Phase groups and the origin of non-locality for qubits

QPL 2009 Phase groups and the origin of non-lo calit y for qubits Bob Co ec k e, Bill Edwards and Robert W. Sp ekk ens Oxfor d University Computing L ab or atory and Perimeter Institute for The or etic al Physics Abstract W e describe a general framew ork in which w e can precisely compare the structures of quantum-lik e theories whic h ma y initially b e form ulated in quite differen t mathematical terms. W e then use this framework to compare t wo theories: quan tum mechanics restricted to qubit stabiliser states and op erations, and Sp ekkens’s toy theory . W e discov er that viewed within our framework these theories are v ery similar, but differ in one key asp ect - a four element group w e term the phase gr oup whic h emerges naturally within our framew ork. In the case of the stabiliser theory this group is Z 4 while for Sp ekkens’s toy theory the group is Z 2 × Z 2 . W e further show that the structure of this group is intimately inv olved in a key ph ysical difference b etw een the theories: whether or not they can b e mo delled by a lo cal hidden v ariable theory . This is done b y establishing a connection b et w een the phase group, and an abstract notion of GHZ state correlations. W e go on to formulate precisely how the stabiliser theory and toy theory are ‘similar’ by defining a notion of ‘m utually un biased qubit theory’, noting that all such theories hav e four elemen t phase groups. Since Z 4 and Z 2 × Z 2 are the only such groups we conclude that the GHZ correlations in this type of theory can only take tw o forms, exactly those app earing in the stabiliser theory and in Sp ekkens’s to y theory . The results p oint at a classification of lo cal/non-lo cal b ehaviours by finite Abelian groups, extending b eyond qubits to finitary theories whose observ ables are all mutually unbiased. 1 In tro duction Muc h interest recen tly has fo cused on pic king out the k ey features of quantum mec hanics whic h mak e it sp ecial (for example, incompatible observ ables, non- lo calit y , computational sp eed-up, no-cloning), in v estigating the relationships 1 W e thank Joachim Ko ck for carefully pro ofreading this manuscript and providing v ery useful feedbac k. This w ork is supp orted b y EPSR C Adv anced Research F ellowship EP/D072786/1, by US Office of Nav al Research (ONR) Grant N00014-09-1-0248, by EU- FP6-FET STREP QICS and b y a Large Grant of the F oundational Questions Institute. This pap er is electronically published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs Coecke, Edw ards and Spekkens b et w een these features, and identifying the mathematical asp ects of the the- ory which embo dy these physical features. Quan tum-lik e theories hav e b een constructed, whic h displa y certain features of quantum mec hanics but not others, allo wing us to see which of these features are in terlinked, and which are essentially indep endent. These theories had diverse motiv ations and are expressed in a range of mathematical forms. Quantum mec hanics uses Hilb ert space. Another theory whic h has recently attracted m uch in terest [ 25 ] employs subsets of certain sets to represen t states, and relations b et ween these sets to represen t the op erations of the theory . Other quan tum-lik e theories use quite differen t mathematical formalisms again [ 5 , 4 ]. The task of comparing these theories w ould b e sim- plified if we had a single mathematical framework in which they could all b e expressed. W e could then pinpoint asp ects of the framework where theo- ries differed, and iden tify these asp ects with differing ph ysical features of the theories. This pap er will outline such a framework, developed in [ 1 , 23 , 12 , 13 , 8 ], and then use it to analyse some k ey examples. In this case the ph ysical prop erty we will b e interested in is non-lo calit y . T o this end we extend the existing frame- w ork to encompass an abstract definition of GHZ state, and a corresp onding notion of correlations. What is nonlocality? The name tells us that “it’s not lo calit y .” The tec hnical definition tells us that “there is no lo cal hidden v ariable theory .” By Bell’s Theorem this means that “some inequalit y is not satisfied.” All this tells us what nonlo cality is not, but what actually “is” nonlo calit y? It is our goal in this pap er to identify the pie c e of structur e of Hilb ert space quantum mec hanics that gener ates non-lo c ality . T o this end we will use our framework to analyse t wo theories whic h make v ery similar predictions, but differ principally in that one is lo cal and the other is non-lo cal. W e will express b oth standard quantum mechanics, and a quan tum-like to y theory prop osed b y one of the authors [ 25 ], called Sp ekkens’s to y theory , in the unifying framework. The toy theory replicates man y fea- tures of QM (e.g. incompatible observ ables, telep ortation, no-cloning), but it is essen tially a lo cal hidden v ariable theory , and so it lacks other t ypically quan tum features, sp ecifically violation of Bell inequalities, and other ‘non- lo cal’ b ehaviour. W e will identify a key piece of the structure of the unifying framew ork where QM and the Sp ekk ens’s toy theory differ (an Ab elian group w e term the phase gr oup ). F urthermore we will show explicitly that it is this piece of structure which in the QM case facilitates a ‘no-go theorem’ which rules out a hidden v ariable in terpretation. Conv ersely , in the toy theory case, the phase group does not allow construction of suc h a no-go theorem. W e spec- ulate that this key piece of structure is resp onsible for the lo calit y/non-lo cality of any quantum-lik e theory that our framew ork is capable of encompasing. 2 Coecke, Edw ards and Spekkens 2 The framew ork: Dagger comp ositional theories T o mak e a comparison b et ween qubit stabiliser formalism and Sp ekk ens’s to y theory we need a framew ork with concepts that are sufficien tly general to accommo date b oth of them. In particular, w e need to b e able to sp eak ab out GHZ states and observ ables for theories other than Hilb ert space quan- tum mec hanics. Suc h a general accoun t of ph ysical theories w as initiated by Abramsky and one of the authors in [ 1 ], and further developed by sev eral oth- ers [ 23 , 12 , 13 , 8 ]. W e refer the reader for physicist friendly introductions and tutorials on symmetric monoidal categories to [ 6 , 3 ] and [ 11 , 2 ] resp ectiv ely . The op erational foundation for these structures is as follo ws – detailed discussions are in [ 6 , 11 ]. Systems are represented by their names A, B , C, . . . Pr o c esses (or op er ations ) are represen ted b y arro ws A f - B or f : A → B indicating the initial system A and the resulting system B . States are sp e- cial arrows ψ : I → A where I is the ‘unsp ecified’ system. W e can sequen- tially comp ose pro cesses if the intermediate systems matc h i.e. A h ◦ g - C = A g - B h - C . There also are pro cesses whic h lea v e the system inv ariant: A 1 A - A . Comp ound systems are denoted A ⊗ B and separate pro cesses thereon A ⊗ C f ⊗ g - B ⊗ D . W e refer to the arro ws I s - I as numb ers . In addition w e assume that each pro cess A f - B comes equipp ed with an adjoin t process B f † - A . The precise mathematical notion whic h accounts for ho w se quential c omp osition , denoted ◦ , and the tensor , denoted ⊗ , in teract is that of a dagger symmetric monoidal category . Definition 2.1 A dagger symmetric monoidal c ate gory C is a category with a bifunctor − ⊗ − : C × C → C , asso ciativit y , unit and symmetry natural isomorphisms, sub ject to the usual coherence conditions, and a con tra v ariant in volutiv e functor − † : C → C whic h coherently preserv es the monoidal struc- ture. An arrow f : A → B is unitary if w e ha v e f † ◦ f = 1 A and f ◦ f † = 1 B . W e assume asso ciativity , unit and symmetry natural isomorphisms to b e unitary . Eac h such dagger symmetric monoidal category moreov er admits a purely diagrammatic calculus [ 6 , 11 , 23 , 24 ], for example: f ≡ f 1 A ≡ g ◦ f ≡ g f f ⊗ g ≡ f f g These diagrams are not merely denotation but are truly equiv alen t to the algebraic symbolic presentation in the following sense. Theorem 2.2 [ 23 ] A n e quation expr essible in the language of dagger sym- metric monoidal c ate gories is pr ovable fr om the axioms of a dagger symmetric monoidal c ate gory if and only if it is derivable in the c orr esp onding diagr am- 3 Coecke, Edw ards and Spekkens matic c alculus. Definition 2.3 A dagger c omp ositional the ory , or in short, † C-the ory , is a dagger symmetric monoidal category in whic h w e in terpret ob jects as systems, morphisms as pro cesses, with states and effects as the particular cases arising from the unit ob ject, composition as p erforming one process after the other, and the tensor as comp osing systems and parallelling pro cesses. Example 2.4 In the † C-theory FHilb the ob jects are finite dimensional Hilb ert spaces, the arrows are linear maps, the tensor is the tensor pro duct, and the dagger is the linear algebraic adjoint. States are of the form | ψ i : C → H :: 1 7→ ψ , and hence corresp ond to vectors, and the num b ers are of the form h c i : C → C :: 1 7→ c , and hence corresp ond to complex num b ers. One can interpret the morphisms in FHilb as pure quantum pro cesses with p ostselection, that is, all the vectors, dual vectors and linear maps (b ecause p ostselected logic gate telep ortation allo ws us to pro duce an y linear map up to a probabilistic weigh t). 2 Example 2.5 In the † C-theory FRel , the ob jects are finite sets, the arro ws are relations, the tensor is the cartesian pro duct, the dagger is the relational con verse. The identit y ob ject is the single element set {∗} . No w states are of the form | r i : {∗} → X :: ∗ 7→ Y ⊆ X , and hence corresp ond to subsets, and the num b ers are of the form h b i : {∗} → {∗} :: ∗ 7→ ∅ or ∗ , and hence corresp ond to the b o oleans. One can interpret these relational op erations as ‘p ossibilistic’ (classical) pro cesses. Example 2.6 [F rom v ectors to ra ys] The states of the † C-theory FHilb as defined ab ov e are vectors in a Hilbert space, not one-dimensional subspaces. In other w ords, it con tains physically redundant ‘global phases’. One w ay to eliminate these global phases is by considering equiv alence classes of linear maps that are equal up to a global phase. Another wa y applies to arbitrary † C-theories: Definition 2.7 [ 7 ] [ W -c onstruction ] Given a † C-theory C we define a † C- theory W C to hav e the same ob jects as C , with W C ( A, B ) := { f ⊗ f † | f ∈ C ( A, B ) } , and with ( f ◦ g ) ⊗ ( f ◦ g ) † as the comp osite of f ⊗ f † and g ⊗ g † . 2 In the language of quantum information theory , these include the pure density op erators, the rank-1 effects, and the completely p ositive maps with a single Kraus op erator, as well as unnormalized versions thereof. 4 Coecke, Edw ards and Spekkens F or f a morphism in C w e set W f := f ⊗ f † for the corresp onding morphism in W C . Example 2.8 [F rom vectors to ra ys con tin ued] This W -construction has the added adv antage that expressions of the form h ψ | φ i := ψ † ◦ φ , after application of the W -construction, b ecome |h ψ | φ i| 2 = ( ψ † ◦ φ ) † ◦ ( ψ † ◦ φ ), that is, transition probabilities according to the Born rule. F or states in FHilb , applying the W -construction essentially b oils down to the same thing as passing from k ets | ψ i to pro jectors | ψ ih ψ | in the densit y matrix form ulation. The num b ers in W FHilb are p ositive real num b ers, whic h w e interpret as probabilistic w eights. W e ha ve W ( W FHilb ) ' W FHilb , and W FRel ' FRel . Definition 2.9 [ 7 ] A † C-theory C is without glob al phases iff W C ' C Belo w, we will alw a ys assume that we hav e eliminated the global phases from FHilb , ev en when w e write FHilb . Hence the num b ers in this category are the p ositive reals, whic h we in terpret as probabilistic w eigh ts. Inner- pro ducts then pro vide the correct transition probabilities according to the Born rule. More generally , w e will interpret the num b ers in † C-theories as probabilistic weigh ts and inner-pro ducts h−|−i := ( − ) † ◦ ( − ) as transition probabilities. 3 Key features of the † C-theory framew ork 3.1 Observables in † C-the ories In this section we explain ho w the usual notion of a non-degenerate observ able can b e generalised from Hilb ert spaces to other † C-theories. Definition 3.1 [ 12 ] Let C b e a † C-theory . By a(n) (non-de gener ate) observ- able for an ob ject X we mean an y comm utative isometric dagger F robenius comonoid ( X , δ,  ). Elsewhere we hav e referred to these non-degenerate observ ables as b asis structur es [ 8 ] or classic al structur es [ 12 ]. What this mathematical concept stands for exactly will b e explained b elow. Their name is justified b y the follo wing theorem. Theorem 3.2 [ 13 ] In FHilb , observables in the sense of Definition 3.1 on a Hilb ert sp ac e H ar e in bije ctive c orr esp ondenc e with the orthonormal b ases of H . Mor e pr e cisely, e ach (unor der e d) orthonormal b asis {| i i} i yields an observable ( H , δ,  ) with    δ : H → H ⊗ H :: | i i 7→ | ii i  : H 7→ C :: | i i 7→ 1 . 5 Coecke, Edw ards and Spekkens Conversely, al l observables in the sense of Definition 3.1 arise uniquely in this manner. Hence in FHilb an orthonormal basis can b e equiv alen tly defined as a comm utative isometric dagger F rob enius comonoid. The orthonormal basis is ‘enco ded’ as the linear map which c opies the vectors of that basis together with the linear map whic h uniformly er ases them. Of course, in quantum theory observ ables correspond to ra ys spanned b y an elemen t of an orthonormal basis rather than to the basis itself. F or a discussion of observ ables in the sense of Definition 3.1 within W FHilb we refer the reader to [ 8 ]. The definition of an observ able in a † C-theory has an equiv alen t, purely diagrammatic, incarnation which suffices for our purp oses in this pap er. W e set: δ =  = Theorem 3.3 [ 17 , 18 , 10 ] A l l c onne cte d diagr ams built fr om ‘c opying’ ( δ ), ‘er as- ing’ (  ), their daggers and str aight wir es, which have the same numb er of inputs and outputs, ar e e qual. We r epr esent these diagr ams by a ‘spider’: ......... .... Conversely, the defining e quations of an observable in a † C-the ory (Definition 3.1 ) ar e al l implie d by the assumption that c onne cte d diagr ams with the same numb er of inputs and outputs ar e e qual. Observ ables on ‘subsystems’ alwa ys lift to the whole system: Prop osition 3.4 [ 8 ] Two observables ( A, δ X ,  X ) and ( B , δ X 0 ,  X 0 ) in a † C- the ory c anonic al ly induc e an observable ( A ⊗ B , δ X ⊗ X 0 ,  X ⊗ X 0 ) with δ X ⊗ X 0 = (1 A ⊗ σ A,B ⊗ 1 B ) ◦ ( δ X ⊗ δ X 0 )  X ⊗ X 0 =  X ⊗  X 0 , wher e σ A,B is the morphism that swaps obje cts A and B . That is, diagr am- matic al ly, δ X ⊗ X 0 =  X ⊗ X 0 = Prop osition 3.5 [ 12 ] Each observable ( A, δ ,  ) in † C-the ory induc es a self- dual dagger c omp act structur e ( A, η := δ ◦  † : I → A ⊗ A ) , diagr ammatic al ly, = : , wher e ‘c omp actness’ me ans that: ( η † ⊗ 1 A ) ◦ (1 A ⊗ η ) = 1 A σ A,A ◦ η = η , 6 Coecke, Edw ards and Spekkens that is, diagr ammatic al ly, = = and . Giv en tw o suc h induced compact structures ( A, η A ) and ( B , η B ), and an arbitrary morphism f : A → B , we can define abstract notions of the tr ans- p ose morphism f ∗ : B → A and the c onjugate morphism f ∗ : B → A [ 1 ], resp ectiv ely as follows f ∗ := ( η † B ⊗ 1 A ) ◦ (1 B ⊗ f ⊗ 1 A ) ◦ (1 B ⊗ η A ) f ∗ := ( η † A ⊗ 1 B ) ◦ (1 A ⊗ f † ⊗ 1 B ) ◦ (1 A ⊗ η B ) . diagrammatically , f ∗ := f f ∗ := f † W e also refer to a dagger compact structure ( A, η ) as a Bel l state . A graphical in terpretation of Bell states can b e found b elow in Definition 3.8 . Let λ I : I ' I ⊗ I. Now we define abstract counterparts of the basis v ectors whic h are copied in FHilb : Definition 3.6 [ 8 ] The eigenstates of an observ able ( X, δ ,  ) in a † C-theory are all states x : I → X which satisfy δ ◦ x = ( x ⊗ x ) ◦ λ I ,  ◦ x = 1 I and x ∗ = x , that is, x = x , x x = † = x † x . The first of these conditions tells us that ‘eigenstates comm ute through dots’. Eigenstates moreov er lift from subsystems to the whole system: Prop osition 3.7 [ 8 ] If x is an eigenstate for observable ( A, δ X ,  X ) and x 0 is an eigenstate for observable ( B , δ X 0 ,  X 0 ) then x ⊗ x 0 is an eigenstate for ( A ⊗ B , δ X ⊗ X 0 ,  X ⊗ X 0 ) as define d in Pr op osition 3.4 . 3.2 GHZ states in † C-the ories Definition 3.8 A GHZ structur e for an ob ject X in a † C-theory is a triple ( X , Ψ : I → X ⊗ X ⊗ X ,  : X → I) where Ψ is called a GHZ state , with • Ψ symmetric i.e. = = • (  ⊗ 1 X ⊗ X ) ◦ Ψ is a Bell state i.e = : = such that = and 7 Coecke, Edw ards and Spekkens • Ψ and  are b oth self-c onjugate, i.e. = = • when ‘tracing out’ tw o subsystems we obtain the maximally mixed state: = Theorem 3.9 GHZ structur es in a † C-the ory ar e in bije ctive c orr esp ondenc e with observables in that † C-the ory via the c orr esp ondenc e: = = which assigns to e ach observable a GHZ structur e, and its c onverse: = = which assigns to e ach GHZ structur e an observable. Pro of: This can straightforw ardly b e v erified using Theorem 16.2 in [ 12 ]. 2 3.3 Phase gr oups in † C-the ories Prop osition 3.10 [ 8 ] F or ( X , δ,  ) an observable in a † C-the ory let • states X := { x : I → X } • x  y := δ † ◦ ( x ⊗ y ) = x y • actions X := { U x := δ † ◦ ( x ⊗ 1 X ) = x | x : I → X } then ( states X ,  ,  ) and ( actions X , ◦ , 1 X ) ar e isomorphic c ommutative monoids. This follows straightforw ardly from Theorem 3.3 (the spider theorem). Example 3.11 In FHilb , for v ectors ψ = ( ψ 1 , . . . , ψ n ) and φ = ( φ 1 , . . . , φ n ) in the basis corresp onding to ( H , δ ,  ) via Theorem 3.2 , w e hav e ψ  φ = ( ψ 1 · φ 1 , . . . , ψ n · φ n ), that is, ψ  φ is the comp onen t-wise pro duct of ψ and φ . Prop osition 3.12 [ 8 ] In FHilb a state ψ (normalise d so that | ψ | 2 = dim ( H ) ) is unbiase d with r esp e ct to the orthonormal b asis c orr esp onding to ( H , δ,  ) via The or em 3.2 iff ψ ∗  ψ =  † . Returning now to a general † C, let dim ( X ) =  ◦  † for observ able ( X , δ ,  ). 8 Coecke, Edw ards and Spekkens Definition 3.13 [ 8 ] In an y † C-theory a state ψ : I → X with ψ † ◦ ψ = dim ( X ) is unbiase d for an observ able ( X , δ ,  ) iff ψ ∗  ψ =  † that is ψ ψ ∗ = . By choosing ψ † ◦ ψ = dim ( X ) rather than ψ † ◦ ψ = 1 I as our normalisation con ven tion w e substan tially simplify the expressions in this paper. W e refer to states ψ : I → X whic h satisfy ψ † ◦ ψ = dim ( X ) as states of length p dim ( X ). Definition 3.14 [ 8 ] Two observ ables are mutual ly unbiase d if the eigenstates of one are unbiased for the other. The diagrammatic significance of this definition is studied in detail in [ 8 ]. Theorem 3.15 [ 8 ] L et now • U - states X b e al l states in states X unbiase d with r esp e ct to the observable ( X , δ,  ) • U - actions X b e al l unitary actions in actions X then ( U - states X ,  ,  ) and ( U - actions X , ◦ , 1 X ) ar e isomorphic ab elian gr oups. F or U - states X the inverses ar e pr ovide d by the adjoint, and for U - actions X the inverses ar e pr ovide d by the c onjugates for the induc e d c omp act structur e. Definition 3.16 [ 8 ] W e call the isomorphic groups of Theorem 3.15 the phase gr oup . Example 3.17 In the case of a qubit the phase group is the circle of ‘relative phases’. Concretely , when expressed in the standard basis, the un biased states and the unitary actions ha ve resp ective matrices: ψ α =   1 e iα   U α = δ ◦ ( ψ α ⊗ 1 Q ) =   1 0 0 e iα   . 3.4 GHZ c orr elations in † C-the ories In Theorem 3.9 , we sho wed the correspondence b etw een observ ables and GHZ states. It comes as no great surprise then, that the measuremen t correlations of our GHZ states are closely related to the phase groups describ ed in the previous section. Definition 3.18 Let ( X , δ,  ) b e an observ able in a † C-theory and let ( X , Ψ ,  ) b e the corresp onding GHZ state. By a GHZ c orr elation triple we mean a triple ( x, x 0 , x 00 ) of states x, x 0 , x 00 : I → X of length p dim ( X ) whic h is such that x 00 = ( x ⊗ x 0 ⊗ 1 X ) † ◦ Ψ = x x , . 9 Coecke, Edw ards and Spekkens By GHZ c orr elations w e mean the set Γ ⊆ C (I , X ) × C (I , X ) × C (I , X ) consisting of all GHZ correlation triples. W e can in terpret these GHZ correlation triples in operational terms: when, in a measurement of the first and second qubit of the GHZ state Ψ, the effects x † and x 0† o ccur then the third qubit is necessarily in state x 00 . If x 00 = 0 this means that effects x † and x 0† can never o ccur together. Prop osition 3.19 F or GHZ c orr elations Γ we have: i. F or states x, x 0 , x 00 : I → X , ( x, x 0 ; x 00 ) ∈ Γ iff x 00 ∗ = x  x 0 ; in other wor ds, c orr elation triples ar e exactly al l triples of the form ( x, x 0 , ( x  x 0 ) ∗ ) wher e x, x 0 , ( x  x 0 ) ∗ 6 = 0 . ii. If ( ψ , ψ 0 , ψ 00 ) is a c orr elation triple and ψ , ψ 0 , ψ 00 ar e in the phase gr oup then any triple obtaine d by p ermuting ψ , ψ 0 and ψ 00 is also a c orr elation triple. Pro of: P art i. follo ws from (reversed triangles are the transp osed): = x x , x x , P art ii. W e hav e ψ 0 = ( ψ ∗  ψ )  ψ 0 = ψ ∗  ( ψ  ψ 0 ) = ( ψ  ( ψ  ψ 0 ) ∗ ) ∗ so ( ψ , ( ψ  ψ 0 ) ∗ ; ψ 0 ) = ( ψ , ( ψ  ψ 0 ) ∗ ; ( ψ  ( ψ  ψ 0 ) ∗ ) ∗ ) is indeed a correlation triple by part i of this prop osition. 2 4 The k ey examples: Stab and Sp ek Ha ving survey ed our unifying categorical framework we no w proceed to con- sider tw o sp ecific examples. The first is stabiliser qubit QM, a restricted ver- sion of standard qubit QM. The second is Sp ekkens’s to y theory , whic h closely mo dels man y features of stabiliser QM, despite b eing essen tially a lo cal hidden v ariable theory . When considered within the categorical framework the simi- larit y b et ween the tw o is striking; and the precise difference b etw een the tw o can b e clearly pin-p ointed. F urthermore the difference is to b e found precisely in a certain categorical structure whic h is in timately in volv ed in describing the ph ysical phenomena where the t wo theories differ most significan tly - lo cality v. non-lo calit y . 4.1 Stabiliser qubit quantum me chanics This is a subset of standard QM. The only systems in the theory are qubits, or collections of qubits. The states which these ‘qubits’ can o ccupy are the sta- biliser states of standard QM (these are the +1 eigenstates of tensor pro ducts 10 Coecke, Edw ards and Spekkens of P auli operators). F or the single qubit there are six suc h states, the standard | 0 i , | 1 i , | + i , |−i , | i i and | − i i . F or t wo qubits we hav e all 36 possible tensor pro ducts of these single qubit states, plus 24 maximally en tangled states, all related to the Bell state 1 √ 2 | 00 i + | 11 i b y lo cal unitary op erations. F or three qubits we hav e many more states, including the GHZ state 1 √ 2 | 000 i + | 111 i . The time ev olution of states is giv en by those unitary op erations which preserv e stabiliser states. Suc h op erations are called Cliffor d unitaries and form a group. In fact, all n -qubit Clifford op erations can b e simulated using the CNOT gate (whic h is itself a Clifford unitary), and the single qubit Clifford unitaries. These single qubit op erations themselves form a group, isomorphic to the p erm utation group S 4 . The only measuremen ts allow ed in the theory are pro jectiv e Pauli measuremen ts. Though a restricted v ersion of QM, qubit stabiliser theory exhibits most of the k ey features of full QM. It has incompatible observ ables. There is a no- cloning theorem. Lo cal hidden v ariable no-go pro ofs hold, as we shall soon see (although in the case of stabiliser QM we need to employ three qubit states, as in the GHZ no-go pro of: although we ha ve the Bell state, making P auli measuremen ts alone cannot violate Bell inequalities). W e ha ve c hosen to in vestigate stabiliser QM rather than the full theory , b ecause it is muc h closer to the second theory which we will consider. 4.2 Sp ekkens’s toy the ory W e don’t ha ve space here to give full details of Sp ekk ens’s to y theory , these can b e found in Ref. [ 25 ]. A brief description of the k ey p oin ts will suffice. The theory attempts to approximate stabiliser qubit QM: there is only one t yp e of system, which is something like a qubit, and the states are discrete. The theory do es not employ v ector space. Instead a single system is describ ed b y a four state phase space. The actual state o ccupied in the phase space is called the ontic state. Ho wev er, the theory p osits a fundamen tal restriction on our knowledge of the ontic state. This restriction is the fundamental principle of the theory , called the ‘know le dge b alanc e principle’ . In full generality this principle is a bit awkw ard to state, but in the case of a single ‘qubit’ it b oils do wn to sa ying that w e can at best know that the system is in one of t wo on tic states, with e qual pr ob ability . Our state of kno wledge - the epistemic state - is the to y theory’s analogue of the quan tum state. The theory is clearly , by construction, a lo cal hidden v ariable theory . Because of the equal probability ca veat, mathematically the epistemic states of the ‘qubit’ system are subsets of a four elemen t set, hence there are six suc h states, just as in the case of stabiliser qubit QM. Inv oking the kno wledge balance principle, one can go on to derive the allo w ed states of comp osite systems, and all the op erations on systems whic h are allo wed in the 11 Coecke, Edw ards and Spekkens theory . There turns out to b e a one-to-one corresp ondence b et w een the states and op erations of the toy theory and those of the stabiliser theory , although ho w the op erations com bine together is not homomorphic. The op erations of the to y theory transform b etw een subsets of sets which represen t the phase spaces of the v arious systems - thus they are most naturally describ ed by r elations on these sets. 4.3 The † C-the ories Stab and Sp ek W e now express both these theories within our † C-theory framework. In ter- estingly , b oth theories can b e defined in a constructive fashion: Definition 4.1 [ Stab ] The † C-theory Stab is the sub- † C-theory of FHilb (recall example 2.4 ) generated b y: • n th tensor p ow ers of qubits Q := C 2 • the single qubit Clifford unitaries • the linear map δ stab : Q → Q ⊗ Q ::    | 0 i 7→ | 00 i | 1 i 7→ | 11 i together with the (necessarily unique) counit of this comultiplication,  S tab . That this collection of op erations is enough to generate all the states and op erations of the stabiliser theory can b e seen as follows: • The Hadamard op eration H is a single qubit Clifford unitary . • CNOT := (1 Q ⊗ ( H ◦ δ † S tab ◦ ( H ⊗ H ))) ◦ ( δ S tab ⊗ 1 Q ) • Arbitrary n -qubit Clifford unitaries U Clifford can b e generated from the single qubit Clifford unitaries and CNOT . • An arbitrary n -qubit stabiliser state Ψ stabilizer = U Clifford (  † ⊗  † ⊗ . . . ⊗  † ) Note that a similar construction actually applies to FHilb if w e substitute the single qubit unitaries for the single qubit Clifford unitaries. It is straigh tforw ard to v erify that ( Q, δ S tab ,  S tab ) is an observable as de- fined in Section 3.1 . The abstract GHZ state deriv ed via Theorem 3.9 is exactly the standard GHZ state 1 √ 2 | 000 i + | 111 i , which, as men tioned earlier, is a stabiliser state. All the results of Section 3 , on phase groups, correlation triples etc. apply . Prop osition 4.2 The ob ject Q in Stab has three observ ables in total: the one men tioned in Definition 4.1 , and t w o others whic h copy the v ectors | + i and |−i , and | i i and | − i i resp ectiv ely . All three observ ables are m utually un biased. 12 Coecke, Edw ards and Spekkens Pro of: That these are the only other observ ables on Q follows as a corollary of Theorem 3.2 , and the fact that Stab is a sub-category of FHilb . That they are all mutually un biased follows from straightforw ard computation. 2 Definition 4.3 [ 9 ][ Sp ek ] The † C-theory Sp ek is the sub- † C-theory of FRel (recall example 2.5 ) generated b y: • n th p o wers of qubits I V := { 1 , 2 , 3 , 4 } • all p erm utations on I V • the relation δ S pek : I V → I V × I V ::                1 7→ { (1 , 1) , (2 , 2) } 2 7→ { (1 , 2) , (2 , 1) } 3 7→ { (3 , 3) , (4 , 4) } 4 7→ { (3 , 4) , (4 , 3) } together with the (necessarily unique) unit of this com ultiplication,  S pek . That these relations are sufficient to generate all the states and op erations of Sp ekk ens’s to y theory (and no more) is not at all ob vious, and is pro v ed in [ 9 ]. P erhaps unsurprisingly , giv en our c hoice of notation, (I V , δ S pek ,  S pek ) turns out to b e an observ able. All the results of Section 3 , on phase groups, correlation triples etc. again apply . Prop osition 4.4 The ob ject I V in Sp ek has three observ ables in total. All three observ ables are mutually unbiased. Pro of: The three observ ables are detailed in [ 9 ]. That these are the only observ ables is sho wn in [ 21 ]. That they are mutually un biased follo ws from straigh tforward computation. 2 Remark 4.5 The use of relations in our construction actually leads to some- thing we would term a p ossibilistic theory . The scalars in FRel and th us in Sp ek are the Bo oleans. Such a theory can’t really tell us the probability of an y measurement outcomes, only whether such outcomes are p ossible or not. This is actually adequate for our later discussions of non-lo cality , since the kind of non-lo calit y pro ofs we will inv ok e only inv olv e measuremen t proba- bilities of 0 and 1. Ho w ever, it should b e noted that there is a well-defined pro cedure for mo difying Sp ek so that its scalars are positive real n umbers, and we can discuss probabilities. 4.4 Pinp ointing the differ enc e b etwe en Sp ek and Stab Our definitions of Stab and Sp ek are in terms of concrete v ector spaces and linear maps, sets and relations. This allo ws us to mak e a clear connection with 13 Coecke, Edw ards and Spekkens the w a y in whic h the theories were originally formulated. F rom our categori- cal p ersp ectiv e ho w ever the internal structure of the ob jects of a category is irrelev an t, only the algebra of comp osition of morphisms is imp ortant. F rom this p ersp ectiv e, b oth Stab and Sp ek are generated b y: • n th p o wers of qubit ob jects Q • the group S 4 acting on Q • an observ able: δ : Q → Q ⊗ Q and its unit  : Q → I By definition, we know that the δ and  morphisms alw ays combine in the same w ay: according to Theorem 3.3 . And b y sp ecifying the group S 4 w e hav e ensured that the group elements com bine with one another in the same wa y in b oth cases. F rom this p oint of view it lo oks like Stab and Sp ek might b e the same theory viewed in abstract categorical grounds. But this can’t b e the case: they describ e quite different ph ysical theories! In fact the difference lies in the wa y that the group elemen ts interact with the observ able. One key example of suc h an in teraction is the phase group. And indeed it is straightforw ard to v erify that the phase groups of the qubit observ ables of Stab and Sp ek differ: Theorem 4.6 The phase gr oup for qubits in Stab is the four element cyclic gr oup Z 4 and the phase gr oup for qubits in Sp ek is the Klein four gr oup Z 2 × Z 2 . Pro of: Straigh tforward computation. 2 In the next section w e will sho w that this mathematical difference b etw een the theories is in timately related to one of their k ey physical differences: the presence or absence of non-lo cality . 5 Mutually un biased qubit theories W e ha ve mentioned ho w Sp ekkens’s toy theory and stabiliser qubit QM are similar kinds of theory: in b oth cases there is a discrete collection of states; in both cases the ‘qubit’ system’s observ ables (of which there are three) are all m utually un biased. W e next try to formally pin down the features which these theories share, within our categorical framework. Definition 5.1 A mutual ly unbiase d qubit the ory , or MUQT, is a dagger sym- metric monoidal category with basis structures, which satisfies the following additional conditions: (i) The ob jects of the category are I , Q (whic h will represen t a qubit-lik e system), and n -fold tensor pro ducts of Q , i.e. Q ⊗ Q ⊗ . . . ⊗ Q . 14 Coecke, Edw ards and Spekkens (ii) The observ ables on an y giv en ob ject are all alike : that is to say , they ha ve the same num b er of eigenstates, and the same phase groups. (iii) The observ ables of Q are all m utually unbiased (recall Definition 3.14 ). (iv) All states of Q (i.e. morphisms of t yp e I → Q ) are eigenstates of some observ able. (v) Q has thr e e observ ables, each with two eigenstates. V arious results follow directly from this definition. (iv) and (v) together imply that Q has six states. (iii) and (iv) together imply that, with resp ect to any observ able on Q , all states are either eigenstates or unbiased. W e can further conclude that each observ able on Q has tw o eigenstates and four un biased states. Prop osition 5.2 Stab and Sp ek are b oth MUQTs. Pro of: This follows from the definitions of the categories, and Prop ositions 4.2 and 4.4 . 2 5.1 Classific ation W e will show that in a MUQT the p ossibilities for the basis structures on Q are quite limited. More precisely the GHZ correlations can take one of t wo forms, and Stab and Sp ek cov er these tw o p ossibilities. The outline of this argumen t is fairly straigh tforward. Firstly , we recall the connection established in 3.19 b et ween GHZ correlations and the monoid generated by the corresp onding observ able. W e will shortly show that in a MUQT the monoid generated b y the basis structures on Q is completely determined b y their phase group. Next we note that the phase group is an Ab elian group, and has as many members as the basis structure has unbiased states, in this case four. Finally we recall that there are only t w o Ab elian groups of four elements, the cyclic group Z 4 and the Klein four-group Z 2 × Z 2 . So it simply remains to pro ve the first step, that in a MUQT the GHZ correlations on Q are completely determined b y the phase group. Recall Def- inition 3.6 of an eigenstate. F rom the axioms of an eigenstate it immediately follo ws that x † ◦ x = 1 I . More sp ecifically , if δ ◦ x = ( x ⊗ x ) ◦ λ I and x ∗ = x , then we hav e that  ◦ x = x † ◦ x . Lemma 5.3 F or x, x 0 : I → X eigenstates we have ( x † ◦ x 0 ) 2 = x † ◦ x 0 . Pro of: ( x † ◦ x 0 ) 2 = λ † I ◦ ( x ⊗ x ) † ◦ ( x 0 ⊗ x 0 ) ◦ λ I = x † ◦ δ † ◦ δ ◦ x 0 = x † ◦ x 0 . 2 Lemma 5.4 If for x, x 0 : I → X eigenstates x † ◦ x 0 = 1 I then x = x 0 . Pro of: Ignoring natural isomorphisms, (1 X ⊗ x † ) ◦ δ ◦ x 0 = (1 X ⊗ x † ) ◦ ( x 0 ⊗ x 0 ) = ( x † ◦ x 0 ) · x 0 = x 0 (where w e use · in place of ⊗ when the ob jects are num b ers) 15 Coecke, Edw ards and Spekkens from whic h it follows by x ∗ = x that x  x 0 = x 0 . By symmetry we also hav e x  x 0 = x and hence x 0 = x  x 0 = x . 2 Hence the inner pro duct of t wo eigenstates is alw ays an idemp otent and for non-equal eigenstates this idemp otent cannot b e 1 I . Definition 5.5 A † C-theory has a zer o if it has exactly t wo idemp otent n um- b ers. The idempotent n um b er 0 which is not the iden tit y is referred to as zer o . Prop osition 5.6 If two states x 6 = x 0 ar e eigenstates for an observable in a † C-the ory with zer o then we have x † ◦ x 0 = 0 and x † ◦ x = 1 I . Pro of: F ollo ws from Lemma 5.3 and Lemma 5.4 . 2 In R , R + and C the only idempotents are 0 and 1. W e will furthermore assume that an y 0-m ultiple of a state ψ : I → A is a unique (trivial) state whic h we also denote by 0 A . W e will indicate reliance on this assumption that there is a unique ‘absorbing idemp oten t num b er’ by (0). This assumption is conceptually justified b y the in terpretation of num b ers as probabilistic w eigh ts – see Example 2.6 . Lemma 5.7 [ 8 ] F or x : I → X an eigenstate and ψ : I → X unbiasse d we have: dim ( X ) · ( x † ◦ ψ ) † · ( x † ◦ ψ ) = 1 I . Setting |h x | ψ i| 2 := ( x † ◦ ψ ) † · ( x † ◦ ψ ) and assuming that dim ( X ) admits an in verse 1 /D , i.e. dim ( X ) · 1 /D = 1 I , results in the familiar form |h x | ψ i| 2 = 1 /D . When we now sub ject a † C-theory to the W -construction of [ 7 ] discussed in Example 2.6 , then in the newly constructed category w e hav e hW x |W ψ i = ( W x ) † ◦ W ψ = ( x † ◦ ψ ) † · ( x † ◦ ψ ) = 1 /D . W e will assume b elow that w e alw ays are in a ‘ † C-theory without global phases’ i.e. a † C-theory which is in v arian t under the W - construction. W e will indicate reliance on this assumption by ( W ). This assumption is again conceptually justified b y the interpretation of n umbers as probabilistic weigh ts – see Ex- ample 2.6 . Remark 5.8 Note that while Sp ek , as a sub category of FRel , obviously has no global phases, it do es ha ve non-trivial relativ e phases, namely Z 2 × Z 2 . Lemma 5.9 L et ( X , δ,  ) b e an observable in a † C-the ory, let ψ , φ : I → X b e unbiase d for it and let x 6 = x 0 : I → X b e eigenstates for it. Then we have: (1) x  x = x and x  x 0 = ( x † ◦ x 0 ) · x (0) = 0 ; (2) x  ψ = ( x † ◦ ψ ) · x ( W ) = 1 /D · x ; 16 Coecke, Edw ards and Spekkens (3) ψ  φ is c ompletely determine d by the phase gr oup. Pro of: F or (1) we ha ve: = x x , x x , = x x , x = x x , x where the last step follows b y x = x ∗ . Hence x  x 0 = ( x † · x 0 ) · x = 0 · x = 0. If rather than x 0 w e w ould hav e considered x itself then this graphical argument yields x  x = ( x † · x ) · x = x . F or (2) the same graphical argument, no w substituting ψ for x 0 , results in x  x 0 = ( x † ◦ ψ ) · x = 1 /D · x . (3) is simply a consequence of the definition of the phase group (definition 3.16 ). 2 Corollary 5.10 Consider a † C-the ory which ob eys (0) and ( W ) and c onsider an observable in it for which al l states on the underlying obje ct ar e either eigenstates or unbiase d. Then, the choic e of phase gr oup c onstitutes the only de gr e e of axiomatic fr e e dom for how the multiplic ation −  − of the observable acts on states. Next we can use Prop osition 3.19 to make the link to GHZ correlations: Lemma 5.11 L et ( X, Ψ ,  ) b e a GHZ state in a † C-the ory, let ψ , φ : I → X b e unbiase d for it and let x 6 = x 0 : I → X b e eigenstates for it. Then we have: (1a) ( x, x ; x ) is a c orr elation triple ; (1b) ther e ar e no c orr elation triples involving b oth x and x 0 ; (2) ( x, ψ ; x ) is a c orr elation triple ; (3) al l c orr elation triples involving at le ast two phase gr oup elements ar e of the form ( ψ , φ ; ( ψ  φ ) ∗ ) – which by Pr op. 3.19 ii. includes p ermutations ther e of. Pro of: Using Prop osition 3.19 , each of these items follo ws from the similarly n umbered item of lemma 5.9 . 2 Corollary 5.12 Consider a † C-the ory which ob eys (0) and ( W ) and c onsider a GHZ state in it for which al l states on the underlying obje ct ar e either eigenstates or unbiase d. Then, the choic e of phase gr oup c onstitutes the only de gr e e of axiomatic fr e e dom for the c orr esp onding GHZ c orr elations. Finally considering that in a MUQT the phase group must ha ve four ele- men ts, and that there are only t w o four element groups Z 4 and Z 2 × Z 2 , we can state our main result: Theorem 5.13 The GHZ c orr elations of the ‘qubit’ obje ct in a MUQT c an take only two forms, c orr esp onding to the two four-element gr oups, Z 4 (as in the c ase of Stab ) and Z 2 × Z 2 (as in the c ase of Sp ek ). 17 Coecke, Edw ards and Spekkens W e conclude that, whilst there is a v ast n umber of p ossible MUQTs, their GHZ correlations can tak e only one of tw o forms, and Stab and Sp ek exem- plify the tw o p ossibilities. 5.2 Link to non-lo c ality The GHZ correlations in the theories are of particular in terest, b ecause these correlations are in vok ed in one of the most elegant ‘no-go’ proofs showing that quan tum mec hanics cannot b e explained b y a lo cal realist theory . F or the full details of this famous pro of the reader is referred to [ 20 ]. Note the following k ey p oints: • This no-go pr o of also applies to stabiliser the ory . The pro of b egins with a GHZ state. The key ingredients are the probabilities of outcomes when we measure the v ariables X ⊗ X ⊗ X , X ⊗ Y ⊗ Y , Y ⊗ X ⊗ Y , and Y ⊗ Y ⊗ X . GHZ states and P auli measuremen ts b oth surviv e the restriction from full QM to the stabiliser theory , so the proof applies equally w ell in this case, i.e. it is imp ossible to mo del stabiliser theory with a lo cal realist theory . • The key structur al ingr e dients of the pr o of ar e al l pr esent in any MUQT 3 . W e can depict all the key ingredien ts of the pro of in our categorical frame- w ork. Diagrammatically the relev ant probabilities are given b y: x x x i j k y y x i j k y x y i j k x y y i j k In our abstract terminology we would say that the pro of is employing a basis structure, and four of its un biased states. An analogue of the argument could b e reconstructed in an y dagger symmetric monoidal category with these features, and with scalars which are num b ers or Bo oleans. Certainly an y MUQT will hav e an analogue of the pro of, where the scalars pictured ab o v e are the GHZ correlations. • No Z 4 MUQT c an have a lo c al r e alist mo del . F or MUQTs with a Z 4 qubit basis structure the pro of will b e identical to the quantum stabiliser case, ruling out a lo cal realist mo del. • A lo c al r e alist mo del c an b e c onstructe d for the GHZ state in any Z 2 × Z 2 MUQT . Hence, in the case of general MUQTs with Z 2 × Z 2 correlations, we cannot rule out such a mo del, b ecause w e ha v e a concrete example of a lo cal realist theory , Sp ek , which exhibits exactly these correlations. Put another w ay , if w e w ere presen ted with the data of a set of Z 2 × Z 2 correlations, we could alwa ys explain them via the hidden v ariables of Sp ek . Th us w e can conclude that no MUQT of the Z 4 t yp e can ha ve a local realist 3 In fact, in this section w e restrict atten tion to pr ob abilistic and p ossibilistic MUQTs, since these are the only † -C theories where it makes sense to discuss lo cality/non-locality . 18 Coecke, Edw ards and Spekkens in terpretation, since at least one of its states (the GHZ) do es not ha v e such an in terpretation. W e cannot conclude that all MUQTs of the Z 2 × Z 2 t yp e will ha ve a local realist in terpretation, since they migh t ha ve other states which had no such in terpretation. W e can at least conclude though, that GHZ-type no-go arguments will not work for them. T urning this on its head, we can see that the Z 4 t yp e basis structure, within our framew ork, is a structural fragmen t whic h embo dies non-lo cality . If your theory has a basis structure of this t yp e, then y our theory has ‘got non-lo calit y’. The Z 2 × Z 2 structure has no non-locality . Whilst a Z 2 × Z 2 t yp e MUQT might ha v e some other non-lo cal piece of structure, the Z 2 × Z 2 t yp e basis structure cannot itself endow a theory with non-lo calit y . 6 Non-lo calit y directly from abstract argumen ts The argumen ts ab o ve are sligh tly round-ab out: we sho w that certain phase groups are exhibited by either Stab or Sp ek , which we know b y other argu- men ts to b e non-lo cal and lo cal resp ectiv ely . W e then conclude that Z 4 GHZ states m ust hav e non-lo cality , whereas Z 2 × Z 2 GHZ states can not. In fact, in the case of Z 4 w e can pro vide a more general argumen t directly from abstract reasoning. Z 4 o o / / s s abstract , , Stab o o / / non-lo cal Let O A b e the set of observ ables on an ob ject A in a † C-theory . Let E o A b e the set of eigenstates of an observ able o A ∈ O A . W e no w define a notion of lo cal realist represen tation which applies to arbitrary † C-theories with R + as n umbers. This can b e extended to † C-theories with more general n um b ers, as w e sho w at the end of this section for the case of purely qualitativ e relational theories. Definition 6.1 Let C b e a † C-theory with R + as n umbers. A state Ψ : I → A 1 ⊗ . . . ⊗ A n in C admits a lo c al r e alist r epr esentation if there exist: • a set of hidden states Ξ ⊆ Q o 1 ∈O 1 E o 1 × . . . × Q o n ∈O n E o n eac h of which assigns an eigenstate in E o i to each observ able o i ∈ O i on each subsystem A i , and we denote this eigenstate for ξ ∈ Ξ by ξ ( o i ) • a σ -additive measure µ : B (Ξ) → R + with µ (Ξ) = 1 and these are suc h that for each c hoice of observ ables o 1 ∈ O 1 , . . . , o n ∈ O n and each choice of eigenstates x 1 ∈ E o 1 , . . . , x n ∈ E o n w e hav e µ ( { ξ ∈ Ξ | x 1 = ξ ( o 1 ) , . . . , x n = ξ ( o n ) } ) =  x † 1 ⊗ . . . ⊗ x † 1  ◦ Ψ . The † C-theory C admits a lo c al r e alist r epr esentation if each of its states admits a lo cal realist representation 19 Coecke, Edw ards and Spekkens W e pro vide a no-go argument for GHZ states that applies to the GHZ states on qubits in Stab and FHilb . This argument is not v ery differen t from the usual one [ 20 ], except for the fact that there is no reference to Hilb ert space anymore and that a contradiction is directly drawn from the structure of the Z 4 phase group. Definition 6.2 Let ( A, Ψ ,  ) b e a GHZ state in a † C-theory . A forbidden triple is a triple of states ( x, x 0 ; x 00 ) suc h that x 00 and x  x 0 are distinct eigen- states for the same observ able. Prop osition 6.3 If ( x, x 0 ; x 00 ) is a forbidden triple for GHZ state ( A, Ψ ,  ) in a † C-the ory with zer o then we have ( x ⊗ x 0 ⊗ x 00 ) † ◦ Ψ = 0 . Pro of: Since x 00 and x  x 0 are distinct eigenstates for the same observ able, ignoring natural isomorphisms, w e ha v e ( x ⊗ x 0 ⊗ x 00 ) † ◦ Ψ = x 00† ◦ ( x ⊗ x 0 ⊗ 1 A ) † ◦ Ψ = x 00† ◦ ( x  x 0 ) = 0 by Prop osition 5.6 . 2 Theorem 6.4 L et ( A, Ψ Z ,  Z ) b e a GHZ state in a † C-the ory with R + as numb ers, which c ontains Z 4 as a sub gr oup of the phase gr oup, and let the identity and the involutive element of this sub gr oup c onstitute the eigenstates of an observable ( A, Ψ X ,  X ) , and its other two elements the eigenstates of an observable ( A, Ψ Y ,  Y ) . Then the state Ψ Z : I → A ⊗ A ⊗ A do es not admit a lo c al r e alist r epr esentation. Pro of: W e denote the identit y of the phase group by | + i and the inv olutiv e elemen t by |−i , and the tw o other elements b y | ] i and | = i . By the Z 4 structure w e hav e: | + i  | + i = | + i | + i  |−i = |−i |−i  |−i = | + i . Hence, b y Prop osition 3.19 ii we ha v e that each correlation triple in volving only states {| + i , |−i} m ust hav e an ev en num b er of o ccurences of |−i ’s, and hence those with an o dd n umber of |−i ’s are forbidden triples. Also: | ] i  | = i = | + i | = i  | = i = |−i | ] i  | ] i = |−i . Hence, b y Prop osition 3.19 ii we ha v e that each correlation triple in volving t wo states in {| ] i , | = i} and one state in {| + i , |−i} must hav e an o dd num b er of o ccurrences of elements in {|−i , | = i} , and hence those with an ev en n um b er of elements in {|−i , | = i} are forbidden triples. Assume that Ψ admits a realist represen tation (Ξ , µ ). T o distinguish betw een the three factors in A ⊗ A ⊗ A we will denote them by A 1 , A 2 , A 3 resp ectiv ely . Using the notation of Definition 6.1 , we hav e for o + / − the observ able with eigenstates {| + i , |−i} that µ n ξ ∈ Ξ    x 1 = ξ ( o + / − 1 ) , x 2 = ξ ( o + / − 2 ) , x 3 = ξ ( o + / − 3 ) o = 0 whenev er the n um b er of |−i ’s in ( x 1 , x 2 , x 3 ) is o dd b y Prop osition 6.3 . Hence µ  ∆ (1 , 2 , 3) odd := n ξ ∈ Ξ    o dd |−i ’s in  ξ ( o + / − 1 ) , ξ ( o + / − 2 ) , ξ ( o + / − 3 ) o = 0 . 20 Coecke, Edw ards and Spekkens and so for ∆ (1 , 2 , 3) ev en = Ξ \ ∆ (1 , 2 , 3) odd w e hav e µ (∆ (1 , 2 , 3) ev en ) = 1. Similarly , for ∆ (1) odd := n ξ ∈ Ξ    o dd |−i ’s & | = i ’s in  ξ ( o + / − 1 ) , ξ ( o ]/ = 2 ) , ξ ( o ]/ = 3 ) o ∆ (2) odd := n ξ ∈ Ξ    o dd |−i ’s & | = i ’s in  ξ ( o ]/ = 1 ) , ξ ( o + / − 2 ) , ξ ( o ]/ = 3 ) o ∆ (3) odd := n ξ ∈ Ξ    o dd |−i ’s & | = i ’s in  ξ ( o ]/ = 1 ) , ξ ( o ]/ = 2 ) , ξ ( o + / − 3 ) o w e ha ve µ (∆ (1) odd ) = µ (∆ (2) odd ) = µ (∆ (3) odd ) = 1 so µ (∆ (1) odd ∩ ∆ (2) odd ∩ ∆ (3) odd ) = 1. It follo ws that there m ust b e an o dd num b er of |−i ’s and | = i ’s in  ξ ( o + / − 1 ) , ξ ( o ]/ = 2 ) , ξ ( o ]/ = 3 ) , ξ ( o ]/ = 1 ) , ξ ( o + / − 2 ) , ξ ( o ]/ = 3 ) , ξ ( o ]/ = 1 ) , ξ ( o ]/ = 2 ) , ξ ( o + / − 3 )  . But due to the double o ccurrences of ξ ( o ]/ = 1 ) , ξ ( o ]/ = 2 ) , ξ ( o ]/ = 3 ), this means an o dd n um b er of |−i ’s in  ξ ( o + / − 1 ) , ξ ( o + / − 2 ) , ξ ( o + / − 3 )  so ∆ (1) odd ∩ ∆ (2) odd ∩ ∆ (3) odd ⊆ ∆ (1 , 2 , 3) odd and hence 1 = µ (∆ (1) odd ∩ ∆ (2) odd ∩ ∆ (3) odd ) ≤ µ (∆ (1 , 2 , 3) odd ) = 0, hence a con- tradiction. 2 7 Conclusions and further work W e hav e describ ed a categorical framework whic h is sufficiently flexible to accommo date b oth stabiliser QM, and Sp ekkens’s toy theory , 4 and whic h helps to cast ligh t on the essen tial difference b et ween the t wo. Structurally this difference is in the phase group: Z 4 in the case of Stab and Z 2 × Z 2 in the case of Sp ek . Physically the difference b et ween the theories is that one is non-local whilst the other is lo cal. W e w ent on to show that it is the presence of the Z 4 phase group that mak es stabiliser QM non-lo cal. In fact, this structure suffices to sho w that full QM is non-lo cal. W e ha ve furthermore defined a special class of to y theories, in whic h all ‘qubit’ observ ables are mutually unbiased, whic h can all b e mo delled in the categorical framework. W e hav e shown that the GHZ-correlations in these theories hav e phase groups Z 4 and Z 2 × Z 2 . W e could extend the definition of a MUQT beyond ‘qubits’, by allowing our basic system to hav e more observ ables, and its observ ables to ha ve more eigenstates, while still insisting that the observ ables are all m utually un biased. W e would then ha ve a more general mutual ly unbiase d the ory or MUT. The result that the GHZ correlations in suc h a theory are completely determined b y the phase group, established in Corollary 5.12 , would still hold. F or example, in the case of qu trits , there are four mutually unbiased ob- serv ables, eac h with three eigenstates. Phase groups in this case would hav e 4 While here w e only considered the pure fragment of both theories, mixed states and op erations can be straightforw ardly adjoined by means of Selinger’s CPM-construction [ 23 ]. 21 Coecke, Edw ards and Spekkens nine elemen ts. There are tw o nine-elemen t groups, Z 9 and Z 3 × Z 3 . There is a well-defined w ay to extend stabiliser QM to higher dimensional systems and recen tly a ‘trit’ version of Sp ekk ens’s toy theory w as prop osed [ 26 ]. In this case the t wo theories coincide and their phase group is Z 3 × Z 3 . The to y theory is lo cal b y construction, and, as it turns out, so is the stabilizer formalism for qutrits. So is there a theory with phase group Z 9 , and what kind of theory is it? Solving questions of this kind is one a v enue for future researc h. Several other av enues suggest themselves, including: • An ob vious line of work b eyond this is to consider ‘higher-dimensional’ MUTs, beyond qubits and qutrits. The lo cality/non-locality prop erties of suc h theories will still be parametrised b y Ab elian groups. What sorts of lo calit y/non-lo cality do w e find? There is a well-defined w a y to extend stabiliser theory to any finite dimensional system. What is the phase group in each case? Can Sp ekk ens’s toy theory b e extended b eyond trits? What w ould its phase group b e? • W e hav e shown that the phase group is imp ortant in determining whether theories are lo cal or exhibit quantum non-lo cal correlations. In fact, theories ha ve b een proposed whose non-locality go es beyond that of quan tum me- c hanics [ 5 ]. Can these b e accommo dated within our framework? 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