |1|000and one easily verifies that the other three are:

|0 (|000 + |101 + |010) SLOCC

|W

+|1(|0 (|01 + |10) SLOCC

|Bell

)

|0|000 + |1(|1 (|01 + |10) SLOCC

|Bell

)

|0 (|000 + |111) SLOCC

|GHZ

+|1|010

respectively, from which we can read off the corresponding right singular vectors.We can also obtain examples of fully parametrized SLOCC-superclasses. That

is, the values of the variables yield all SLOCC-classes that the superclass con-tains.

corresponds with the parametrized SLOCC-superclass:

|0((|00 + |1) SLOCC

|Bell

|) + |1|0|Bell

So instead of relying on a heavy-handed inductive method to generate ar-

bitrary multi-partite 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/X-calculus

Z/X-pairs (also called complementary classical structures) provide a graphicalmeans to reason about the behaviour of interacting, maximally non-commutingobservables in a quantum system. They also provide a handle on embedding clas-sical data in a quantum state space [5]. Here, we show how (in two dimensions)the theory of GHZ/W-pairs subsumes the theory of Z/W-pairs.

Definition 4. A Z/X-pair 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/X-pair on C2 in FHilb are the SCFAs correspondingto the Z- and the X-eigenstates, i.e. a pair of complementary observables. Bycomposition one for example obtains:

CNOT = := =

and the calculus then enables to reason about circuits, measurement-based quan-tum computing, QKD and many other things [5,13,6,14,10]. Meanwhile there isthe quantomatic software which semi-automates reasoning within Z/X-calculus.

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/W-pair induces a Z/X-pair, in particular:

=

-

-

-

--

-

=

-

-- =

- =

(5)

One particular application of Z/X-calculus is that it enables to write downgraph states and reason about them. Graph states constitute the main resourcein measurement-based 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/X-graph graph state

The results in this section tell us that the GHZ/W-calculus 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/W-calculus.

6 Conclusion and outlook

In the light of the results in Sec 5, and given the power of the graphical cal-culus of complementary observables, we expect even more to come from ourGHZ/W-calculus. As the W-state and the GHZ-state 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 measurement-based 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 semi-automated.

We can now ask which fragment of equations that hold for Hilbert spacequantum mechanics can be reproduced with GHZ/W-calculus, augmented withthe plugging rule Eq (4). Does there exist an extension which provides a com-pleteness result? This question is less ambitious then it may sound at first, givenSelingers theorem that dagger compact closed categories are complete with re-spect to Hilbert spaces [26]. The obvious next step is to explore the states rep-resentable in this theory, and the types of (provably correct) protocols they canimplement.

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A Proofs

A.1 Graphical Lemmas

We begin by developing several lemmas based upon the following axioms for anGHZ/W-pair. First, we recall the definition of GHZ/W-pair from section 4.

Definition. A GHZ/W-pair 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 operation-transpose 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/W-pair.

Lemma 1. = -

Proof. From axiom (i.) we conclude that the - is self-inverse. 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 N-qubitstates such that| and| are SLOCC-equivalent, then there exists an invertiblelinear map L : C2 C2 such that | = LN|.

(; SCFA-GHZ) 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

(; SCFA-GHZ) 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.

(; ACFA-W) ( , , , ) 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 = a|tt. So, for some |:

= a|tt t| + | tTake the right partial trace of both sides:

|t = a|t + 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 = b|t + ct,

| = b|tt + ctt+ (1 a)ttPlugging in to , letting d = (1 a):

= a|tt t| + b|tt + ctt+ dtt td = 0, otherwise is separable. Let |s = 1

d(b|t + ct):

= a|tt t| +

d|st + d

tt

t

Note that |s = k|t, otherwise is separable. Also, by definition of Frobenius al-gebra, is non-degenerate (i.e. entangled). So, choose non-proportional |s, |tsuch that:

(t| 1) = 1a|s (t 1) = 1

d|t

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

(; ACFA-W) For the converse, we mirror the construction from the proof ofthe SCFA-GHZ 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):= L|1

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 non-degenerate. Hence it either mustbe SLOCC-equivalent 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 non-degenerate 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 non-degenerate pairing. In the case where M = W, let = 1| (L -11 ).Then = (10| + 01|)(L2 L3), which is also a non-degenerate 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 CFA-morphism is always of the form:

...

...

...

(8)

A.5 Proof of Proposition 1.

We can show this using Thm 8 and anti-speciality:

= = =

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(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.


First, we fix a SCFA ( , , , ) and show that the axioms of an GHZ/W-pair

given in Def 3 can uniquely determine the ACFA ( , , , ). By [9], we know

that the vectors |ei such that |ei = |ei|ei 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 non-zero 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 anti-specialness, 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.


For A defined as follows:

L :=

-

-

-

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any 1-qubit 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 unit-diagonal lower-triangular andupper-triangular 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).


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 interac-tion properties:

(I.) = (II .) =

(III.) = (IV .) =

and their upside-down counterparts. It shall suffice however to show just (I-IV.),as -transpose induces the suitable functor to turn any Z/X-pair graph upside-down. 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 anti-specialness 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 straight-forward plugging arguments on the basis { , -}.

B Classical multiplexers versus QMUX

The classical multiplexer (MUX) is a device that acts as a two-way 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 3-state 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 junc-tions, 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 finite-dimensional 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 func-tor that preserves braidings (resp. symmetries).

There are many examples of symmetric monoidal categories. Our main ex-ample 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 finite-dimensional 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 self-dual.

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 2-dimensional 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 2-dimensional 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 2-dimensionalby nature expressible as a (1-dimensional) 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 well-formedequation 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 well-formed equa-tion 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 dia-grams that look like (14) and (15). This is because quantum circuit diagrams are

a special case of the graphical representation of monoidal categories. Tradition-ally, 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 well-formed equa-tion 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 controlled-not 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 Bell-states and Bell-effects [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 multi-plicative 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 k-algebras.

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, upside-down:

= = =

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:quant-ph/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:quant-ph/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.Springer-Verlag.

a5. P. Mathonet, S. Krins, M. Godefroid, L. Lamata, E. Solano, T. Bastin (2009)

Entanglement Equivalence of n-qubit 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, Springer-Verlag. arXiv:0908.3347

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