Quantum Graph Homomorphisms via Operator Systems
aa r X i v : . [ m a t h . OA ] F e b QUANTUM GRAPH HOMOMORPHISMS VIA OPERATORSYSTEMS
CARLOS M. ORTIZ AND VERN I. PAULSEN
Abstract.
We explore the concept of a graph homomorphism throughthe lens of C ∗ -algebras and operator systems. We start by studyingthe various notions of a quantum graph homomorphism and examinehow they are related to each other. We then define and study a C ∗ -algebra that encodes all the information about these homomorphismsand establish a connection between computational complexity and therepresentation of these algebras. We use this C ∗ -algebra to define anew quantum chromatic number and establish some basic properties ofthis number. We then suggest a way of studying these quantum graphhomomorphisms using certain completely positive maps and describetheir structure. Finally, we use these completely positive maps to definethe notion of a “quantum” core of a graph. Introduction
Let G = ( V ( G ) , E ( G )) and H = ( V ( H ) , E ( H )) be graphs on vertices V ( G ) = { , ..., n } and V ( H ) = { , ..., m } . The theory of graph homo-morphisms is one of the central tools of graph theory and is used in thedevelopment of the concept of the core of a graph. More recently, work inquantum information theory has led to quantum versions of many conceptsin graph theory and there is an extensive literature ([1], [3], [15]). In partic-ular, D. Roberson[17] and L. Mancinska [11] developed an extensive theoryof quantum homomorphisms of graphs. D. Stahlke[18] interpreted graphhomomorphisms in terms of “completely positive(CP) maps on the tracelessoperator space of a graph”.These papers motivate us to consider quantum and classical graph ho-momorphisms as special families of completely positive maps between theoperator systems of the graphs.There is not just a single quantum theory of graphs, but there are re-ally possibly several different quantum theories depending on the validityof certain conjectures of Connes and Tsirelson. In earlier work on quan-tum chromatic numbers[16, 15], we studied the differences and similaritiesbetween the properties of the quantum chromatic numbers defined by the Date : February 23, 2016.2000
Mathematics Subject Classification.
Primary 46L15; Secondary 47L25.
Key words and phrases. graph homomorphisms; operator systems; non-locality; entan-gled games. possibly different quantum theories. We wish to parallel those ideas forquantum graph homomorphisms. One technique of [15] and [6] was to showthat the existence of quantum colorings was equivalent to the existence ofcertain types of traces on a C*-algebra affiliated with the graph and we wishto expand upon that topic here. This leads us to introduce the C*-algebra ofa graph homomorphism and we will show that the existence or non-existenceof various types of quantum graph homomorphisms are related to proper-ties of this C*-algebra, e.g., whether or not it has any finite dimensionalrepresentations or has any traces.Finally, we wish to use our correspondence between quantum graph ho-momorphisms and CP maps to introduce a quantum analogue of the coreof a graph. 2.
The Homomorphism Game
Given graphs G and H a graph homomorphism from G to H is amapping f : V ( G ) → V ( H ) such that( v, w ) ∈ E ( G ) = ⇒ ( f ( v ) , f ( w )) ∈ E ( H ) . When a graph homomorphism from G to H exists we write G → H .Paralleling the work on quantum chromatic numbers [16], we study agraph homomorphism game, played by Alice, Bob, and a referee. Givengraphs G and H , the referee gives Alice and Bob a vertex of G , say v and w , respectively, and they each respond with a vertex from H , say x and y ,respectively. Alice and Bob win provided that: v = w = ⇒ x = y,v ∼ G w = ⇒ x ∼ H y. If they have some random strategy and we let p ( x, y | v, w ) denote theprobability that we get outcomes x and y given inputs v and w , then theseequations translate as:(1) p ( x = y | v = w ) = 0(2) p ( x ≁ H y | v ∼ G w ) = 0Now say G has n vertices and H has m vertices. We consider the sets ofcorrelations studied in [15] and [16]: Q l ( n, m ) ⊆ Q q ( n, m ) ⊆ Q qa ( n, m ) ⊆ Q qc ( n, m ) ⊆ Q vect ( n, m ) . In the appendix, we review the definition and some known facts aboutthese sets.For t ∈ { l, q, qa, qc, vect } we define: G t −→ H, provided that there exists p ( x, y | v, w ) ∈ Q t ( n, m ) UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 3 satisfying (1) and (2). Notice that when we write p ( x, y | v, w ) ∈ Q t ( n, m )we really mean (cid:0) p ( x, y | v, w ) (cid:1) v,w,x,y ∈ Q t ( n, m ). Any p ( x, y | v, w ) ∈ Q t ( n, m )satisfying these conditions we call a winning t-strategy and say that thereexists a quantum t-homomorphism from G to H. The condition (1) is easily seen to be the synchronous condition de-fined in [15] and the subset of correlations satisfying this condition wasdenoted Q st ( n, m ) . Thus, p ( x, y | v, w ) is a winning t-strategy if and only if p ( x, y | v, w ) ∈ Q st ( n, m ) and satisfies (2).The following result is known, but we provide a proof since we are usinga slightly different (but equivalent) characterization of Q l ( n, m ) . Theorem 2.1.
Let G and H be graphs. Then G → H if and only if G l −→ H. Proof.
First assume that G → H. Let f : V ( G ) → V ( H ) be a graph ho-momorphism. Let Ω = { t } be the singleton probability space. For each v ∈ V ( G ) let Alice have the “random variable”, f v ( t ) = f ( v ) and for each w ∈ V ( G ) let Bob have the random variable g w ( t ) = f ( w ) . Then p ( x, y | v, w ) := P rob ( x = f v ( t ) , y = g w ( t )) = ( , when x = f ( v ) , y = f ( w )0 , else . From this it easily follows that p ( x, y | v, w ) satisfies (1) and (2).Conversely, assume that we have a probability space (Ω , P ) and randomvariables f v , g w : Ω → V ( H ) = { , ..., m } so that p ( x, y | v, w ) = P ( x = f v ( ω ) , y = g w ( ω )) satisfies (1) and (2). By (1), for each v the set B v = { ω : f v ( ω ) = g v ( ω ) } has probability 1. Similarly, for each ( v, w ) ∈ E ( G ) the set Q v,w = { ω : ( f v ( ω ) , g w ( ω )) ∈ E ( H ) } has probability 1. Thus, D = (cid:0) ∩ v ∈ V ( G ) B v (cid:1) ∩ (cid:0) ∩ ( v,w ) ∈ E ( G ) Q v,w (cid:1) has measure 1, and so in particular is non-empty. Fix any ω ∈ D and define f : V ( G ) → V ( H ) by f ( v ) := f v ( ω ) = g v ( ω ) . Then whenever ( v, w ) ∈ E ( G )we have that ( f ( v ) , f ( w )) = ( f v ( ω ) , g w ( ω )) ∈ E ( H ) . Thus, f is a graphhomomorphism. (cid:3) Thus, quantum l-homomorphisms correspond to classical graph homo-morphisms.
Remark 2.2. In [1] several notions of graph homomorphisms were alsointroduced, including G B −→ H, G V −→ H and G + −→ H. A look at theirdefinition shows that G vect −→ H if and only if G V −→ H Corollary 2.3.
Let G and H be graphs. Then G −→ H = ⇒ G q −→ H = ⇒ G qa −→ H = ⇒ G qc −→ H = ⇒ G vect −→ H Proof.
This is a direct consequence of the above definitions, Theorem 2.1,and the corresponding set containments. (cid:3)
C. M. ORTIZ AND V. I. PAULSEN Quantum Homomorphisms and CP Maps
Recall [12] that the operator system of a graph G on n vertices is thesubspace of the n × n complex matrices M n given by S G = span { E v,w : v = w or ( v, w ) ∈ E ( G ) } , where E v,w denotes the n × n matrix that is 1 in the ( v, w )-entry and 0elsewhere.We now wish to use a winning x-strategy for the homomorphism gameto define a CP map from S G to S H . It will suffice to do this in the case ofwinning vect-strategies since every other strategy is a subset.
Proposition 3.1.
Let p ( x, y | v, w ) ∈ Q svect ( n, m ) , let E v,w ∈ M n and E x,y ∈ M m denote the canonical matrix unit bases. Then the linear map φ p : M n → M m defined on the basis by φ p ( E v,w ) = X x,y p ( x, y | v, w ) E x,y , is completely positive.Proof. By Choi’s theorem [14], to prove that φ p is CP it is enough to provethat the Choi matrix, P := X v,w E v,w ⊗ φ p ( E v,w ) = X v,w,x,y p ( x, y | v, w ) E v,w ⊗ E x,y ∈ M n ⊗ M m = M nm is positive semidefinite.Recall that by the definition and characterization of vector correlationssatisfying the synchronous condition in [16] there exists a Hilbert space andvectors { h v,x } satisfying: • h v,x ⊥ h v,y for all x = y , • P x h v,x = P x h w,x for all v, w , • k P x h v,x k = 1 , such that p ( x, y | v, w ) = h h v,x , h w,y i . Now let { e v } and { f x } denote the canonical orthonormal bases for C n and C m , respectively, let a v,x ∈ C be arbitrary complex numbers, so that k = P v,x a v,x e v ⊗ f x is an arbitrary vector in C n ⊗ C m . We have that h P k, k i = X v,w,x,y a v,x a w,y p ( x, y | v, w ) = X v,w,x,y a v,x a w,y h h v,x , h w,y i = h h, h i , where h = P v,x a v,x h v,x . Thus, P is positive semidefinite and φ p is CP. (cid:3) Theorem 3.2.
Let G and H be graphs, let p ( x, y | v, w ) ∈ Q svect ( n, m ) bea winning vect-strategy for a quantum vect-homomorphism from G to H and let φ p : M n → M m be the CP map defined in Proposition 3.1. Then φ p ( S G ) ⊆ S H and φ p is trace-preserving on S G . UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 5
Proof.
To see that φ p is trace preserving on S G it will be enough to show that tr (cid:0) φ p ( E v,v ) (cid:1) = tr ( E v,v ) = 1 , and for v ∼ G w, tr (cid:0) φ p ( E v,w ) (cid:1) = tr ( E v,w ) = 0 . When v = w we have that tr (cid:0) φ p ( E v,v ) (cid:1) = tr (cid:0) X x,y p ( x, y | v, v ) E x,y (cid:1) = X x p ( x, x | v, v ) = 1 = tr ( E v,v ) , by the definition of p .Finally, if v = w and E v,w ∈ S G , then tr (cid:0) φ p ( E v,w ) (cid:1) = X x p ( x, x | v, w ) = 0 = tr ( E v,w ) , by (2) and the fact that x ≁ H x. Hence, φ p is trace-preserving on S G . Now we prove that φ ( S G ) ⊆ S H . First, φ p ( E v,v ) = P x,y p ( x, y | v, v ) E x,y , but since p is synchronous, p ( x, y | v, v ) = 0 for x = y. Hence, φ p ( E v,v ) isa diagonal matrix and so in S H . To finish the proof it will be enough toshow that when v ∼ G w, we have φ p ( E v,w ) ∈ S H . But by property (2), p ( x, y | v, w ) = 0 when x ≁ H y. Thus, φ p ( E v,w ) ∈ S H . In fact, it is a matrixwith 0-diagonal in S H . (cid:3) Corollary 3.3.
Let x ∈ { l, q, qa, qc, vect } . If p ( x, y | v, w ) ∈ Q sx ( n, m ) isa winning x-strategy, then the map φ p : M n → M m is CP, φ p ( S G ) ⊆ S H and φ p is trace-preserving on S G . We say that the correlation p ( x, y | v, w ) implements the quantum x-homomorphism. Example 3.4.
Suppose we have a graph homomorphism G → H givenby f : V ( G ) → V ( H ) . If we let
Ω = { t } be a one point probability spaceand define Alice and Bob’s random variables f v , g w : Ω → V ( H ) by f v ( t ) = f ( v ) , g w ( t ) = f ( w ) as in the proof of Theorem 2.1, then we obtain p ( x, y | v, w ) ∈ Q sl ( n, m ) with p ( x, y | v, w ) = P rob ( f v = x, g w = y ) = ( x = f ( v ) , y = f ( w )0 else . The corresponding CP map satisfies φ p ( E v,w ) = E f ( v ) ,f ( w ) . We now wish to turn our attention to the composition of quantum graphhomomorphisms. First we need a preliminary result.
Proposition 3.5.
Let x ∈ { l, q, qa, qc, vect } , let p ( x, y | v, w ) ∈ Q x ( n, m ) andlet q ( a, b | x, y ) ∈ Q x ( m, l ) . Then r ( a, b | v, w ) := X x,y q ( a, b | x, y ) p ( x, y | v, w ) ∈ Q x ( n, l ) . Moreover, if p and q are synchronous, then r is synchronous. C. M. ORTIZ AND V. I. PAULSEN
Proof.
First we show the synchronous condition is met by r . Suppose that v = w and a = b. Since p is synchronous, all the terms p ( x, y | v, v ) vanishunless x = y. Thus, r ( a, b | v, v ) = P x q ( a, b | x, x ) p ( x, x | v, v ) . But because q is synchronous, each q ( a, b | x, x ) = 0 . Hence, if a = b, then r ( a, b | v, v ) = 0 . The cases when x = l, q, qa, qc are shown in [15, Lemma 6.5]Finally we tackle the case when x = vect. In this case, we are given Hilbertspaces H , H , unit vectors η ∈ H , η ∈ H , and vectors h v,x , k w,y ∈ H , f x,a , g y,b ∈ H such that: h v,x ⊥ h v,y , k v,x ⊥ k v,y , ∀ x = y, f x,a ⊥ f x,b , g x,a ⊥ g x,b , ∀ a = b, X x h v,x = X x k v,x = η , ∀ v, X a f x,a = X a g x,a = η , ∀ x such that p ( x, y | v, w ) = h h v,x , k w,y i and q ( a, b | x, y ) = h f x,a , g y,b i . We set α v,a = P x h v,x ⊗ f x,a and β w,b = P y k w,y ⊗ g y,b . Now one checksthat these vectors satisfy all the necessary conditions, e.g., α v,a ⊥ α v,b , ∀ a = b and P a α v,a = η ⊗ η , ∀ v, and that h α v,a , β w,b i = X x,y h h v,x , k w,y ih f x,a , g y,b i = r ( a, b | x, y ) . (cid:3) Corollary 3.6.
Let x ∈ { l, q, qa, qc, vect } , let p ( x, y | v, w ) ∈ Q x ( n, m ) ,q ( a, b | x, y ) ∈ Q x ( m, l ) and let r ( a, b | v, w ) = P x,y q ( a, b | x, y ) p ( x, y | v, w ) ∈ Q x ( n, l ) . If φ p : M n → M m , φ q : M m → M l and φ r : M n → M l are thecorresponding linear maps, then φ r = φ q ◦ φ p . The following is now immediate:
Theorem 3.7.
Let x ∈ { l, q, qa, qc, vect } , let G, H and K be graphs on n, m and l vertices, respectively, and assume that G x → H , H x → K . If p ( x, y | v, w ) ∈ Q x ( n, m ) and q ( a, b | x, y ) ∈ Q x ( m, l ) are winning quantum x-strategies for homomorphisms from G to H and H to K , respectively, then r ( a, b | v, w ) = P x,y q ( a, b | x, y ) p ( x, y | v, w ) ∈ Q x ( n, l ) is a winning x-strategyfor a homomorphism from G and K , so that G x → K. In summary,if G x → H and H x → K , then G x → K. C*-algebras and Graph Homomorphisms
We wish to define a C*-algebra A ( G, H ) generated by the relations arisingfrom a winning strategy for the graph homomorphism game.
Definition 4.1.
Let G and H be graphs. A set of projections { E v,x : v ∈ V ( G ) , x ∈ V ( H ) } on a Hilbert space H satisfying the following relations: (1) for each v ∈ V ( G ) , P x E v,x = I H , (2) if ( v, w ) ∈ E ( G ) and ( x, y ) / ∈ E ( H ) then E v,x E w,y = 0 , UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 7 is called a representation of the graph homomorphism game from G to H . If no set of projections on any Hilbert space exists satisfying theserelations, then we say that the graph homomorphism game from G to H is not representable . Definition 4.2.
Let G and H be graphs. If a representation of the graphhomomorphism game exists, then we let A ( G, H ) denote the “universal” C*-algebra generated by such sets of projections. If the graph homomorphismgame from G to H is not representable, then we say that A ( G, H ) does notexist . We write G C ∗ −→ H if and only if A ( G, H ) exists. By “universal” we mean that A ( G, H ) is a unital C*-algebra generatedby projections { e v,x : v ∈ V ( G ) , x ∈ V ( H ) } satisfying(1) for each v ∈ V ( G ) , P x e v,x = 1 , (2) if ( v, w ) ∈ E ( G ) and ( x, y ) / ∈ E ( H ) , then e v,x e w,y = 0 , with the property that for any representation of the graph homomorphismgame on a Hilbert space H by projections { E v,x } satisfying the above rela-tions, there exists a *-homomorphism π : A ( G, H ) → B ( H ) with π ( e v,x ) = E v,x . Here is one result that relates to existence. Let E m be the “empty” graphon m vertices, i.e., the graph with no edges. Proposition 4.3.
Let G be a graph with at least one edge, ( v, w ) ∈ E ( G ) .Then A ( G, E m ) does not exist.Proof. By definition we have that e v,x e w,y = 0 for all x, y. Thus,0 = X x,y e v,x e w,y = (cid:0) X x e v,x (cid:1)(cid:0) X y e w,y (cid:1) = 1 , a contradiction. (cid:3) In [1, Definition 2] another type of graph homomorphism was defined,denoted by G B → H . Briefly, if in our definition of Q vect ( n, m ) we haddropped the requirement that all the inner products be non-negative, thenwe would obtain a larger set of tuples and G B → H if and only if thereexists a p ( x, y | v, w ) in this larger set satisfying the conditions (1) and (2) ofa winning strategy for the graph homomorphism game. Note that in thiscase, since these numbers need not be non-negative, we cannot interpretthem as probabilities. Proposition 4.4. If G C ∗ −→ H or G vect → H , then G B −→ H , as defined in [1] .Proof. The vect case is obvious from the remarks above. Let { E v,x : v ∈ V ( G ) , x ∈ V ( H ) } be a set of projections that yields a representation of thegraph homomorphism game on a Hilbert space H and let h ∈ H be any unitvector.If we set h vx = E v,x h, then set of vectors { h vx } satisfies all the propertiesof the definition of G B −→ H in [1, Definition 2]. (cid:3) C. M. ORTIZ AND V. I. PAULSEN
Remark 4.5.
We do not know necessary and sufficient conditions for A ( G, H ) to exist. In particular, we do not know if G B → H implies G C ∗ −→ H . Proposition 4.6. If G C ∗ → H and H C ∗ → K , then G C ∗ → K .Proof. Since G C ∗ → H and H C ∗ → K , then we know that there exist projections { E v,x } and { F y,a } with v ∈ V ( G ), x, y ∈ V ( H ) and a ∈ V ( K ) on Hilbertspaces H and K , respectively, satisfying (1) and (2). Consider the set ofself-adjoint operators on H ⊗ K defined by G v,a = P x ∈ V ( H ) E v,x ⊗ F x,a for x ∈ V ( G ) and a ∈ V ( K ). Notice that, G v,a G v,a = ( X x E v,x ⊗ F x,a )( X y E v,y ⊗ F y,a ) = X x,y E v,x E v,y ⊗ F x,a F y,a = X x E v,x ⊗ F x,a = G v,a by (2) and the fact that E v,x and F x,a are projections. Thus, each G v,a is aprojection. Furthermore, for each v ∈ V ( G ), X a G v,a = X a X x E v,x ⊗ F x,a = X x E v,x ⊗ ( X a F x,a ) = ( X x E v,x ) ⊗ I K = I H ⊗ I K by (1). Moreover, for each ( v, w ) ∈ E ( G ) and ( a, b ) E ( K ), G v,a G w,b = ( X x E v,x ⊗ F x,a )( X y E w,y ⊗ F y,b ) = X x X y ( E v,x ⊗ F x,a )( E w,y ⊗ F y,b )= X x X y E v,x E w,y ⊗ F x,a F y,b = X x ∼ y E v,x E w,y ⊗ F x,a F y,b = 0by (2). Hence, { G v,a : v ∈ V ( G ) , a ∈ V ( K ) } is a representation of a graphhomomorphism game from G to K . (cid:3) Recall that a trace on a unital C*-algebra B is any state τ such that τ ( ab ) = τ ( ba ) for all a, b ∈ B . Theorem 4.7.
Let G be a graph and let x ∈ { l, q, qa, qc, vect } . (1) G qc → H if and only if there exists a tracial state on A ( G, H ) , (2) if G qc → H , then G C ∗ → H, (3) G q → H if and only if A ( G, H ) has a finite dimensional representa-tion, (4) G → H if and only if A ( G, H ) has an abelian representation.Proof. We have that G qc → H if and only if there exists a winning qc -strategy p ( x, y | v, w ) ∈ Q sqc ( n, m ). By [15] this strategy must be given by a trace onthe algebra generated by Alice’s operators with p ( x, y | v, w ) = τ ( A v,x A w,y ) . Moreover, in the GNS representation, this trace will be faithful.We now wish to show that these operators satisfy the necessary relationsto induce a representation of A ( G, H ) . UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 9
By the original hypotheses, we will have that A v,x A v,y = 0 for x = y. When ( v, w ) ∈ E ( G ) and ( x, y ) E ( H ) , we will have that τ ( A v,x A w,y ) = p ( x, y | v, w ) = 0 and hence, A v,x A w,y = 0 . Thus, Alice’s operators give rise to a representation of A ( G, H ) and com-posing this *-homomorphism with the tracial state on the algebra generatedby Alice’s operators gives the trace on A ( G, H ) . The converse follows bysetting p ( x, y | v, w ) = τ ( A v,x A w,y ) . Clearly, (2) follows from (1).The proof of (3) is similar to the proof of (1). In this case since p ( x, y | v, w ) ∈ Q sq ( n, m ) the operators all live on a finite dimensional space and hence gen-erate a finite dimensional representation.The proof of (4) first uses the fact that G → H if and only if G l → H (2.1). If we let (Ω , λ ) be the corresponding probability space and let f v , g w :Ω → V ( H ) be the random variables for Alice and Bob, respectively, thenthe conditions imply that f v = g v a.e. If we let E v,x denote the charac-teristic function of the set f − ( { x } ) , then it is easily checked that theseprojections in L ∞ (Ω , λ ) satisfy all the conditions needed to give an abelianrepresentation of A ( G, H ) . (cid:3) Note that saying that A ( G, H ) has an abelian representation is equivalentto requiring that it has a one-dimensional representation.We now apply these results to coloring numbers. Let K c denote thecomplete graph on c vertices. Proposition 4.8.
Let x ∈ { l, q, qa, qc, vect } , then χ x ( G ) is the least integer c for which G x → K c . Proof.
Any winning x -strategy for a homomorphism from G to H is a win-ning strategy for a x -coloring. (cid:3) The above result motivates the following definition.
Definition 4.9.
Define χ C ∗ ( G ) to be the least integer c for which G C ∗ → K c . Similarly, define ω C ∗ ( G ) to be the biggest integer c for which K c C ∗ → G. We let ϑ ( G ) denote the Lovasz theta function of a graph G and we let G denote the graph with the same vertex set as G and edges defined by( v, w ) ∈ E ( G ) ⇐⇒ v = w and ( v, w ) / ∈ E ( G ). Proposition 4.10.
Let G be a graph, then ω C ∗ ( G ) ≤ ϑ ( G ) ≤ χ C ∗ ( G ) . Proof.
Let c := χ C ∗ ( G ). If we combine 4.4 with [1, Theorem 6] we knowthat G C ∗ → K c = ⇒ G B → K c ⇐⇒ ϑ ( G ) ≤ ϑ ( K n ) = c. Similarly, if you apply the above proof to K d C ∗ → G , where d := ω C ∗ ( G ), youget the remaining inequality. (cid:3) Remark 4.11.
Since G qc → K c = ⇒ G C ∗ → K c , we have that χ qc ( G ) ≥ χ C ∗ ( G ) , but we don’t know the relation between χ C ∗ ( G ) and χ vect ( G ) . This leads to the following results:
Theorem 4.12.
Let G be a graph. (1) χ ( G ) is the least integer c for which there is an abelian representationof A ( G, K c ) . (2) χ q ( G ) is the least integer c for which A ( G, K c ) has a finite dimen-sional representation. (3) χ qc ( G ) is the least integer c for which A ( G, K c ) has a tracial state. (4) χ C ∗ ( G ) is the least integer c for which A ( G, K c ) exists. Theorem 4.13.
Let G be a graph. (1) The problem of determining if A ( G, K ) has an abelian representa-tion is NP-complete. (2) The problem of determining if A ( G, K ) has a finite dimensionalrepresentation is NP-hard. (3) The problem of determining if A ( G, K c ) has a trace is solvable by asemidefinite programming problem.Proof. We have shown that A ( G, K ) has an abelian representation if andonly if G has a 3-coloring and this latter problem is NP-complete [2].In [8, Theorem 1], it is proven that an NP-complete problem is polyno-mially reducible to determining if χ q ( G ) = 3. Hence, this latter problem isNP-hard.In [15], it is proven that for each n and c there is a spectrahedron S n,c ⊆ R n c such that for each graph G on n vertices there is a linear functional L G : R n c → R with the property that χ qc ( G ) ≤ c if and only if there is apoint p ∈ S n,c with L G ( p ) = 0 . Thus, determining if χ qc ( G ) ≤ c is solvableby a semidefinite programming problem. But we have seen that χ qc ( G ) ≤ c if and only if A ( G, K c ) has a trace. (cid:3) Remark 4.14.
Currently, there are no known algorithms for determiningif χ q ( G ) ≤ , i.e., for determining if A ( G, K ) has a finite dimensionalrepresentation. Remark 4.15.
We do not know the complexity level of determining if A ( G, H ) exists. In particular, we do not know the complexity level of deter-mining if G C ∗ → K , or any algorithm. Remark 4.16. In [1] it is proven that χ vect ( G ) = ⌈ ϑ + ( G ) ⌉ , which is solvableby an SDP. Remark 4.17.
There is a family of finite input, finite output games that arecalled synchronous games [6] , of which the graph homomorphism game is aspecial case. For any synchronous game G we can construct the C ∗ -algebra ofthe game A ( G ) and there are analogues of many of the above theorems. For UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 11 instance, the game will have a winning qc-strategy, q-strategy or l-strategy ifand only if A ( G ) has a trace, finite dimensional, or abelian representation,respectively. Factorization of Graph Homomorphisms
In this section, we show that the CP maps that arise from graph homo-morphisms have a canonical factorization involving A ( G, H ) . Proposition 5.1.
Let G and H be graphs on n and m vertices, respectively.The map Γ : M n → M m ( A ( G, H )) defined on matrix units by Γ( E v,w ) = P x,y E x,y ⊗ e v,x e w,y is CP.Proof. Let E v,x , v ∈ V ( G ) , x ∈ V ( H ) denote the n × m matrix units. Let Z = P w,y E w,y ⊗ e w,y ∈ M n,m ( A ( G, H )) . ThenΓ( X v,w c v,w E v,w ) = Z ∗ (cid:0) c v,w E v,w ⊗ I (cid:1) Z, where I denotes the identity of A ( G, H ) and (cid:0) c v,w E v,w ⊗ I (cid:1) ∈ M n ( A ( G, H )) . (cid:3) Let p ( x, y | v, w ) ∈ Q sqc ( n, m ) be a winning qc -strategy for a graph homo-morphism from G to H . Then there is a tracial state τ : A ( G, H ) → C suchthat p ( x, y | v, w ) = τ ( e v,x e w,y ) and hence φ p factors as φ p = ( id m ⊗ τ ) ◦ Γ , where id m ⊗ τ : M m ( A ( G, H )) → M m . Conversely, if τ : A ( G, H ) → C is any tracial state, then ( id m ⊗ τ ) ◦ Γ = φ p for some winning qc -strategy p ( x, y | v, w ) ∈ Q sqc ( n, m ) . Similarly, this map φ p arises from a winning q -strategy if and only if itarises from a τ that has a finite dimensional GNS representation and froma winning l -strategy if and only if it arises from a τ with an abelian GNSrepresentation.This factorization leads to the following result. Recall that ϑ ( G ) denotesthe Lovasz theta function of a graph and let k φ k cb denote the completelybounded norm of a map. Lemma 5.2.
Let G be a graph on n vertices, let H be a Hilbert space,let P v,w ∈ B ( H ) , ∀ v, w ∈ V ( G ) and regard P = ( P v,w ) as an operator on H ⊗ C n . If (1) P = ( P v,w ) ≥ , (2) P v,v = I H , (3) ( v, w ) ∈ E ( G ) = ⇒ P v,w = 0 , then k P k ≤ ϑ ( G ) . Proof.
Any vector k ∈ H⊗ C n has a unique representation as k = P v k v ⊗ e v , where k v ∈ H and e v ∈ C n denotes the standard orthonormal basis. Set h v = k v / k k v k (with h v = 0 when k v = 0), and λ v = k k v k . Let y = P v λ v e v ∈ C n so that k y k C n = k k k . Set B k = (cid:0) h P v,w h w , h v i (cid:1) ∈ M n = B ( C n ) , so that h P k, k i H⊗ C n = h B k y, y i C n . This observation shows that if for any h v ∈ H , ∀ v ∈ V ( G ) with k h v k = 1we let (cid:0) h P v,w h w , h v i (cid:1) ∈ M n = B ( C n ) , then k P k = sup {k ( h P v,w h w , h v i ) k M n : k h v k = 1 } . Now by the above hypotheses each matrix ( h P v,w h w , h v i ) ≥ , has alldiagonal entries equal to 1 and ( v, w ) ∈ E ( G ) = ⇒ h P v,w h w , h v i = 0 . Thus,by [10], k ( h P v,w h w , h v i ) k ≤ ϑ ( G ) . (cid:3) Proposition 5.3.
Let p ( x, y | v, w ) ∈ Q sqc ( n, m ) be a winning qc -strategy fora graph homomorphism from G to H. Then k φ p k cb ≤ ϑ ( G ) . Proof.
Since id m ⊗ τ is a completely contractive map, we have that k φ p k cb ≤k Γ k cb . Since this map is CP, by [14] we have that k Γ k cb = k Γ( I ) k = k Z ∗ Z k = k ZZ ∗ k . Since e ∗ w,y = e w,y , we have ZZ ∗ = X v,w,x,y ( E v,x ⊗ e v,x )( E w,y ⊗ e w,y ) ∗ = X v,w E v,w ⊗ (cid:0) X x e v,x e w,x (cid:1) . Now if we let p v,w denote the ( v, w )-entry of the above matrix in M n ( A ( G, H )) , then p v,v = P x e v,x = I. When ( v, w ) ∈ E ( G ) , then by Definition 4.1(3), wehave that p v,w = 0 . Hence, by the above lemma, k ZZ ∗ k ≤ ϑ ( G ) . (cid:3) Quantum Cores of Graphs A retract of a graph G is a subgraph H of G such that there existsa graph homomorphism f : G → H , called a retraction with f ( x ) = x for any x ∈ V ( H ). A core is a graph which does not retract to a propersubgraph [7].Note that if f : G → G is an idempotent graph homomorphism and wedefine a graph H by setting V ( H ) = f ( V ( G )) and defining ( x, y ) ∈ E ( H ) ifand only if there exists ( v, w ) ∈ E ( G ) with f ( v ) = x, f ( w ) = y, then H is asubgraph of G and f is a retraction onto H. We denote H by f ( G ) . The following result is central to proofs of the existence of cores of graphs.
Theorem 6.1 ([7]) . Let f be an endomorphism of a graph G . Then thereis an n such that f n is idempotent and a retraction onto R = f n ( G ) . Our goal in this section is to attempt to define a quantum analogue ofthe core using completely positive maps, in particular we will use the abovetheorem as a guiding principle.For A = ( a ij ) ∈ M n , denote || A || = P i,j | a ij | and σ ( A ) = P i,j a ij . Let φ p : M n → M m , φ p ( E vw ) = P x,y p ( x, y | v, w ) E xy , for some p ( x, y | v, w ) ∈ Q svect ( n, m ). Before we continue our discussions on cores we will need thefollowing facts: Lemma 6.2. σ ( φ p ( A )) = σ ( A ) UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 13
Proof.
By linearity it is enough to show the claim for matrix units, σ ( φ p ( E vw )) = X x,y p ( x, y | v, w ) = X x,y h h v,x , h w,y i = h X x h v,x , X y h w,y i = h η, η i = 1 = σ ( E vw ) (cid:3) Lemma 6.3.
Let A = ( a vw ) be a matrix, then || φ p ( A ) || ≤ || A || If the entries of A are non-negative, then k φ p ( A ) k = k A k .Proof. We have || φ p ( A ) || = X x,y | X v,w p ( x, y | v, w ) a v,w | ≤ X v,w | a v,w | ( X x,y p ( x, y | v, w ))= X v,w | a vw | = || A || When the entries of A are all non-negative, the first inequality is an equality. (cid:3) For the next step in our construction we need to recall the concept ofa
Banach generalized limit . A Banach generalized limit is a positive linearfunctional f on ℓ ∞ ( N ), such that: • if ( a k ) ∈ ℓ ∞ ( N ) and lim k a k exists, then f (( a k )) = lim k a k , • if b k = a k +1 , then f (( b k )) = f (( a k )) . The existence and construction of these are presented in [4], along withmany of their other properties. Often a Banach generalized limit functionalis written as glim .Now fix a Banach generalized limit glim , assume that n = m , and that φ p : M n → M n , φ p ( E vw ) = P x,y p ( x, y | v, w ) E xy , for some p ( x, y | v, w ) ∈ Q sqc ( n, n ). Fix a matrix A ∈ M n and set a x,y ( k ) = h φ kp ( A ) e y , e x i so that φ kp ( A ) = P x,y a x,y ( k ) E x,y . By Lemma 6.3, for every pair, ( x, y ) thesequence ( a x,y ( k )) ∈ ℓ ∞ ( N ) . We define a map, ψ p : M n → M n by setting ψ p ( A ) = X x,y glim (( a x,y ( k ))) E x,y . Alternatively, we can write this as ψ p ( A ) = ( id n ⊗ glim ) φ kp ( A ) . Proposition 6.4.
Let (cid:0) p ( x, y | v, w ) (cid:1) ∈ Q svect ( n, n ) and let ψ p : M n → M n bethe map obtained as above via some Banach generalized limit, glim. Then: (1) ψ p is CP, (2) σ ( ψ p ( A )) = σ ( A ) for all A ∈ M n , (3) k ψ p ( A ) k ≤ k A k , (4) ψ p ◦ φ p = φ p ◦ ψ p = ψ p , (5) ψ p ◦ ψ p = ψ p . Proof.
The first two properties follow from the linearity of the glim func-tional. For example, if A = ( a x,y ) and h = ( h , ..., h n ) ∈ C n , then h ψ p ( A ) h, h i = X x,y glim (( a x,y ( k ))) h y h x = glim (cid:0) X x,y a x,y ( k ) h y h x (cid:1) = glim (cid:0) h φ kp ( A ) h, h i (cid:1) If A ≥
0, then φ k ( A ) ≥ k, and so is the above function of k . Since glim is a positive linear functional, we find A ≥ h ψ p ( A ) h, h i ≥ , for all h. This shows that ψ p is a positive map. The proof that it is CP issimilar, as is the proof that it preserves σ. The proof of the third property is similar to the proof of Lemma 6.3.For the next claim, we have that ψ p ( φ p ( A )) = ( id ⊗ glim )( φ k +1 p ( A )) = ( id ⊗ glim )( φ kp ( A )) = ψ p ( A ) . If we set ψ p ( A ) = P v,w b v,w E v,w , with b v,w = glim ( a v,w ( k )) , then φ p ( ψ p ( A )) = X x,y,v,w p ( x, y | v, w ) b v,w E x,y = X x,y glim (cid:0) X v,w p ( x, y | v, w ) a v,w ( k ) (cid:1) E x,y = X x,y glim (cid:0) a x,y ( k +1) (cid:1) E x,y = ψ p ( A )Finally, to see the last claim, we have that ψ p ( ψ p ( A )) = ( id ⊗ glim )( φ kp ( ψ p ( A ))) = ( id ⊗ glim )( ψ p ( A )) = ψ p ( A ) , since the glim of a constant sequence is equal to the constant. (cid:3) Theorem 6.5.
Let G be a graph on n vertices, let x ∈ { l, qa, qc, vect } andlet p ( x, y | v, w ) ∈ Q sx ( n, n ) be a winning x -strategy implementing a quantumgraph x -homomorphism from G to G . Set p ( x, y | v, w ) = p ( x, y | v, w ) andrecursively define p k +1 ( x, y | v, w ) = X a,b p ( x, y | a, b ) p k ( a, b | v, w ) . If we set r ( x, y | v, w ) = glim (cid:0) p k ( x, y | v, w ) (cid:1) , then r ( x, y | v, w ) ∈ Q sx ( n, n ) isa winning x -strategy implementing a graph x -homomorphism from G to G such that: (1) ψ p = φ r , (2) r ( x, y | v, w ) = P a,b r ( x, y | a, b ) r ( a, b | v, w ) . UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 15
Proof.
By Theorem 3.7, φ kp = φ p k , and p k is a winning x -strategy for a graph x -homomorphism from G to G . Thus, ψ p ( E v,w ) = ( id ⊗ glim )( φ kp ( E v,w )) = ( id ⊗ glim )( φ p k ( E v,w ))= X x,y glim (cid:0) p k ( x, y | v, w ) (cid:1) E x,y = φ r ( E v,w ) . Thus, (1) follows.Since φ r ◦ φ r = ψ p ◦ ψ p = ψ p = φ r , (2) follows from Proposition 3.6.Finally, if a bounded sequence of matrices A k = (cid:0) a v,w ( k ) (cid:1) ∈ M n all belongto a closed set, then it is not hard to see that A = (cid:0) glim ( a v,w ( k )) (cid:1) alsobelongs to the same closed set. Thus, since (cid:0) p k ( x, y | v, w ) (cid:1) is in the closedset Q sx ( n, n ) for all k, we have that (cid:0) r ( x, y | v, w ) (cid:1) ∈ Q sx ( n, n ) . Also, since p k is a winning x -strategy for a graph x -homomorphism of G , for all k , we havethat for all k , (cid:0) p k ( x, y | v, w ) (cid:1) is zero in certain entries. Since the glim ofthe 0 sequence is again 0, we will have that (cid:0) r ( x, y | v, w ) (cid:1) is also 0 in theseentries. Hence, r is a winning x -strategy for a graph x -homomorphism. (cid:3) Remark 6.6.
In the case that p is a winning q -strategy implementing agraph q -homomorphism, all we can say about r is that it is a winning qa -strategy implementing a graph qa -homomorphism, since we do not know ifthe set Q sq ( n, n ) is closed. There is a natural partial order on idempotent CP maps on M n . Giventwo idempotent maps φ, ψ : M n → M n we set ψ ≤ φ if and only if ψ ◦ φ = φ ◦ ψ = ψ. Theorem 6.7.
Let x ∈ { l, qa, qc, vect } , then there exists r ( x, y | v, w ) ∈ Q sx ( n, n ) implementing a quantum x -homomorphism, such that φ r : M n → M n is idempotent and is minimal in the partial order on idempotent mapsof the form φ p implemented by a quantum x -homomorphism of G .Proof. Quantum x -homomorphisms always exist, since the identity map on G belongs to the l -homomorphisms, which is the smallest set. By the lasttheorem we see that beginning with any correlation p implementing a quan-tum x -homomorphism, there exists a correlation r implementing a quantum x -homomorphism with φ r idempotent.It remains to show the minimality claim. We will invoke Zorn’s lemmaand show that every totally ordered set of such correlations has a lowerbound. Let (cid:8) p t ( x, y | v, w ) : t ∈ T (cid:9) ⊂ Q sx ( n, n ) with T a totally ordered set,where all p t ( x, y | v, w ) implement a quantum x -homomorphisms, with φ p t idempotent, and φ p t ≤ φ p s , whenever s ≤ t. These define a net in the compact set Q sx ( n, n ) and so we may choose aconvergent subnet. Now it is easily checked that if we define p ( x, y | v, w )to be the limit point of this subnet, then it implements a quantum x -homomorphism, φ p is idempotent, and φ p ≤ φ p t for all t ∈ T. (cid:3) Remark 6.8.
It is important to note that we are not claiming that φ r canbe chosen minimal among all idempotent CP maps, just minimal among allsuch maps that implement a quantum x -homomorphism of G . Definition 6.9.
Let x ∈ { l, qa, qc, vect } , then a quantum x -core for G is any r ( x, y | v, w ) ∈ Q sx ( n, n ) that implements a quantum x -homomorphismsuch that φ r is idempotent and minimal among all φ p implemented by aquantum x -homomorphism of G. Appendix: Background Material
Let I and O be two finite sets called the input set and output set, respec-tively. Definition 6.10.
A set of real numbers p ( x, y | v, w ) , v, w ∈ I, x, y ∈ O iscalled a local or classical correlation if there is a probability space (Ω , µ ) and random variables, f v , g w : Ω → O for each v, w ∈ I such that p ( x, y | v, w ) = µ ( { ω | f v ( ω ) = x, g w ( ω ) = y } )To motivate this definition, imagine that there are two people, Alice andBob, when Alice receives input v she uses the random variable f v and whenBob receives input w he uses the random variable g w . In this case p ( x, y | v, w )represents the probability of getting outcomes x and y respectively, giventhat they received inputs v and w , respectively. Definition 6.11.
Given a Hilbert space H , a collection { E x : x ∈ O } ofbounded operators on H is called a projection valued measure(PVM) on H , provided that each E x is an orthogonal projection and P x ∈ O E x = I H . The set is called a positive operator valued measure(POVM) on H , provided that each E x is a positive semidefinite operator on H and P x ∈ O E x = I H . Definition 6.12.
A density p is called a quantum correlation if it arisesas follows:Suppose Alice and Bob have finite dimensional Hilbert spaces H A , H B and for each input v ∈ I Alice has PVMs { F v,x } x ∈ O on H A and for eachinput w ∈ I Bob has PVMs { G w,y } y ∈ O on H B and they share a state ψ ∈H A ⊗ H B , then p ( x, y | v, w ) = h F v,x ⊗ G w,y ψ, ψ i This is the probability of getting outcomes x, y given that they conductedexperiments v, w.
Definition 6.13.
A density p is called a quantum commuting correla-tion if there is a single Hilbert space H , such that for each v ∈ I Alice has
UANTUM GRAPH HOMOMORPHISMS VIA OPERATOR SYSTEMS 17
PVMs { F v,x } x ∈ O on H and for each w ∈ I Bob has PVMs { G w,y } y ∈ O on H satisfying F v,x G w,y = G w,y F v,x , ∀ v, w, x, y and p ( x, y | v, w ) = h F v,x G w,y ψ, ψ i where ψ ∈ H is a shared state. Remark 6.14.
Suppose we have projection valued measures { P v,i } mi =1 and { Q w,j } mj =1 on H as in 6.13. Set X v,i = P v,i k , Y w,j = Q w,j k . Then(1) X v,i ⊥ X v,j for every i = j .(2) Y w,i ⊥ Y w,j for every i = j .(3) P i X v,i = P j Y w,j for every v, w and k P i X v,i k = 1 . (4) hX v,i , Y w,j i ≥ since hX v,i , Y w,j i = D P v,i , Q w,j E = h Q w,j P v,i k, Q w,j P v,i k i = k Q w,j P v,i k k ≥ where the second equality results from the fact that Q w,j and P v,i are commuting projections. Definition 6.15.
A density p is called a vectorial correlation if p ( i, j | v, w ) = hX v,i , Y w,j i for sets of vectors {X v,i } , {Y w,j } satisfying (1) through (4) in6.14. Letting n := | I | and m := | O | , we let: • Q loc ( n, m ) denote the set of all densities that are local correlations. • Q q ( n, m ) denote the set of all densities that are quantum correla-tions. • Set Q qa ( n, m ) := Q q ( n, m ), the closure of Q q ( n, m ). • Q qc ( n, m ) denote the set of all densities that are quantum commutingcorrelations. • Q vect ( n, m ) denote the set of all densities that are vectorial correla-tions. • For x ∈ { loc, q, qa, qc } , we let Q sx ( n, m ) denote the set of synchronouscorrelations in Q x ( n, m ). Remark 6.16.
Results in [16] and [15] show that the possibly larger sets thatone obtains by using the larger collection of all POVMs in the definitionsof Q q , Q qa and Q qc in place of PVMs, yield the same sets. These equalitiesessentially follow from Stinespring’s theorem. Also, while earlier versionsof [15] use the notation Q t ( n, m ) , which we have adopted here, this notationwas changed to C t ( n, m ) in later versions. Remark 6.17.
In addition to Q vect ( n, m ) being a natural relaxation of theother sets, determining membership in this set reduces to standard prob-lems in linear algebra. Another important reason for studying Q vect ( n, m ) is Tsirelson’s 1980 [9] attempted proof that Q q ( n, m ) = Q qc ( n, m ) . He at-tempted to show that Q q ( n, m ) = Q vect ( n, m ) , from which the other equality would follow, by starting with vectors satisfying (1) through (4) and attempt-ing to build projections { P v,i } , { Q w,j } on finite dimensional Hilbert space,and a vector k such that X v,i = P v,i k and Y w,j = Q w,j k commuted. In [1] agraph on 15 vertices is constructed for which χ q ( G ) = 8 = χ vect ( G ) = 7 , giv-ing a definitive proof that Q q (15 , = Q vect (15 , , hence showing that forsome such set of vectors, one cannot construct corresponding projections.Later, for this same graph [15] proved that χ qc ( G ) = 8 = χ vect ( G ) showingthat Q qc (15 , = Q vect (15 , . Here are some further facts and open problems about these sets that showtheir importance. • Q loc ( n, m ) ⊆ Q q ( n, m ) ⊆ Q qa ( n, m ) ⊆ Q qc ( n, m ) ⊆ Q vect ( n, m ). • Q loc ( n, m ), Q qa ( n, m ), Q qc ( n, m ), and Q vect ( n, m ) are closed. • Bounded entanglement conjecture: Q q ( n, m ) = Q qa ( n, m ) ∀ n, m ,i.e., is Q q ( n, m ) closed. • Tsirelson conjecture [9]: Q q ( n, m ) = Q qc ( n, m ) ∀ n, m . • Ozawa [13] proved that Connes’ embedding conjecture [5] is true ifand only if Q qa ( n, m ) = Q qc ( n, m ) , ∀ n, m . • Paulsen and Dykema [6] proved that Connes’ embedding conjectureis true if and only if Q sq ( n, m ) = Q sqc ( n, m ) , ∀ n, m . • The synchronous approximation conjecture: Q sq ( n, m ) = Q sqa ( n, m ) ∀ n, m . • If Tsirelson’s conjecture is true, then the Connes’ embedding con-jecture and the bounded entanglement conjecture are true. • If Connes’ embedding conjecture is true, then the synchronous ap-proximation conjecture is true.
Acknowledgements
The authors wish to thank S. Severini and D. Stahlke for several valu-able comments that led to improvements in the paper. This research wassupported in part by NSF grant DMS-1101231.
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Pacific Northwest National Laboratory, Richland, WA 99352, U.S.A.
E-mail address : [email protected] IQC and Department of Pure Mathematics, University of Waterloo, Wa-terloo, Ontario, N2L 3G1, Canada
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