The quantum-to-classical graph homomorphism game
aa r X i v : . [ m a t h . OA ] S e p THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME
MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS
Abstract.
Motivated by non-local games and quantum coloring problems, we introduce a graph homo-morphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum-classical game”–that is, a non-local game of two players involving quantum questions and classical answers.This game generalizes the graph homomorphism game between classical graphs. We show that winningstrategies in the various quantum models for the game directly generalize the notion of non-commutativegraph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context thatgeneralizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen andM. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yield-ing the surprising fact that the algebra of the 4-coloring game for a quantum graph is always non-trivial,extending a result of [22].
In recent years, the theory of non-local games has risen to a level of great prominence in quantuminformation theory and related parts of physics and mathematics. In quantum information theory, non-local games provide a convenient framework in which one can exhibit the advantages of using quantumentanglement as a resource to accomplish certain tasks. In physics, non-local games are intimately tiedto the study Tsirelson’s correlation sets and Bell’s work on local hidden variable models [48]. Withinmathematics, the theory of non-local games has led to some spectacular developments in the field of operatoralgebras. Most notable here is the work of Junge-Navascues-Palazuelos-Perez-Garcia-Scholz-Werner [24], T.Fritz [16] and N. Ozawa [35] connecting the Connes-Kirchberg conjecture to Tsirelson’s correlation sets inquantum information. Very recently, Ji-Natarajan-Vidick-Wright-Yuen [23] used non-local games to providea counterexample to the Connes-Kirchberg conjecture. Another recent and quite remarkable application ofnon-local games in mathematics is the work of Manˇcinska-Roberson [33] which uses a non-local game, calledthe graph isomorphism game, to provide a quantum interpretation of pairs of graphs that admit the samenumber of homomorphisms from planar graphs.The general setup of a (classical input, classical output) two player non-local game is given in termsof a tuple G = ( I, O, V ), where I and O are finite sets and V : O × O × I × I → { , } is a predicatefunction which determines the rules of the game. The game is played by two cooperating players, Alice andBob, and a verifier (Referee). Each round proceeds by the verifier (randomly) selecting a pair of questions( x, y ) ∈ I × I and sending x to Alice and y to Bob. Alice and Bob then respond with answers ( a, b ) ∈ O × O .The verifier declares the round won if V ( a, b, x, y ) = 1 and declares it lost if V ( a, b, x, y ) = 0. The term non-local refers to the fact that during each round, Alice and Bob are spatially separated and are unableto communicate; neither Alice nor Bob knows which questions/answers the other received/returned. Thisnon-locality of G makes winning each round of the game (with high probability) generally very difficult. Itis in these scenarios that “quantum strategies” (which make use of some shared entangled resource betweenAlice and Bob) can allow the players to drastically improve their performance by better correlating theirbehaviors.In this paper, we are mainly interested in a non-local game called the graph homomorphism gameand certain extensions of it. The graph homomorphism game is a well studied example of a non-local game[22, 32, 39, 47]. This game is described by a pair of finite simple graphs G, H , with input set I = V ( G ) (thevertex set of G ) and output set O = V ( H ). The goal of Alice and Bob in this game is to convince thereferee that there exists a homomorphism G → H . In particular, the rules of the game are determined bythe following two requirements:(1) Alice and Bob’s answers must be synchronous , meaning that if they receive the same vertex x ∈ V ( G ),then they must return the same vertex a ∈ V ( H ).(2) If the referee supplies an edge ( x, y ) ∈ E ( G ) to Alice and Bob, then they must respond with an edge( a, b ) ∈ E ( H ). The graph homomorphism game (in particular, the special case of the graph coloring game) has led to manydevelopments in the operator algebraic aspects of non-local games. A particular notion of interest here is thenotion of a synchronous non-local game and synchronous strategies for such games [22]. Winning strategiesfor synchronous games turn out to be completely described in terms of traces on a certain ∗ -algebra associatedto the game, bringing to bear many powerful operator algebraic techniques in the theory of non-local games.Within information theory (both quantum and classical) graph theory plays a central role, appearingquite naturally in the theory of zero-error communication in the form of confusability graphs of noisy com-munications channels. If the channel at hand is classical, the confusability graph is a finite simple graph onthe input alphabet whose edges indicate which letters can be confused after passing through the channel.If the channel is genuinely quantum, it was shown in [13] that the role of the confusability graph in thiscase must be played by more general structure called a quantum graph . Quantum graphs are an operatorspace generalization of classical graphs, which have emerged in different disguises in operator systems theory,non-commutative topology and quantum information theory. Traditionally, a quantum graph is viewed asan operator system that serves as a quantum generalization of the adjacency matrix. It was first introducedin [13] for studying a zero-error channel capacity problem and arose independently in the study of quantumrelations [51, 52] around the same time. An alternate approach was used in [34] to define a quantum graphusing a quantum adjacency matrix acting on a finite-dimensional C ∗ -algebra which plays the role of functionson the vertex set. Both these perspectives are shown to be essentially equivalent [34, Theorem 7.7] and offerdifferent advantages and perspectives.In the present work, motivated by several recent works extending the notion of chromatic number fromgraphs to the setting of quantum graphs [29, 38, 39, 44], our aim is to develop a non-local game that capturesthe coloring problem for quantum graphs. To this end, we study homomorphisms from quantum graphs toclassical graphs, using a non-local game with quantum inputs and classical outputs. The inputs are quantuminputs, in the sense that the referee initializes the state space C n ⊗ C n , where Alice has access to the left copyand Bob has access to the right copy of C n . Alice and Bob are allowed to share a(n entanglement) resourcespace H in some prepared state ψ . After receiving the input ϕ on C n ⊗ C n , they can perform measurementson the triple tensor product C n ⊗ H ⊗ C n , and respond to the referee with classical outputs based on theirmeasurements.The winning strategies for this game give rise to a notion of quantum graph homomorphism thatconsolidates and generalizes several notions of quantum graph homomorphism in the literature [34, 44, 52].We also construct a game ∗ -algebra for this and show that this game algebra extends the game algebra forgraph homomorphisms given in [22]. Further, we consider the coloring game for quantum graphs and studythe associated chromatic numbers. We show interesting extensions of classical results in this framework.In particular, we use unitary error basis tools to show that every quantum graph admits a finite chromaticnumber in the quantum model (but not necessarily the local model), and the fact that every quantum graphis 4-colorable in the algebraic model.The organization of the paper is as follows: Section 1 develops the general theory of quantum input-classical output correlations and the various quantum models which give rise to such correlations. Herewe also introduce and study the universal operator system Q n,c associated to such correlations and its C ∗ -envelope. In section 2, we introduce a generalization of synchronous correlations to our quantum framework.In particular, we establish in this section characterizations of synchronous correlations in terms of tracialstates on C ∗ -algebras, and we also establish an extension of the well-known equality of the quantum andquantum-spatial correlation sets for synchronous correlations, extending a result of [28]. In section 3 weconsider the structure of quantum approximate correlations in our context, extending the result of [28] byidentifying synchronous quantum approximate correlations with the closure of the synchronous quantumcorrelations. In section 4 we define the homomorphism game from quantum graphs to classical graphs andstudy the corresponding winning strategies and game ∗ -algebra. Finally, in section 5, we study the coloringproblem for quantum graphs, demonstrating explicit colorings of all quantum graphs in the q -model with thehelp of some quantum teleportation-like schemes, as well as extending classical results on algebraic coloringsto this framework. Acknowledgements.
The authors are grateful to Marius Junge, Vern Paulsen, Ivan Todorov, LyudmilaTurowska, Nik Weaver, and Andreas Winter for fruitful discussions related to the content of this paper. Weare especially grateful to Ivan Todorov and Lyudmila Turowska, who shared with us an early draft of their
HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 3 preprint [47], which independently obtains some of the results in this paper. MB and PG were partiallysupported by NSF Grant DMS-2000331 and a T3 Grant from Texas A&M University. SH was partiallysupported by an NSERC postdoctoral fellowship.1.
Quantum Input, Classical Output Correlations
In this section, we develop some general theory on non-local games with quantum questions andclassical answers. These have already been used in the two-output context of quantum XOR games [19, 41].To motivate things, first recall that in the classical setup of n classical input, c classical outputtwo-player non-local games, the main objects of study are the bipartite correlation sets C ( n, c ) ⊂ R n c which model the players’ behavior. Namely, any element P = ( p ( a, b | x, y )) ≤ a,b ≤ c ≤ x,y ≤ n ∈ C ( n, c ) specifies theprobability p ( a, b | x, y ) that the players Alice and Bob return answers a and b (respectively), given thatthey received questions x and y (respectively). The correlations (behaviors) P ∈ C ( n, c ) that are physicallyrelevant are the ones that can be realized by a (quantum) strategy , that is, by Alice and Bob performing jointmeasurements on a quantum mechanical system prepared in some initial state. Mathematically, a quantumstrategy amounts to the data of two finite-dimensional Hilbert spaces H A and H B , and families of positiveoperator-valued measure (POVMs) { P x , ..., P xc } on H A , { Q y , ..., Q yc } on H B , and a state χ ∈ H A ⊗ H B .From this data, one obtains a correlation P ∈ C ( n, c ) via the formula p ( a, b | x, y ) = h χ | P xa ⊗ Q yb | χ i . The subset of all correlations obtainable from quantum strategies as above is dented by C q ( n, c ). In asimilar manner, one can define other classes of correlations (local, quantum spatial, quantum approximate,quantum commuting) that are built from of the corresponding classes of strategies. (See, for example, [28]for a review of all of these models.)Our goal now is to develop the analogous notion of the correlation set C ( n, c ) and its various subclassesarising from quantum strategies. The main idea is quite simple – in order to allow for quantum questions,we replace the question set [ n ] × [ n ] with the set of quantum states on the bipartite system C n ⊗ C n . In thefollowing, our approach is somewhat backwards, in that we first define the different strategies associated toa two-player scenario with quantum questions (on C n ⊗ C n ) and classical answers in { , , ..., c } . Afterwards,we consider the associated correlations. For our purposes it is easiest to begin with the quantum (i.e.,finite-dimensional tensor product) strategies.A quantum strategy , or a q -strategy , is given by two finite-dimensional Hilbert spaces H A and H B , a POVM { P , ..., P c } on C n ⊗ H A , a POVM { Q , ..., Q c } on H B ⊗ C n , and a state χ ∈ H A ⊗ H B .A quantum spatial strategy , or a qs -strategy , is given in the same way as a q -strategy, exceptthat we no longer assume that H A and H B are finite-dimensional.A quantum commuting strategy , or a qc -strategy , is given by a single Hilbert space H , a POVM { P , ..., P c } on C n ⊗ H , a POVM { Q , ..., Q c } on H ⊗ C n , and a state χ ∈ H , with the property that( P a ⊗ I n )( I n ⊗ Q b ) = ( I n ⊗ Q b )( P a ⊗ I n ) for all a, b . Remark 1.1.
It is helpful to understand the above commutation condition in terms of block matrices.For ≤ a ≤ c , one may write P a = ( P a,ij ) ∈ M n ( B ( H )) with P a,ij ∈ B ( H ) . Similarly, we may write Q b = ( Q b,kℓ ) ∈ M n ( B ( H )) with Q b,kℓ ∈ B ( H ) . With this in mind, the above commutation relation is easilyseen to be equivalent to the requirement that [ P a,ij , Q b,kl ] = 0 ∈ B ( H ) for each a, b, i, j, k, l . (See, e.g., [8] and [21] .) Finally, in view of the above remark, we define a local strategy , or a classical strategy , as a quan-tum commuting strategy with the property that the set of operators P a,ij and Q b,kℓ generate a commutative C ∗ -algebra.Suppose now that the referee initializes C n ⊗ C n in the state ϕ . For a quantum strategy, the probabilitythat Alice outputs a and Bob outputs b is given by p ( a, b | ϕ ) = h ( P a ⊗ Q b )( ϕ ⊙ χ ) , ϕ ⊙ χ i , where by ϕ ⊙ χ we mean the (permuted) state in C n ⊗ ( H A ⊗H B ) ⊗ C n rather than on C n ⊗ C n ⊗ ( H A ⊗H B ). Fora quantum commuting strategy, we simply replace H A ⊗ H B with H and ( P a ⊗ Q b ) with ( P a ⊗ I n )( I n ⊗ Q b ).We note that this definition of the probability of outputs can easily be extended to other (e.g. mixed)states in C n ⊗ C n that may not be included in the definition of the game. This is because the probabilities MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS corresponding to Alice and Bob’s strategy are encoded entirely in the correlation associated to their strategy.The correlation associated to the strategy ( P , ..., P c , Q , ..., Q c , χ ) with n -dimensional quantum inputs and c classical outputs is given by the tuple X := ( X ( a,b )( i,j ) , ( k,ℓ ) ) = (( h ( P a,ij ⊗ Q b,kℓ ) χ, χ i ) i,j,k,ℓ ) a,b ∈ ( M n ⊗ M n ) c , in the case when the entanglement resource space for Alice and Bob is of the form H A ⊗ H B . In the casewhen their resource space is a single Hilbert space H , we replace P a,ij ⊗ Q b,kℓ with P a,ij Q b,kℓ .We will let Q q ( n, c ) be the set of all correlations of this form that arise from quantum strategies. Inother words, Q q ( n, c ) = { ( h ( P a,ij ⊗ Q b,kℓ ) χ, χ i ) ≤ i,j,k,ℓ ≤ n, ≤ a,b ≤ c ⊆ ( M n ⊗ M n ) c , where H A and H B are finite-dimensional Hilbert spaces; P a,ij ∈ B ( H A ) are such that P a = ( P a,ij ) ∈ M n ( B ( H A )) is positive with P ca =1 P a = I ; Q b,kℓ ∈ B ( H B ) are such that Q b = ( Q b,kℓ ) ∈ M n ( B ( H B )) arepositive with P cb =1 Q b = I , and χ ∈ H A ⊗ H B is a state.Similarly, we will let Q qs ( n, c ) be the set of all quantum spatial correlations (where H A and H B maynot be finite-dimensional), and we let Q qc ( n, c ) be the set of all quantum commuting correlations of the aboveform (where we replace the tensor product space H A ⊗ H B with a single Hilbert space H , and P a,ij ⊗ Q b,kℓ with P a,ij Q b,kℓ ). Keeping the analogy with the sets C t ( n, k ) corresponding to classical inputs, we will alsodefine Q qa ( n, c ) as the closure of Q q ( n, c ) in the norm topology. Lastly, we define Q loc ( n, c ) as the set of allquantum commuting correlations where C ∗ ( { P a,ij , Q b,kℓ : 1 ≤ a, b ≤ c, ≤ i, j, k, ℓ ≤ n } ) is a commutative C ∗ -algebra.Since each of the correlation sets above are defined in terms of POVMs, an argument involving directsums shows that Q t ( n, c ) is convex for all t ∈ { loc, q, qs, qa, qc } . Moreover, Q qa ( n, c ) is closed (by definition)and an application of Theorem 1.12 shows that Q qc ( n, c ) is closed. Similarly, Proposition 1.13 shows that Q loc ( n, c ) is closed.Next, we define a universal operator system that encodes the above correlation sets. Define Q n,c as the universal operator system generated by c sets of n entries q a,ij with the property that the matrix Q a = ( q a,ij ) is positive in M n ( Q n,c ) for each 1 ≤ a ≤ c and P ca =1 Q a = I n . The correlations above aredirectly related to states on certain operator system tensor products of Q n,c . For these results, we will usesome facts about Q n,c and its C ∗ -envelope. For convenience, we define P n,c to be the universal unital C ∗ -algebra generated by c sets of n entries p a,ij such that P a = ( p a,ij ) is an orthogonal projection in M n ( P n,c )for each 1 ≤ a ≤ c and P ca =1 P a = I n . Similarly, we define B n,c as the universal unital C ∗ -algebra generatedby n entries u ij with the property that U = ( u ij ) ∈ M n ( B n,c ) is a unitary of order c . The latter algebra isan obvious quotient of the Brown algebra B n , which is the universal C ∗ -algebra generated by the entries ofan n × n unitary. The algebra B n first appeared in [3].Our goal is to show a quantum-classical version of the disambiguation theorems; that is, we will showthat all correlations in Q t ( n, c ) can be achieved using projection-valued measures (PVMs) instead of themore general notion of POVMs. First, we will show that POVMs in our context dilate to PVMs. Proposition 1.2.
Let H be a Hilbert space, and let { Q a } ca =1 be a POVM in B ( H ) . Then there is a PVM { P a } ca =1 in M c +1 ( B ( H )) such that, if E is the first diagonal matrix unit in M c +1 , then ( E ⊗ I H ) P a ( E ⊗ I H ) = Q a for all ≤ a ≤ c .Proof. We define V = Q ... Q c ∈ M c, ( B ( H )). Then V is an isometry, so U = (cid:18) V √ I − V V ∗ − V ∗ (cid:19) ∈ M c +1 ( B ( H ))is a unitary. Define P a = U ∗ ( E aa ⊗ I H ) U for 1 ≤ a ≤ c −
1, and define P c = U ∗ (( E cc + E c +1 ,c +1 ) ⊗ I H ) U .Then { P a } ca =1 is a PVM in M c +1 ( B ( H )). Write U = ( U kℓ ) c +1 k,ℓ =1 where each U kℓ ∈ B ( H ). The (1 ,
1) entry of P a is given by ( P a ) = U ∗ a U a = ( Q a )( Q a ) = Q a , as desired. (cid:3) HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 5
As a result of Proposition 1.2, we obtain the desired dilation property for POVMs over M n ( B ( H )). Proposition 1.3.
Let H be a Hilbert space, and let q a,ij ∈ B ( H ) for ≤ i, j ≤ n and ≤ a ≤ c be suchthat { Q a } ca =1 is a POVM in M n ( B ( H )) , where Q a = ( q a,ij ) . Let V : H → H ( c +1) be the isometry sending H to the first direct summand of H ( c +1) . Then there are operators p a,ij ∈ M c +1 ( B ( H )) such that { P a } ca =1 is a PVM in M n ( M c +1 ( B ( H ))) , where P a = ( p a,ij ) , and V ∗ p a,ij V = q a,ij for all ≤ i, j ≤ n and ≤ a ≤ c .Proof. We can regard { Q a } ca =1 as a POVM in M n ( B ( H )). By Proposition 1.2, there is a PVM { S a } ca =1 in M c +1 ( M n ( B ( H ))) such that the (1 ,
1) entry of S a is Q a . Performing a canonical shuffle M c +1 ( M n ( B ( H ))) ≃ M n ( M c +1 ( B ( H ))) [36, p. 97] on each S a , we obtain operators p a,ij ∈ M c +1 ( B ( H )) such that the (1 , p a,ij is q a,ij , and P a = ( p a,ij ) ∈ M n ( M c +1 ( B ( H ))) are projections with P ca =1 P a = I , completing theproof. (cid:3) Remark 1.4.
In the case of classical inputs and outputs, one would consider n POVMs in B ( H ) with c outputs each. It is a standard fact that such systems of POVMs can be dilated to a system of n PVMs with c outputs on a larger Hilbert space, which remains finite-dimensional whenever H is finite-dimensional.Alternatively, one can consider n POVMs { P a,x } ca =1 for ≤ x ≤ n on H as a single POVM on C n ⊗ H by setting Q a = P a, ⊕ · · · ⊕ P a,n . Then one applies Proposition 1.3 to obtain a single PVM in M n ( H ⊗ C c +1 ) ; however, the projections may no longer be block-diagonal, so they may not induce a familyof n PVMs in B ( H ⊗ C c +1 ) . In the case that n = 1 , one can dilate a POVM with c outputs in B ( H ) to aPVM with c outputs in B ( H ⊗ C c ) , which is more optimal than Proposition 1.3. On the other hand, as soonas n ≥ , the dilation of Proposition 1.3 will be more optimal, since the general dilation of n POVMs to n PVMs requires an inductive argument.
In the following, we let C ∗ env ( S ) be the C ∗ -envelope of an operator system, first shown to exist by M.Hamana [18]. Proposition 1.5.
Let n, c ∈ N . (1) C ∗ env ( Q n,c ) is canonically ∗ -isomorphic to the universal C ∗ -algebra P n,c . (2) There is a ∗ -isomorphism P n,c ≃ B n,c given by the map c X a =1 ω a P a ← U, where ω is a primitive c -th root of unity.Proof. We only prove the first claim; the second claim is analogous to the fact that C ∗ ( Z c ) ≃ ℓ c ∞ (see, forexample, [24], [16] or [35]). Let p a,ij be the canonical generators of P n,c , for 1 ≤ i, j ≤ n and 1 ≤ a ≤ c .Since P a = ( p a,ij ) is a projection in M n ( P n,c ), it is positive. Since P ca =1 P a = I n , there is a ucp map ϕ : Q n,c → P n,c such that ϕ ( q a,ij ) = p a,ij . If we represent Q n,c ⊆ B ( H ) for some Hilbert space H , then byProposition 1.3, there is a unital ∗ -homomorphism π : P n,c → M c +1 ( B ( H )) such that compressing to thefirst coordinate yields the map p a,ij → q a,ij . Hence, ϕ is a complete order isomorphism. This shows that P n,c is a C ∗ -cover for Q n,c , in the sense that there is a unital complete order embedding of Q n,c into P n,c ,whose range generates P n,c as a C ∗ -algebra.By the universal property of the C ∗ -envelope [18], there is a unique, surjective unital ∗ -homomorphism ρ : P n,c → C ∗ env ( Q n,c ) such that ρ ( p a,ij ) = q a,ij for all 1 ≤ i, j ≤ n and 1 ≤ a ≤ c . As each P a is a projectionin P n,c , the matrix Q a = ( q a,ij ) ∈ M n ( C ∗ env ( Q n,c )) is a projection as well. We will show that ρ is injectiveby constructing an inverse. We assume that P n,c is faithfully represented as a C ∗ -algebra of operators ona Hilbert space K . Then the map ϕ : Q n,c → P n,c above extends to a ucp map σ : C ∗ env ( Q n,c ) → B ( K ) byArveson’s extension theorem [1]. We let σ = V ∗ β ( · ) V be a minimal Stinespring representation of σ , where V : K → L is an isometry and β : C ∗ env ( Q n,c ) → B ( L ) is a unital ∗ -homomorphism. With respect to thedecomposition L = K ⊕ K ⊥ , one has β ( q a,ij ) = (cid:18) ϕ ( q a,ij ) ∗∗ ∗ (cid:19) = (cid:18) p a,ij ∗∗ ∗ (cid:19) . Thus, after a shuffle, one may write β ( n ) ( Q a ) = ( β ( q a,ij )) as (cid:18) ϕ ( n ) ( Q a ) ∗∗ ∗ (cid:19) = (cid:18) P a ∗∗ ∗ (cid:19) . MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS As Q a is a projection in M n ( C ∗ env ( Q n,c )), so is β ( n ) ( Q a ) in M n ( B ( L )). But P a is a projection as well, so theoff-diagonal blocks must be 0. Therefore, reversing the shuffle yields β ( q a,ij ) = (cid:18) p a,ij ∗ (cid:19) . Considering β ( q ∗ a,ij q a,ij ) and β ( q a,ij q ∗ a,ij ), it follows that the multiplicative domain of σ contains q a,ij foreach 1 ≤ i, j ≤ n and 1 ≤ a ≤ c ; as these elements generate C ∗ env ( Q n,c ), σ must be a ∗ -homomorphism.Since ρ and σ are mutual inverses on the generators, they must be mutual inverses on the whole algebras.Hence, ρ is injective, so that C ∗ env ( Q n,c ) ≃ P n,c . (cid:3) Combining part (2) of Proposition 1.5 and Proposition 1.3, one can obtain a similar dilation corre-sponding to a block unitary of order c . Indeed, if T = ( T ij ) ∈ M n ( B ( H )) is a contraction that can be writtenas T = P ca =1 ω a Q a , where ω is a primitive c -th root of unity and { Q a } ca =1 is a POVM in M n ( B ( H )), thenone can dilate T to a unitary U = ( U ij ) ∈ M n ( M c +1 ( B ( H ))) of order c , such that the (1 ,
1) block of each U ij is T ij . It is sometimes convenient to use this form of the dilation, rather than the dilation of the POVMto a PVM.We now study some of the structure of P n,c . First, we show that P n,c has the lifting property. Recallthat a C ∗ -algebra A has the lifting property if, whenever B is a C ∗ -algebra, J is an ideal in B , and ϕ : A → B / J is a contractive completely positive map, then there exists a contractive completely positivelift e ϕ : A → B of ϕ . As noted in [4, Lemma 13.1.2], when A is unital, one need only deal with the case when B is unital and ϕ, e ϕ are unital.On the way to proving that P n,c has the lifting property, we will need the following fact. We includea proof for convenience. Proposition 1.6.
Let B be a unital C ∗ -algebra, J be an ideal in B , and p , ..., p c ∈ B / J be projections with P ca =1 p a = 1 B / J . Let q : B → J be the canonical quotient map. Then there are positive elements e p , ..., e p c in B such that P ca =1 e p a = 1 B and q ( e p a ) = p a for all ≤ a ≤ c .Proof. The assumption implies that there is a unital ∗ -homomorphism π : ℓ ∞ c → B / J such that π ( e a ) = p a for each a . As ℓ ∞ c is separable and nuclear, the Choi-Effros lifting theorem [7] gives a ucp lift ϕ : ℓ ∞ c → B of π . Defining e p a = ϕ ( e a ) concludes the proof. (cid:3) Theorem 1.7. P n,c has the lifting property.Proof. This proof is similar in nature to results from [4,24]. First, suppose that B is a unital C ∗ -algebra, J isan ideal in B and π : P n,c → B / J is a ∗ -homomorphism. Then π ( n ) = id n ⊗ π : M n ( P n,c ) → M n ( B ) /M n ( J )is a ∗ -homomorphism. Define Q a = π ( n ) ( P a ). By Proposition 1.6, one can find a POVM e Q , ..., e Q c in M n ( B )that is a lift of Q , ..., Q c . Next, we apply Proposition 1.3 and compress to the (1 ,
1) corner to obtain a ucpmap γ : P n,c → B given by γ ( p a,ij ) = e Q a,ij for all a, i, j . As γ is a lift of π on the generators, a multiplicativedomain argument establishes that γ is a lift of π .Now we deal with the general case. Let ϕ : P n,c → B / J be a ucp map. Since P n,c is separable, onecan restrict if necessary and assume without loss of generality that B is separable. Then we apply Kasparov’sdilation theorem [25] to ϕ : letting K = K ( ℓ ) denote the compact operators and M ( A ) denote the multiplieralgebra of a (separable) C ∗ -algebra A , there is a ∗ -homomorphism ρ : P n,c → M ( K ⊗ min ( B / J )) satisfying ρ ( x ) = ϕ ( x ) for all x ∈ P n,c . (Here, ρ ( x ) refers to the (1 ,
1) entry of ρ ( x ).) If q : B → B / J is thecanonical quotient map, then id ⊗ q : K ⊗ min
B → K ⊗ min ( B / J ) extends to a surjective ∗ -homomorphism σ : M ( K⊗ min B ) → M ( K⊗ min ( B / J )) by the non-commutative Tietze extension theorem [31, Proposition 6.8].Therefore, we can lift the ∗ -homomorphism ρ to a ucp map η : P n,c → M ( K⊗ min B ). Defining e ϕ ( x ) = η ( x ) ,we obtain a lift of ϕ . (cid:3) Next, we establish residual finite-dimensionality of P n,c . Recall that a C ∗ -algebra A is called resid-ually finite-dimensional (RFD) if, for any x ∈ A\{ } , there exists k ∈ N and a finite-dimensionalrepresentation π : A → M k with π ( x ) = 0. Theorem 1.8. P n,c is RFD.Proof. This proof is very similar to the proofs that C ∗ ( F ) and B n are RFD, respectively [6, 20]. As P n,c is separable, we may represent it faithfully as a subalgebra of B ( H ), where H is separable and infinite-dimensional. Let ( E m ) ∞ m =1 be an increasing sequence of projections in B ( H ) such that rank( E m ) = m and HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 7
SOT-lim m →∞ E m = I H . For each 1 ≤ a ≤ c and 1 ≤ i, j ≤ n , we let p ( m ) a,ij = E m p a,ij E m . Then thematrices P ( m ) a = ( p ( m ) a,ij ) ∈ M n ( B ( E m H )) define a POVM with c outputs in B ( E m H ). Applying Proposition1.3, we obtain a unital ∗ -homomorphism ρ m : P n,c → M c +1 ( B ( E m H )) which, after a shuffle of the form M n ( M c +1 ( B ( E m H ))) → M c +1 ( M n ( B ( E m H ))), can be written as P a ( U ∗ m ( E aa ⊗ I E m H ) U m ≤ a ≤ c − U ∗ m (( E cc + E c +1 ,c +1 ) ⊗ I E m H ) U m a = c , where U m = ( P ( m )1 ) ...( P ( m ) c ) (cid:16) ( δ ab I E m H − ( P ( m ) a ) ( P ( m ) b ) ) ca,b =1 (cid:17) − (cid:16) ( P ( m )1 ) · · · ( P ( m ) c ) (cid:17) ∈ M c +1 ( M n ( B ( E m H ))) . The key point is that, considering ρ m ( p a,ij ) ∈ M c +1 ( B ( E m H )), each block from B ( E m H ) belongs to the C ∗ -subalgebra of B ( E m H ) generated by the set { p ( m ) a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } . This set of blocks is closedunder the adjoint since ( E m p a,ij E m ) ∗ = E m p ∗ a,ij E m = E m p a,ji E m . Since SOT-lim m →∞ E m = I H , we haveSOT ∗ -lim m →∞ p ( m ) a,ij = p a,ij . By joint continuity of multiplication in the unit ball with respect to SOT ∗ , itfollows that SOT ∗ -lim m →∞ P ( m ) a = SOT ∗ -lim m →∞ ( P ( m ) a ) = P a for each a . One can check thatSOT ∗ - lim m →∞ U m = P ... P c I H − P . . . I H − P c − ( (cid:0) P · · · P c (cid:1) ) . Applying a shuffle, we see that, for each a, i, j , SOT ∗ -lim m →∞ ρ m ( p a,ij ) exists in M c +1 ( B ( H )); moreover, its(1 , p a,ij . As P a = ( p a,ij ) is a projection, another shuffle argument shows thatSOT ∗ - lim m →∞ ρ m ( p a,ij ) = (cid:18) p a,ij ∗ (cid:19) ∈ M c +1 ( B ( H )) . Therefore, if W is a linear combination of finite words in the generators of P n,c , by considering the (1 , ρ m ( W ), it follows that SOT ∗ -lim m →∞ ρ m ( W ) = (cid:18) W ∗ (cid:19) . By passing to a subsequence if necessary,this forces lim m →∞ k ρ m ( W ) k M c +1 ( B ( E m H )) = k W k B ( H ) . Hence, L ∞ m =1 ρ m : P n,c → L ∞ m =1 M c +1 ( B ( E m H ))is a ∗ -homomorphism that is isometric on the dense ∗ -subalgebra spanned by finite words in the generators p a,ij . It follows that L ∞ m =1 ρ m is an isometry on the whole algebra. As each E m H is finite-dimensional, weconclude that P n,c is RFD. (cid:3) A standard fact is that minimal tensor products of RFD C ∗ -algebras remain RFD. Hence, P n,c ⊗ min P n,c is RFD. We can use this fact to relate Q qa ( n, c ) to states on the minimal tensor product. First, we needthe fact that quantum commuting correlations with a finite-dimensional entanglement space must belong to Q q ( n, c ). Lemma 1.9.
Suppose that X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ Q qc ( n, c ) can be written as X = ( h P a,ij Q b,kℓ χ, χ i ) , where P a = ( P a,ij ) and Q b = ( Q b,kℓ ) are positive in M n ( B ( H )) , P ca =1 P a = P ca =1 Q a = I n , [ P a,ij , Q b,kℓ ] = 0 forall i, j, k, ℓ, a, b and χ ∈ H is a unit vector. If H is finite-dimensional, then X ∈ Q q ( n, c ) .Proof. Let A be the C ∗ -algebra generated by the set { P a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } and let B be the C ∗ -algebra generated by the set { Q b,kℓ : 1 ≤ b ≤ c, ≤ k, ℓ ≤ n } . Then A and B are unital C ∗ -subalgebrasof B ( H ), and every element of A commutes with every element of B . By a theorem of Tsirelson [49], there arefinite-dimensional Hilbert spaces H A and H B , an isometry V : H → H A ⊗ H B , and unital ∗ -homomorphisms π : A → B ( H A ) and ρ : B → B ( H B ) such that V ∗ ( π ( P a,ij ) ⊗ ρ ( Q b,kℓ )) V = P a,ij Q b,kℓ for all a, b, i, j, k, ℓ .Defining the unit vector ξ = V χ ∈ H A ⊗ H B , we see that X ( a,b )( i,j ) , ( k,ℓ ) = h ( π ( P a,ij ) ⊗ ρ ( Q b,kℓ )) ξ, ξ i . MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS
Therefore, X ∈ Q q ( n, c ). (cid:3) Now, we can prove the disambiguation theorems for Q t ( n, c ). We note that, by Proposition 1.3, anyelement of Q q ( n, c ) can be represented using a finite-dimensional tensor product framework H A ⊗ H B andPVMs { P a } ca =1 on H A and { Q b } cb =1 on H B , respectively. This fact holds because, given a POVM { Q b } cb =1 in B ( H ), the dilation in Proposition 1.3 is in M c +1 ( B ( H )) ≃ B ( H ( c +1) ); in particular, the Hilbert spaceremains finite-dimensional if H is finite-dimensional. Similarly, it is easy to see that all elements of Q qs ( n, c )can be represented using PVMs.Next, we show that every element Q qa ( n, c ) can be represented by PVMs, which arise from the minimaltensor product of P n,c . Theorem 1.10.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ ( M n ⊗ M n ) c . The following are equivalent: (1) X belongs to Q qa ( n, c ) . (2) There is a state s : P n,c ⊗ min P n,c → C satisfying s ( p a,ij ⊗ p b,kℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) for all ≤ a, b ≤ c and ≤ i, j, k, ℓ ≤ n . (3) There is a state s : Q n,c ⊗ min Q n,c → C satisfying s ( q a,ij ⊗ q b,kℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) for all ≤ a, b ≤ c and ≤ i, j, k, ℓ ≤ n .Proof. We recall that the minimal tensor product of operator spaces (in particular, operator systems) isinjective (see, for example, [26]). Since Q n,c ⊆ P n,c via the mapping q a,ij p a,ij , injectivity of the minimaltensor product shows that Q n,c ⊗ min Q n,c ⊆ P n,c ⊗ min P n,c completely order isomorphically. Using theHahn-Banach theorem, it then follows that (2) and (3) are equivalent.If (1) holds, then X is in Q qa ( n, c ), so it is a pointwise limit of elements of Q q ( n, c ). Since elementsof Q q ( n, c ) can be represented by PVMs, X is a limit of elements which correspond to finite-dimensionaltensor product representations of P n,c ⊗ min P n,c , which are automatically continuous. Hence, (1) implies(2). Lastly, suppose that (2) holds. Since P n,c ⊗ min P n,c is RFD, a theorem of R. Exel and T.A. Loring [15]shows that s is a w ∗ -limit of states s λ on P n,c ⊗ min P n,c whose GNS representations are finite-dimensional.Applying Lemma 1.9, each s λ applied to the generators p a,ij ⊗ p b,kℓ of P n,c ⊗ min P n,c yields an element X λ of Q q ( n, c ); moreover, lim λ X λ = X pointwise. This shows that X ∈ Q q ( n, c ) = Q qa ( n, c ), which shows that(2) implies (1). (cid:3) To establish the same disambiguation theorem for qc -correlations, we will show that the commuting tensor product Q n,c ⊗ c Q n,c is completely order isomorphic to the copy of Q n,c ⊗Q n,c inside of P n,c ⊗ max P n,c .We recall that, if S and T are operator systems, then an element Y in M n ( S ⊗ c T ) is defined as positive inthe commuting tensor product provided that Y = Y ∗ and, whenever ϕ : S → B ( H ) and ψ : T → B ( H ) areucp maps with commuting ranges, then ( ϕ · ψ ) ( n ) ( Y ) is positive in M n ( B ( H )), where ϕ · ψ : S ⊗ T → B ( H )is the linear map defined by ( ϕ · ψ )( x ⊗ y ) = ϕ ( x ) ψ ( y ) for all x ∈ S and y ∈ T .The next lemma is an adaptation of [20, Proposition 4.6]. Lemma 1.11.
Let S be an operator system. Then the canonical map Q n,c ⊗ c S → P n,c ⊗ max S is a completeorder embedding.Proof. Since P n,c is a unital C ∗ -algebra, we have P n,c ⊗ c S = P n,c ⊗ max S [26, Theorem 6.7]. The canonicalmap Q n,c ⊗ c S → P n,c ⊗ c S is a tensor product of canonical inclusion maps, which are ucp. By functorialityof the commuting tensor product [26], the inclusion Q n,c ⊗ c S → P n,c ⊗ c S is ucp. Hence, it suffices to showthat this map is a complete order embedding.To this end, suppose that Y = Y ∗ ∈ M m ( Q n,c ⊗ S ) is a positive element of M m ( P n,c ⊗ c S ). Let ϕ : Q n,c → B ( H ) and ψ : S → B ( H ) be ucp maps with commuting ranges; we will show that ( ϕ · ψ ) ( m ) ( Y )is positive in M m ( B ( H )). For convenience, we define Q a,ij = ϕ ( q a,ij ). By Proposition 1.3, there is a unital ∗ -homomorphism π : P n,c → M c +1 ( B ( H )) such that the (1 ,
1) corner of π ( p a,ij ) is Q a,ij for all 1 ≤ a ≤ c and1 ≤ i, j ≤ n . Moreover, for each x ∈ P n,c , each block of π ( x ) in B ( H ) belongs to the C ∗ -algebra generatedby the set { Q a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } . We extend ϕ to a ucp map on P n,c by defining ϕ ( · ) = ( π ( · )) .Define e ψ : S → M c +1 ( B ( H )) by e ψ ( s ) = I c +1 ⊗ ψ ( s ). Since ψ ( s ) commutes with the range of ϕ , ψ ( s ) mustcommute with the C ∗ -algebra generated by the range of ϕ . Hence, ψ ( s ) commutes with every block of π ( p a,ij ), for all a, i, j . By the multiplicativity of π , ψ ( s ) commutes with the range of π . By definition of thecommuting tensor product, this means that π · e ψ : P n,c ⊗ c S → M c +1 ( B ( H )) is ucp; moreover, the (1 ,
1) block
HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 9 of π · e ψ is ϕ · ψ . This means that ϕ · ψ is ucp on P n,c ⊗ c S . Restricting to the copy of the algebraic tensorproduct Q n,c ⊗ S , it follows that ( ϕ · ψ ) ( m ) ( Y ) is positive, making the canonical map Q n,c ⊗ c S → P n,c ⊗ c S a complete order embedding. (cid:3) Theorem 1.12.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ ( M n ⊗ M n ) c . The following are equivalent. (1) X belongs to Q qc ( n, c ) . (2) There is a state s : P n,c ⊗ max P n,c → C satisfying s ( p a,ij ⊗ p b,kℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) for all ≤ a, b ≤ c and ≤ i, j, k, ℓ ≤ n . (3) There is a state s : Q n,c ⊗ c Q n,c → C satisfying s ( q a,ij ⊗ q b,kℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) for all ≤ a, b ≤ c and ≤ i, j, k, ℓ ≤ n .Proof. Since Q qc ( n, c ) is defined in terms of POVMs where Alice’s entries commute with Bob’s, we see that(1) is equivalent to (3). Based on two applications of Lemma 1.11, we see that Q n,c ⊗ c Q n,c is completelyorder isomorphic to the image of Q n,c ⊗ Q n,c in P n,c ⊗ max P n,c . Hence, (2) and (3) are equivalent. (cid:3) When considering the quantum-to-classical graph homomorphism game, the local model will be ofinterest because of its link to the usual notion of a (classical) homomorphism from a quantum graph to aclassical graph. It is helpful to note that all strategies in Q loc ( n, c ) can be obtained using PVMs instead ofjust POVMs. We use a bit of a different direction for proving this fact. First, we show the following simplefact: Proposition 1.13. Q loc ( n, c ) is a closed set.Proof. Let X m = ( X ( a,b ) m, ( i,j ) , ( k,ℓ ) ) ∈ Q loc ( n, c ) be a sequence of correlations such that lim m →∞ X m = X pointwisein ( M n ⊗ M n ) c . For each m , there is a unital commutative C ∗ -algebra A m , POVMs P ( m )1 , ..., P ( m ) c and Q ( m )1 , ..., Q ( m ) c in M n ( A m ), and a state s m on A m such that X ( a,b ) m, ( i,j ) , ( k,ℓ ) = s m ( P ( m ) a,ij Q ( m ) b,kℓ ) ∀ a, b, i, j, k, ℓ. Define A = L ∞ m =1 A m , P a,ij = L ∞ m =1 P ( m ) a,ij and Q b,kℓ = L ∞ m =1 Q ( m ) b,kℓ . Then P a = ( P a,ij ) and Q b = ( Q b,kℓ )define two POVMs in M n ( A ) with c outputs. Using the canonical compression from A onto A m , we canextend s m to a state e s m on A . As the state space of A is w ∗ -compact, we choose a w ∗ -limit point s of thesequence ( e s m ) ∞ m =1 . Then X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) = ( s ( P a,ij Q b,kℓ )), which shows that X ∈ Q loc ( n, c ). (cid:3) We note that the above proof works just as well for projection-valued measures. A standard argu-ment shows that limits of convex combinations of elements of Q loc ( n, c ) represented by PVMs from abelianalgebras can still be represented by PVMs from abelian algebras. With this fact in hand, we can prove thedisambiguation theorem for Q loc ( n, c ). Theorem 1.14.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ ( M n ⊗ M n ) c . The following are equivalent: (1) X belongs to Q loc ( n, c ) ; (2) There is a commutative C ∗ -algebra A , a state s on A and POVMs { P , ..., P c } , { Q , ..., Q c } ⊆ M n ( A ) such that X ( a,b )( i,j ) , ( k,ℓ ) = s ( P a,ij Q b,kℓ );(3) There is a commutative C ∗ -algebra A , a state s on A , and PVMs { P , ..., P c } , { Q , ..., Q c } ⊆ M n ( A ) such that X ( a,b )( i,j ) , ( k,ℓ ) = s ( P a,ij Q b,kℓ ) . Proof.
Clearly (1) and (2) are equivalent by the definition of Q loc ( n, c ). Since every PVM is a POVM, (3)implies (2). Hence, we need only show that (2) implies (3). Suppose that X ( a,b )( i,j ) , ( k,ℓ ) = s ( P a,ij Q b,kℓ )for a state s on a commutative C ∗ -algebra A and a POVMs P , ..., P c and Q , ..., Q c in M n ( A ). Then A ≃ C ( Y ) for a compact Hausdorff space Y . The extreme points of the state space of Y are simply evaluationfunctionals δ y for y ∈ Y , which are multiplicative. Hence, δ ( n ) y ( Q a ) ∈ M n ( C ) defines a POVM with c outputs in M n ( C ), where δ ( n ) y = id n ⊗ δ y . Recall that the extreme points of the set of positive contractions in a vonNeumann algebra are precisely the projections in the von Neumann algebra. An easy application of thisargument shows that the extreme points of the set of POVMs with c outputs in a von Neumann algebraare precisely the PVMs with c outputs. Hence, { δ ( n ) y ( Q ) , ..., δ ( n ) y ( Q c ) } lies in the closed convex hull of theset of PVMs in M n ( C ) with c outputs. Applying a similar argument to { δ ( n ) y ( P ) , ..., δ ( n ) y ( P c ) } , it followsthat the correlation ( δ y ( P a,ij Q b,kℓ )) is a convex combination of elements of Q loc ( n, c ) obtained by tensoringprojections from M n ( C ). Taking the closed convex hull, we obtain the original correlation X . In this way,we can write X using projection-valued measures, which shows that (2) implies (3). (cid:3) For t ∈ { loc, q, qs, qa, qc } , we let C t ( n, c ) denote the set of correlations with classical inputs andclassical outputs in the t -model, where Alice and Bob now possess n PVMs (equivalently, POVMs) with c outputs each. These sets embed into Q t ( n, c ) in a natural way. Proposition 1.15.
Let t ∈ { loc, q, qs, qa, qc } . Then C t ( n, c ) is affinely isomorphic to { X ∈ Q t ( n, c ) : X ( a,b )( i,j ) , ( k,ℓ ) = 0 if i = j or k = ℓ } ⊆ Q t ( n, c ) . Moreover, the compression map X ( δ ij δ kℓ X ( a,b )( i,j ) , ( k,ℓ ) ) : Q t ( n, c ) → C t ( n, c ) is a continuous affine map.Proof. All of the claims follow from the following observations: if { E a,x } is a collection of positive operatorssuch that { E a,x } ca =1 is a POVM in B ( H ) for each 1 ≤ x ≤ n , then the operators P a := L nx =1 E a,x define aPOVM in M n ( B ( H )). Similarly, if { Q a } ca =1 is a POVM in M n ( B ( H )), then setting F a,x = Q a,xx ∈ B ( H ),we see that { F a,x } ca =1 is a POVM in B ( H ) for each 1 ≤ x ≤ n . We leave the verification of the claims aboveto the reader. (cid:3) Using what we have established so far, we see that the sets Q t ( n, c ) satisfy Q loc ( n, c ) ⊆ Q q ( n, c ) ⊆ Q qs ( n, c ) ⊆ Q qa ( n, c ) ⊆ Q qc ( n, c ) . The sets Q loc ( n, c ), Q qa ( n, c ) and Q qc ( n, c ) are all closed, and Q qa ( n, c ) = Q qs ( n, c ) = Q q ( n, c ). Using theprevious proposition, all of the containments above are strict in general. Indeed, Q loc (2 , = Q q (2 ,
2) bythe CHSH game [54, Chapter 3]. By a theorem of A. Coladangelo and J. Stark [9], Q q (5 , = Q qs (5 , Q qs (5 , = Q qa (5 , T quantum XOR game and a result of R. Cleve, L. Liu and V.I. Paulsen [8], one can show that Q qs (3 , = Q qa (3 , C t (3 , synchronous versions are equal; in fact, C sq (3 ,
2) = C sqc (3 ,
2) [42].) Lastly, due tothe negative resolution to Connes’ embedding problem [23], it follows that Q qa ( n, c ) = Q qc ( n, c ) for some(likely very large) values of n and c .We close this section with the following isomorphism between P n,c and its opposite algebra. Thisisomorphism will be used in our discussion of synchronous correlations in the next few sections. Recall that,if A is a C ∗ -algebra, then A op is the C ∗ -algebra with the same norm structure as A , but with multiplicationgiven by a op b op = ( ba ) op . Lemma 1.16.
The map p a,ij p opa,ji extends to a unital ∗ -isomorphism π : P n,c → P opn,c .Proof. In P opn,c , one has n X k =1 p opa,kj p opa,ik = n X k =1 ( p a,ik p a,kj ) op = n X k =1 p a,ik p a,kj ! op = p opa,ij , where we have used the fact that P a = ( p a,ij ) is a projection in P n,c . Evidently ( p opa,ij ) ∗ = ( p ∗ a,ij ) op = p opa,ji , sothe above calculations show that P opa := ( p opa,ji ) ni,j =1 is a projection in M n ( P opn,c ). Moreover, P ca =1 P opa = I n . HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 11
By the universal property of P n,c , there is a unital ∗ -homomorphism π : P n,c → P opn,c such that π ( p a,ij ) = p opa,ji . One can show that P opn,c is the universal C ∗ -algebra generated by entries p opa,ij with the property that P opa = ( p opa,ji ) ni,j =1 is a projection in M n ( P opn,c ) with P ca =1 P opa = I n . By a similar argument to the above, themap p opa,ji p a,ij extends to a ∗ -homomorphism ρ : P opn,c → P n,c . Since π and ρ are mutual inverses on thegenerators of the respective algebras, they both extend to isomorphisms, yielding the desired result. (cid:3) Synchronous Quantum Input–Classical Output Correlations
We now generalize the notion of synchronous correlations. Recall that a correlation P = ( p ( a, b | x, y )) ∈ C ( n, k ) is called synchronous if p ( a, b | x, x ) = 0 whenever a = b [22].In the following considerations, we fix once and for all an orthonormal basis { e , ..., e n } for C n . Definition 2.1.
Let S ⊆ [ n ] . We define the maximally entangled Bell state corresponding to S asthe vector ϕ S = 1 p | S | X j ∈ S e j ⊗ e j . Definition 2.2.
Let X ∈ Q t ( n, c ) be a correlation in n -dimensional quantum inputs and c classical outputs,where t ∈ { loc, q, qs, qa, qc } . We say that X is synchronous provided that there is a partition S ˙ ∪ · · · ˙ ∪ S ℓ of [ n ] with the property that, if a = b , then p ( a, b | ϕ S r ) = 0 for all ≤ r ≤ ℓ. We define the subset Q st ( n, c ) = { X ∈ Q t ( n, c ) : X is synchronous } . The following proposition gives a very useful description of synchronicity in terms of the entries of thematrices involved in the correlation.
Proposition 2.3.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ Q t ( n, c ) for t ∈ { loc, q, qs, qa, qc } . The following are equivalent: (1) X is synchronous. (2) X satisfies the equation (2.1) 1 n c X a =1 n X i,j =1 X ( a,a )( i,j ) , ( i,j ) = 1 . (3) If a = b , then (2.2) n X i,j =1 X ( a,b )( i,j ) , ( i,j ) = 0 . Proof.
Suppose that X can be represented using the PVMs { P a } ca =1 in B ( C n ⊗H ) and { Q b } cb =1 in B ( H⊗ C n )and the state χ ∈ H . We observe that, if S ⊆ [ n ], then p ( a, b | ϕ S ) = 1 | S | X i,j ∈ S h ( P a ⊗ I n )( I n ⊗ Q b )( e j ⊗ χ ⊗ e j ) , e i ⊗ χ ⊗ e i i = 1 | S | X i,j ∈ S h P a,ij Q b,ij χ, χ i = 1 | S | X i,j ∈ S X ( a,b )( i,j ) , ( i,j ) . Suppose that X is synchronous, and let S , ..., S ℓ be a partition of [ n ] for which p ( a, b | ϕ S r ) = 0 whenever a = b and 1 ≤ r ≤ ℓ . Then the above calculation shows that P i,j ∈ S r X ( a,b )( i,j ) , ( i,j ) = 0 for all r . Summing overall r , it follows that P ni,j =1 X ( a,b )( i,j ) , ( i,j ) = 0 whenever a = b . Hence, (1) implies (3). Next, we show that (3) implies (2). Notice that, for any X ∈ Q qc ( n, c ), c X a,b =1 n X i,j =1 X ( a,b )( i,j ) , ( k,ℓ ) = c X a,b =1 n X i,j =1 h P a,ij Q b,ij χ, χ i = n X i,j =1 * c X a =1 P a,ij ! c X b =1 Q b,ij ! χ, χ + = n X i =1 h χ, χ i = n, where we have used the fact that P ca =1 P a = P cb =1 Q b = I n implies that P ca =1 P a,ij = P cb =1 Q b,ij is I when i = j and 0 otherwise. Therefore, 1 n n X i,j =1 X ( a,b )( i,j ) , ( i,j ) = 1 , which shows that (2) holds.Lastly, if (2) holds, then (1) immediately follows using the single-set partition S = [ n ]. (cid:3) Remark 2.4.
In the case of a correlation p ( a, b | x, y ) ∈ C t ( n, c ) with n classical inputs and c classical outputs,using the [ n ] = { } ∪ { } ∪ · · · ∪ { n } , we see that any synchronous correlation in C t ( n, c ) is a synchronouscorrelation in the sense of the definition above. In this way, we see that C st ( n, c ) ⊆ Q st ( n, c ) . We wish to show the analogue of [39, Theorem 5.5]; that is, synchronous correlations with n -dimensionalinputs and c outputs arise from tracial states on the algebra generated by Alice’s operators (respectively,Bob’s operators). We will also see that, in any realization of a synchronous correlation, Bob’s operators canbe described naturally in terms of Alice’s operators. By a realization of X ∈ Q qc ( n, c ), we simply mean a4-tuple ( { P a } ca =1 , { Q b } cb =1 , H , ψ ), where { P a } ca =1 is a PVM on C n ⊗ H , { Q b } cb =1 is a PVM on H ⊗ C n , ψ isa state in H , and [ P a ⊗ I n , I n ⊗ Q b ] = 0 for all a, b . Theorem 2.5.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ Q sqc ( n, c ) . Let ( { P a } ca =1 , { Q b } cb =1 , H , ψ ) be a realization of X . Then: (1) Q a,ij ψ = P ∗ a,ij ψ for all ≤ a ≤ c and ≤ i, j ≤ n . (2) The state ρ = h ( · ) ψ, ψ i is a tracial state on the C ∗ -algebra A generated by { P a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } , and on the C ∗ -algebra B generated by { Q b,kℓ : 1 ≤ b ≤ c, ≤ k, ℓ ≤ n } .Conversely, if P a,ij are operators in a tracial C ∗ -algebra A with a trace τ , such that the operators P a =( P a,ij ) ∈ M n ( A ) form a PVM with c outputs, then ( τ ( P a,ij P ∗ b,kℓ )) defines an element of Q sqc ( n, c ) . HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 13
Proof.
Suppose X ∈ Q sqc ( n, c ), with realization ( { P a } ca =1 , { Q b } cb =1 , H , ψ ). By Proposition 2.3, we have1 = 1 n c X a =1 n X i,j =1 X ( a,a )( i,j ) , ( i,j ) (2.3) = 1 n c X a =1 n X i,j =1 h P a,ij Q a,ij ψ, ψ i≤ n c X a =1 n X i,j =1 |h P a,ij Q a,ij ψ, ψ i| (2.4) = 1 n c X a =1 n X i,j =1 |h Q a,ij ψ, P ∗ a,ij ψ i|≤ n c X a =1 n X i,j =1 k Q a,ij ψ kk P ∗ a,ij ψ k (2.5) ≤ n c X a =1 n X i,j =1 k Q a,ij ψ k c X a =1 n X i,j =1 k P ∗ a,ij ψ k (2.6) = 1 n c X a =1 n X i,j =1 h Q ∗ a,ij Q a,ij ψ, ψ i c X a =1 n X i,j =1 h P a,ij P ∗ a,ij ψ, ψ i = 1 n c X a =1 n X i,j =1 h Q a,ji Q a,ij ψ, ψ i c X a =1 n X i,j =1 h P a,ij P a,ji ψ, ψ i Since P a and Q a are projections, the last line is equal to1 n c X a =1 n X j =1 h Q a,jj ψ, ψ i c X a =1 n X i =1 h P a,ii ψ, ψ i ! = 1 n n X j =1 h I H ψ, ψ i n X i =1 h I H ψ, ψ i ! = 1 n · √ n · √ n = 1 . Therefore, all of these inequalities are equalities. Then (2.4) implies that X ( a,a )( i,j ) , ( i,j ) ≥ ≤ a ≤ c, ≤ i, j ≤ n. The equality case of (2.5) shows that(2.7) Q a,ij ψ = α a,ij P ∗ a,ij ψ for some α a,ij ∈ T . Then equation (2.7) yields X ( a,a )( i,j ) , ( i,j ) = α a,ij h P a,ij P ∗ a,ij ψ, ψ i = α a,ij k P ∗ a,ij ψ k . Since X ( a,a )( i,j ) , ( i,j ) ≥ k P ∗ a,ij ψ k ≥
0, we either have P ∗ a,ij ψ = 0 or α a,ij = 1. In either case, we obtain Q a,ij ψ = P ∗ a,ij ψ, as desired.To prove (2), it suffices to show that it holds for A = C ∗ ( { P a,ij } a,i,j ); a similar argument works for B = C ∗ ( { Q b,kℓ } b,k,ℓ ). Let ρ : A → C be the state given by ρ ( X ) = h Xψ, ψ i . Let W = P m a ,i j · · · P m k a k ,i k j k bea word in { P a,ij , P ∗ a,ij } a,i,j , where we denote by P − a ℓ ,i ℓ j ℓ the operator P ∗ a ℓ ,i ℓ j ℓ and let m ℓ ∈ {− , } . We will first show that W ψ = Q − m k a k ,i k j k · · · Q − m a ,i j ψ , where Q − a ℓ ,i ℓ j ℓ := Q ∗ a ℓ ,i ℓ j ℓ . Using the fact that P a,ij and Q b,kℓ ∗ -commute for each a, b, i, j, k, ℓ , we obtain W ψ = P m a ,i j · · · P m k a k ,i k j k ψ = P m a ,i j · · · P m k − a k − ,i k − j k − Q − m k a k ,i k j k ψ = Q − m k a k ,i k j k ( P m a ,i j · · · P m k − a k − ,i k − j k − ) ψ. and the desired equality easily follows by induction on k . For 1 ≤ a ≤ c and 1 ≤ i, j ≤ n , ρ ( P a,ij W ) = h P a,ij W ψ, ψ i = h P a,ij ( Q m a ,i j · · · Q m k a k ,i k j k ) ∗ ψ, ψ i = h ( Q m a ,i j · · · Q m k a k ,i k j k ) ∗ P a,ij ψ, ψ i = h P a,ij ψ, Q m a ,i j · · · Q m k a k ,i k j k ψ i = h P a,ij ψ, ( P m a ,i j · · · P m k a k ,i k j k ) ∗ ψ i = h P a,ij ψ, W ∗ ψ i = h W P a,ij ψ, ψ i = ρ ( W P a,ij ) . In the same way, ρ ( P a,ij P b,kℓ W ) = ρ ( W P a,ij P b,kℓ ). It follows by induction, linearity and continuity that ρ is tracial on A , as desired.For the converse direction, we recall the standard fact that, if A is a unital C ∗ -algebra and τ is atrace on A , then there is a state s : A ⊗ max A op → C satisfying s ( x ⊗ y op ) = τ ( xy ) for all x, y ∈ A . Thus, if P , ..., P c ∈ M n ( A ) is a projection-valued measure, then s ( P a,ij ⊗ P opb,kℓ ) = τ ( P a,ij P b,kℓ ) ∀ ≤ a, b ≤ c, ≤ i, j, k, ℓ ≤ n. Applying the universal property of P n,c , we obtain a state γ : P n,c ⊗ max P opn,c → C satisfying γ ( p a,ij ⊗ p opb,kℓ ) = τ ( P a,ij P b,kℓ ) . By Lemma 1.16, the map p a,ij ⊗ p b,kℓ τ ( P a,ij P b,ℓk ) = τ ( P a,ij P ∗ b,kℓ ) defines a state on P n,c ⊗ max P n,c . ThenTheorem 1.12 shows that X := τ ( P a,ij P ∗ b,kℓ )defines an element of Q qc ( n, c ). If a = b , then n X i,j =1 X ( a,b )( i,j ) , ( i,j ) = n X i,j =1 τ ( P a,ij P ∗ b,ij )= Tr ⊗ τ ( P a P b ) = 0 , since P a P b = 0. By Proposition 2.3, X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ Q sqc ( n, c ). (cid:3) In light of Theorem 2.5, we may refer to a synchronous t -strategy ( { P a } ca =1 , χ ) when referring toa t -strategy ( { P a } ca =1 , { Q b } cb =1 , χ ) where the associated correlation is synchronous. Corollary 2.6.
Let ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ Q st ( n, c ) where t ∈ { loc, q, qs, qa, qc } . Then: (1) X ( a,b )( i,i ) , ( j,j ) ≥ for all ≤ a, b ≤ c and ≤ i, j ≤ n . (2) X ( a,b )( i,j ) , ( k,ℓ ) = X ( a,b )( j,i ) , ( ℓ,k ) . (3) For any ≤ a = b ≤ c and ≤ i, j ≤ n , we have n X k =1 X ( a,b )( i,k ) , ( j,k ) = n X k =1 X ( a,b )( k,i ) , ( k,j ) = 0 . (4) For any ≤ i, j ≤ n , we have c X a =1 n X k =1 X ( a,a )( i,k ) , ( j,k ) = c X a =1 n X k =1 X ( a,a )( k,i ) , ( k,j ) = δ ij . HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 15
Proof.
By Theorem 2.5, we may choose projections P , ..., P c ∈ M n ( A ), for a unital C ∗ -algebra A , alongwith a tracial state τ on A such that X ( a,b )( i,j ) , ( k,ℓ ) = τ ( P a,ij P ∗ b,kℓ ) for all 1 ≤ a, b ≤ c, ≤ i, j, k, ℓ ≤ n. Since P a is a projection, it defines a positive element of M n ( A ). Compressing to any diagonal block preservespositivity, which implies that P a,ii ∈ A + for any i . Since τ is a trace, it follows that τ ( P a,ii P b,jj ) ≥ i, j, a, b . Hence, (1) follows.We note that (2) follows easily from the fact that τ is a trace and that, since τ is a state, one has τ ( Y ∗ ) = τ ( Y ) for all Y ∈ A .To show (3), we observe that n X k =1 X ( a,b )( i,k ) , ( j,k ) = n X k =1 τ ( P a,ik P ∗ b,jk )= n X k =1 τ ( P a,ik P b,kj )= τ n X k =1 P a,ik P b,kj ! = τ (( P a P b ) ij ) = 0 , since P a P b = 0. Similarly, P nk =1 X ( a,b )( k,i ) , ( k,j ) = 0 when a = b .A similar argument establishes (4). Indeed, we have c X a =1 n X k =1 X ( a,a )( i,k ) , ( j,k ) = c X a =1 n X k =1 τ ( P a,ik P a,kj ) = τ c X a =1 P a,ij ! , and this latter sum is δ ij , since P ca =1 P a = I . The other equation in (4) follows similarly. (cid:3) Remark 2.7.
It makes sense (and we will have occasion) to discuss synchronicity of a strategy with respectto a different orthonormal basis v = { v , ..., v n } of C n . In this case, a qc -strategy ( { P a } ca =1 , { Q b } cb =1 , χ ) issaid to be synchronous with respect to { v , ..., v n } if there is a partition S ∪ · · · ∪ S s of [ n ] such that foreach r and ϕ S r , v := √ | S r | P j ∈ S r v j ⊗ v j , we have p ( a, b | ϕ S r , v ) = 0 if a = b. One can then write down an analogue of Theorem 2.5 in this context. Alternatively, one can simply let e P a = U ∗ P a U and e Q b = U ∗ Q b U , where U is the unitary satisfying U e i = v i for all i . Then applying Theorem2.5 relates the entries of e Q a to the entries of e P a , while showing that the state h ( · ) χ, χ i is a trace on the algebragenerated by the entries of the operators e Q a (respectively, e P a ). Since P a = U e P a U ∗ and Q b = U e Q b U ∗ , theentries of P a (respectively, Q b ) are linear combinations of the entries of e P a (respectively, e Q b ), so it followsthat the algebra generated by the entries of the operators P a (respectively, Q b ) is the same as the algebragenerated by the entries of e P a (respectively, e Q b ). It is helpful to describe the simplest ways to realize synchronous correlations. To that end, we spendthe rest of this section describing the simplest realizations for t ∈ { loc, q, qs } . We start with the case of Q sloc ( n, c ). Corollary 2.8.
Let X ∈ ( M n ⊗ M n ) c . Then X belongs to Q sloc ( n, c ) if and only if there is a unital,commutative C ∗ -algebra A , a projection-valued measure { P a } ca =1 ⊆ M n ( A ) for ≤ a ≤ c , and a faithfulstate ψ ∈ S ( A ) such that, for all ≤ a, b ≤ c and ≤ i, j, k, ℓ ≤ n , X ( a,b )( i,j ) , ( k,ℓ ) = ψ ( P a,ij P ∗ b,kℓ ) . Moreover, if X is an extreme point in Q sloc ( n, c ) , then we may take A = C . Proof. If X ∈ Q sloc ( n, c ), then by definition of loc-correlations, X can be written using projection-valued mea-sures { P a } ca =1 and { Q b } cb =1 in M n ( B ( H )), along with a state χ ∈ H , such that X ( a,b )( i,j ) , ( k,ℓ ) = h P a,ij Q b,kℓ χ, χ i and the C ∗ -algebra A generated by the set of all entries P a,ij and Q b,ℓ is a commutative C ∗ -algebra. Ap-plying Theorem 2.5, we can write X ( a,b )( i,j ) , ( k,ℓ ) = ψ ( P a,ij P ∗ b,kℓ ), where ψ ( · ) = h ( · ) χ, χ i . As this state is tracial,by replacing A with its quotient by the kernel of the GNS representation of ψ if necessary, we may assumewithout loss of generality that ψ is faithful, which establishes the forward direction. The converse follows bythe converse of Theorem 2.5 and the definition of Q loc ( n, c ).To establish the claim about extreme points, we note that the proof of Proposition 1.13 shows thatevery element of Q loc ( n, c ) is a limit of convex combinations of correlations arising from PVMs in M n ( C ).Evidently the set of elements of Q loc ( n, c ) that have realizations using PVMs in M n ( C ) is a closed set. As Q loc ( n, c ) is compact and convex, the converse of the Krein-Milman theorem shows that extreme points in Q loc ( n, c ) must have a realization using PVMs in M n ( C ). Now, the proof of the forward direction of Theorem2.5 shows that n P ca =1 P ni,j =1 Y ( a,b )( i,j ) , ( i,j ) ≤ Y ∈ Q qc ( n, c ). Moreover, this inequality is an equality ifand only if Y is synchronous, by Proposition 2.3. Hence, Q sloc ( n, c ) is a face in Q loc ( n, c ), so extreme pointsin Q sloc ( n, c ) are also extreme points in Q loc ( n, c ). This shows that X has a realization using the algebra A = C . (cid:3) Corollary 2.9.
Let X ∈ ( M n ⊗ M n ) c . Then X belongs to Q sq ( n, c ) if and only if there is a finite-dimensional C ∗ -algebra A , a projection-valued measure { P a } ca =1 ⊆ M n ( A ) for ≤ a ≤ c , and a faithful tracial state ψ ∈ S ( A ) such that, for all ≤ a, b ≤ c and ≤ i, j, k, ℓ ≤ n , X ( a,b )( i,j ) , ( k,ℓ ) = ψ ( P a,ij P ∗ b,kℓ ) . Moreover, if X is an extreme point in Q sq ( n, c ) , then we may take A = M d for some d , and hence ψ = tr d ,where tr d is the normalized trace on M d .Proof. If X belongs to Q sq ( n, c ), then one can write X = ( h ( P a,ij ⊗ Q b,kℓ ) χ, χ i ) for projection-valued measures { P a } ca =1 ⊆ M n ( B ( H A )) and { Q b } cb =1 ⊆ M n ( B ( H B )) on finite-dimensional Hilbert spaces H A and H B , alongwith a unit vector χ ∈ H A ⊗H B . By Theorem 2.5, we may write X = ψ ( P a,ij P ∗ b,kℓ ) where ψ is the (necessarilyfaithful) tracial state on the finite-dimensional C ∗ -algebra A generated by the set { P a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } .Conversely, if X can be written as X = ( ψ ( P a,ij P ∗ b,kℓ )) for a projection-valued measure { P a } ca =1 ⊆ M n ( A ), where A is a finite-dimensional C ∗ -algebra with a faithful trace ψ on A , then the proof of Theorem2.5 yields a finite-dimensional realization of X as an element of Q sqc ( n, c ). By Lemma 1.9, we must have X ∈ Q sq ( n, c ).Now, assume that X is extreme in Q sq ( n, c ). Since A is finite-dimensional, it is ∗ -isomorphic to L mr =1 M k r for some r and numbers k , ..., k r ∈ N . Since ψ is a trace on A , there must be t , ..., t m ≥ P mr =1 t r = 1 and ψ ( · ) = P mr =1 t r tr k r ( · ), where tr k r is the normalized trace on M k r . Writing P a,ij = L mr =1 P ( r ) a,ij for each 1 ≤ a ≤ c and 1 ≤ i, j ≤ n , where P ( r ) a,ij ∈ M k r , we have X ( a,b )( i,j ) , ( k,ℓ ) = m X r =1 t r tr k r ( P ( r ) a,ij ( P ( r ) b,kℓ ) ∗ ) . Since P ( r ) a = ( P ( r ) a,ij ) ∈ M n ( M k r ) must define an orthogonal projection and P ca =1 P ( r ) a = I n ⊗ I k r , it fol-lows that X ( a,b ) r, ( i,j ) , ( k,ℓ ) = tr k r ( P ( r ) a,ij ( P ( r ) b,kℓ ) ∗ ) ∈ Q sq ( n, c ), and P mr =1 t r X ( a,b ) r, ( i,j ) , ( k,ℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) . Therefore, X ( a,b ) r, ( i,j ) , ( k,ℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) for each r . This shows that we may take A to be a matrix algebra, completing theproof. (cid:3) We will end this section by showing that Q sqs ( n, c ) = Q sq ( n, c ). To prove this fact, we use a similarapproach to [28]. In fact, by an application of Proposition 1.15, the following theorem is a direct generalizationof the analogous result in [28]. Theorem 2.10.
For each n, c ∈ N , we have Q sqs ( n, c ) = Q sq ( n, c ) . HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 17
Proof.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) ∈ Q sqs ( n, c ), and write X ( a,b )( i,j ) , ( k,ℓ ) = h ( P a,ij ⊗ Q b,kℓ ) ψ, ψ i , where P a = ( P a,ij ) is a projection in C n ⊗ H A for each 1 ≤ a ≤ c , Q b = ( Q b,ij ) is a projection in H B ⊗ C n for each 1 ≤ b ≤ c , P ca =1 P a = I C n ⊗H A , P cb =1 Q b = I H B ⊗ C n , and ψ ∈ H A ⊗ H B is a state. We can arrangeto have dim( H A ) = dim( H B ). For example, if dim( H A ) < dim( H B ), then we choose a Hilbert space H C with dim( H A ⊕ H C ) = dim( H B ), and extend P a by defining e P a,ij = P a,ij ⊕ δ ij I H C . Then h ( e P a,ij ⊗ Q b,kℓ ) ψ, ψ i = h ( P a,ij ⊗ Q b,kℓ ) ψ, ψ i = X ( a,b )( i,j ) , ( k,ℓ ) . In this way, we may assume without loss of generality that dim( H A ) = dim( H B ).We write down a Schmidt decomposition ψ = ∞ X p =1 α p e p ⊗ f p , where { e p } ∞ p =1 ⊆ H A and { f p } ∞ p =1 ⊆ H B are orthonormal, and α ≥ α ≥ ... ≥ P ∞ p =1 α p = 1.If one extends these orthonormal sets to orthonormal bases for H A and H B respectively, and defines additional α p ’s to be 0, then after direct summing a Hilbert space on one side if necessary, we may assume thatdim( H A ) = dim( H B ) and that { e r } r ∈ I is an orthonormal basis for H A , and { f s } s ∈ I is an orthonormal basisfor H B .We rewrite the (at most) countable set { α q : α q = 0 } = { β v : v ∈ V } , where V = { , , ... } and β v > β v +1 for all v ∈ V . We define K v = { e q : α q = β v } and L v = { f q : α q = β v } , and define subspaces K v = span( K v ) and L v = span( L v ) of H A and H B , respectively. Since P ∞ q =1 | α q | = 1, it follows thateach K v and L v must be finite, so that K v and L v are finite-dimensional. We will show that each K v isinvariant for the operators { P a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } , and that each L v is invariant for the operators { Q b,kℓ : 1 ≤ b ≤ c, ≤ k, ℓ ≤ n } . To this end, let ω be a primitive c -th root of unity, and define order c unitaries U = P ca =1 ω a P a ∈ B ( C n ⊗ H A ) and V = P cb =1 ω − b Q b ∈ B ( H A ⊗ C n ). Since X is synchronous, byTheorem 2.5, we know that( I H A ⊗ Q ∗ a,ij ) ψ = ( P a,ij ⊗ I H B ) ψ and ( I H A ⊗ Q a,ij Q ∗ b,ij ) ψ = ( P b,ij P ∗ a,ij ⊗ I H B ) ψ. Since U ij U ∗ ij = P ca,b =1 ω a − b P a,ij P ∗ b,ij and V ij V ∗ ij = P ca,b =1 ω b − a Q a,ij Q ∗ b,ij , it follows that( I H A ⊗ V ∗ ij ) ψ = ( U ij ⊗ I H B ) ψ and ( I H A ⊗ V ij V ∗ ij ) ψ = ( U ij U ∗ ij ⊗ I H B ) ψ. Using this fact and the decomposition of ψ , α q h U ij e q , e p i = h ( U ij ⊗ I H B ) ψ, e p ⊗ f q i = h ( I H A ⊗ V ∗ ij ) ψ, e p ⊗ f q i = α p h V ∗ ij f p , f q i . Since U and V are unitary, it follows that, for all p , n X i,j =1 k U ∗ ij e p k = n X i,j =1 h U ij U ∗ ij e p , e p i = n and n X i,j =1 k U ij e p k = n X i,j =1 h U ∗ ij U ij e p , e p i = n. Similarly, P ni,j =1 k V ∗ ij f q k = P ni,j =1 k V ij f q k = n . Suppose that q is such that e q ∈ K . Then using the factthat α q = α and that α p ≤ α for all p yields n | α | = n X i,j =1 | α | k V ∗ ij f q k ≥ n X i,j =1 ∞ X p =1 | α p | |h V ∗ ij f p , f q i| = n X i,j =1 ∞ X p =1 | α q | |h U ij e q , e p i| = | α | n X i,j =1 ∞ X p =1 |h U ij e q , e p i| = | α | n X i,j =1 ∞ X p =1 |h U ∗ ij e p , e q i| = | α | n X i,j =1 k U ∗ ij e q k = n | α | . Therefore, we must have equality at all lines. If p is such that e p K , then since α p < α , we must have0 = P ni,j =1 | α p | |h V ∗ ij f p , f q i| = P ni,j =1 | α q | |h U ij e q , e p i| . Therefore, h U ij e q , e p i = 0 for each such p , whichshows that U ij e q ⊥ e p for all p with e p K . Since this happens whenever α q = α , the subspace K mustbe invariant for every U ij . By the same argument as above with the quantity P ni =1 | α | k V ij f q k , it followsthat K is invariant for every U ∗ ij . Therefore, K is reducing for the operators U ij , for all 1 ≤ i, j ≤ n . Asimilar argument proves that L is reducing for the operators V kℓ , for all 1 ≤ k, ℓ ≤ n .Now, choose q such that e q ∈ K and f q ∈ L . If α p > α q , then α p = α , so that e p ∈ K and f p ∈ K . The above shows that h U ij e q , e p i = 0 and h U ∗ ij e q , e p i = 0, so that U ij e q ⊥ K and U ∗ ij e q ⊥ K .Similarly, V kℓ f q ⊥ L and V ∗ kℓ f q ⊥ L . Then using a similar string of inequalities as before, one obtains U ij e q ⊥ e p whenever p is such that e p K and q is such that e q ∈ K . Therefore, one finds that K isinvariant for each U ij . A similar argument shows that K is invariant for U ∗ ij , so that K is reducing for { U ij : 1 ≤ i, j ≤ n } . The same argument shows that { V kℓ : 1 ≤ k, ℓ ≤ n } reduces K .It follows by induction that K v is reducing for { U ij : 1 ≤ i, j ≤ n } for all v and that L v is reducingfor { V kℓ : 1 ≤ k, ℓ ≤ n } for all v . By construction of the unitaries U and V , we know that P a = 1 c c X d =1 ω − ad U d and Q b = 1 c c X d =1 ω bd V d . Therefore, K v is reducing for each P a,ij , and L v is reducing for each Q b,kℓ , as desired.Finally, we will exhibit X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) as a countable convex combination of elements of Q sq ( n, c ).One can regard elements of Q sq ( n, c ) as elements of C n c , or as elements of R n c ) . Then by a countablyinfinite version of Carath´eodory’s Theorem [10], this will show that X belongs to Q sq ( n, c ), which willcomplete the proof. (As mentioned in [28], this result from [10] holds even with non-closed convex sets, ofwhich Q sq ( n, c ) is an example.)For each v ∈ V , we let d v = dim( K v ) = dim( L v ) = | K v | = | L v | , which is finite. Define the state ψ v = 1 √ d v X p : e p ∈ K v e p ⊗ f p , and define P v,a,ij = P a,ij | K v and Q v,b,kℓ = Q b,kℓ | L v . Since K v is reducing for P a,ij , and since P a is a projection, the operator P v,a = ( P v,a,ij ) ni,j =1 is a projectionon C n ⊗ K v . Similarly, Q v,b = ( Q v,b,kℓ ) nk,ℓ =1 is a projection on L v ⊗ C n . Moreover, P ca =1 P v,a = I C n ⊗ I K v HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 19 and P cb =1 Q v,b = I L v ⊗ I C n . Therefore, the correlation X v = ( X ( a,b ) v, ( i,j ) , ( k,ℓ ) ) = ( h ( P v,a,ij ⊗ Q v,b,kℓ ) ψ v , ψ v i )belongs to Q q ( n, c ) for each v . Set t v = β v d v . Then t v ≥ P v ≥ t v = P ∞ p =1 | α p | = 1. Finally, for each1 ≤ a, b ≤ c and 1 ≤ i, j ≤ n , we compute X ( a,b )( i,j ) , ( k,ℓ ) = h ( P a,ij ⊗ Q b,kℓ ) ψ, ψ i = X v X p,q : e p ,e q ∈ K v β v h ( P a,ij ⊗ Q b,kℓ )( e p ⊗ f p ) , e q ⊗ f q i = X v β v d v h ( P v,a,ij ⊗ Q v,b,kℓ ) ψ v , ψ v i = X v t v X ( a,b ) v, ( i,j ) , ( k,ℓ ) . It follows that X = P v t v X v . Since each X v ∈ Q q ( n, c ), it follows that X ∈ Q q ( n, c ). Since X is alsosynchronous, we obtain X ∈ Q sq ( n, c ), completing the proof. (cid:3) Approximately finite-dimensional correlations
In this section, we will show that elements of Q sqa ( n, c ) arise from amenable traces. Equivalently,elements of Q sqa ( n, c ) can be represented using the trace on R U and projection-valued measures with c outputs in M n ( R U ), where R U denotes an ultrapower of the hyperfinite II -factor R by a free ultrafilter U on N . The proof is similar to [28, Section 3], and the main result is a generalization of [28, Theorem 3.6].Relevant details on R U can be found in [4].For amenable traces, we will use the following result of Kirchberg [30, Proposition 3.2], which can alsobe found in [4, Theorem 6.2.7]. Theorem 3.1.
Let A be a separable C ∗ -algebra and let τ be a tracial state on A . The following statementsare equivalent: (1) The trace τ is amenable ; i.e., whenever A ⊆ B ( H ) is a faithful representation, then there is a state ρ on B ( H ) such that ρ |A = τ and ρ ( u ∗ T u ) = ρ ( T ) for all T ∈ B ( H ) and unitaries u ∈ A ; (2) There is a ∗ -homomorphism π : A → R U along with a completely positive contractive lift ϕ : A → ℓ ∞ ( R ) such that tr R U ◦ π = τ ; (3) There is a sequence of natural numbers N ( k ) and completely positive contractive maps ϕ k : A → M N ( k ) satisfying lim k →∞ k ϕ k ( ab ) − ϕ k ( a ) ϕ k ( b ) k = 0 and lim k →∞ | tr N ( k ) ( ϕ k ( a )) − τ ( a ) | = 0 for all a, b ∈ A ; (4) The linear functional γ : A ⊗ A op → C given by γ ( a ⊗ b op ) = τ ( ab ) extends to a continuous linearmap on A ⊗ min A op . As pointed out in [28], as soon as γ is continuous on A ⊗ min A op in condition (4) above, it is auto-matically a state on the minimal tensor product.In what follows, we will let k · k denote the 2-norm with respect to the trace. For the convenience ofthe reader, we recall the following perturbation result. Lemma 3.2. (Kim-Paulsen-Schafhauser, [28])
Let ε > and c ∈ N . Then there exists a δ > suchthat, if n, N ∈ N and P , ..., P c ∈ M n ( M N ) are positive contractions with k P a P b k < δ for all a = b and k P a − P a k < δ for all a , then there exist orthogonal projections Q , ..., Q c ∈ M n ( M N ) with Q a Q b = 0 for a = b and k P a − Q a k < ε . Moreover, if k P ca =1 P a − I n ⊗ I N k < δ , then we may arrange for the projections Q , ..., Q c to satisfy P ca =1 Q a = I n ⊗ I N . Note that this lemma is stated slightly differently than in [28]; however, it is easy to see that theirresult is equivalent to the above result. Notice that, in the above lemma, if we write P a = ( P a,ij ) ∈ M n ( M N )and Q a = ( Q a,ij ) ∈ M n ( M N ), then one has k Q a,ij − P a,ij k ≤ k Q a − P a k < ε , where the first 2-norm is in M N and the second 2-norm is in M n ( M N ). Theorem 3.3.
Let X = ( X ( a,b )( i,j ) , ( k,ℓ ) ) be an element of ( M n ⊗ M n ) c . The following statements are equivalent: (1) X belongs to Q sqa ( n, c ) ; (2) X belongs to Q sq ( n, c ) ; (3) There is a separable unital C ∗ -algebra A , a PVM { P , ..., P c } in M n ( A ) , and an amenable trace τ on A such that, for all ≤ i, j, k, ℓ ≤ n and ≤ a, b ≤ c , X ( a,b )( i,j ) , ( k,ℓ ) = τ ( P a,ij P ∗ b,kℓ );(4) There are elements q a,ij in R U such that q a = ( q a,ij ) are projections in M n ( R U ) with P ca =1 q a = I n and X ( a,b )( i,j ) , ( k,ℓ ) = tr R U ( q a,ij q ∗ b,kℓ ) . Proof.
First, we show that (1) implies (3). Since Q qa ( n, c ) is the closure of Q q ( n, c ), this means that there arecorrelations X r = ( X ( a,b ) r, ( i,j ) , ( k,ℓ ) ) ∈ Q q ( n, c ) with lim r →∞ X r = X pointwise. We may choose natural numbers N ( r ) and M ( r ) along with projection-valued measures { P ( r )1 , ..., P ( r ) c } ⊆ M n ( M N ( r ) ) and { Q ( r )1 , ..., Q ( r ) c } ⊆ M n ( M M ( r ) ) and unit vectors χ r ∈ C N ( r ) ⊗ C M ( r ) such that, for all r ∈ N and for all 1 ≤ a, b ≤ c and1 ≤ i, j, k, ℓ ≤ n , X ( a,b ) r, ( i,j ) , ( k,ℓ ) = h ( P ( r ) a,ij ⊗ Q ( r ) b,kℓ ) χ r , χ r i . Then for each r , by Theorem 1.10 and Theorem 2.5, there is a state ϕ r on P n,c ⊗ min P opn,c satisfying ϕ r ( p a,ij ⊗ p opb,kℓ ) = h ( P ( r ) a,ij ⊗ Q ( r ) b,kℓ ) χ r , χ r i . As the state space of any unital C ∗ -algebra is w ∗ -compact, we may take a w ∗ -limit point ϕ of the sequence( ϕ r ) ∞ r =1 . By construction, we note that ϕ ( p a,ij ⊗ p opb,kℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ℓ ≤ n . We write ϕ = h π ( · ) χ, χ i in its GNS representation, where π : P n,c ⊗ min P opn,c → B ( H ) is a unital ∗ -homomorphism and χ ∈ H is a unit vector. Applying Theorem 2.5 and restricting to P n,c , we see that τ ( y ) := h π ( y ⊗ χ, χ i defines a trace on P n,c . To establish (3), we need to show that τ ( x ⊗ y op ) = τ ( xy ) forall x, y ∈ P n,c . Notice that for each a and i, j , by Theorem 2.5 we have π ( x ⊗ p opa,ij ) χ = π ( x ⊗ op )( π (1 ⊗ p opa,ij )) χ = π ( xp a,ij ⊗ op ) χ. Then for each b, k, ℓ , we have π ( x ⊗ p opa,ij p opb,kℓ ) χ = π ( x ⊗ p opa,ij ) π (1 ⊗ p opb,kℓ ) χ = π ( x ⊗ p opa,ij ) π ( p b,kℓ ⊗ χ = π ( xp b,kℓ ⊗ p opa,ij ) χ = π ( xp b,kℓ p a,ij ⊗ χ. Since p opa,ij p opb,kℓ = ( p b,kℓ p a,ij ) op and the elements p a,ij generate P n,c , one can see that π ( x ⊗ y op ) χ = π ( xy ⊗ χ .Therefore, ϕ ( x ⊗ y op ) = τ ( xy ) for all x, y ∈ P n,c , showing that τ is an amenable trace. Setting P a,ij = π ( p a,ij ⊗ { P , ..., P c } in M n ( A ), where A = π ( P n,c ⊗
1) and P a = ( P a,ij ). Moreover, X ( a,b )( i,j ) , ( k,ℓ ) = τ ( P a,ij P ∗ b,kℓ ), so (1) implies (3).Next, we show that (3) implies (2). Let { P , ..., P c } be a PVM in M n ( A ) for a separable unital C ∗ -algebra, and let τ be an amenable trace on A such that X ( a,b )( i,j ) , ( k,ℓ ) = τ ( P a,ij P ∗ b,kℓ ) for all a, b, i, j, k, ℓ .By Theorem 3.1, we may choose a sequence of completely positive contractive maps ϕ r : A → M N ( r ) with lim r →∞ k ϕ r ( xy ) − ϕ r ( x ) ϕ r ( y ) k = 0 and lim r →∞ | tr N ( r ) ( ϕ r ( x )) − τ ( x ) | = 0 for all x, y ∈ A . Define p ( r ) a,ij = ϕ r ( P a,ij ) and set p ( r ) a = ( p ( r ) a,ij ) ∈ M n ( M N ( r ) ). Using the 2-norm on M n ( M N ( r ) ) and the fact that HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 21 P a = P a implies p ( r ) a,ij = P nk =1 ϕ r ( P a,ik P a,kj ), one sees that k ( p ( r ) a ) − p ( r ) a k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 p ( r ) a,ik p ( r ) a,kj − p ( r ) a,ij ! i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 ( ϕ r ( P a,ik ) ϕ r ( P a,kj ) − ϕ r ( P a,ik P a,kj )) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n X i,j,k =1 k ϕ r ( P a,ik ) ϕ r ( P a,kj ) − ϕ r ( P a,ik P a,kj ) k r →∞ −−−→ . Similarly, one can show that lim r →∞ k p ( r ) a p ( r ) b k = 0 whenever a = b and (cid:13)(cid:13)(cid:13)P ca =1 p ( r ) a − n (cid:13)(cid:13)(cid:13) →
0. ApplyingLemma 3.2 and dropping to a subsequence if necessary, we obtain a sequence of PVMs { q ( r )1 , ..., q ( r ) c } ⊆ M n ( M N ( r ) ) with c outputs with k p ( r ) a,ij − q ( r ) a,ij k r →∞ −−−→ a, i, j . There is a unital ∗ -homomorphism ψ r : P n,c → M N ( r ) with ψ r ( p a,ij ) = q ( r ) a,ij . As P n,c is generated by { p a,ij } a,i,j , a standard argument shows thatlim r →∞ k ϕ r ( x ) − ψ r ( x ) k = 0 for every x ∈ P n,c . This implies that | tr N ( r ) ( ϕ r ( x )) − tr N ( r ) ( ψ r ( x )) | →
0, so thatlim r →∞ | tr N ( r ) ( ψ r ( x )) − τ ( x ) | = 0. Hence, lim r →∞ tr N ( r ) ( q ( r ) a,ij ( q ( r ) b,kℓ ) ∗ ) = τ ( P a,ij P ∗ b,kℓ ) for all a, b, i, j, k, ℓ . As eachcorrelation X ( a,b ) r, ( i,j ) , ( k,ℓ ) = tr N ( r ) ( q ( r ) a,ij ( q ( r ) b,kℓ ) ∗ ) defines an element of Q sq ( n, c ), we see that X = ( τ ( P a,ij P ∗ b,kℓ ))belongs to Q sq ( n, c ). Hence, (3) implies (2).Since Q qa ( n, c ) is the closure of Q q ( n, c ), it is easy to see that (2) implies (1).To show that (3) implies (4), we use Theorem 3.1. If { P , ..., P c } is a PVM in M n ( A ) and τ is anamenable trace on A satisfying X ( a,b )( i,j ) , ( k,ℓ ) = τ ( P a,ij P ∗ b,kℓ ), then there is a unital ∗ -homomorphism ρ : A → R U that preserves τ . Setting q a,ij = ρ ( P a,ij ), we obtain projections q a = ( q a,ij ) ∈ M n ( R U ) summing to theidentity and satisfying X ( a,b )( i,j ) , ( k,ℓ ) = tr R U ( q a,ij q ∗ b,kℓ ) , which establishes (4).Lastly, we prove that (4) implies (3). Given the elements q a,ij in (4), there is a unital ∗ -homomorphism σ : P n,c → R U satisfying σ ( p a,ij ) = q a,ij . By Theorem 1.7, P n,c has the lifting property, so there is a ucpmap ζ : P n,c → ℓ ∞ ( R ) that is a lift of σ . Then Theorem 3.1 shows that τ := tr R U ◦ σ is an amenable traceon P n,c . Since τ ( p a,ij p ∗ b,kℓ ) = tr R U ( q a,ij q ∗ b,kℓ ) = X ( a,b )( i,j ) , ( k,ℓ ) , we see that (3) holds. (cid:3) The game for quantum-to-classical graph homomorphisms
In this section, we define the quantum-to-classical game for quantum-classical graph homomorphisms.Throughout our discussion, we use the bimodule perspective of quantum graphs considered by N. Weaver[51, 52] (which is a direct generalization of the non-commutative graphs considered by R. Duan, S. Severiniand A. Winter in [13], and D. Stahlke in [44]). In addition, we will see later how our framework also relatesnicely to other perspectives as well (e.g., the quantum adjacency matrix formalism of quantum graphsintroduced by B. Musto, S. Reuter and D. Verdon in [34]).For our purposes, we refer to a quantum graph as a triple ( S , M , M n ), where • M is a (non-degenerate) von Neumann algebra and M ⊆ M n ; • S ⊆ M n is an operator system; and • S is an M ′ - M ′ -bimodule with respect to matrix multiplication.In our discussion below, one can just as well use the “traceless” version of quantum graphs along thelines of D. Stahlke [44]; i.e., one replaces the second condition with the condition that S is a self-adjointsubspace of M n with Tr( X ) = 0 for every X ∈ S . This condition, combined with the bimodule property,would force S ⊆ ( M ′ ) ⊥ . Our use of the operator system approach is generally cosmetic: one can easilyadapt the quantum-classical game to traceless self-adjoint operator spaces that are M ′ - M ′ -bimodules withrespect to matrix multiplication.We begin by exhibiting a certain orthonormal basis for S with respect to the (unnormalized) trace on M n . It is from this (preferred) basis for S that we will extract our input states for the homomorphism game. Proposition 4.1.
Let K , ..., K m be non-zero subspaces of C n with K ⊕ · · · ⊕ K m = C n , such that M actsirreducibly on each K r . Let E r be the orthogonal projection of C n onto K r , for each ≤ r ≤ m . Then thereexists an orthonormal basis F of S ⊆ M n with respect to the unnormalized trace, such that • √ dim( K r ) E r ∈ F for each ≤ r ≤ m ; • F contains an orthonormal basis for M ′ ; and • For each Y ∈ F , there are unique r, s with E r Y E s = Y .Proof. Since M acts irreducibly on K r , it follows that E r ∈ M ′ . Let X be an element of S . As S is an M ′ - M ′ -bimodule, it follows that E r XE s ∈ S for all 1 ≤ r, s ≤ m . Moreover, since P mr =1 E r = 1, we have X = P mr,s =1 E r XE s . Given X, Y ∈ S , we have h E r XE s , E p Y E q i = 0 whenever r = p or s = q , where h· , ·i is the inner product with respect to the unnormalized trace on M n . We choose an orthonormal basis F r,s for E r S E s with respect to this inner product as follows. We start with an orthonormal basis for E r M ′ E s ; if r = s , then we arrange for this orthonormal basis to contain √ dim( K r ) E r . Then we extend the orthonormalbasis for E r M ′ E s to an orthonormal basis for E r S E s . We may do this since, if X ∈ S ∩ ( M ′ ) ⊥ and Y ∈ M ′ ,then h E r XE s , Y i = Tr( Y ∗ E r XE s ) = Tr( XE s Y ∗ E r ) = h X, E r Y E s i = 0 , which shows that E r ( S ∩ ( M ′ ) ⊥ ) E s ⊥ M ′ . Then F = S r,s F r,s is an orthonormal basis for S , which evidentlysatisfies all three properties. (cid:3) Definition 4.2.
We call a basis for S satisfying Proposition 4.1 as a quantum edge basis for ( S , M , M n ) . Alternatively, one could arrange for a quantum edge basis for S to also contain a normalized systemof matrix units for M ′ , since a quantum edge basis must already contain the normalized diagonal matrixunits. We will see in Theorem 4.7 that the game is independent of the quantum edge basis chosen.Once an orthonormal basis for C n has been fixed, one can define the inputs for the game using thefollowing well-known correspondence between vectors in C n ⊗ C n and matrices in M n . With respect to abasis { v , ..., v n } , this correspondence is given by the assignment v i ⊗ v j v i v ∗ j , where v i v ∗ j is the rank-oneoperator in M n such that v i v ∗ j ( x ) = h x, v j i v i for all x ∈ C n . Proposition 4.3.
Let ( S , M , M n ) be a quantum graph with quantum edge basis F . Let { v , ..., v n } be anorthonormal basis for C n that can be partitioned into bases for the subspaces K , ..., K m . For each Y α ∈ F ,write Y α = P p,q y α,pq v p v ∗ q for y α,pq ∈ C . Then the set (X p,q y α,pq v p ⊗ v q ) α ⊂ C n ⊗ C n is orthonormal.Proof. This result immediately follows from the fact that the correspondence v i ⊗ v j v i v ∗ j preserves innerproducts, when using the canonical inner product on C n ⊗ C n and the (unnormalized) Hilbert-Schmidt innerproduct on M n . (cid:3) With the notion of quantum edge bases in hand, we now define the homomorphism game for thequantum graph ( S , M , M n ) and the classical graph G . Definition 4.4.
Let ( S , M , M n ) be a quantum graph, and let { v , ..., v n } be a basis for C n that can bepartitioned into bases for the subspaces K , ..., K r . Let G be a classical (undirected) graph on c vertices,with no multiple edges and no loops. The quantum-to-classical graph homomorphism game for (( S , M , M n ) , G ) , with respect to the basis { v , ..., v n } and the quantum edge basis F , is defined as follows: • The inputs are of the form P p,q y α,pq v p ⊗ v q , where Y α := P p,q y α,pq v p v ∗ q is an element of F . Theoutputs are vertices a, b ∈ { , ..., c } = V ( G ) . There are two rules to the game: • (Adjacency rule) If Y α ⊥ M ′ , then Alice and Bob must respond with an edge in G ; i.e., a ∼ b . • (Same “vertex” rule) If Y α ∈ M ′ , then Alice and Bob must respond with the same vertex; i.e., a = b . Notice that the second rule will include a synchronicity condition: the inputs corresponding to √ dim( K r ) E r will arise in the second rule. We will see that the rule applied to these inputs will force Bob’s HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 23 projections to arise from Alice’s projections; the rule applied to the other basis elements of M ′ will be whatforces the projections to live in M ⊗ B ( H ), rather than M n ⊗ B ( H ).While the above definition of the game seems heavily basis-dependent, we will see that the existenceof winning strategies in the various models is independent of the basis { v , ..., v n } , and independent of thequantum edge basis F chosen for ( S , M , M n ). This will be a direct consequence of Theorem 4.7.We would also like to relate winning strategies for the homomorphism game to the non-commutativegraph homomorphisms in the sense of D. Stahlke [44]. For this, we first review Kraus operators in the infinite-dimensional case. Recall that a von Neumann algebra N is finite if every isometry in N is a unitary; i.e., v ∗ v = 1 implies vv ∗ = 1 in N . In this case, it is well-known that N is equipped with a normal tracial state.We will be dealing with the case when N is a finite von Neumann algebra equipped with a faithful normaltrace τ . One may always choose a faithful normal representation N ⊆ B ( H ) such that τ ( · ) = h ( · ) χ, χ i forsome unit vector χ ∈ H .Suppose that L ⊆ B ( K ) is another von Neumann algebra with faithful normal trace ρ . If Φ : L → N is a normal ucp map, then Φ ∗ : N ∗ → L ∗ is a CPTP map. In our context, L will be a finite-dimensionalvon Neumann algebra, so a ucp map Φ : L → N is automatically normal. One may choose K to be finite-dimensional and extend Φ to a ucp map from B ( K ) to B ( H ), which is still (automatically) normal. Then onemay choose Kraus operators F i such that Φ( · ) = P mi =1 F ∗ i ( · ) F i , where m is either finite or countably infinite.In the latter case, the sum converges in the SOT ∗ -topology. Then Φ ∗ : N ∗ → L ∗ = L can be written asΦ ∗ ( · ) = m X i =1 F i ( · ) F ∗ i . The interested reader can consult [12] (and the references therein) for more information on these topics.Now, we address some of the basis dependence of the game before the main theorem. The next lemmashows that, up to a unitary conjugation, the basis for C n in Definition 4.4 does not matter. Lemma 4.5.
Let ( S , M , M n ) be a quantum graph, and write C n = K ⊕ · · · ⊕ K m , where M acts irreduciblyon each K r . let G be a classical graph on c vertices, and let { v , ..., v n } be an orthonormal basis for C n that can be partitioned into bases for the subspaces K , ..., K m . Define U ∈ M n to be the unitary such that U e i = v i for all i , where { e , ..., e n } is another orthonormal basis for C n . Suppose that X ∈ Q qc ( n, c ) , and let { Y α } α be a quantum edge basis for ( S , M , M n ) . Then X is a winning strategy for the homomorphism gamefor (( S , M , M n ) , G ) with respect to { Y α } α if and only if Z := ( U ⊗ U ) ∗ X ( U ⊗ U ) is a winning strategy forthe homomorphism game for (( U ∗ S U, U ∗ M U, M n ) , G ) with respect to the quantum edge basis { U ∗ Y α U } α .Proof. Suppose that we can write X = ( h ( P a ⊗ I n )( I n ⊗ Q b )( e j ⊗ χ ⊗ e ℓ ) , e i ⊗ χ ⊗ e k i ), where ( { P a } ca =1 , { Q b } cb =1 , χ )is a qc -strategy on a Hilbert space H . Then h ( P a ⊗ I n )( I n ⊗ Q b )( v j ⊗ χ ⊗ v ℓ ) , v i ⊗ χ ⊗ v k i = h ( U ∗ P a U ⊗ I n )( I n ⊗ U ∗ Q b U )( e j ⊗ χ ⊗ e ℓ ) , e i ⊗ χ ⊗ e k i . In other words, the element Z = ( Z ( a,b ) ) := (( U ⊗ U ) ∗ X ( a,b ) ( U ⊗ U )) is a qc -correlation with respect to thebasis { v , ..., v n } . It is not hard to see that, if F is a quantum edge basis for ( S , M , M n ), then U ∗ F U isa quantum edge basis for ( U ∗ S U, U ∗ M U, M n ), since U ∗ M ′ U = ( U ∗ M U ) ′ and the Hilbert-Schmidt innerproduct is invariant under unitary conjugation. Therefore, if Y α = P p,q y α,pq v p v ∗ q belongs to F , then itsassociated input vector is P p,q y α,pq v p ⊗ v q . Then U ∗ Y α U = P p,q y α,pq U ∗ v p v ∗ q U has associated input vector P p,q y α,pq U ∗ v p ⊗ U ∗ v q = P p,q y α,pq e p ⊗ e q .Therefore, the probability of Alice and Bob outputting ( a, b ) given the input vector P p,q y α,pq v p ⊗ v q ,with respect to the correlation X , is the same as the probability of outputting ( a, b ) given the input vector P p,q y α,pq e p ⊗ e q , with respect to the correlation Z . As this equality occurs for any element of the quantumedge basis F , the desired result follows. (cid:3) Remark 4.6.
The previous remark, along with the adjacency rule, forces any winning strategy to be synchro-nous with respect to the basis { v , ..., v n } . Thus, in our main theorem, we may assume that we are dealingwith a synchronous t -strategy ( { P a } ca =1 , χ ) , where { P a } ca =1 is a PVM and χ is a faithful normal tracialstate on the von Neumann algebra generated by the entries of { P a } ca =1 . Note that conjugating { P a } ca =1 by aunitary in M n does not change the von Neumann algebra generated by the entries of the operators P a . Theorem 4.7.
Let ( S , M , M n ) be a quantum graph, let G be a classical graph on c vertices, and let t ∈{ loc, q, qa, qc } . Let N ⊆ B ( H ) be a (non-degenerate) finite von Neumann algebra, and χ ∈ H be a unit vectorsuch that τ = h ( · ) χ, χ i is a faithful (normal) trace on N . The following are equivalent: (1) There is a winning strategy ( { P a } ca =1 , χ ) from N for the homomorphism game for (( S , M , M n ) , G ) with respect to any quantum edge basis. (2) There is a winning strategy ( { P a } ca =1 , χ ) from N for the homomorphism game for (( S , M , M n ) , G ) with respect to some quantum edge basis. (3) There is a PVM { P a } ca =1 in M ⊗ N satisfying the following: if ≤ a, b ≤ c and a b in G , then P a (( S ∩ ( M ′ ) ⊥ ) ⊗ P b = 0 . (4) There is a CPTP map
Φ :
M ⊗ N ∗ → D c of the form Φ( · ) = P mi =1 F i ( · ) F ∗ i such that F i (( S ∩ ( M ′ ) ⊥ ) ⊗ N ) F ∗ j ⊆ S G ∩ ( D c ) ⊥ for all i,j , and F i ( M ′ ⊗ N ) F ∗ j ⊆ D c for all i, j. Proof.
Clearly (1) implies (2). We will show that (2) = ⇒ (3) = ⇒ (4) = ⇒ (1). Let { v , ..., v n } be anorthonormal basis for C n . Let U be the unitary such that U e i = v i for all i . Suppose that we can establish(3) for the PVM { ( U ⊗ N ) ∗ P a ( U ⊗ N ) } ca =1 and the quantum graph ( U ∗ S U, U ∗ M U, M n ). Using the factthat ( U ∗ M U ) ′ = U ∗ M ′ U , the condition in (3) can be written as( U ⊗ N ) ∗ P a ( U ⊗ N )(( U ∗ S U ) ∩ ( U ∗ M ′ U ) ⊥ ⊗ N )( U ⊗ N ) ∗ P b ( U ⊗ N ) = 0 if a b. It is not hard to see that ( U ∗ M ′ U ) ⊥ = U ∗ ( M ′ ) ⊥ U , so that the above reduces to( U ⊗ N ) ∗ P a (( S ∩ ( M ′ ) ⊥ ) ⊗ N ) P b ( U ⊗ N ) = 0 . Since U is a unitary, we obtain the desired condition for { P a } ca =1 with respect to the quantum graph( S , M , M n ). Hence, we may assume without loss of generality that v i = e i for all i .Then, given a matrix Y = P p,q y pq v p v ∗ q with associated unit vector y = P p,q y pq v p ⊗ v q , the probabilityof Alice and Bob outputting a and b respectively, given y and using the synchronous strategy ( { P a } ca =1 , χ ),is p ( a, b | y ) = * ( P a,ij P ∗ b,kℓ ) ( i,j ) , ( k,ℓ ) X p,q y pq v p ⊗ χ ⊗ v q ! , X r,s y rs v r ⊗ χ ⊗ v s + = * X i,j,k,ℓ v i ⊗ P a,ij y jℓ P ∗ b,kℓ χ ⊗ v k , X r,s y rs v r ⊗ χ ⊗ v s + = X i,j,k,ℓ (cid:10) P a,ij y jℓ P ∗ b,kℓ y ik χ, χ (cid:11) = X i,j,k,ℓ τ ( P a,ij y jℓ P b,ℓk y ik )= Tr ⊗ τ X j,ℓ P a,ij y jℓ P b,ℓk i,k ( Y ∗ ⊗ N ) = Tr ⊗ τ ( P a ( Y ⊗ N ) P b ( Y ∗ ⊗ N ))= Tr ⊗ τ ( P a ( Y ⊗ N ) P b ( Y ∗ ⊗ N ) P a ) , (4.1)where we have used the fact that P a is an orthogonal projection. Now, suppose that F = { Y α } α is a quantumedge basis for ( S , M , M n ), and suppose that ( { P a } ca =1 , χ ) is a winning strategy with respect to this quantumedge basis. If Y α ∈ M ′ , then Equation 4.1 and faithfulness of the trace gives P a ( Y α ⊗ N ) P b = 0 whenever a = b . Then P a ( Y α ⊗ N ) P a = c X b =1 P a ( Y α ⊗ N ) P b = P a ( Y α ⊗ N ) c X b =1 P b ! = P a ( Y α ⊗ N ) . HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 25
Similarly, P a ( Y α ⊗ N ) P a = ( Y α ⊗ N ) P a . Hence, P a commutes with Y α ⊗ N whenever Y α ∈ M ′ . Thisshows that P a ∈ ( M ′ ⊗ N ) ′ ∩ ( M n ⊗ N ) = M ⊗ B ( H ) ∩ ( M n ⊗ N ) = M ⊗ N .Similarly, if Y α ⊥ M ′ , then the rules of the game and the faithfulness of the trace force P a ( Y α ⊗ N ) P b =0 whenever a b , which shows that (3) holds.Now we show that (3) implies (4). If (3) holds, then there is a projection-valued measure { P a } ca =1 in M ⊗ N such that P a ( Y ⊗ N ) P b = 0 for all X ∈ S ∩ ( M ′ ) ⊥ and a b . Then the map Ψ : D c → M ⊗ N given by Ψ( E kk ) = P k is a unital ∗ -homomorphism. Since D c is finite-dimensional, Ψ is normal. Hence, wemay find Kraus operators F , F , ... in B ( C n ⊗ H , C c ) such thatΨ( · ) = m X i =1 F ∗ i ( · ) F i , where m is either finite or ℵ . In the infinite case, these sums converge in the SOT ∗ -topology. Then Ψ = Θ ∗ for a CPTP map Θ : M ∗ ⊗ N ∗ = M ⊗ N ∗ → D c given byΘ( · ) = m X i =1 F i ( · ) F ∗ i . Given Y ∈ S , we set Z a,b,i,j = E aa F i ( Y ⊗ N ) F ∗ j E bb . Then we haveTr( Z a,b,i,j Z ∗ a,b,i,j ) = Tr( E aa F i ( Y ⊗ N ) F ∗ j E bb F j ( Y ∗ ⊗ N ) F ∗ i E aa )= Tr( E aa F i ( Y ⊗ N ) F ∗ j E bb F j ( Y ∗ ⊗ N ) F ∗ i )= Tr( F ∗ i E aa F i ( Y ⊗ N ) F ∗ j E bb F j ( Y ∗ ⊗ N ))= h F ∗ i E aa F i ( Y ⊗ N ) F ∗ j E bb F j , ( Y ⊗ N ) i . Summing over all i, j , we obtain m X i,j =1 Tr( Z a,b,i,j Z ∗ a,b,i,j ) = * m X i =1 F ∗ i E aa F i ! ( Y ⊗ N ) m X i =1 F ∗ j E bb F j ! , Y ⊗ N + = h Ψ( E aa )( Y ⊗ N )Ψ( E bb ) , ( Y ⊗ N ) i = h P a ( Y ⊗ N ) P b , Y ⊗ N i . First, suppose that Y ∈ S ∩ ( M ′ ) ⊥ . Since each term in the sum is positive, Tr( Z a,b,i,j Z ∗ a,b,i,j ) = 0. Byfaithfulness of the trace, we obtain E aa F i ( Y ⊗ N ) F ∗ j E bb = 0 for all i, j and a b . Since F i ( Y ⊗ N ) F ∗ j ∈ M c ,we can write F i ( Y ⊗ N ) F ∗ j = c X a,b =1 E aa F i ( Y ⊗ N ) F ∗ j E bb = X a ∼ b E aa F i ( Y ⊗ N ) F ∗ j E bb . Note that S G ∩ ( D c ) ⊥ = P a ∼ b C E ab = P a ∼ b E aa M c E bb , so we see that F i ( Y ⊗ N ) F ∗ j ∈ S G ∩ ( D c ) ⊥ for all i, j and X ∈ ( S ∩ ( M ′ ) ⊥ ).Next, consider the case when Y ∈ M ′ . For a = b , letting Z a,b,i,j = E aa F i ( Y ⊗ N ) F ∗ j E bb as before,we have that m X i,j =1 Tr( Z a,b,i,j Z ∗ a,b,i,j ) = h P a ( Y ⊗ N ) P b , Y ⊗ N i = 0 , since P b commutes with Y ⊗ N and P a P b = 0. Therefore, E aa F i ( Y ⊗ N ) F ∗ j E bb = 0 for a = b . One findsthat F i ( Y ⊗ N ) F ∗ j = c X a,b =1 E aa F i ( Y ⊗ N ) F ∗ j E bb = c X a =1 E aa F i ( Y ⊗ N ) F ∗ j E aa ∈ D c . Thus, (4) holds.Lastly, suppose that (4) holds; we will obtain a winning strategy for the game. Suppose that Φ : M ∗ ⊗ N ∗ → D c is a CPTP map of the form Φ( · ) = P mi =1 R i ( · ) R ∗ i , such that R i ( Y ⊗ N ) R ∗ j ∈ S G ∩ D ⊥ c forall 1 ≤ i, j ≤ m and Y ∈ S ∩ ( M ′ ) ⊥ , and R i ( Y ⊗ N ) R ∗ j ∈ D c for all Y ∈ M ′ . Then Φ ∗ ( · ) = P mi =1 R ∗ i ( · ) R i defines a normal ucp map from D c to M ⊗ N . Let P a = Φ ∗ ( E aa ) = P mi =1 R ∗ i E aa R i for each 1 ≤ a ≤ c .Since Φ ∗ is UCP, { P a } ca =1 is a POVM in M ⊗ N . As in the proof of (2) = ⇒ (3), by considering the POVM { U ∗ P a U } ca =1 and the quantum graph ( U ∗ S U, U ∗ M U, M n ), we may assume without loss of generality that v i = e i for all i . We will show that X ( a,b )( i,j ) , ( k,ℓ ) = ( τ ( P a,ij P ∗ b,kℓ )) defines a winning t -strategy for the quantumgraph homomorphism game for (( S , M , M n ) , G ). Let Y ∈ S . Then for each a, b , m X i,j =1 Tr( E aa ( R i ( Y ⊗ N ) R ∗ j E bb R j ( Y ∗ ⊗ N ) R ∗ i E aa ) = m X i,j =1 h R ∗ i E aa R i ( Y ⊗ N ) R ∗ j E bb R j , Y ⊗ N i = h Φ ∗ ( E aa )( Y ⊗ N )Φ ∗ ( E bb ) , Y ⊗ N i = h P a ( Y ⊗ N ) P b , Y ⊗ N i . (4.2)By condition (4), if Y ∈ M ′ and a = b , then h P a ( Y ⊗ N ) P b , Y ⊗ N i = 0. The special case of Y = 1 M andfaithfulness of the trace shows that P a P b = 0 for a = b . This forces { P a } ca =1 to be a PVM. In the case when Y ∈ S ∩ ( M ′ ) ⊥ and a b in G , then Equation (4.2) also evaluates to 0, by condition (4).Therefore, using Equation (4.1), if { Y α } α is a quantum edge basis for ( S , M , M n ), Y α has associatedunit vector y α and Y α ⊥ M ′ , then by equation (4.1), p ( a, b | y α ) = h P a ( Y α ⊗ N ) P b , Y α i = 0 . if a b. If Y α belongs to M with associated unit vector y α , then p ( a, b | y α ) = h P a ( Y ⊗ N ) P b , Y ⊗ N i = 0 as well.This shows that ( { P a } ca =1 , χ ) defines a winning strategy for the homomorphism game for ( S , M , M n ) and G with respect to any quantum edge basis, completing the proof. (cid:3) The next theorem will show that, in the loc model, condition (4) of Theorem 4.7 is a direct gen-eralization of Stahlke’s notion of graph homomorphism from a non-commutative graph (i.e. a quantumgraph with M = M n ) into a classical graph [44]. Hence, Theorems 4.7 and 4.9 show that these two notionsare equivalent in this context. Moreover, these definitions are equivalent in the q model, and have naturalgeneralizations to the qa and qc models. In fact, we have proved something stronger: if we start with aprojection-valued measure { P a } ca =1 whose block entries are in a tracial von Neumann algebra ( N , τ ), where τ is faithful and normal, then either all four conditions of Theorem 4.7 are satisfied by the PVM, or none ofthe four conditions are satisfied. Notice that we needed to start with a PVM and a faithful trace for this tohappen.Working towards Theorem 4.9, we first show that, if the ancillary algebra N is finite-dimensional,then the second part of condition (4) in Theorem 4.7 can be dropped. Lemma 4.8.
Let ( S , M , M n ) be a quantum graph, and let G be a classical graph on c vertices. Let N ⊆ B ( H ) be a finite-dimensional von Neumann algebra with faithful trace τ = h ( · ) χ, χ i for some χ ∈ H . Suppose that Φ :
M ⊗ N ∗ → D c is a CPTP map given by Φ( · ) = P mi =1 F i ( · ) F ∗ i , such that F i (( S ∩ ( M ′ ) ⊥ ) ⊗ N ) F ∗ j ⊆ S G ∩ ( D c ) ⊥ for all i,j . Then there exists a PVM { P a } ca =1 in M ⊗ N such that P a (( S ∩ ( M ′ ) ⊥ ) ⊗ N ) P b = 0 if a b, and P a ( M ′ ⊗ N ) P b = 0 if a = b. Proof.
As in the proof of (4) = ⇒ (1) in Theorem 4.7, Φ ∗ ( · ) = P mi =1 F ∗ i ( · ) F i is a normal ucp map from D c to M ⊗ N , and Q a = Φ ∗ ( E aa ) = P mi =1 F ∗ i E aa F i defines a POVM { Q a } ca =1 ⊆ M ⊗ N such that, whenever Y ∈ S and 1 ≤ a, b ≤ c , m X i,j =1 Tr( E aa F i ( Y ⊗ N ) F ∗ j E bb F j ( Y ∗ ⊗ N ) F ∗ i E aa ) = h Q a ( Y ⊗ N ) Q b , Y ⊗ N i . By assumption, whenever a b in G , the above quantity is 0. Now, the set of POVMs in M ⊗ N with c outputs is convex and closed in ( M ⊗ N ) c ; moreover, the extreme points are the PVMs in M ⊗ N with c outputs. Since dim( M ⊗ N ) < ∞ , an application of Carath´eodory’s theorem shows that we can write Q a = s X v =1 t v P ( v ) a , HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 27 for some s ∈ N , where t , ..., t s > P sv =1 t v = 1, and the set { P ( v ) a } ca =1 is a PVM in M ⊗ N , for each1 ≤ v ≤ s . Thus, whenever Y ∈ S ∩ ( M ′ ) ⊥ and a b in G , we have0 = h Q a ( Y ⊗ N ) Q b , Y ⊗ N i = s X v,w =1 t v t w h P ( v ) a ( Y ⊗ N ) P ( w ) b , Y ⊗ N i . Since P ( v ) a and P ( w ) b are projections, each quantity in the sum must be non-negative. As t v > v , wemust have h P ( v ) a ( Y ⊗ N ) P ( w ) b , Y ⊗ N i = 0 for all 1 ≤ v, w ≤ s , for all Y ∈ S ∩ ( M ′ ) ⊥ , and for all a b .Choosing some value of v and setting P a = P ( v ) a for all 1 ≤ a ≤ c , we obtain a PVM in M ⊗ N with theproperty that h P a ( Y ⊗ N ) P b , Y ⊗ N i = 0 for all Y ∈ S ∩ ( M ′ ) ⊥ and for all a b. Since P a and P b are projections, this quantity is equal to Tr ⊗ τ ( P a ( Y ⊗ N ) P b ( Y ⊗ N ) ∗ P a ) = 0. Byfaithfulness of trace, P a ( Y ⊗ N ) P b = 0 whenever Y ∈ S ∩ ( M ′ ) ⊥ and a b in G .Lastly, since { P a } ca =1 is a PVM in M ⊗ N , each P a commutes with M ′ ⊗ N . Hence, if a = b and Y ∈ M ′ , then P a ( Y ⊗ N ) P b = P a P b ( Y ⊗ N ) = 0 , since P a P b = 0. Hence, the PVM { P a } ca =1 satisfies all of the required properties. (cid:3) In the following discussion, we write ( S , M , M n ) t −→ G to mean that there is a winning t -strategy for thegraph homomorphism game for (( S , M , M n ) , G ).We will also write ( S , M , M n ) → G if ( S , M , M n ) loc −−→ G .Our choice of notation is since, from a loc-homomorphism, one can always obtain a graph homomorphism.Using Theorem 4.7 and the characterizations of synchronous correlations, we obtain the followingtheorem: Theorem 4.9.
Let ( S , M , M n ) be a quantum graph and let G be a classical graph on c vertices. (1) ( S , M , M n ) loc −−→ G if and only if there is a CPTP map Φ :
M → D c of the form Φ( · ) = P mi =1 F i ( · ) F ∗ i such that F i ( S ∩ ( M ′ ) ⊥ ) F ∗ j ⊆ S G ∩ ( D c ) ⊥ for all i, j. (2) ( S , M , M n ) q −→ G if and only if there is d ∈ N and a CPTP map Φ :
M ⊗ M d → D c of the form Φ( · ) = P mi =1 F i ( · ) F ∗ i such that F i (( S ∩ ( M ′ ) ⊥ ) ⊗ I d ) F ∗ j ⊆ S G ∩ ( D c ) ⊥ for all i, j. (3) ( S , M , M n ) qa −→ G if and only if there is a CPTP map Φ :
M ⊗ ( R U ) ∗ → D c of the form Φ( · ) = P mi =1 F i ( · ) F ∗ i such that F i (( S ∩ ( M ′ ) ⊥ ) ⊗ R U ) F ∗ j ⊆ S G ∩ ( D c ) ⊥ for all i, j, and F i ( M ′ ⊗ R U ) F ∗ j ⊆ D c for all i, j. (4) ( S , M , M n ) qc −→ G if and only if there is a von Neumann algebra N , a faithful normal trace τ on N ,and a CPTP map Φ :
M ⊗ N ∗ → D c of the form Φ( · ) = P mi =1 F i ( · ) F ∗ i such that F i (( S ∩ ( M ′ ) ⊥ ) ⊗ N ) F ∗ j ⊆ S G ∩ ( D c ) ⊥ for all i, j, and F i ( M ′ ⊗ N ) F ∗ j ⊆ D c for all i, j. Proof.
We consider the case t = loc first. If ( S , M , M n ) loc −−→ G , then there is a winning loc -strategy for thehomomorphism game from ( S , M , M n ) into G . Since Q sloc ( n, c ) is convex and non-empty, one may obtainan extreme point in Q sloc ( n, c ) that wins the game with probability 1. Applying Corollary 2.8, there is arealization of this correlation using a PVM { P a } ca =1 in M = M ⊗ C . Then the result follows by condition(4) of Theorem 4.7 with N = C . The converse of (1) holds by an application of Lemma 4.8 with N = C ,which yields a PVM in M that satisfies condition (3) of Theorem 4.7.The argument is similar for t = q . Indeed, if there is a winning strategy for the homomorphismgame in the q -model, then an application of Corollary 2.9 shows that there is a winning q -strategy using an extreme point in Q sq ( n, c ), which can be realized using projections whose entries are in M d , for some d .Then condition (4) of Theorem 4.7 with N = M d yields the desired CPTP map. The converse holds by anapplication of Lemma 4.8 with N = M d .We note that (3) holds because of Theorem 3.3. Condition (4) is achieved using the following well-known trick: if A is a unital, separable C ∗ -algebra with tracial state τ , and if π τ : A → B ( H τ ) is the GNSrepresentation of τ with cyclic vector ξ , then π τ ( A ) ′′ is a finite von Neumann algebra and h ( · ) ξ, ξ i is a faithfulnormal trace on π τ ( A ) ′′ . We leave the details to the reader. (cid:3) Remark 4.10.
Condition (4) of Theorem 4.7 describes quantum homomorphisms between ( S , M , M n ) and G in terms of a quantum channel (i.e. a CPTP map). In the cases of t = loc and t = q , we can drop thesecond requirement of the channel concerning M ′ . However, we are not sure if this condition is necessaryfor t = qa or t = qc , where one might require infinitely many extreme points to describe the POVM. Weleave this question open, and hope to address it in future work. For synchronous games with classical inputs and classical outputs, J.W. Helton, K.P. Meyer, V.I.Paulsen and M. Satriano constructed a universal ∗ -algebra for the game, generated by self-adjoint idempo-tents whose products were 0 when the related pair of outputs was not allowed [22]. One can define a game ∗ -algebra in our context as follows. Definition 4.11.
Let ( S , M , M n ) be a quantum graph and let G be a classical graph on c vertices. The game ∗ -algebra for the homomorphism game for (( S , M , M n ) , G ) , denoted A ( Hom (( S , M , M n ) , G )) , is theuniversal ∗ -algebra generated by entries { p a,ij : 1 ≤ a ≤ c, ≤ i, j ≤ n } subject to the relations • p a = ( p a,ij ) i,j satisfies p a = p a = p ∗ a and P ca =1 p a = I n , where I n is the n × n identity matrix; • p a (( S ∩ ( M ′ ) ⊥ ) ⊗ p b = 0 for each a b ; and • p a ( M ′ ⊗ p b = 0 for each a = b .We say that the algebra exists if = 0 in the algebra. As one might expect, we obtain the following characterizations of the various flavors of winningstrategies for the homomorphism game in terms of ∗ -homomorphisms of the game algebra. Theorem 4.12.
Let ( S , M , M n ) be a quantum graph and let G be a classical graph. (1) ( S , M , M n ) loc −−→ G ⇐⇒ there is a unital ∗ -homomorphism A ( Hom (( S , M , M n ) , G )) → C . (2) ( S , M , M n ) q −→ G if and only if there is a unital ∗ -homomorphism A ( Hom (( S , M , M n ) , G )) → M d for some d ∈ N . (3) ( S , M , M n ) qa −→ G if and only if there is a unital ∗ -homomorphism A ( Hom (( S , M , M n ) , G )) → R U . (4) ( S , M , M n ) qc −→ G if and only if there is a unital ∗ -homomorphism A ( Hom (( S , M , M n ) , G )) → C ,where C is a tracial C ∗ -algebra. (5) ( S , M , M n ) alg −−→ G if and only if A ( Hom (( S , M , M n ) , G )) = 0 . One can also define C ∗ -homomorphisms and hereditary homomorphisms of quantum graphs intoclassical graphs. We write ( S , M , M n ) C ∗ −−→ G provided that there is a unital ∗ -homomorphism π : A (Hom(( S , M , M n ) , G )) → B ( H ) , for some Hilbert space H . (Equivalently, by the Gelfand-Naimark theorem, one may simply require that thegame algebra has a representation into some unital C ∗ -algebra.)For hereditary homomorphisms, we recall the concept of a hereditary (unital) ∗ -algebra. Recall that aunital ∗ -algebra A is said to be hereditary if, whenever x , ..., x n ∈ A are such that x ∗ x + · · · + x ∗ n x n = 0,then x = x = · · · = x n = 0. If one defines A + as the cone generated by all elements of the form x ∗ x for x ∈ A , then A being hereditary is equivalent to having A + ∩ ( −A + ) = { } . Every unital C ∗ -algebra ishereditary as a unital ∗ -algebra.With this background in hand, we will write ( S , M , M n ) hered −−−−→ G provided that there is a unital ∗ -homomorphism from A (Hom(( S , M , M n ) , G )) into a (non-zero) hereditary unital ∗ -algebra. One has thefollowing sequence of implications for these types of homomorphism:( S , M , M n ) → G = ⇒ ( S , M , M n ) q −→ G = ⇒ ( S , M , M n ) qa −→ G = ⇒ ( S , M , M n ) qc −→ G = ⇒ ( S , M , M n ) C ∗ −−→ G = ⇒ ( S , M , M n ) hered −−−−→ G = ⇒ ( S , M , M n ) alg −−→ G. (4.3) HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 29
Our notions of homomorphisms reduce to the analogous types of homomorphisms for classical graphsin the case when ( S , M , M n ) is a classical graph. Recall [50] that, for a classical graph G on n vertices, thegraph operator system S G (or classical quantum graph) is defined as S G = span ( { E ii : 1 ≤ i ≤ n } ∪ { E ij : i ∼ j in G } ) . Note that S G is naturally a quantum graph in the previous sense if we regard it as a bimodule over thediagonal algebra D n = D ′ n ⊆ M n . In [51] it is shown that quantum graphs of the form ( S , D n , M n ) are inone-to-one correspondence with classical graphs on n vertices. Corollary 4.13.
Let G and H be classical graphs on n and c vertices, respectively. Suppose that t ∈{ loc, q, qa, qc, C ∗ , hered, alg } . Then G t −→ H if and only if ( S G , D n , M n ) t −→ H .Proof. We will show that the algebra A (Hom( G, H )) from [22] is isomorphic to A (Hom(( S G , D n , M n ) , H )).The former algebra is the universal unital ∗ -algebra generated by self-adjoint idempotents e x,a such that P nx =1 e x,a = 1, e x,a e x,b = 0 if a = b , and e x,a e y,b = 0 if x ∼ y in G but a b in H . Since D n = D ′ n , thelatter algebra is the universal unital ∗ -algebra generated by elements p a,ij such that p a = ( p a,ij ) ∈ M n ( A )is a self-adjoint idempotent with P ca =1 p a = I n , p a (( S G ∩ ( D n ) ⊥ ) ⊗ p b = 0 whenever a b in H , and p a ( D n ⊗ p b = 0 whenever a = b . Since E ii ∈ D n , using the fact that p a ( D n ⊗ p b = 0 for a = b , we obtain p a ( E ii ⊗ p a = c X b =1 p a ( E ii ⊗ p b = p a ( E ii ⊗ . Similarly, p a ( E ii ⊗ p a = ( E ii ⊗ p a , so that E ii ⊗ p a . It follows that p a,ij = 0 whenever i = j . Since p a = p a = p ∗ a , we see that p a,ii = p a,ii = p ∗ a,ii . For 1 ≤ a ≤ c and 1 ≤ x ≤ n , we define q x,a = p a,xx . Then q x,a is a self-adjoint idempotent and P ca =1 q x,a = 1 for all 1 ≤ x ≤ n . Note that, if x ∼ y in G but a b in H , then q x,a q y,b = p a,xx p b,yy = p a ( E xy ⊗ p b = 0 , since E xy ∈ S G ∩ ( D n ) ⊥ and a b in H . Similarly, if a = b , then q x,a q x,b = p a ( E xx ⊗ p b = 0 since E xx ∈ D n .By the universal property of A (Hom( G, H )), there is a unital ∗ -homomorphism π : A (Hom( G, H )) →A (Hom(( S G , D n , M n ) , H )) such that π ( e x,a ) = q x,a for all x, a .Conversely, in A (Hom( G, H )), one can construct the n × n matrices f a = ( f a,ij ) with f a,ij = 0 for i = j and f a,ii = e a,i . Then evidently f a = f a = f ∗ a and P ca =1 f a = I n . Since e x,a e x,b = 0 for a = b , wesee that f a ( E xx ⊗ f b = 0 if a = b . Since D n = span { E xx : 1 ≤ x ≤ n } , it follows that f a ( D n ⊗ f b = 0for a = b . Similarly, it is not hard to see that f a ( E xy ⊗ f b = 0 whenever x ∼ y in G but a b in H . Bythe universal property, there is a unital ∗ -homomorphism ρ : A (Hom(( S G , D n , M n ) , H )) → A (Hom( G, H ))such that ρ ( p a,ij ) = f a,ij . Evidently ρ and π are mutual inverses on the generators, so we conclude that thealgebras are ∗ -isomorphic. The result follows. (cid:3) It is known that some of the implications in (4.3) cannot be reversed. While there are many examplesof classical graphs G and H with G q −→ H but G H , Theorem 5.11 will show that ( M n , M , M n ) q −→ K dim( M ) but ( M n , M , M n ) K dim( M ) whenever M is non-abelian. Here, K dim( M ) denotes the (classical) completegraph on dim( M ) vertices. Using the work of S.-J. Kim, V.I. Paulsen and C. Schafhauser on synchronousbinary constraint (syncBCS) games, there is a graph G and a number m such that K m qa −→ G holds, but K m q −→ G does not hold, where G denotes the graph complement of G [28, Corollary 5.5]. The other knownseparation is that alg −−→ does not imply hered −−−−→ . For example, K alg −−→ K holds, but K hered −−−−→ K does nothold [22]. This result will be generalized to quantum graphs.5. Coloring quantum graphs
A special case of the homomorphism game is when the target graph is the classical complete graph K c on c vertices. In this case, the resulting game is a generalization of the coloring game for classical graphs. Definition 5.1.
Let t ∈ { loc, q, qs, qa, qc, C ∗ , hered, alg } . Let ( S , M , M n ) be a quantum graph. We define χ t (( S , M , M n )) = min { c ∈ N : ( S , M , M n ) t −→ K c } , and we define χ t (( S , M , M n )) = ∞ if ( S , M , M n ) t −→ K c for all c ∈ N . Due the inclusions of the models, we always have χ loc (( S , M , M n )) ≥ χ q (( S , M , M n )) ≥ χ qa (( S , M , M n )) ≥ χ qc (( S , M , M n )) ≥ χ C ∗ (( S , M , M n )) ≥ χ hered (( S , M , M n )) ≥ χ alg (( S , M , M n )) . As a consequence of Corollary 4.13, whenever G is a classical graph, we have χ t ( G ) = χ t (( S G , D n , M n )).This result is well known (see e.g., [38]). As χ loc ( G ) is the (classical) chromatic number of a classical graph G , we sometimes use the notation χ (( S , M , M n )) for χ loc (( S , M , M n )). Example 5.2.
Let S = span { I, E ij : i = j } ⊆ M n , which is a quantum graph on M n . It is known [27] that χ (( S , M n , M n )) = n . Here, we will show that χ qc (( S , M n )) = n as well, which shows that χ t (( S , M n )) = n for any t ∈ { loc, q, qa, qc } .Evidently the basis F = { I, E ij : i = j } is a quantum edge basis for ( S , M n , M n ). Now, suppose that P , ..., P c are projections in M n ( B ( H )) with P a ( E kℓ ⊗ I ) P a = 0 for all 1 ≤ a ≤ c and 1 ≤ k = ℓ ≤ n . Awinning strategy in the qc -model for coloring ( S , M n ) with c colors would mean that there is a trace τ onthe algebra generated by the P a,ij ’s and that p ( a, a | e i ⊗ e j ) = 0 if i = j. This implies that τ ( P a,ii P ∗ a,jj ) = 0 for all i = j. By taking a quotient by the kernel of the GNS representation of the trace, we may assume that τ is faithful.Then by faithfulness of τ and positivity of P a,jj , we have P a,ii P a,jj = 0 for all i = j . Now, choose i = j .Notice that, for each i , the set { P a,ii } is a POVM on H . Moreover, for any a, b ∈ { , ..., c } , p ( a, b | e i ⊗ e j ) = τ ( P a,ii P ∗ b,jj ) = τ ( P a,ii P b,jj ) . Thus, the only information relevant to Alice and Bob winning the game is the correlation ( τ ( P a,ii P b,jj )) a,b,i,j ∈ C sqc ( n, c ). By faithfulness, this forces each P a,ii to be a projection. By the synchronous condition, the pre-vious equation and faithfulness of the trace, we obtain P a,ii P a,jj = 0 = P a,ii P b,ii whenever a = b and i = j . Therefore, ( τ ( P a,ii P b,jj )) a,b,i,j ∈ C bsqc ( n, c ); that is, the correlation is bisynchronous in the sense of [37]. By [37], we must have c ≥ n . Therefore, χ qc ( S , M n , M n ) ≥ n . It follows that χ t ( S , M n , M n ) = n for every t ∈ { loc, q, qa, qc } .5.1. Quantum Complete Graphs and Algebraic Colorings.
In this section, we consider quantumcomplete graphs ; that is, graphs of the form ( M n , M , M n ), where M ⊆ M n is a non-degenerate vonNeumann algebra. We show that χ t (( M n , M , M n )) = dim( M ) for all t ∈ { q, qa, qc, C ∗ , hered } . In contrast,we will see that χ loc (( M n , M , M n )) is finite if and only if M is abelian; in the case when M is abelian,we recover known results on colorings for the (classical) complete graph on dim( M ) vertices. The algebraicmodel for colorings is known to be very wild. At the end of this section, we will extend a surprising resultof [22]: in the algebraic model, any quantum graph can be 4-colored.We start with a simple proposition on unitary equivalence that we will use throughout this section. Proposition 5.3.
Let
M ⊆ M n be a non-degenerate von Neumann algebra. Then there is a unitary U ∈ M n such that U ∗ M U = L mr =1 C I n r ⊗ M k r . Moreover, for any t ∈ { loc, q, qa, qs, qc, C ∗ , hered, alg } , we have χ t (( M n , M , M n )) = χ t M n , m M r =1 C I n r ⊗ M k r , M n ! . Proof.
The existence of the unitary U is a consequence of the theory of finite-dimensional C ∗ -algebras. Itis not hard to see that ( U ∗ M U ) ′ = U ∗ M ′ U . Now, an element X ∈ M n belongs to M ′ if and only ifTr( XY ) = 0 for all Y ∈ M ′ . This statement is equivalent to having Tr(( U ∗ XU )( U ∗ Y U )) = 0 for all Y ∈ M ′ , since U is unitary. It follows that U ∗ ( M ′ ) ⊥ U = ( U ∗ M ′ U ) ⊥ .Now, suppose that { P a } ca =1 ⊆ M n ⊗ A is a collection of self-adjoint idempotents summing to I n ⊗ A ,where A is a unital ∗ -algebra. Then it is evident that P a (( M ′ ) ⊥ ⊗ A ) P a = 0 if and only if e P a (( U ∗ M ′ U ) ⊥ ⊗ A ) e P a = 0, where e P a = ( U ∗ ⊗ A ) P a ( U ⊗ A ). Similarly, if a = b , then P a ( M ′ ⊗ A ) P b = 0 if and HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 31 only if e P a (( U ∗ M ′ U ) ⊗ A ) e P b = 0. Thus, there is a bijective correspondence between algebraic c -colorings of( M n , M , M n ) and algebraic c -colorings of ( M n , L mr =1 C I n r ⊗ M k r , M n ). This yields the equality of chromaticnumbers for t = alg ; the other cases are similar. (cid:3) The different chromatic numbers satisfy a certain monotonicity as well.
Proposition 5.4. If ( S , M , M n ) and ( T , M , M n ) are quantum graphs with S ⊆ T , then χ t (( S , M , M n )) ≤ χ t (( T , M , M n )) . Proof.
We deal with the t = alg case; all the other cases are similar. If ( T , M , M n ) has no algebraic coloring,then χ alg (( T , M , M n )) = ∞ , so the desired result holds. Otherwise, let A be a (non-zero) unital ∗ -algebra.Suppose that { P a } ca =1 are self-adjoint idempotents in M n ( A ) such that P ca =1 P a = I n , P a (( T ∩ ( M ′ ) ⊥ ) ⊗ A ) P a = 0 for all a , and P a ( M ′ ⊗ A ) P b = 0 for all a = b . Then evidently P a (( S ∩ ( M ′ ) ⊥ ) ⊗ A ) P a =0 as well, so the self-adjoint idempotents form an algebraic c -coloring of ( S , M , M n ). This shows that χ alg (( S , M , M n )) ≤ χ alg (( T , M , M n )). The proof for the other models is the same. (cid:3) By Proposition 5.4, to establish that every quantum graph has a finite quantum coloring, it suffices toconsider quantum complete graphs. First, we look at ( M n , M n , M n ), the quantum complete graph. Whilewe will have an alternative quantum coloring of this quantum graph from Theorem 5.6, the protocol givenin Theorem 5.5 is minimal for ( M n , M n , M n ) in terms of the dimension of the ancillary algebra. Moreover,it gives a foretaste of the protocol that we use for the quantum complete graph ( M n , M , M n ) when M isnot isomorphic to a matrix algebra. Theorem 5.5.
Let d, k ∈ N , and let n = dk . Let M = C I d ⊗ M k . Then χ q (( M n , M , M n )) ≤ k .Proof. We construct our projections from the canonical orthonormal basis for C k ⊗ C k that consists ofmaximally entangled vectors; that is, the basis of the form ϕ a,b = 1 √ k k − X p =0 exp (cid:18) πia ( p + b ) k (cid:19) e b + p ⊗ e p , where addition in the indices of the vectors is done modulo k . (See [17] for example.) We define projectionsin M ⊗ M , for all 1 ≤ a, b ≤ n , by P ( a,b ) = 1 k k − X p,q =0 exp (cid:18) πia ( p − q ) k (cid:19) I d ⊗ E b + p,b + q ⊗ I d ⊗ E pq . Since the set { ϕ ( a,b ) } na,b =1 is orthonormal, it is not hard to see that { P ( a,b ) } na,b =1 is a family of mutuallyorthogonal projections. Moreover, P na,b =1 P ( a,b ) = I d ⊗ I k ⊗ I d ⊗ I k . With respect to M n , ( M ′ ) ⊥ is spannedby elements of the form E xy ⊗ E vw and E xy ⊗ ( E vv − E ww ) for 1 ≤ x, y ≤ d and 1 ≤ v, w ≤ k with v = w .For Y = E xy ⊗ E vw ⊗ ( I d ⊗ I k ), one computes P ( a,b ) Y P ( a,b ) and obtains1 k k − X p,q,p ′ ,q ′ =0 exp (cid:18) πia ( p + p ′ − q − q ′ ) k (cid:19) E xy ⊗ E b + p,b + q E vw E b + p ′ ,b + q ′ ⊗ I d ⊗ E pq E p ′ q ′ . For a term in the above sum to be non-zero, one requires that b + q = v , w = b + p ′ , and q = p ′ . Equivalently,a term in the sum is non-zero only when q = p ′ and b + q = v = w . Hence, if v = w , then the above sum is0. In the case when v = w , one obtains1 k k − X p,q ′ =0 exp (cid:18) πia ( p − q ′ ) k (cid:19) E xy ⊗ E b + p,b + q ′ ⊗ I d ⊗ E pq ′ . The above expression does not depend on v , so we conclude that, for all 1 ≤ v, w ≤ k , P ( a,b ) ( E xy ⊗ E vv ⊗ I d ⊗ I k ) P ( a,b ) = P ( a,b ) ( E xy ⊗ E ww ⊗ I d ⊗ I k ) P ( a,b ) . This shows that P ( a,b ) ( X ⊗ I d ⊗ I k ) P ( a,b ) = 0 whenever X = E xy ⊗ E vw or X = E xy ⊗ ( E vv − E ww ) for v = w . As such elements span ( M ′ ) ⊥ , we see that P ( a,b ) ( X ⊗ I d ⊗ I k ) P ( a,b ) = 0 ∀ X ∈ ( M ′ ) ⊥ . Finally, we show that P ( a,b ) ( M ′ ⊗ I d ⊗ I k ) P ( a ′ ,b ′ ) = 0 whenever ( a, b ) = ( a ′ , b ′ ). If Y ∈ M ′ , then Y ⊗ ( I d ⊗ I k )commutes with each P ( a,b ) , since P ( a,b ) ∈ M ⊗ ( I d ⊗ M k ). Therefore, if ( a, b ) = ( a ′ , b ′ ), we have P ( a,b ) ( Y ⊗ ( I d ⊗ I k )) P ( a ′ ,b ′ ) = P ( a,b ) P ( a ′ ,b ′ ) ( Y ⊗ ( I d ⊗ I k )) = 0 . Putting all of these equations together, we see that there is a representation of the game algebra π : A (Hom(( M n , M , M n ) , K k )) → C I d ⊗ M k ⊗ M k . Therefore, χ q (( M n , M , M n )) ≤ k , which yields theclaimed result. (cid:3) For a general complete quantum graph ( M n , M , M n ), we require a slightly different approach. Theprotocol in the previous proof is used in the context of quantum teleportation, and essentially arises from theuse of a “shift and multiply” unitary error basis for M n [17, 53]. To give a dim( M )-coloring for ( M n , M , M n )in the q -model, we will use what we refer to as a “global shift and local multiply” framework. Theorem 5.6.
Let M be a non-degenerate von Neumann algebra in M n . For the quantum complete graph ( M n , M , M n ) , we have χ q (( M n , M , M n )) ≤ dim( M ) .Proof. Up to unitary equivalence in M n , we may write M = L mr =1 ( C I n r ⊗ M k r ), where n = P mr =1 n r k r .We will exhibit a PVM in M ⊗ M d , with d = dim( M ) = P mr =1 k r , satisfying the properties of a quantumcoloring for ( M n , M , M n ). For notational convenience, we set k = 0, and index our colors a by the set { , , ..., d − } . For 1 ≤ r ≤ m and 1 ≤ i ≤ k r , we set γ ( r, i ) = r − X p =0 k p + ( i − k r . For 0 ≤ a ≤ d −
1, we define P a = L mr =1 I n r ⊗ P ( r ) a , where P ( r ) a = ( P ( r ) a,ij ) k r i,j =1 ∈ M k r ( M d ) is given by P ( r ) a,ij = 1 k r ω ( i − j ) ar k r − X ℓ =0 E γ ( r,i )+ ℓ + a,γ ( r,j )+ ℓ + a . Here, the addition in the indices is done modulo d , while ω r is a primitive k r -th root of unity. By our choiceof the operators P a , it is immediate that P a belongs to M ⊗ M d for each 0 ≤ a ≤ d − P da =1 P a = I n ⊗ I d . For each 1 ≤ r ≤ d and 1 ≤ i, j ≤ k r , d − X a =0 P ( r ) a,ij = 1 k r d − X a =0 k r − X ℓ =0 ω ( i − j ) ar E γ ( r,i )+ ℓ + a,γ ( r,j )+ ℓ + a . If 1 ≤ p, q ≤ d and p = q , then E pp (cid:16)P da =1 P ( r ) a,ij (cid:17) E qq will either be 0, or it will be a sum of terms where γ ( r, i ) + ℓ + a = p and γ ( r, j ) + ℓ + a = q , which yields ℓ + a = p − γ ( r, i ) and ℓ + a = q − γ ( r, j ).To each of the k r values of ℓ , there is a unique a for which ℓ + a = p − γ ( r, i ) and ℓ + a = q − γ ( r, j );moreover, the possible values of a are all distinct modulo k r . Therefore, the coefficient on E pq will be ofthe form k r P k r − f =0 ω ( i − j ) fr . This sum is 0 if i = j . Hence, P da =1 P ( r ) a,ij = 0 if i = j . By a similar argument, E pp ( P da =1 P ( r ) a,ii ) E pp = k r (cid:16) k r E pp (cid:17) = E pp . Hence, P da =1 P ( r ) a,ii = I k r . Putting these facts together, we seethat P da =1 P a = I n ⊗ I d .Next, we check that each P a is an orthogonal projection. By definition of P ( r ) a,ij , it is easy to see that P ∗ a = P a for all a . To compute P a , we note that P a = m M r =1 I n r ⊗ ( P ( r ) a ) , HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 33 so it suffices to show that each P ( r ) a is an idempotent in M k r . We compute P ( r ) a,vj P ( r ) a,jw = 1 k r ω ( v − w ) ar k r − X ℓ,p =0 E γ ( r,v )+ ℓ + a,γ ( r,j )+ ℓ + a E γ ( r,j )+ p + a,γ ( r,w )+ p + a = 1 k r ω ( v − w ) ar k r − X ℓ =0 E γ ( r,v )+ ℓ + a,γ ( r,w )+ ℓ + a = 1 k r P ( r ) a,vw . Since this happens for all 1 ≤ v, w ≤ k r , it follows that P ( r ) a,vw = P k r j =1 P ( r ) a,vj P ( r ) a,jw . Therefore, P ( r ) a is anidempotent, so P a is an orthogonal projection.Now, we show that P a (( M ′ ) ⊥ ⊗ I d ) P a = 0 for all a . We note that M ′ = L mr =1 M n r ⊗ C I k r . Hence,( M ′ ) ⊥ is spanned by all elements of the form E ij , where E ij L mr =1 M n r ⊗ M k r , and elements from each M n r ⊗ M k r of the form E ij ⊗ E vw and E ij ⊗ ( E vv − E ww ), where v = w . By a consideration of blocks, if E ij does not belong to L mr =1 M n r ⊗ M k r , then for any 1 ≤ k, ℓ ≤ n , either P a,ki is 0 or P a,jℓ is 0, which yields P a,ki P a,jℓ = 0. Since the ( k, ℓ ) entry of P a ( E ij ⊗ I d ) P a is P a,ki P a,jℓ , we see that P a ( E ij ⊗ I d ) P a = 0 in thiscase. Next, we suppose that 1 ≤ v, w ≤ k r with v = w , and consider an element E ij ⊗ E vw ∈ M n r ⊗ M k r .Note that | γ ( r, v ) − γ ( r, w ) | ≥ k r . Since the projection P a acts as the identity on M n r , we obtain P a ( E ij ⊗ E vw ⊗ I d ) P a = E ij ⊗ k r X k,ℓ =1 E kℓ ⊗ P ( r ) a,kv P ( r ) a,wℓ . We observe that, if 1 ≤ v, w, k, ℓ ≤ k r and v = w , then P ( r ) a,kv P ( r ) a,wℓ = 1 k r ω ( k − v + w − ℓ ) ar k r − X p,q =0 E γ ( r,k )+ p + a,γ ( r,v )+ p + a E γ ( r,w )+ q + a,γ ( r,ℓ )+ q + a = 0 , since 0 ≤ p, q ≤ k r − | γ ( r, v ) − γ ( r, w ) | ≥ k r . Thus, P a ( E ij ⊗ E vw ⊗ I d ) P a = 0 in this case. Lastly, wedeal with E ij ⊗ ( E vv − E ww ) ∈ M n r ⊗ M k r . We can write P a ( E ij ⊗ ( E vv − E ww ) ⊗ I d ) P a = E ij ⊗ k r X k,ℓ =1 ( P ( r ) a,kv P ( r ) a,vℓ − P ( r ) a,kw P ( r ) a,wℓ ) = 0 , since P ( r ) a,kv P ( r ) a,vℓ = k r P ( r ) a,kℓ for any r and 1 ≤ k, v, ℓ ≤ k r .Putting all of these facts together, we conclude that { P a } da =1 ⊆ M ⊗ M d is a quantum d -coloring of( M n , M , M n ), as desired. (cid:3) Remark 5.7.
In general, the above ancillary algebra is not the minimal choice. Indeed, in the case of thequantum complete graph ( M n , M n , M n ) , one can verify that the smallest possible ancillary algebra is M n ,whereas the above construction would have used M n when M = M n . In the abelian case, one can alwaysuse C as the ancilla, using our results on correlations in the loc model. In general, the ancillary algebra canalways be taken to be M d for some d ; however, we are not sure what the minimal choice for d is in general. Next, we will show that χ hered (( M n , M , M n )) ≥ dim( M ), which will show that, for every t ∈{ q, qa, qc, C ∗ , hered } , we have χ t (( M n , M , M n )) = dim( M ). Moreover, we will show that dim( M )-coloringsof ( M n , M , M n ) in the hereditary model must arise from trace-preserving ∗ -homomorphisms Ψ : D dim( M ) →M⊗A . More precisely, we equip D dim( M ) with its canonical uniform trace ψ D dim( M ) satisfying ψ D dim( M ) ( e a ) = M ) for all 1 ≤ a ≤ dim( M ). We also equip the von Neumann algebra M ≃ L mr =1 C I n r ⊗ M k r with itscanonical “Plancherel” trace given by ψ M = m M r =1 k r n r dim( M ) Tr n r k r ( · ) . Then we will show that the ∗ -homomorphism Ψ satisfies the following trace covariance condition:( ψ M ⊗ id)Ψ( x ) = ψ D dim( M ) ( x )1 A ( x ∈ D dim( M ) ) . We thus establish that the hereditary coloring number for any complete quantum graph ( M n , M , M n ) isdim( M ), and moreover, the above trace-preserving condition shows that any minimal hereditary coloringinduces a quantum version of isomorphism between ( M n , M , M n ) and the complete graph K dim( M ) ondim( M ) vertices. Here, the notion of a “quantum isomorphism” means a quantum isomorphism betweenquantum graphs in the sense of [2], when using an ancillary hereditary unital ∗ -algebra A . This result can beinterpreted as a quantum analogue of the (classically obvious) fact that any minimal coloring of a completegraph K c is automatically a graph isomorphism K c → K c .We consider the case when M ≃ C I d ⊗ M k first. Lemma 5.8.
Let d, k ∈ N and let n = dk . Consider the quantum graph ( M n , M , M n ) with M = C I d ⊗ M k .Let A be a unital ∗ -algebra, and let { P , ..., P c } ∈ M ⊗ A be a family of mutually orthogonal projections suchthat P ca =1 P a = I dk ⊗ A and P a ( X ⊗ A ) P a = 0 for all X ∈ ( M ′ ) ⊥ . Then for each a , the element R a = kd dim( M ) ( Tr dk ⊗ id A )( P a ) is a self-adjoint idempotent in A , and P ca =1 R a = k A .Proof. Since M = I d ⊗ M k , we have M ′ = M d ⊗ I k and n = dk . Now, let 1 ≤ v, w ≤ k with x = y , and let1 ≤ i, j ≤ d . Then E ij ⊗ ( E vv − E ww ) belongs to ( M ′ ) ⊥ , so we must have P a ( E ij ⊗ ( E vv − E ww ) ⊗ A ) P a = 0 ∀ ≤ a ≤ c. Similarly, E ij ⊗ E vw is in ( M ′ ) ⊥ , so P a ( E ij ⊗ E vw ⊗ A ) P a = 0 ∀ ≤ a ≤ c. Note that P a ∈ M ⊗ A = I d ⊗ M k ⊗ A , so P a = P kp,q =1 P dx =1 E xx ⊗ E pq ⊗ P a,x,pq , with the property that P a,x,pq = P a,y,pq for any 1 ≤ x, y ≤ d . For simplicity, we set P a,pq = P a,x,pq for any 1 ≤ x ≤ d . The quantityon the left of the above is exactly k X p,q =1 E ij ⊗ E pq ⊗ P a,pv P a,wq so this says that P a,pv P a,wq = 0 and P a,pv P a,vq = P a,pw P a,wq . Now, since P a is a projection, we have P a,pq = P kv =1 P a,pv P a,vq = kP a,pv P a,vq for all p, q . In particular, P a,vv = kP a,vv . By scaling, we see that kP a,vv is a self-adjoint idempotent. Similarly, since P a,pv P a,wq = 0 if v = w , we see that P a,vv P a,ww = 0.Therefore, { kP a,vv } nv =1 is a collection of mutually orthogonal projections in A .Next, we set R a = P kv =1 kP a,vv for each 1 ≤ a ≤ c . Then R a is a self-adjoint idempotent. We seethat c X a =1 R a = c X a =1 k X v =1 kP a,vv = k X v =1 k A = k A , which completes the proof. (cid:3) Now, we deal with the case of a general quantum complete graph.
Theorem 5.9.
Let ( M n , M , M n ) be a quantum complete graph. Let A be a hereditary ∗ -algebra, and let { P a } ca =1 ⊆ M⊗A be a hereditary c -coloring of ( M n , M , M n ) . Then c ≥ dim( M ) . Moreover, if c = dim( M ) ,then for each ≤ a ≤ dim( M ) we have ( ψ M ⊗ id A )( P a ) = 1dim( M ) 1 A . Proof.
Up to unitary equivalence, we may write M = L mr =1 C I n r ⊗ M k r . Then M ′ = m M r =1 M n r ⊗ C I k r . Define E r = 0 ⊕· · ·⊕ I n r ⊗ I k r ⊕ ⊕· · ·⊕
0, which belongs to M ′ ∩M . Then defining e P a = ( E r ⊗ A ) P a ( E r ⊗ A ) ∈ ( E r ME r ) ⊗ A , we obtain a family of mutually orthogonal projections whose sum is E r . Since E r is central in M , we see that ( E r ME r ) ′ = E r M ′ E r , while E r M n E r = M n r k r . It is evident that X ∈ B ( E r C n ) ∩ ( E r M ′ E r ) ⊥ HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 35 if and only if X = E r X E r and X ⊥ M ′ in M n . Therefore, for X ∈ B ( E r C n ) ∩ ( E r M ′ E r ) ⊥ and 1 ≤ a ≤ c ,one has e P a ( X ⊗ A ) e P a = ( E r ⊗ A ) P a ( E r X E r ⊗ A ) P a ( E r ⊗ A ) = 0 , using the fact that E r X E r = X and X belongs to M ′ . Therefore, { e P a } ca =1 is a hereditary coloring of thequantum complete graph ( M n r k r , E r ME r , M n r k r ).Since E r ME r = C I n r ⊗ M k r , by Lemma 5.8, we see that R ( r ) a := k r n r (Tr n r k r ⊗ id A )( e P a ) is a self-adjointidempotent in A for each 1 ≤ a ≤ c and 1 ≤ r ≤ m . Moreover, P ca =1 R ( r ) a = k r A .Next, we claim that R ( r ) a R ( s ) a = 0 if r = s . To show this orthogonality relation, it suffices to show that P a,xx P a,yy = 0 whenever P a,xx is a block from ( E r ME r ) ⊗ A and P a,yy is a block from ( E s ME s ) ⊗ A . If x and y are chosen in this way, then the matrix unit E xy in M n satisfies E r ( E xy ) E s = E xy and E p E xy E q = 0for all other pairs ( p, q ). It is not hard to see that E xy belongs to ( M ′ ) ⊥ , so that P a ( E xy ⊗ A ) P a = 0.Considering the ( x, y )-block of this equation gives P a,xx P a,yy = 0. It follows that R ( r ) a R ( s ) a = 0 for r = s .Since { R ( r ) a } mr =1 is a collection of mutually orthogonal projections in A , the element R a := P mr =1 R ( r ) a is a self-adjoint idempotent in A for each a . Considering blocks, it is not hard to see that c X a =1 R a = c X a =1 m X r =1 R ( r ) a = m X r =1 k r A = dim( M )1 A . Since R a is a self-adjoint idempotent, so is 1 A − R a . Their sum is given by c X a =1 (1 A − R a ) = c A − c X a =1 R a = ( c − dim( M ))1 A . It follows that c ≥ dim( M ), since the sum above is a sum of positives and A is hereditary.Now, if c = dim( M ), then the above sum of positives in A is 0, which forces 1 A − R a = 0 for all a .Hence, R a = 1 A . Since R a = P mr =1 R ( r ) a and R ( r ) a = k r n r (Tr n r k r ⊗ id A )( P a ), we see that m X r =1 k r n r (Tr n r k r ⊗ id A )( P a ) = 1 A . Therefore, ( ψ M ⊗ id A )( P a ) = m X r =1 k r dim( M ) n r (Tr n r k r ⊗ id A )( P a ) = 1dim( M ) 1 A . (cid:3) Remark 5.10.
In essence, Theorem 5.9 proves that any q -coloring of ( M n , M , M n ) with dim( M ) colorsinduces a quantum isomorphism between the quantum graph ( M n , M , M n ) and the classical graph K dim( M ) .This isomorphism occurs because any such coloring with ancillary algebra A yields a (necessarily injective)unital ∗ -isomorphism π : D dim( M ) → M ⊗ A satisfying the properties of a quantum graph homomorphism,with the additional property that ( ψ M ⊗ id A ) ◦ π = π ◦ ψ D dim( M ) . In contrast to the case of q -colorings, the existence of a loc -coloring for a complete quantum graph isequivalent to the von Neumann algebra being abelian. Theorem 5.11.
Let
M ⊆ M n be a non-degenerate von Neumann algebra. Then χ loc (( M n , M , M n )) is finiteif and only if M is abelian. In particular, if M is non-abelian, then χ (( M n , M , M n )) = χ q (( M n , M , M n )) .Proof. Suppose that there is a c -coloring of ( M n , M , M n ) in the loc -model. Up to unitary equivalence,we write M = L mr =1 C I n r ⊗ M k r . We may choose projections P a ∈ M such that P ca =1 P a = I n and P a (( M ′ ) ⊥ ) P a = 0 for all a . Let R a = P mr =1 k r n r Tr n r k r ( P a ) as in the proof of the last theorem. Each R a is anidempotent in C ; hence, either R a = 0 or R a = 1. We know that P ca =1 R a = dim( M ), so exactly dim( M )of the R a ’s are non-zero. Since R a is given by a trace on M which is faithful, having R a = 0 implies that P a = 0. Hence, by discarding any projections P a for which R a = 0, we may assume without loss of generalitythat R a = 1 for all a , and that c = dim( M ).Let E r be the orthogonal projection onto the copy of C I n r ⊗ M k r inside of M = L mr =1 C I n r ⊗ M k r . Then, as before, the PVM {E r P a E r } dim( M ) a =1 yields a classical dim( M )-coloring for ( M n r k r , C I n r ⊗ M k r , M n r k r ). We will show that k r = 1. By the same argument as above, by discarding values of a for which k r n r Tr n r k r ( E r P a E r ) = 0, we may assume that there are exactly k r non-zero projections E r P a E r that yield a k r -classical coloring for ( M n r k r , C I n r ⊗ M k r , M n r k r ). Set e P a = E r P a E r . By Theorem 5.9, for each a , we have k r n r Tr n r k r ( e P a ) = 1. Notice that k r e P a = I n r ⊗ k r Q a for some projection k r Q a ∈ M k r . Hence, Tr k r ( k r Q a ) = 1.Let λ , ..., λ k r be the eigenvalues of k r Q a in M k r . Since each λ i ∈ { , } and P k r i =1 λ i = Tr k r ( k r Q a ) = 1,there is exactly one λ i that is non-zero. Hence, Q a is rank one. The sum over all non-zero Q a gives I k r , andeach Q a is rank one. Hence, the number of a for which Q a is non-zero must be k r . Since we assumed thatthis number is k r , we must have k r = k r . Since k r >
0, we have k r = 1. Since r was arbitrary, we see that M = L mr =1 C I n r ⊗ M k r = L mr =1 C I n r is abelian.Conversely, suppose that M is abelian. Then the proof of Theorem 5.9 yields projections P a ∈M ⊗ M d , where d = dim( M ), such that P da =1 P a = I n ⊗ I d and P a ( X ⊗ I d ) P a = 0 whenever X ∈ ( M ′ ) ⊥ .Moreover, the projections obtained in this case satisfy P a,ij P b,kℓ = P b,kℓ P a,ij for all 1 ≤ a, b ≤ d and1 ≤ i, j, k, ℓ ≤ n . Thus, the entries of the projections P a must ∗ -commute with each other, so the C ∗ -algebrathey generate is abelian. Since there is a d -coloring for ( M n , M , M n ) with an abelian ancilla, this impliesthat χ loc (( M n , M , M n )) ≤ d . (cid:3) Using the monotonicity of colorings and the results above on quantum complete graphs, we see thatevery quantum graph has a finite quantum coloring. As a result, we obtain the following generalization of atheorem from [22].
Theorem 5.12.
Let ( S , M , M n ) be any quantum graph. Then χ alg (( S , M , M n )) ≤ .Proof. Suppose that χ alg ( S , M , M n ) ≤ c for some c < ∞ . Then A (Hom(( S , M , M n ) , K c )) exists. We willlet p , ..., p c be the canonical self-adjoint idempotents in the matrix algebra M n ( A (Hom(( S , M , M n ) , K c )).By [22], there is an algebraic homomorphism K c → K . Thus, there are self-adjoint idempotents f a,v in A (Hom( K c , K )) for 1 ≤ a ≤ c and 1 ≤ v ≤ P v =1 f a,v = 1 for all a and f a,v f b,v = 0 whenever a = b . Define q v,ij = c X a =1 p a,ij ⊗ f a,v ∈ A (Hom(( S , M , M n ) , K c )) ⊗ A (Hom( K c , K )) . Then n X k =1 q v,ik q v,kj = n X k =1 c X a =1 p a,ik ⊗ f a,v ! c X b =1 p b,kj ⊗ f b,v ! = n X k =1 c X a,b =1 p a,ik p b,kj ⊗ f a,v f b,v = c X a =1 n X k =1 p a,ik p a,kj ⊗ f a,v = c X a =1 p a,ij ⊗ f a,v = q v,ij . Therefore, q v = ( q v,ij ) is an idempotent for each v . Similarly, one can see that q ∗ v = q v (that is, q ∗ v,ij = q v,ji )and P v =1 q v,ij is 0 if i = j and 1 if i = j . Let X = ( x ij ) ∈ M n . Letting 1 ⊗ q v ( X ⊗ ⊗ q w = n X k,ℓ =1 q v,ik x kℓ q w,ℓj i,j = n X k,ℓ =1 c X a,b =1 p a,ik x kℓ p b,ℓj ⊗ f a,v f b,w i,j . If X ∈ ( M ′ ) ⊥ and a = b , then the above sum becomes q v ( X ⊗ ⊗ q v = n X k,ℓ =1 c X a =1 p a,ik x kℓ p a,ℓj ⊗ f a,v = c X a =1 p a ( X ⊗ p a ⊗ f a,v = 0 , HE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 37 by definition of A (Hom(( S , M , M n ) , K c )). If X ∈ M ′ and a = b , then P nk,ℓ =1 p a,ik x kℓ p b,ℓj is the ( i, j ) entryof p a ( X ⊗ p b = 0. Thus, if v = w , then Equation (5.1) reduces to q v ( X ⊗ ⊗ q w = n X k,ℓ =1 c X a =1 p a,ik x kℓ p a,ℓj ⊗ f a,v f a,w i,j = 0 , since f a,v f a,w = 0 for v = w . Therefore, letting r v,ij be the canonical generators of A (Hom(( S , M , M n ) , K )),we obtain a unital ∗ -homomorphism π : A (Hom(( S , M , M n ) , K )) → A (Hom(( S , M , M n ) , K c )) ⊗ A (Hom( K c , K r )) ,r v,ij q v,ij . The latter algebra is non-zero, so A (Hom(( S , M , M n ) , K )) = { } . Thus, χ alg (( S , M , M n )) ≤ (cid:3) References [1] W. Arveson,
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