THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME Introduction Required Here. in This Work, We Will Extend the Graph Homomorphism

THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME

MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

Abstract. We consider a graph homomorphism 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 winning strategies in the various quantum models for the game correspond to quantum homomorphisms in the sense of B. Musto, D.J. Reutter and D. Verdon [26], and directly generalize the notion of non-commutative graph homomorphisms due to D. Stahlke [33]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [18]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the 4-coloring game for a quantum graph is always non-trivial, extending a result of [18].

Introduction required here. In this work, we will extend the graph homomorphism game to the case of homomorphisms from a quantum graph to a classical graph. This game will have quantum inputs, in the sense that the referee will initialize the state space Cn ⊗ Cn; Alice has access to the left copy and Bob has access to the right copy of Cn. Alice and Bob are allowed to share an (entanglement) resource space H in some prepared state ψ. After receiving the input ϕ on Cn ⊗ Cn, they can perform measurements on the triple tensor product Cn ⊗ H ⊗ Cn, and respond with classical outputs to the referee based on the measurement.

1. Quantum Input, Classical Output Correlations In this section, we develop a bit of general theory on non-local games with quantum questions and classical answers. These have already been used in the two-output context of quantum XOR games [15,31]. Suppose that we have a non-local game with quantum questions (on Cn ⊗ Cn) and classical answers in {1, 2, ..., c}. We will define the different notions of strategies for the different models. It is easiest, for our purposes, 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 n HA and HB, a positive operator-valued measure (POVM) {P1, ..., Pc} on C ⊗ HA, a POVM n {Q1, ..., Qc} on HB ⊗ C , and a state χ ∈ HA ⊗ HB. A quantum spatial strategy, or a qs-strategy, is given in the same way as a q- strategy, except that we no longer assume that HA and HB are finite-dimensional. A quantum commuting strategy, or a qc-strategy, is given by a single Hilbert space n n H, a POVM {P1, ..., Pc} on C ⊗ H, a POVM {Q1, ..., Qc} on H ⊗ C , and a state χ ∈ H, with the property that (Pa ⊗ In)(In ⊗ Qb) = (In ⊗ Qb)(Pa ⊗ In) for all a. Note: it is helpful to understand the above commutation condition in terms of block matrices. For 1 ≤ a ≤ c, one may write Pa = (Pa,ij) where each Pa,ij ∈ B(H). Similarly, 1 2 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

we may write Qb = (Qb,k`) where each Qb,k` ∈ B(H). With this in mind, we can deﬁne a local strategy, or a classical strategy, as a quantum commuting strategy where the set ∗ of entries Pa,ij and Qb,k` generate a commutative C -algebra. The following proposition gives a convenient form of the commutation relation. This result can be found in other contexts (regarding “unitary correlations”) in [7] and [17].

n n Proposition 1.1. Let P = (Pij) ∈ B(C ⊗ H) and Q = (Qk`) ∈ B(H ⊗ C ) be positive operators. Then [P ⊗ In,In ⊗ Q] = 0 if and only if [Pij,Qk`] = 0 for all 1 ≤ i, j, k, ` ≤ n. Suppose that the referee initializes Cn ⊗ Cn in the state ϕ. For a quantum strategy, the probability that Alice outputs a and Bob outputs b is given by

p(a, b|ϕ) = h(Pa ⊗ Qb)(ϕ ⊗ χ), ϕ ⊗ χi, n n where by ϕ ⊗ χ we mean the (permuted) state in C ⊗ (HA ⊗ HB) ⊗ C rather than on n n C ⊗C ⊗(HA ⊗HB). For a quantum commuting strategy, we simply replace HA ⊗HB with H and (Pa ⊗ Qb) with (Pa ⊗ In)(In ⊗ Qb). We note that this deﬁnition of the probability of outputs can easily be extended to other states in Cn ⊗ Cn that may not be included in the deﬁnition of the game. This fact is because the probabilities corresponding to Alice and Bob’s strategy are encoded entirely in the correlation associated to their strategy. The correlation associated to the strategy (P1, ..., Pc,Q1, ..., Qc, χ) with n-dimensional quantum inputs and c classical outputs is given by the tuple

(a,b) c2 X := (X(i,j),(k,`)) = ((h(Pa,ij ⊗ Qb,k`)χ, χi)i,j,k,`)a,b ∈ (Mn ⊗ Mn) .

We will let Qq(n, c) be the set of all correlations of this form that arise from quantum strategies. In other words, c2 Qq(n, c) = {(h(Pa,ij ⊗ Qb,k`)χ, χi)1≤i,j,k,`≤n, ⊆ (Mn ⊗ Mn) , 1≤a,b≤c

where HA and HB are finite-dimensional Hilbert spaces; Pa,ij ∈ B(HA) are such that Pa = Pc (Pa,ij) ∈ Mn(B(HA)) is positive with a=1 Pa = In; Qb,k` ∈ B(HB) are such that Qb = Pc (Qb,k`) ∈ Mn(B(HB)) are positive with b=1 Qb = In, and χ ∈ HA ⊗ HB is a state. Similarly, we will let Qqs(n, c) be the set of all quantum spatial correlations (where HA and HB may not be finite-dimensional), and we let Qqc(n, c) be the set of all quantum commuting correlations of the above form (where, as usual, we replace the tensor product framework with the commuting operator framework). Keeping the analogy with the sets Ct(n, k) corresponding to classical inputs, we will also define Qqa(n, c) as the closure of Qq(n, c) in the norm topology. Lastly, we define Qloc(n, c) as the set of all quantum com- ∗ muting correlations where C ({Pa,ij,Qb,k` : 1 ≤ a, b ≤ c, 1 ≤ i, j, k, ` ≤ n}) is a commutative C∗-algebra. Since each of the correlation sets above are defined in terms of POVMs, an argument involving direct sums shows that Qt(n, c) is convex for all t ∈ {loc, q, qs, qa, qc}. Moreover, Qqa(n, c) is closed (by definition) and an application of Theorem 1.12 shows that Qqc(n, c) is closed. One can also show that Qloc(n, c) is closed, but we will not need this fact. Define the operator system Qn,c as the universal operator system generated by c sets of 2 n entries qa,ij with the property that the matrix Qa = (qa,ij) is positive in Mn(Qn,c) for each Pc 1 ≤ k ≤ c and a=1 Qa = In. The correlations above are directly related to states on tensor ∗ squares of Qn,c. For these facts, we will use some facts about Qn,c and its C -envelope. For ∗ 2 convenience, we define Pn,c to be the universal unital C -algebra generated by c sets of n THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 3

entries pa,ij such that Pa = (pa,ij) is an orthogonal projection in Mn(Pn,c) for each 1 ≤ a ≤ c. ∗ 2 Similarly, we define Bn,c as the universal unital C -algebra generated by n entries uij with the property that U = (uij) ∈ Mn(Bn,c) is a unitary of order c. The latter algebra is an ∗ obvious quotient of the Brown algebra Bn, which is the universal C -algebra generated by the entries of an n × n unitary. The algebra Bn first appeared in [2]. c Proposition 1.2. Let H be a Hilbert space, and let {Qa}a=1 be a POVM in B(H). Then c there is a PVM {Pa}a=1 in Mc+1(B(H)) such that, if E11 is the first diagonal matrix unit in Mc+1, then (E11 ⊗ IH)Pa(E11 ⊗ IH) = Qa for all 1 ≤ a ≤ c.  1  2 Q1  .  Proof. We define V =  .  ∈ Mc,1(B(H)). Then V is an isometry, so  1  2 Qc √ V I − VV ∗ U = ∈ M (B(H)) 0 −V ∗ c+1 ∗ ∗ is a unitary. Define Pa = U (Eaa ⊗ IH)U for 1 ≤ a ≤ c − 1, and define Pc = U ((Ecc + c c+1 Ec+1,c+1) ⊗ IH)U. Then {Pa}a=1 is a PVM in Mc+1(B(H)). Write U = (Uk`)k,`=1 where each Uk` ∈ B(H). The (1, 1) entry of Pa is given by 1 1 ∗ 2 2 (Pa)11 = Ua1Ua1 = (Qa )(Qa ) = Qa, as desired. As a result of Proposition 1.2, we obtain the desired dilation property for POVMs over Mn(B(H)).

Proposition 1.3. Let H be a Hilbert space, and let qa,ij ∈ B(H) for 1 ≤ i, j ≤ n and 1 ≤ a ≤ c (c+1) c be such that {Qa}a=1 is a POVM in Mn(B(H)), where Qa = (qa,ij). Let V : H → H be the isometry sending H to the first direct summand of H(c+1). Then there are operators c pa,ij ∈ Mc+1(B(H)) such that {Pa}a=1 is a PVM in Mn(Mc+1(B(H))), where Pa = (pa,ij), ∗ and V pa,ijV = qa,ij for all 1 ≤ i, j ≤ n and 1 ≤ a ≤ c. c Proof. We can regard {Qa}a=1 as a POVM in Mn(B(H)). By Proposition 1.2, there is a PVM c {Sa}a=1 in Mc+1(Mn(B(H))) such that the (1, 1) entry of Sa is Qa. Performing a canonical shuffle Mc+1(Mn(B(H))) ' Mn(Mc+1(B(H))) on each Sa, we obtain operators pa,ij ∈ Mc+1(B(H)) such that the (1, 1)-entry of pa,ij is qa,ij, and Pa = (pa,ij) ∈ Mn(Mc+1(B(H))) Pc are projections with a=1 Pa = I, completing the proof. 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. c Alternatively, one can consider n POVMs {Pa,x}a=1 for 1 ≤ x ≤ n on H as a single n POVM on C ⊗H by setting Qa = Pa,1⊕· · ·⊕Pa,n. Then one applies Proposition 1.3 to obtain c+1 a single PVM in Mn(H ⊗ C ); however, the projections may no longer be block-diagonal, so they may not induce a family of n PVMs in B(H ⊗ Cc+1). In the case that n = 1, one can dilate a POVM with c outputs in B(H) to a PVM with c outputs in B(H ⊗ Cc), which is more optimal than Proposition 1.3. On the other hand, as soon as n ≥ 2, the dilation 4 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

of Proposition 1.3 will be more optimal, since the general dilation of n POVMs requires an inductive argument. Proposition 1.5. Let n, c ∈ N. ∗ ∗ (1) Cenv(Qn,c) is canonically ∗-isomorphic to the universal C -algebra Pn,c generated by 2 c sets of n entries pa,ij with the property that Pa = (pa,ij) is a projection in Mn(Pn,c) Pc with a=1 Pa = In. (2) Pn,c 'Bn,c via the map c X a ω Pa ← U, a=1 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 (Zc) ' c ∗n`∞ (see, for example, [19], [12] or [27]). Since Pa is a projection in Pn,c, it is positive. Hence, there is a ucp map ϕ : Qn,c → Pn,c such that ϕ(qa,ij) = pa,ij. If we represent Qn,c ⊆ B(H) for some Hilbert space H, then by Proposition 1.3, there is a unital ∗-homomorphism π : Pn,c → Mc+1(B(H)) such that compressing to the first coordinate yields the map pa,ij → qa,ij. ∗ Hence, ϕ is a complete order isomorphism. This shows that Pn,c is a C -cover for Qn,c. By the universal property of C∗-envelope [14], there is a unique, surjective unital ∗- ∗ homomorphism πe : Pn,c → Cenv(Qn,c) such that ρ(pa,ij) = qa,ij for all 1 ≤ i, j ≤ n and ∗ 1 ≤ a ≤ c. As each Pa is a projection in Pn,c, the matrix Qa = (qa,ij) ∈ Mn(Cenv(Qn,c)) is a projection as well. We will show that ρ is injective by constructing an inverse. We assume ∗ that Pn,c is faithfully represented as a C -algebra of operators on a Hilbert space K. Then ∗ the map ϕ : Qn,c → Pn,c above extends to a ucp map σ : Cenv(Qn,c) → B(K) by Arveson’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 : Cenv(Qn,c) → B(L) is a unital ∗-homomorphism. With respect to the decomposition L = K ⊕ K⊥, one has ϕ(q ) ∗ p ∗ β(q ) = a,ij = a,ij . a,ij ∗ ∗ ∗ ∗ (n) Thus, after a shuffle, one may write β (Qa) = (β(qa,ij)) as ϕ(n)(Q ) ∗ P ∗ a = a . ∗ ∗ ∗ ∗ (n) As Qa is a projection, so is β (Qa). But Pa is a projection as well, so the off-diagonal blocks must be 0. Therefore, reversing the shuffle yields p 0 β(q ) = a,ij . a,ij 0 ∗

It follows that the multiplicative domain of ϕ contains qa,ij for each 1 ≤ i, j ≤ n and ∗ 1 ≤ a ≤ c; as these elements generate Cenv(Qn,c), σ must be a ∗-homomorphism. Since πe and σ are mutual inverses on the generators, they must be mutual inverses on the whole ∗ algebras. Hence, πe is injective, so that Cenv(Qn,c) 'Pn,c. Combining part (2) of Proposition 1.5 and Proposition 1.3, one can obtain a similar dilation corresponding to a block unitary of order c. Indeed, if T = (Tij) ∈ Mn(B(H)) is Pc a a contraction that can be written as T = a=1 ω Qa, where ω is a primitive c-th root of c unity and {Qa}a=1 is a POVM in Mn(B(H)), then one can dilate T to a unitary U = (Uij) ∈ THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 5

Mn(Mc+1(B(H))) of order c, such that the (1, 1) block of each Uij is Tij. It will sometimes be more convenient to use this form of the dilation, rather than the dilation of the POVM to a PVM. ∗ First, we show that Pn,c has the lifting property. Recall that 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 positive lift ϕe : A → B of ϕ. As noted in [3, Lemma 13.1.2], when A is unital, one need only deal with the case when B is unital and ϕ, ϕe are unital. On the way to proving that Pn,c has the lifting property, we will need the following fact. We include a proof for convenience.

∗ Proposition 1.6. Let B be a unital C -algebra, J be an ideal in B, and p1, ..., pc ∈ B/J be Pc projections with a=1 pa = 1B/J . Let q : B → J be the canonical quotient map. Then there Pc are positive elements pe1, ..., pec in B such that a=1 pea = 1B and q(pea) = pa for all 1 ≤ a ≤ c. ∞ Proof. The assumption implies that there is a unital ∗-homomorphism π : `c → B/J such ∞ that π(ea) = pa for each a. As `c is separable and nuclear, the Choi-Eﬀros lifting theorem ∞ [6] gives a ucp lift ϕ : `c → B of π. Deﬁning pea = ϕ(ea) concludes the proof.

Theorem 1.7. Pn,c has the lifting property. Proof. This proof is similar in nature to results from [3, 19]. First, suppose that B is a ∗ unital C -algebra, J is an ideal in B and π : Pn,c → B/J is a ∗-homomorphism. Then (n) (n) π = idn ⊗ π : Mn(Pn,c) → Mn(B)/Mn(J ) is a ∗-homomorphism. Define Qa = π (Pa). By Proposition 1.6, one can find a POVM Qe1, ..., Qec in Mn(B) that is a lift of Q1, ..., Qc. Next, we apply Proposition 1.3 and compress to the (1, 1) corner to obtain a ucp map γ : Pn,c → B given by γ(pa,ij) = Qea,ij for all a, i, j. As γ is a lift of π on the generators, a multiplicative domain argument establishes that γ is a lift of π. Now we deal with the general case. Let ϕ : Pn,c → B/J be a ucp map. Since Pn,c is separable, one can restrict if necessary and assume without loss of generality that B is separable. Then we apply Kasparov’s dilation theorem [20] to ϕ: letting K = K(`2) denote the compact operators and M(A) denote the multiplier algebra of a (separable) C∗-algebra A, there is a ∗-homomorphism ρ : Pn,c → M(K ⊗min (B/J )) satisfying ρ(x)11 = ϕ(x) for all x ∈ Pn,c. (Here, ρ(x)11 refers to the (1, 1) entry of ρ(x).) If q : B → B/J is the canonical 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 [25, Proposition 6.8]. Therefore, we can lift the ∗-homomorphism ρ to a ucp map η : Pn,c → M(K ⊗min B). Defining ϕe(x) = η(x)11, we obtain a lift of ϕ.

Theorem 1.8. Pn,c is RFD.

∗ Proof. This proof is very similar to the proofs that C (F2) and Bn are RFD, respectively [5, 16]. As Pn,c is separable, we may represent it faithfully as a subalgebra of B(H), ∞ where H is separable and infinite-dimensional. Let (Em)m=1 be an increasing sequence of projections in B(H) such that rank(Em) = m and SOT-limm→∞ Em = IH. For each (m) (m) (m) 1 ≤ a ≤ c and 1 ≤ i, j ≤ n, we let pa,ij = Empa,ijEm. Then the matrices Pa = (pa,ij ) ∈ Mn(B(EmH)) define a POVM with c outputs in B(EmH). Applying Proposition 1.3, we obtain a unital ∗-homomorphism ρm : Pn,c → Mc+1(B(EmH)) such that, after a shuffle, 6 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

Mn(Mc+1(B(EmH))) → Mc+1(Mn(B(EmH))), Pa gets sent to

( ∗ Um(Eaa ⊗ IEmH)Um 1 ≤ a ≤ c − 1 Pa 7→ ∗ , Um((Ecc + Ec+1,c+1) ⊗ IEmH)Um a = c where  (m) 1   2 (P1 ) 1 . (m) 1 (m) 1 c 2  .  (δ I − (P ) 2 (P ) 2 )   .  ab EmH a b a,b=1  Um =  (m) 1  ∈ Mc+1(Mn(B(EmH))).  (Pc ) 2   (m) 1 (m) 1  0 − (P1 ) 2 ··· (Pc ) 2

The key point is that, considering ρm(pa,ij) ∈ Mc+1(B(EmH)), each from B(EmH) belongs ∗ (m) to the C -subalgebra of B(EmH) generated by the set {pa,ij : 1 ≤ a ≤ c, 1 ≤ i, j ≤ ∗ ∗ n}. This set is closed under the adjoint since (Empa,ijEm) = Empa,ijEm = Empa,jiEm. ∗ (m) Since SOT-limm→∞ Em = IH, we have SOT -limm→∞ pa,ij = pa,ij. By joint continuity of ∗ ∗ (m) multiplication in the unit ball with respect to SOT , it follows that SOT -limm→∞ Pa = ∗ (m) 1 SOT -limm→∞(Pa ) 2 = Pa for each a. One can check that

P1 IH − P1  . . ∗  .   ..  SOT - lim Um =   m→∞  Pc IH − Pc  0 −( P1 ··· Pc ) ∗ Applying a shuﬄe, we see that SOT -limm→∞ ρm(pa,ij) exists for all a, i, j in Mc+1(B(H)); moreover, its (1, 1)-entry is exactly pa,ij. As Pa = (pa,ij) is a projection, another shuﬄes argument shows that ∗ pa,ij 0 SOT - lim ρm(pa,ij) = ∈ Mc+1(B(H)). m→∞ 0 ∗

Therefore, if W is a linear combination of ﬁnite words in the generators of Pn,c, by considering W 0 the (1, 1) entry of ρ (W ), it follows that SOT∗-lim ρ (W ) = . By passing to m m→∞ m 0 ∗

a subsequence if necessary, this forces limm→∞ kρm(W )kMc+1(B(EmH)) = kW kB(H). Hence, L∞ L∞ m=1 ρm : Pn,c → m=1 Mc+1(B(EmH)) is a ∗-homomorphism that is isometric on the L∞ dense ∗-subalgebra spanned by ﬁnite words in the generators pa,ij. It follows that m=1 ρm is an isometry on the whole algebra. As each EmH is ﬁnite-dimensional, we conclude that Pn,c is RFD. A standard fact is that minimal tensor products of RFD C∗-algebras remain RFD. Hence, Pn,c ⊗min Pn,c is RFD. We can use this fact to relate Qqa(n, c) to states on the minimal tensor product. First, we need the following fact.

(a,b) Lemma 1.9. Suppose that X = (X(i,j),(k,`)) ∈ Qqc(n, c) can be written as X = (hPa,ijQb,k`χ, χi), Pc where Pa = (Pa,ij) and Qb = (Qb,k`) are projections in Mn(B(H)) such that a=1 Pa = Pc a=1 Qa = In, [Pa,ij,Qb,k`] = 0 for all i, j, k, ` and a, b and χ ∈ H is a unit vector. If H is ﬁnite-dimensional, then X ∈ Qq(n, c). THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 7

∗ Proof. Let A be the C -algebra generated by the set {Pa,ij : 1 ≤ a ≤ c, 1 ≤ i, j ≤ n} and ∗ let B be the C -algebra generated by the set {Qb,k` : 1 ≤ b ≤ c, 1 ≤ k, ` ≤ n}. Then A and B are unital C∗-subalgebras of B(H), and every element of A commutes with every element of B. By a theorem of Tsirelson [37], there are ﬁnite-dimensional Hilbert spaces HA and HB, an isometry V : H → HA ⊗ HB, and unital ∗-homomorphisms π : A → B(HA) and ∗ ρ : B → B(HB) such that V (π(Pa,ij) ⊗ ρ(Qb,k`))V = Pa,ijQb,k` for all a, b, i, j, k, `. Deﬁning the unit vector ξ = V χ ∈ HA ⊗ HB, we see that (a,b) X(i,j),(k,`) = h(π(Pa,ij) ⊗ ρ(Qb,k`))ξ, ξi.

Therefore, X ∈ Qq(n, c). (a,b) c2 Theorem 1.10. Let X = (X(i,j),(k,`)) ∈ (Mn ⊗ Mn) . The following are equivalent:

(1) X belongs to Qqa(n, c). (a,b) (2) There is a state s : Pn,c ⊗min Pn,c → C satisfying s(pa,ij ⊗ pb,k`) = X(i,j),(k,`) for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n. (a,b) (3) There is a state s : Qn,c ⊗min Qn,c → C satisfying s(qa,ij ⊗ qb,k`) = X(i,j),(k,`) for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n.

Proof. Since Pn,c ⊆ Qn,c, injectivity of the minimal tensor product shows that Pn,c⊗minPn,c ⊆ Qn,c ⊗min Qn,c completely order isomorphically. Using the Hahn-Banach theorem, it then follows that (2) and (3) are equivalent. If (1) holds, then X is in Qqa(n, c), so it is a pointwise limit of elements of Qq(n, c), which correspond to ﬁnite-dimensional tensor product representations of Qn,c⊗minQn,c, which are automatically continuous. Hence, (1) implies (3). Lastly, suppose that (3) holds. Since ∗ Qn,c ⊗min Qn,c is RFD, a theorem of R. Exel and T.A. Loring [11] shows that s is a w -limit of states sλ on Qn,c ⊗min Qn,c whose GNS representations are ﬁnite-dimensional. Applying Lemma 1.9, each sλ applied to the generators pa,ij ⊗ pb,k` of Qn,c ⊗min Qn,c yields an element Xλ of Qq(n, c); moreover, limλ Xλ = X pointwise. This shows that X ∈ Qq(n, c) = Qqa(n, c), which shows that (3) implies (1). To establish the same disambiguation theorem for qc-correlations, we will show that Pn,c ⊗c Pn,c is completely order isomorphic to the copy of Pn,c ⊗ Pn,c inside of Qn,c ⊗max Qn,c. We use the next result, which is an adaptation of [16, Proposition 4.6].

Lemma 1.11. Let S be an operator system. Then the canonical map Pn,c ⊗c S → Qn,c ⊗max S is a complete order embedding.

∗ Proof. Since Qn,c is a unital C -algebra, we have Qn,c ⊗c S = Qn,c ⊗max S [21, Theorem 6.7]. The canonical map Pn,c ⊗c S → Qn,c ⊗c S is a tensor product of canonical inclusion maps, which are ucp. By functoriality of the commuting tensor product [21], the inclusion Pn,c ⊗c S → Qn,c ⊗c S is ucp. Hence, it suffices to show that this map is a complete order embedding. ∗ Now, let Y = Y ∈ Mm(Qn,c ⊗c S) be positive. Let ϕ : Pn,c → B(H) and ψ : Pn,c → B(H) be ucp maps with commuting ranges; we will show that (ϕ · ψ)(m)(Y ) is positive in Mm(B(H)). For convenience, we define Qa,ij = ϕ(pa,ij). By Proposition 1.3, there is a unital ∗-homomorphism π : Qn,c → Mc+1(B(H)) such that the (1, 1) corner of π(pa,ij) is Qa,ij for all 1 ≤ a ≤ c and 1 ≤ i, j ≤ n. Moreover, each block of π(pa,ij) in B(H) belongs to the ∗ C -algebra generated by the set {Qa,ij : 1 ≤ a ≤ c, 1 ≤ i, j ≤ n}. We extend ϕ to a ucp map on Qn,c by defining ϕ(·) = (π(·))11. Define ψe : S → Mc+1(B(H)) by ψe(s) = Ic+1 ⊗ ψ(s). 8 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

Since ψ(s) commutes with the range of ϕ, ψ(s) must commute with the C∗-algebra generated by the range of ϕ. Hence, ψ(s) commutes with every block of π(pa,ij), for all a, i, j. By the multiplicativity of π, ψ(s) commutes with the range of π. By deﬁnition of the commuting tensor product, this means that π · ψe : Qn,c ⊗c S → Mc+1(B(H)) is ucp; moreover, the (1, 1) block of π · ψe is ϕ · ψ. This means that ϕ · ψ is ucp on Qn,c ⊗c S. Restricting to the copy of (m) the algebraic tensor product Pn,c ⊗ S, it follows that (ϕ · ψ) (Y ) is positive.

(a,b) c2 Theorem 1.12. Let X = (X(i,j),(k,`)) ∈ (Mn ⊗ Mn) . The following are equivalent.

(1) X belongs to Qqc(n, c). (a,b) (2) There is a state s : Pn,c ⊗c Pn,c → C satisfying s(pa,ij ⊗ pb,k`) = X(i,j),(k,`) for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n. (a,b) (3) There is a state s : Qn,c ⊗max Qn,c → C satisfying s(qa,ij ⊗ qb,k`) = X(i,j),(k,`) for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n.

Proof. Since Qqc(n, c) is deﬁned in terms of POVMs where Alice’s entries commute with Bob’s, we see that (1) is equivalent to (2). Based on two applications of Lemma 1.11, we see that Pn,c ⊗c Pn,c is completely order isomorphic to the image of Pn,c ⊗ Pn,c in Qn,c ⊗max Qn,c. Hence, (2) and (3) are equivalent. When considering the quantum-to-classical graph homomorphism game, the local model will be of interest because of its link to the usual notion of a (classical) homomorphism from a quantum graph to a classical graph. To that end, it is important to note that all strategies in Qloc(n, c) can be obtained using PVMs instead of just POVMs. We use a bit of a diﬀerent direction for proving this fact. First, we show the following simple fact:

Proposition 1.13. Qloc(n, c) is a closed set.

(a,b) Proof. Let Xm = (Xm,(i,j),(k,`)) ∈ Qloc(n, c) be a sequence of correlations such that limm→∞ Xm = c2 ∗ X pointwise in (Mn ⊗ Mn) . For each m, there is a unital commutative C -algebra Am, (m) (m) (m) (m) POVMs P1 , ..., Pc and Q1 , ..., Qc in Mn(Am), and a state sm on Am such that

(a,b) (m) (m) Xm,(i,j),(k,`) = sm(Pa,ij Qb,k`) ∀a, b, i, j, k, `.

L∞ L∞ (m) L∞ (m) Deﬁne A = m=1 Am, Pa,ij = m=1 Pa,ij and Qb,k` = m=1 Qb,k`. Then Pa = (Pa,ij) and Qb = (Qb,k`) deﬁne two POVMs in Mn(A) with c outputs. Using the canonical compression ∗ from A onto Am, we can extend sm to a state sem on A. As the state space of A is w -compact, ∗ ∞ (a,b) we choose a w -limit point s of the sequence (sem)m=1. Then X = (X(i,j),(k,`)) = (s(Pa,ijQb,k`)), which shows that X ∈ Qloc(n, c). We note that the above proof works just as well for projection-valued measures. A standard argument shows that limits of convex combinations of elements of Qloc(n, c) represented by PVMs from abelian algebras can still be represented by PVMs from abelian algebras. With this fact in hand, we can prove the disambiguation theorem for Qloc(n, c).

(a,b) c2 Theorem 1.14. Let X = (X(i,j),(k,`)) ∈ (Mn ⊗ Mn) . The following are equivalent:

(1) X belongs to Qloc(n, c); THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 9

∗ (2) There is a commutative C -algebra A, a state s on A and POVMs {P1, ..., Pc}, {Q1, ..., Qc} ⊆ Mn(A) such that (a,b) X(i,j),(k,`) = s(Pa,ijQb,k`); ∗ (3) There is a commutative C -algebra A, a state s on A, and PVMs {P1, ..., Pc}, {Q1, ..., Qc} ⊆ Mn(A) such that (a,b) X(i,j),(k,`) = s(Pa,ijQb,k`).

Proof. Clearly (1) and (2) are equivalent by definition of Qloc(n, c). Since every PVM is a POVM, (3) implies (2). Hence, we need only show that (2) implies (3). Suppose that (a,b) X(i,j),(k,`) = s(Pa,ijQb,k`) ∗ for a state s on a commutative C -algebra A and a POVMs P1, ..., Pc and Q1, ..., Qc in Mn(A). Then A' C(Y ) for a compact Hausdorff space Y . The extreme points of the state space of Y are simply evaluation functionals δy for y ∈ Y , which are multiplicative. (n) (n) Hence, δy (Qa) ∈ Mn(C) defines a POVM with c outputs in Mn(C), where δy = idn ⊗ δy. Recall that the extreme points of the set of positive contractions in a von Neumann algebra are precisely the projections in the von Neumann algebra. An easy application of this argument shows that the extreme points of the set of POVMs with c outputs in a von (n) (n) Neumann algebra are precisely the PVMs with c outputs. Hence, {δy (Q1), ..., δy (Qc)} lies in the closed convex hull of the set of PVMs in Mn(C) with c outputs. Applying a similar (n) (n) argument to {δy (P1), ..., δy (Pc)}, it follows that the correlation (δy(Pa,ijQb,k`)) is a convex combination of elements of Qloc(n, c) obtained by tensoring projections from Mn(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).

For t ∈ {loc, q, qs, qa, qc}, we let Ct(n, c) denote the set of correlations with classical inputs and classical outputs in the t-model, where Alice and Bob now possess n PVMs (equivalently, POVMs) with c outputs each. These sets embed into Qt(n, c) in a natural way.

Proposition 1.15. Let t ∈ {loc, q, qs, qa, qc}. Then Ct(n, c) is aﬃnely isomorphic to (a,b) {X ∈ Qt(n, c): X(i,j),(k,`) = 0 if i 6= j or k 6= `}. Moreover, the compression map (a,b) X 7→ (δijδk`X(i,j),(k,`)): Qt(n, c) → Ct(n, c) is a continuous aﬃne map.

Proof. All of the claims follow from the following observations: if {Ea,x} is a collection of c positive operators such that {Ea,x}a=1 is a POVM in B(H) for each 1 ≤ x ≤ n, then the Ln c operators Pa := x=1 Ea,x deﬁne a POVM in Mn(B(H)). Similarly, if {Qa}a=1 is a POVM c in Mn(B(H)), then setting Fa,x = Qa,xx ∈ B(H), we see that {Fa,x}a=1 is a POVM in B(H) for each 1 ≤ x ≤ n. We leave the veriﬁcation of the claims above to the reader.

Using what we have established so far, we see that the sets Qt(n, c) satisfy

Qloc(n, c) ⊆ Qq(n, c) ⊆ Qqs(n, c) ⊆ Qqa(n, c) ⊆ Qqc(n, c). 10 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

The sets Qloc(n, c), Qqa(n, c) and Qqc(n, c) are all closed, and Qqa(n, c) = Qqs(n, c) = Qq(n, c). Using the previous proposition, all of the containments above are strict in general. Indeed, Qloc(2, 2) 6= Qq(2, 2) by the CHSH game (REFERENCE). By a theorem of J. Stark and A. Coladangelo, Qq(5, 3) 6= Qqs(5, 3). A theorem of K. Dykema, V.I. Paulsen and J. Prakash, Qqs(5, 2) 6= Qqa(5, 2). In fact, using the T2 quantum XOR game and a result of R. Cleve, L. Liu and V.I. Paulsen (REFERENCES), one can show that Qqs(3, 2) 6= Qqa(3, 2). (The analogous problem for Ct(3, 2), perhaps surprisingly, is still open, although it has been s s shown that the synchronous versions are equal; in fact, Cq (3, 2) = Cqc(3, 2) [?Ru19].) Lastly, due to the negative resolution to Connes’ embedding problem (REFERENCE), it follows that Qqa(n, c) 6= Qqc(n, c) for some (likely very large) values of n and c. We close this section with the following isomorphism between Pn,c and its opposite algebra. This isomorphism will be used in our discussion of synchronous correlations in the next few sections.

op op Lemma 1.16. The map pa,ij 7→ pa,ji extends to a unital ∗-isomorphism π : Pn,c → Pn,c. op Proof. In Pn,c, one has n n X op op X op pa,kjpa,ik = (pa,ikpa,kj) k=1 k=1 n !op X = pa,ikpa,kj k=1 op = pa,ij, op ∗ where we have used the fact that Pa = (pa,ij) is a projection in Pn,c. Evidently (pa,ij) = ∗ op op op op n (pa,ij) = pa,ji, so the above calculations show that Pa := (pa,ji)i,j=1 is a projection in op Pc op Mn(Pn,c). Moreover, a=1 Pa = In. By the universal property of Pn,c, there is a unital op op ∗-homomorphism π : Pn,c → Pn,c such that π(pa,ij) = pa,ji. op ∗ op One can show that Pn,c is the universal C -algebra generated by entries pa,ij with the op op n op Pc op property that Pa = (pa,ji)i,j=1 is a projection in Mn(Pn,c) with a=1 Pa = In. By a similar op op argument to the above, the map pa,ji 7→ pa,ij extends to a ∗-homomorphism ρ : Pn,c → Pn,c. Since π and ρ are mutual inverses on the generators of the respective algebras, they both extend to isomorphisms, yielding the desired result.

2. The synchronicity condition n Deﬁnition 2.1. Let S ⊆ [n] and let {e1, ..., en} be the canonical orthonormal basis for C . We deﬁne the maximally entangled Bell state corresponding to S as the vector 1 X ϕS = p ej ⊗ ej. |S| j∈S

Deﬁnition 2.2. Let X ∈ Qt(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 S1∪˙ · · · ∪˙ S` of [n] with the property that, if a 6= b, then

p(a, b|ϕSr ) = 0 for all 1 ≤ r ≤ `. THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 11

We deﬁne the subset s Qt (n, c) = {X ∈ Qt(n, c): X is synchronous}. The following proposition gives a very useful description of synchronicity in terms of the entries of the matrices involved in the correlation. (a,b) Proposition 2.3. Let X = (X(i,j),(k,`)) ∈ Qt(n, c) for t ∈ {loc, q, qs, qa, qc}. The following are equivalent: (1) X is synchronous. (2) X satisﬁes the equation c n 1 X X (a,a) (2.1) X = 1. n (i,j),(i,j) a=1 i,j=1 (3) If a 6= b, then n X (a,b) (2.2) X(i,j),(i,j) = 0. i,j=1 Proof. We observe that, if S ⊆ [n], then 1 X p(a, b|ϕ ) = h(P ⊗ I )(I ⊗ Q )(e ⊗ χ ⊗ e ), e ⊗ χ ⊗ e i S |S| a n n b i i j j i,j∈S 1 X = hP Q χ, χi |S| a,ij b,ij i,j∈S

1 X (a,b) = X . |S| (i,j),(i,j) i,j∈S

Suppose that X is synchronous, and let S1, ..., S` be a partition of [n] for which p(a, b|ϕSr ) = 0 whenever a 6= b and 1 ≤ r ≤ `. Then the above calculation shows that P X(a,b) = 0 i,j∈Sr (i,j),(i,j) Pn (a,b) for all r. Summing over all r, it follows that i,j=1 X(i,j),(i,j) = 0 whenever a 6= b. Hence, (1) implies (3). Next, we show that (3) implies (2). Notice that, for any X ∈ Qqc(n, c), c n c n X X (a,b) X X X(i,j),(k,`) = hPa,ijQb,ijχ, χi a,b=1 i,j=1 a,b=1 i,j=1 n * c ! c ! + X X X = Pa,ij Qb,ij χ, χ i,j=1 a=1 b=1 n X = hχ, χi = n, i=1 Pc Pc Pc where we have used the fact that a=1 Pa = b=1 Qb = In implies that a=1 Pa,ij = Pc b=1 Qb,ij is I when i = j and 0 otherwise. Therefore, n 1 X (a,b) X = 1, n (i,j),(i,j) i,j=1 12 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS which shows that (2) holds. Lastly, if (2) holds, then (1) immediately follows using the single-set partition S = [n].

Remark 2.4. In the case of a correlation p(a, b|x, y) ∈ Ct(n, c) with n classical inputs and c classical outputs, using the [n] = {1}∪{2}∪· · ·∪{n}, we see that any synchronous correlation in Ct(n, c) is a synchronous correlation in the sense of the deﬁnition above. In this way, we s s see that Ct (n, c) ⊆ Qt (n, c).

We wish to show the analogue of [29, Theorem 5.5]; that is, that synchronous correlations with n-dimensional inputs and c outputs arise from tracial states on the algebra generated by Alice’s operators (respectively, Bob’s operators). First, we prove that, in any realization of a synchronous correlation, Bob’s operators can be described naturally in terms of Alice’s operators.

(a,b) s c c Theorem 2.5. Let X = (X(i,j),(k,`)) ∈ Qqc(n, c). Let ({Pa}a=1, {Qb}b=1, H, ψ) be a realization of X. Then:

∗ (1) Qa,ijψ = Pa,ijψ for all 1 ≤ a ≤ c and 1 ≤ i, j ≤ n. ∗ (2) The state ρ = h·ψ, ψi is a tracial state on the C -algebra A generated by {Pa,ij : 1 ≤ ∗ a ≤ c, 1 ≤ i, j ≤ n}, and on the C -algebra B generated by {Qb,k` : 1 ≤ b ≤ c, 1 ≤ k, ` ≤ n}.

∗ Conversely, if Pa,ij are operators in a tracial C -algebra A with a trace τ, such that the ∗ operators Pa = (Pa,ij) ∈ Mn(A) form a PVM with c outputs, then (τ(Pa,ijPb,k`)) deﬁnes an s element of Qqc(n, c). THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 13

s Proof. Suppose X ∈ Qqc(n, c) is as above. By Proposition 2.3, this implies that c n 1 X X (a,b) (2.3) 1 = X n (i,j),(i,j) a=1 i,j=1 c n 1 X X = hP Q ψ, ψi n a,ij a,ij a=1 i,j=1 c n 1 X X (2.4) ≤ |hP Q ψ, ψi| n a,ij a,ij a=1 i,j=1 c n 1 X X = |hQ ψ, P ∗ ψi| n a,ij a,ij a=1 i,j=1 c n 1 X X (2.5) ≤ kQ ψkkP ∗ ψk n a,ij a,ij a=1 i,j=1 1 1 c n ! 2 c n ! 2 1 X X X X (2.6) ≤ kQ ψk2 kP ∗ ψk2 n a,ij a,ij a=1 i,j=1 a=1 i,j=1 1 1 c n ! 2 c n ! 2 1 X X X X = hQ∗ Q ψ, ψi hP P ∗ ψ, ψi n a,ij a,ij a,ij a,ij a=1 i,j=1 a=1 i,j=1 1 1 c n ! 2 c n ! 2 1 X X X X = hQ Q ψ, ψi hP P ψ, ψi n a,ji a,ij a,ij a,ji a=1 i,j=1 a=1 i,j=1

Since Pa and Qa are projections, the last line is equal to 1 1 1 1 c n ! 2 c n ! 2 n ! 2 n ! 2 1 X X X X 1 X X hQ ψ, ψi hP ψ, ψi = hI ψ, ψi hI ψ, ψi n a,jj a,ii n H H a=1 j=1 a=1 i=1 j=1 i=1 1 √ √ = · n · n n = 1. Therefore, all of these inequalities are equalities. Then (2.4) implies that (a,a) X(i,j),(i,j) ≥ 0 for all 1 ≤ a ≤ c, 1 ≤ i, j ≤ n. The equality case of (2.5) shows that ∗ (2.7) Qa,ijψ = αa,ijPa,ijψ for some α ∈ T. Then equation (2.7) yields

(a,a) ∗ ∗ 2 X(i,j),(i,j) = αa,ijhPa,ijPa,ijψ, ψi = αa,ijkPa,ijψk .

(a,a) ∗ 2 ∗ Since X(i,j),(i,j) ≥ 0 and kPa,ijψk ≥ 0, we either have Pa,ijψ = 0 or αa,ij = 1. In either case, we obtain ∗ Qa,ijψ = Pa,ijψ, 14 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

as desired. ∗ To prove (2), it suﬃces to show that it holds for A = C ({Pa,ij}a,i,j); a similar argument ∗ works for B = C ({Qb,k`}b,k,`). Let ρ : A → C be the state given by ρ(X) = hXψ, ψi. Let W = P m1 ··· P mk be a word in {P ,P ∗ } , where we denote by P −1 the operator a1,i1j1 ak,ikjk a,ij a,ij a,i,j a`,i`j` P ∗ and let m ∈ {−1, 1}. We will ﬁrst show that W ψ = Q−mk ··· Q−m1 ψ, where a`,i`j` ` ak,ikjk a1,i1j1 Q−1 := Q∗ . Using the fact that P and Q ∗-commute for each a, b, i, j, k, `, we a`,i`j` a`,i`j` a,ij b,k` obtain W ψ = P m1 ··· P mk ψ a1,i1j1 ak,ikjk = P m1 ··· P mk−1 Q−mk ψ a1,i1j1 ak−1,ik−1jk−1 ak,ikjk = Q−mk (P m1 ··· P mk−1 )ψ. ak,ikjk a1,i1j1 ak−1,ik−1jk−1 and the desired equality easily follows by induction on k. For 1 ≤ a ≤ c and 1 ≤ i, j ≤ n,

ρ(Pa,ijW ) = hPa,ijW ψ, ψi = hP (Qm1 ··· Qmk )∗ψ, ψi a,ij a1,i1j1 ak,ikjk = h(Qm1 ··· Qmk )∗P ψ, ψi a1,i1j1 ak,ikjk a,ij = hP ψ, Qm1 ··· Qmk ψi a,ij a1,i1j1 ak,ikjk = hP ψ, (P m1 ··· P mk )∗ψi a,ij a1,i1j1 ak,ikjk ∗ = hPa,ijψ, W ψi

= hWPa,ijψ, ψi = ρ(WPa,ij).

In the same way, ρ(Pa,ijPb,k`W ) = ρ(WPa,ijPb,k`). It follows by induction 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 op op and τ is a trace on A, then there is a state s : A ⊗max A → C satisfying s(x ⊗ y ) = τ(xy) for all x, y ∈ A. Thus, if P1, ..., Pc ∈ Mn(A) is a projection-valued measure, then op s(Pa,ij ⊗ Pb,k`) = τ(Pa,ijPb,k`) ∀1 ≤ a, b ≤ c, 1 ≤ i, j, k, ` ≤ n. op Applying the universal property of Pn,c, we obtain a state γ : Pn,c ⊗max Pn,c → C satisfying op γ(pa,ij ⊗ pb,k`) = τ(Pa,ijPb,k`). ∗ By Lemma 1.16, the map pa,ij ⊗ pb,k` 7→ τ(Pa,ijPb,`k) = τ(Pa,ijPb,k`) deﬁnes a state on Pn,c ⊗max Pn,c. Then Theorem 1.12 shows that ∗ X := τ(Pa,ijPb,k`) deﬁnes an element of Qqc(n, c). If a 6= b, then n n X (a,b) X ∗ X(i,j),(i,j) = τ(Pa,ijPb,ij) i,j=1 i,j=1

= Tr ⊗ τ(PaPb) = 0,

(a,b) s since PaPb = 0. By Proposition 2.3, X = (X(i,j),(k,`)) ∈ Qqc(n, c). c In light of Theorem 2.5, we may refer to a synchronous t-strategy ({Pa}a=1, χ) when c c referring to a t-strategy ({Pa}a=1, {Qb}b=1, χ) where the associated correlation is synchronous. THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 15

(a,b) s Corollary 2.6. Let (X(i,j),(k,`)) ∈ Qt (n, c) where t ∈ {loc, q, qs, qa, qc}. Then: (a,b) (1) X(i,i),(j,j) ≥ 0 for all 1 ≤ a, b ≤ c and 1 ≤ i, j ≤ n. (a,b) (a,b) (2) X(i,j),(k,`) = X(j,i),(`,k). (3) For any 1 ≤ a 6= b ≤ c and 1 ≤ i, j ≤ n, we have n n X (a,b) X (a,b) X(i,k),(j,k) = X(k,i),(k,j) = 0. k=1 k=1 (4) For any 1 ≤ i, j ≤ n, we have n n X (a,a) X (a,a) X(i,k),(j,k) = X(k,i),(k,j) = δij. k=1 k=1 ∗ Proof. By Theorem 2.5, we may choose projections P1, ..., Pc ∈ Mn(A), for a unital C - algebra A, along with a tracial state τ on A such that (a,b) ∗ X(i,j),(k,`) = τ(Pa,ijPb,k`) for all 1 ≤ a, b ≤ c, 1 ≤ i, j, k, ` ≤ n.

Since Pa is a projection, it deﬁnes a positive element of Mn(A). Compressing to any diagonal + block preserves positivity, which implies that Pa,ii ∈ A for any i. Since τ is a trace, it follows that τ(Pa,iiPb,jj) ≥ 0 for any 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 n X (a,b) X ∗ X(i,k),(j,k) = τ(Pa,ikPb,jk) k=1 k=1 n X = τ(Pa,ikPb,kj) k=1 n ! X = τ Pa,ikPb,kj k=1

= τ((PaPb)ij) = 0, Pn (a,b) since PaPb = 0. Similarly, k=1 X(k,i),(k,j) = 0 when a 6= b. A similar argument establishes (4). Indeed, we have c n c n c ! X X (a,a) X X X X(i,k),(j,k) = τ(Pa,ikPa,kj) = τ Pa,ij , a=1 k=1 a=1 k=1 a=1 Pc and this latter sum is δij, since a=1 Pa = I. The other equation in (4) follows similarly. Remark 2.7. We will have occasion to discuss synchronicity of a strategy with respect to a n c c diﬀerent orthonormal basis {v1, ..., vn} of C . In this case, a qc-strategy ({Pa}a=1, {Qb}b=1, χ) is said to be synchronous with respect to {v1, ..., vn} if there is a partition S1 ∪ · · · ∪ Ss of [n] such that, letting ϕ = √1 P v ⊗ v , one has, for all r, that Sr,v j∈Sr j j |Sr|

p(a, b|ϕSr,v) = 0 if a 6= b. 16 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

One can then write down an analogue of Theorem 2.5 in this context. Alternatively, one can ∗ ∗ simply let Pea = U PaU and Qeb = U QbU, where U is the unitary satisfying Uei = vi for all i. Then applying Theorem 2.5 relates the entries of Qea to the entries of Pea, while showing that the state h(·)χ, χi is a trace on the algebra generated by the entries of the operators Qea ∗ ∗ (respectively, Pea). Since Pa = UPeaU and Qb = UQebU , the entries of Pa (respectively, Qb) are linear combinations of the entries of Pea (respectively, Qeb), the algebra generated by the entries of the operators Pa (respectively, Qb) is the same as the algebra generated by the entries of Pea (respectively, Qeb). It is helpful to describe the simplest ways to realize synchronous correlations. To that end, we spend the rest of this section describing the simplest realizations. We start with the s case of Qloc(n, c). c2 s Corollary 2.8. Let X ∈ (Mn ⊗ Mn) . Then X belongs to Qloc(n, c) if and only if there ∗ c is a unital, commutative C -algebra A, a projection-valued measure {Pa}a=1 ⊆ Mn(A) for 1 ≤ a ≤ c, and a faithful state ψ ∈ S(A) such that, for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n,

(a,b) ∗ X(i,j),(k,`) = ψ(Pa,ijPb,k`). s Moreover, if X is an extreme point in Qloc(n, c), then we may take A = C. s Proof. If X ∈ Qloc(n, c), then by definition of loc-correlations, X can be written using c c projection-valued measures {Pa}a=1 and {Qb}b=1 in Mn(B(H)), along with a state χ ∈ H, (a,b) ∗ such that X(i,j),(k,`) = hPa,ijQb,k`χ, χi and the C -algebra A generated by the set of all ∗ entries Pa,ij and Qb,` is a commutative C -algebra. Applying Theorem 2.5, we can write (a,b) ∗ X(i,j),(k,`) = ψ(Pa,ijPb,k`), where ψ(·) = h(·)χ, χi. As this state is tracial, by replacing A with its quotient by the kernel of the GNS representation of ψ if necessary, we may assume without loss of generality that ψ is faithful, which establishes the forward direction. The converse follows by the converse of Theorem 2.5 and the definition of Qloc(n, c). To establish the claim about extreme points, we note that the proof of Proposition 1.13 shows that every element of Qloc(n, c) is a limit of convex combinations of correlations arising from PVMs in Mn(C). Evidently the set of elements of Qloc(n, c) that have realizations using PVMs in Mn(C) is a closed set. As Qloc(n, c) is compact and convex, the converse of the Krein-Milman theorem shows that extreme points in Qloc(n, c) must have a realization using PVMs in Mn(C). Now, the proof of the forward direction of Theorem 2.5 shows that 1 Pc Pn (a,b) n a=1 i,j=1 Y(i,j),(i,j) ≤ 1 for any Y ∈ Qqc(n, c). Moreover, this inequality is an equality if s and only if Y is synchronous, by Proposition 2.3. Hence, Qloc(n, c) is a face in Qloc(n, c), so s extreme points in Qloc(n, c) are also extreme points in Qloc(n, c). This shows that X has a realization using the algebra A = C. c2 s Corollary 2.9. Let X ∈ (Mn ⊗ Mn) . Then X belongs to Qq(n, c) if and only if there ∗ c is a finite-dimensional C -algebra A, a projection-valued measure {Pa}a=1 ⊆ Mn(A) for 1 ≤ a ≤ c, and a faithful tracial state ψ ∈ S(A) such that, for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n, (a,b) ∗ X(i,j),(k,`) = ψ(Pa,ijPb,k`). s Moreover, if X is an extreme point in Qq(n, c), then we may take A = Md for some d, and hence ψ = trd, where trd is the normalized trace on Md. THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 17

s Proof. If X belongs to Qq(n, c), then one can write X = (h(Pa,ij ⊗Qb,k`)χ, χi) for projection- c c valued measures {Pa}a=1 ⊆ Mn(B(HA)) and {Qb}b=1 ⊆ Mn(B(HB)) on finite-dimensional Hilbert spaces HA and HB, along with a unit vector χ ∈ HA ⊗ HB. By Theorem 2.5, ∗ we may write X = ψ(Pa,ijPb,k`) where ψ is the (necessarily faithful) tracial state on the ∗ finite-dimensional C -algebra A generated by the set {Pa,ij : 1 ≤ a ≤ c, 1 ≤ i, j ≤ n}. ∗ Conversely, if X can be written as X = (ψ(Pa,ijPb,k`)) for a projection-valued measure c ∗ {Pa}a=1 ⊆ Mn(A), where A is a finite-dimensional C -algebra with ψ a faithful trace on A, then the proof of Theorem 2.5 yields a finite-dimensional realization of X as an element of s s Qqc(n, c). By Lemma 1.9, we must have X ∈ Qq(n, c). s Now, assume that X is extreme in Qq(n, c). Since A is finite-dimensional, it is ∗- Lm isomorphic to r=1 Mkr for some r and numbers k1, ..., kr ∈ N. Since ψ is a trace on A, Pm Pm there must be t1, ..., tm ≥ 0 such that r=1 tr = 1 and ψ(·) = r=1 trtrkr (·), where trkr is Lm (r) the normalized trace on Mkr . Writing Pa,ij = r=1 Pa,ij for each 1 ≤ a ≤ c and 1 ≤ i, j ≤ n, (r) where Pa,ij ∈ Mkr , we have m (a,b) X (r) (r) ∗ X(i,j),(k,`) = trtrkr (Pa,ij(Pb,k`) ). r=1 (r) (r) Pc (r) Since Pa = (Pa,ij) ∈ Mn(Mkr ) must define an orthogonal projection and a=1 Pa = (a,b) (r) (r) ∗ s Pm (a,b) In ⊗ Ikr , it follows that Xr,(i,j),(k,`) = trkr (Pa,ij(Pb,k`) ) ∈ Qq(n, c), and r=1 trXr,(i,j),(k,`) = (a,b) (a,b) (a,b) X(i,j),(k,`). Therefore, Xr,(i,j),(k,`) = X(i,j),(k,`) for each r. This shows that we may take A to be a matrix algebra, completing the proof. s s We will end this section by showing that Qqs(n, c) = Qq(n, c). To prove this fact, we use a similar approach to [23]. In fact, by an application of Proposition 1.15, the following theorem is a direct generalization of the analogous result in [23].

s s Theorem 2.10. For each n, c ∈ N, we have Qqs(n, c) = Qq(n, c).

(a,b) s Proof. Let X = (X(i,j),(k,`)) ∈ Qqs(n, c), and write (a,b) X(i,j),(k,`) = h(Pa,ij ⊗ Qb,k`)ψ, ψi, n where Pa = (Pa,ij) is a projection in C ⊗ HA for each 1 ≤ a ≤ c, Qb = (Qb,ij) is a projection n c c P n P n in HB ⊗C for each 1 ≤ b ≤ c, a=1 Pa = IC ⊗HA , b=1 Qb = IHB ⊗C , and ψ ∈ HA ⊗HB is a state. We can arrange to have dim(HA) = dim(HB). For example, if dim(HA) < dim(HB), then we choose a Hilbert space HC with dim(HA ⊕ HC ) = dim(HB), and extend Pa by

deﬁning Pea,ij = Pa,ij ⊕ δijIHC . Then (a,b) h(Pea,ij ⊗ Qb,k`)ψ, ψi = h(Pa,ij ⊗ Qb,k`)ψ, ψi = X(i,j),(k,`).

In this way, we may assume without loss of generality that dim(HA) = dim(HB). We write down a Schmidt decomposition ∞ X ψ = αpep ⊗ fp, p=1 ∞ ∞ where {ep}p=1 ⊆ HA and {fp}p=1 ⊆ HB are orthonormal, and α1, α2, ... ≥ 0 are such that P∞ 2 p=1 αp = 1. If one extends these orthonormal sets to orthonormal bases for HA and HB 18 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS respectively, and defines additional αp’s to be 0, then after direct summing a Hilbert space on one side if necessary, we may assume that dim(HA) = dim(HB) and that {er}r∈I is an orthonormal basis for HA, and {fs}s∈I is an orthonormal basis for HB. We rewrite the (at most) countable set {αq : αq 6= 0} = {βv : v ∈ V }, where V = {1, 2, ...} and βv > βv+1 for all v ∈ V . We define Kv = {eq : αq = βv} and Lv = {fq : αq = βv}, and define subspaces Kv = span(Kv) and Lv = span(Lv) of HA and HB, respectively. P∞ 2 Since q=1 |αq| = 1, it follows that each Kv and Lv must be finite, so that Kv and Lv are finite-dimensional. We will show that each Kv is invariant for the operators {Pa,ij : 1 ≤ a ≤ c, 1 ≤ i, j ≤ n}, and that each Lv is invariant for the operators {Qb,k` : 1 ≤ b ≤ c, 1 ≤ k, ` ≤ n}. To this end, let ω be a primitive c-th root of unity, and define order c unitaries Pc a n Pc −b n U = a=1 ω Pa ∈ B(C ⊗HA) and V = b=1 ω Qb ∈ B(HA ⊗C ). Since X is synchronous, by Theorem 2.5, we know that

∗ ∗ ∗ (IHA ⊗ Qa,ij)ψ = (Pa,ij ⊗ IHB )ψ and (IHA ⊗ Qa,ijQa,ij)ψ = (Pa,ijPa,ij ⊗ IHB )ψ. It follows that

∗ ∗ ∗ (IHA ⊗ Vij)ψ = (Uij ⊗ IHB )ψ and (IHA ⊗ UijUij)ψ = (VijVij ⊗ IHB )ψ. Using this fact,

∗ ∗ αqhUijeq, epi = h(Uij ⊗ IHB )ψ, ep ⊗ fqi = h(IHA ⊗ Vij)ψ, ep ⊗ fqi = αphVijfp, fqi. Since U and V are unitary, it follows that n n n n X ∗ 2 X ∗ X 2 X ∗ kUijerk = hUijUijer, eri = 1 and kUijerk = hUijUijep, epi = 1. j=1 j=1 i=1 i=1

Pn ∗ 2 Pn 2 Similarly, j=1 kVijfqk = i=1 kVijfqk = 1. Suppose that q is such that eq ∈ K1. Then using the fact that αq = α1 and that αp ≤ α1 for all p yields n 2 X 2 ∗ 2 |α1| = |α1| kVijfqk j=1 n ∞ X X 2 ∗ 2 ≥ |αp| |hVijfp, fqi| j=1 p=1 n ∞ X X 2 2 = |αq| |hUijeq, epi| j=1 p=1 n ∞ 2 X X 2 = |α1| |hUijeq, epi| j=1 p=1 n ∞ 2 X X ∗ 2 = |α1| |hUijep, eqi| j=1 p=1 n 2 X ∗ 2 = |α1| kUijeqk j=1 2 = |α1| . THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 19

Therefore, we must have equality at all lines. If p is such that ep 6∈ K1, then since αp < α1, we Pn 2 ∗ 2 Pn 2 2 must have 0 = j=1 |αp| |hVijfp, fqi| = j=1 |αq| |hUijeq, epi| . Therefore, hUijeq, epi = 0 for each such p, which shows that Uijeq ⊥ ep for all p with ep 6∈ K1. Since this happens whenever αq = α1, the subspace K1 must be invariant for every Uij. By the same argument Pn 2 2 ∗ as above with the quantity i=1 |α1| kVijfqk , it follows that K1 is invariant for every Uij. Therefore, K1 is reducing for the operators Uij, for all 1 ≤ i, j ≤ n. A similar argument proves that L1 is reducing for the operators Vk`, for all 1 ≤ k, ` ≤ n. Now, choose q such that eq ∈ K2 and fq ∈ L2. If αp > αq, then αp = α1, so that ∗ ep ∈ K1 and fp ∈ K1. The above shows that hUijeq, epi = 0 and hUijeq, epi = 0, so that ∗ ∗ Uijeq ⊥ K1 and Uijeq ⊥ K2. Similarly, Vk`fq ⊥ L1 and Vk`fq ⊥ L1. Then using a similar string of inequalities as before, one obtains Uijeq ⊥ ep whenever p is such that ep 6∈ K2 and q is such that eq ∈ K2. Therefore, one ﬁnds that K2 is invariant for each Uij. A similar ∗ argument shows that K2 is invariant for Uij, so that K2 is reducing for {Uij : 1 ≤ i, j ≤ n}. The same argument shows that {Vk` : 1 ≤ k, ` ≤ n} reduces K2. It follows by induction that Kv is reducing for {Uij : 1 ≤ i, j ≤ n} for all v and that Lv is reducing for {Vk` : 1 ≤ k, ` ≤ n} for all v. By construction of the unitaries U and V , we know that

c c 1 X 1 X P = ω−adU d and Q = ωbdV d. a c b c d=1 d=1

Therefore, Kv is reducing for each Pa,ij, and Lv is reducing for each Qb,k`, as desired. (a,b) Finally, we will exhibit X = (X(i,j),(k,`)) as a countable convex combination of elements s s n4c2 of Qq(n, c). One can regard elements of Qq(n, c) as elements of C , or as elements of 4 2 R2(n c ). Then by a countably infinite version of Carathéodory’s Theorem [8], this will show s that X belongs to Qq(n, c), which will complete the proof. (As mentioned in [23], this result s from [8] holds even with non-closed convex sets, of which Qq(n, c) is an example.) For each v ∈ V , we let dv = dim(Kv) = dim(Lv) = |Kv| = |Lv|, which is finite. Define

1 X ψv = √ ep ⊗ fp, dv ep∈Kv fp∈Lv and deﬁne

Pv,a,ij = Pa,ij|Kv and Qv,b,k` = Qb,k`|Lv .

n Since Kv is reducing for Pa,ij, and since Pa is a projection, the operator Pv,a = (Pv,a,ij)i,j=1 is n n n a projection on C ⊗Kv. Similarly, Qv,b = (Qv,b,k`)k,`=1 is a projection on Lv ⊗C . Moreover, c c P n P n a=1 Pv,a = IC ⊗ IKv and b=1 Qv,b = ILv ⊗ IC . Therefore, the correlation

(a,b) Xv = (Xv,(i,j),(k,`)) = (h(Pv,a,ij ⊗ Qv,b,k`)ψv, ψvi) 20 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

2 P P∞ 2 belongs to Qq(n, c) for each v. Set tv = βv dv. Then tv ≥ 0 and v≥1 tv = p=1 |αp| = 1. Finally, for each 1 ≤ a, b ≤ c and 1 ≤ i, j ≤ n, we compute (a,b) X(i,j),(k,`) = h(Pa,ij ⊗ Qb,k`)ψ, ψi X X 2 = βv h(Pa,ij ⊗ Qb,k`)(ep ⊗ fp), eq ⊗ fqi v p,q:ep,eq∈Kv X 2 = βv dvh(Pv,a,ij ⊗ Qv,b,k`)ψv, ψvi v X (a,b) = tvXv,(i,j),(k,`). v P It follows that X = v tvXv. Since each Xv ∈ Qq(n, c), it follows that X ∈ Qq(n, c). Since s X is also synchronous, we obtain X ∈ Qq(n, c), completing the proof. 3. Approximately finite-dimensional correlations s U In this section, we will show that elements of Qqa(n, c) arise from the trace on R and U U projection-valued measures with c outputs in Mn(R ), where R denotes the ultrapower of the hyperﬁnite II1-factor R by a free ultraﬁlter U on N. Most of the proofs are similar to [23, Section 3]. The main result is a generalization of [23, Theorem 3.6]. Relevant details on RU can be found in [3]. For amenable traces, we will use the following result of Kirchberg [24, Proposition 3.2], which can also be found in [3, 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 statements are 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 → RU along with a completely positive contractive ∞ lift ϕ : A → ` (R) such that trRU ◦ π = τ; (3) There is a sequence of natural numbers N(k) and completely positive contractive maps ϕk : A → MN(k) satisfying

lim kϕk(ab) − ϕk(a)ϕk(b)k2 = 0 and lim |trN(k)(ϕk(a)) − τ(a)| = 0 k→∞ k→∞ for all a, b ∈ A; (4) The linear functional γ : A ⊗ Aop → C given by γ(a ⊗ bop) = τ(ab) extends to a op continuous linear map on A ⊗min A . op As pointed out in [23], as soon as γ is continuous on A ⊗min A in condition (4) above, it is automatically a state on the minimal tensor product. For the convenience of the reader, we recall the following perturbation result. Lemma 3.2. (Kim-Paulsen-Schafhauser, [23]) Let ε > 0 and c ∈ N. Then there exists a δ > 0 such that, if n, N ∈ N and P1, ..., Pc ∈ Mn(MN ) are positive contractions with 2 kPaPbk2 < δ for all a 6= b and kPa − Pak2 < δ for all a, then there exist orthogonal projections Q1, ..., Qc ∈ Mn(MN ) with QaQb = 0 for a 6= b and kPa − Qak2 < ε. Moreover, Pc if k a=1 Pa − In ⊗ IN k2 < δ, then we may arrange for the projections Q1, ..., Qc to satisfy Pc a=1 Qa = In ⊗ IN . THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 21

Note that this lemma is stated slightly diﬀerently than in [23]; however, it is easy to see that their result is equivalent to the above result. Notice that, in the above lemma, if we write Pa = (Pa,ij) ∈ Mn(MN ) and Qa = (Qa,ij) ∈ Mn(MN ), then one has kQa,ij − Pa,ijk2 ≤ kQa − Pak2 < ε.

(a,b) c2 Theorem 3.3. Let X = (X(i,j),(k,`)) be an element of (Mn ⊗Mn) . The following statements are equivalent: s (1) X belongs to Qqa(n, c); s (2) X belongs to Qq(n, c); ∗ (3) There is a separable unital C -algebra A, a PVM {P1, ..., Pc} in Mn(A), and an amenable trace τ on A such that, for all 1 ≤ i, j, k, ` ≤ n and 1 ≤ a, b ≤ c,

(a,b) ∗ X(i,j),(k,`) = τ(Pa,ijPb,k`); U U (4) There are elements qa,ij in R such that qa = (qa,ij) are projections in Mn(R ) with Pc a=1 qa = In and (a,b) ∗ X(i,j),(k,`) = trRU (qa,ijqb,k`).

Proof. First, we show that (1) implies (3). Since Qqa(n, c) is the closure of Qq(n, c), this (a,b) means that there are correlations Xr = (Xr,(i,j),(k,`)) ∈ Qq(n, c) with limr→∞ Xr = X pointwise. We may choose natural numbers N(r) and M(r) along with projection-valued (r) (r) (r) (r) measures {P1 , ..., Pc } ⊆ Mn(MN(r)) and {Q1 , ..., Qc } ⊆ Mn(MM(r)) and unit vectors N(r) M(r) χr ∈ C ⊗ C such that, for all r ∈ N and for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n, (a,b) (r) (r) Xr,(i,j),(k,`) = h(Pa,ij ⊗ Qb,k`)χr, χri. op Then for each r, there is a state ϕr on Pn,c ⊗min Pn,c satisfying op (r) (r) ϕr(pa,ij ⊗ pb,k`) = h(Pa,ij ⊗ Qb,k`)χr, χri. As the state space of any unital C∗-algebra is w∗-compact, we may take a w∗-limit point ϕ ∞ op (a,b) of the sequence (ϕr)r=1. By construction, we note that ϕr(pa,ij ⊗ pb,k`) = X(i,j),(k,`) for all 1 ≤ a, b ≤ c and 1 ≤ i, j, k, ` ≤ n. We write ϕ = hπ(·)χ, χi in its GNS representation, where op π : Pn,c ⊗min Pn,c → B(H) is a unital ∗-homomorphism and χ ∈ H is a unit vector. Applying Theorem 2.5 and restricting to Pn,c, we see that τ(y) := hπ(y ⊗ 1)χ, χi deﬁnes a trace on op Pn,c. To establish (3), we need to show that τ(x ⊗ y ) = τ(xy) for all x, y ∈ Pn,c. Notice that for each a and i, j, by Theorem 2.5 we have op op op op π(x ⊗ pa,ij)χ = π(x ⊗ 1 )(π(1 ⊗ pa,ij))χ = π(xpa,ij ⊗ 1 )χ. Then for each b, k, `, we have op op op op π(x ⊗ pa,ijpb,k`)χ = π(x ⊗ pa,ij)π(1 ⊗ pb,k`)χ op = π(x ⊗ pa,ij)π(pb,k` ⊗ 1)χ op = π(xpb,k` ⊗ pa,ij)χ

= π(xpb,k`pa,ij ⊗ 1)χ. op op op Since pa,ijpb,k` = (pb,k`pa,ij) and the elements pa,ij generate Pn,c, one can see that π(x ⊗ op op y )χ = π(xy ⊗ 1)χ. Therefore, ϕ(x ⊗ y ) = τ(xy) for all x, y ∈ Pn,c, showing that τ is an 22 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

amenable trace. Setting Pa,ij = π(pa,ij ⊗ 1), we obtain a PVM {P1, ..., Pc} in Mn(A), where (a,b) ∗ A = π(Pn,c ⊗ 1) and Pa = (Pa,ij). Moreover, X(i,j),(k,`) = τ(Pa,ijPb,k`), so (1) implies (3). Next, we show that (3) implies (2). Let {P1, ..., Pc} be a PVM in Mn(A) for a separable ∗ (a,b) ∗ unital C -algebra, and let τ be an amenable trace on A such that X(i,j),(k,`) = τ(Pa,ijPb,k`) for all a, b, i, j, k, `. By Theorem 3.1, we may choose a sequence of completely positive contractive maps ϕr : A → MN(r) with limr→∞ kϕr(xy) − ϕr(x)ϕr(y)k2 = 0 and limr→∞ |trN(r)(ϕr(x)) − (r) (r) (r) τ(x)| = 0 for all x, y ∈ A. Deﬁne pa,ij = ϕr(Pa,ij) and set pa = (pa,ij) ∈ Mn(MN(r)). Using 2 (r) Pn the 2-norm on Mn(MN(r)) and the fact that Pa = Pa implies pa,ij = k=1 ϕr(Pa,ikPa,kj), one sees that

n ! (r) 2 (r) X (r) (r) (r) k(p ) − p k2 = p p − p a a a,ik a,kj a,ij k=1 i,j 2 n !

X = (ϕr(Pa,ik)ϕr(Pa,kj) − ϕr(Pa,ikPa,kj)) k=1 2 n X r→∞ ≤ kϕr(Pa,ik)ϕr(Pa,kj) − ϕr(Pa,ikPa,kj)k2 −−−→ 0. i,k=1

(r) (r) Pc (r) Similarly, one can show that limr→∞ kpa pb k2 = 0 whenever a 6= b and a=1 pa − 1n → 2 0. Applying Lemma 3.2 and dropping to a subsequence if necessary, we obtain a sequence of (r) (r) (r) (r) r→∞ PVMs {q1 , ..., qc } ⊆ Mn(MN(r)) with c outputs with kpa,ij − qa,ijk2 −−−→ 0 for all a, i, j. There is a unital ∗-homomorphism ψr : Pn,c → MN(r) with ψr(pa,ij) = qa,ij. As Pn,c is generated by {pa,ij}a,i,j, a standard argument shows that limr→∞ kϕr(x) − ψr(x)k2 = 0 for every x ∈ Pn,c. This implies that |trN(r)(ϕr(x))−trN(r)(ψr(x))| → 0, so that limr→∞ |trN(r)(ψr(x))− (r) (r) ∗ ∗ τ(x)| = 0. Hence, limr→∞ trN(r)(qa,ij(qb,k`) ) = τ(Pa,ijPb,k`) for all a, b, i, j, k, `. As each cor- (a,b) (r) (r) ∗ s relation Xr,(i,j),(k,`) = trN(r)(qa,ij(qb,k`) ) deﬁnes an element of Qq(n, c), we see that X = ∗ s (τ(Pa,ijPb,k`)) belongs to Qq(n, c). Hence, (3) implies (2). Since Qqa(n, c) is the closure of Qq(n, c), it is easy to see that (2) implies (1). To show that (3) implies (4), we use Theorem 3.1. If {P1, ..., Pc} is a PVM in Mn(A) (a,b) ∗ and τ is an amenable trace on A satisfying X(i,j),(k,`) = τ(Pa,ijPb,k`), then there is a unital ∗- U homomorphism ρ : A → R that preserves τ. Setting qa,ij = ρ(Pa,ij), we obtain projections U qa = (qa,ij) ∈ Mn(R ) summing to the identity and satisfying

(a,b) ∗ X(i,j),(k,`) = trRU (qa,ijqb,k`), which establishes (4). Lastly, we prove that (4) implies (3). Given the elements qa,ij in (4), there is a unital U ∗-homomorphism σ : Pn,c → R satisfying σ(pa,ij) = qa,ij. By Theorem 1.7, Pn,c has the ∞ lifting property, so there is a ucp map ζ : Pn,c → ` (R) that is a lift of σ. Then Theorem 3.1 ∗ ∗ shows that τ := trRU ◦ σ is an amenable trace on Pn,c. Since τ(pa,ijpb,k`) = trRU (qa,ijqb,k`) = (a,b) X(i,j),(k,`), we see that (3) holds. THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 23

4. The game for quantum-to-classical graph homomorphisms In this section, we will deﬁne the quantum-classical game for quantum-to-classical graph homomorphisms. Throughout our discussion, our viewpoint of quantum graphs will be in terms of the work of N. Weaver (REFERENCE). However, we will see that our framework relates nicely to other viewpoints as well. For our purposes, we refer to a quantum graph as a triple (S, M,Mn), where

•M is a (non-degenerate) von Neumann algebra and M ⊆ Mn; •S⊆ Mn is an operator system; and •S is an M0-M0-bimodule with respect to matrix multiplication. In our discussion below, one can just as well use the “traceless” version of quantum graphs; i.e., one replaces the second condition with the condition that S is a self-adjoint subspace of Mn with Tr(X) = 0 for every X ∈ S. This condition, combined with the bimodule property, would force S ⊆ (M0)⊥. Our use of the operator system approach is generally cosmetic: one can easily adapt the quantum-classical game to traceless self-adjoint operator spaces that are M0-M0-bimodules with respect to matrix multiplication. We begin by exhibiting a certain orthonormal basis for S with respect to the (unnormalized) trace on Mn. It is from this (preferred) basis for S that we will extract our input states for the homomorphism game. n n Proposition 4.1. Let K1, ..., Km are non-zero subspaces of C with K1 ⊕ · · · ⊕ Km = C , n such that M acts irreducibly on each Kr. Let Er be the orthorgonal projection of C onto Kr, for each 1 ≤ r ≤ m. Then there exists an orthonormal basis F of S with respect to the unnormalized trace, such that 1 • √ Er ∈ F for each 1 ≤ r ≤ m; dim(Kr) •F contains an orthonormal basis for M0; and • For each Y ∈ F, there are unique r, s with ErYEs = Y . 0 0 Proof. Let X be an element of S. As S is an M -M -bimodule, it follows that ErXEs ∈ S Pm Pm for all 1 ≤ r, s ≤ m. Moreover, since r=1 Er = 1, we have X = r,s=1 ErXEs. Moreover, if X,Y ∈ S, then hErXEs,EpYEqi = 0 whenever r 6= p or s 6= q, where h·, ·i is the inner product with respect to the unnormalized trace on Mn. We choose an orthonormal basis Fr,s for ErSEs with respect to this inner product as follows. We start with an orthonormal basis 0 1 for ErM Es; if r = s, then we arrange for this orthonormal basis to contain √ Er. dim(Kr) 0 Then we extend the orthonormal basis for ErM Es to an orthonormal basis for ErSEs. We may do this since, if X ∈ S ∩ (M0)⊥ and Y ∈ M0, then ∗ ∗ hErXEs,Y i = Tr(Y ErXEs) = Tr(XEsY Er) = hX,ErYEsi = 0, 0 ⊥ 0 S which shows that Er(S ∩ (M ) )Es ⊥ M . Then F = r,s Fr,s is an orthonormal basis for S, which evidently satisfies all three properties. Definition 4.2. We will call a basis for S that satisfies Proposition 4.1 a quantum edge basis for (S, M,Mn). Alternatively, one could arrange for a quantum edge basis for S to also contain a normalized system of matrix units for M0, since a quantum edge basis must already contain the normalized diagonal matrix units. We will see in Theorem 4.7 that the game is independent of the quantum edge basis chosen. 24 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

Once a basis for Cn has been chosen, one can deﬁne the inputs for the game using the n n following well-known correspondence between vectors in C ⊗ C and matrices in Mn. With ∗ respect to a basis {v1, ..., vn}, this correspondence is given by the assignment vi ⊗ vj 7→ vivj , ∗ ∗ n where vivj is the rank-one operator in Mn such that vivj (x) = hx, vjivi for all x ∈ C .

Proposition 4.3. Let (S, M) be a quantum graph, where M ⊆ Mn. Let F be a quantum n edge basis for (S, M). Let {v1, ..., vn} be an orthonormal basis for C that can be partitioned P ∗ into bases for the subspaces K1, ..., Km. For each Yα ∈ F, write Yα = p,q yα,pqvpvq for yα,pq ∈ C. Then the set ( ) X yα,pqvp ⊗ vq p,q is orthonormal.

Proof. This result immediately follows from the fact that the correspondence vi ⊗ vj 7→ ∗ n n vivj preserves inner product, when using the canonical inner product on C ⊗ C and the (unnormalized) Hilbert-Schmidt inner product on Mn. With the notion of quantum edge bases in hand, we now define the homomorphism game for the quantum graph (S, M,Mn) and the classical graph G. n Definition 4.4. Let (S, M,Mn) be a quantum graph, and let {v1, ..., vn} be a basis for C that can be partitioned into bases for the subspaces K1, ..., Kr. Let G be a classical (undirected) graph of c vertices, with no multiple edges and no loops. The quantum-to-classical graph homomorphism game for ((S, M,Mn),G), with respect to the basis {v1, ..., vn} and the quantum edge basis F, is defined as follows: P P ∗ • The inputs are of the form p,q yα,pqvp ⊗vq, where Yα := p,q yα,pqvpvq is an element of F. The outputs are vertices a, b ∈ {1, ..., c} = V (G). There are two rules to the game: 0 • (Adjacency rule) If Yα ⊥ M , then Alice and Bob must respond with different vertices in G; i.e., a 6= b. 0 • (Same “vertex” rule) If Yα ∈ 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 corre- 1 sponding to √ Er will arise in the second rule. We will see that the rule applied to dim(Kr) these inputs will force Bob’s projections to arise from Alice’s projections; the rule applied to the other basis elements of M0 will be what forces the projections to live in M ⊗ B(H), rather than Mn ⊗ B(H). While the above definition of the game seems heavily basis-dependent, we will see that the existence of winning strategies in the various models is independent of the basis {v1, ..., vn}, and independent of the quantum edge basis F chosen for (S, M,Mn). This will be a direct consequence of the next theorem. We would also like to relate winning strategies for the homomorphism game to non- commutative graph homomorphisms in the sense of D. Stahlke [33]. For these, we need a bit of background on 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, (REFERENCE) N is equipped with a tracial state. We will THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 25

be dealing with the case when N is a finite von Neumann algebra equipped with a faithful normal trace τ. One may always choose a faithful normal representation N ⊆ B(H) such that τ(·) = h(·)χ, χi for some 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-dimensional von 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 one may choose Kraus operators Fi Pm ∗ such that Φ(·) = i=1 Fi (·)Fi, 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 as m X ∗ Φ∗(·) = Fi(·)Fi . i=1 The interested reader can consult REFERENCES for more information on these topics. Remark 4.5. As one might expect, up to a unitary conjugation of the quantum graph, the basis {v1, ..., vn} does not matter. Indeed, let (S, M,Mn) be a quantum graph, let G be a n classical graph on c vertices, and let {v1, ..., vn} be an orthonormal basis for C , and define U ∈ Mn to be the unitary such that Uei = vi for all i, where {e1, ..., en} is the canonical basis n for C . Suppose that X ∈ Qqc(n, c). Let {Yα}α be a quantum edge basis for (S, M,Mn).We claim that X is a winning strategy for the homomorphism game for ((S, M,Mn),G) with ∗ respect to {Yα}α if and only if (U ⊗U) X(U ⊗U) is a winning strategy for the homomorphism ∗ ∗ ∗ game for ((U SU, U MU, Mn),G) with respect to the quantum edge basis {U YαU}α. Indeed, suppose that we can write X = (h(Pa ⊗ In)(In ⊗ Qb)(ej ⊗ χ ⊗ e`), ei ⊗ χ ⊗ eki), where c c ({Pa}a=1, {Qb}b=1, χ) is a qc-strategy on a Hilbert space H. Then ∗ ∗ h(Pa⊗In)(In⊗Qb)(vj ⊗χ⊗v`), vi⊗χ⊗vki = h(U PaU⊗In)(In⊗U QbU)(ej ⊗χ⊗e`), ei⊗χ⊗eki. In other words, the element Z = (Z(a,b)) := ((U ⊗ U)∗X(a,b)(U ⊗ U)) is a qc-correlation with respect to the basis {v1, ..., vn}. Now, if Yα belongs to a quantum edge basis for (S, M,Mn), ∗ ∗ ∗ P ∗ then U YαU belongs to a quantum edge basis for (U SU, U MU, Mn). If Yα = p,q yα,pqvpvq , P ∗ P ∗ ∗ then its associated input vector is p,q yα,pqvp ⊗ vq. Then U YαU = p,q yα,pqU vpvq U has P ∗ ∗ P associated input vector p,q yα,pqU vp ⊗ U vq = p,q yα,pqep ⊗ eq. Therefore, the probability of Alice and Bob outputting (a, b) given the input vector P p,q yα,pqvp ⊗vq, with respect to the correlation X, is the same as the probability of outputting P (a, b) given the input vector p,q yα,pqep ⊗ eq, with respect to the correlation Z, as desired. Remark 4.6. The previous remark, along with the adjacency rule, forces any winning strategy to be synchronous with respect to the basis {v1, ..., vn}. Thus, in our main theorem, we c c may assume that we are dealing with a synchronous t-strategy ({Pa}a=1, χ), where {Pa}a=1 is a PVM and χ is a faithful normal tracial state on the von Neumann algebra generated by c c the entries of {Pa}a=1. Note that conjugating {Pa}a=1 by a unitary in Mn does not change the von Neumann algebra generated by the entries of the operators Pa.

Theorem 4.7. Let (S, M,Mn) 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 vector such that τ = h(·)χ, χi is a faithful (normal) trace on N . The following are equivalent: 26 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

c (1) There is a winning strategy ({Pa}a=1, χ) from N for the homomorphism game for ((S, M,Mn),G) with respect to any quantum edge basis. c (2) There is a winning strategy ({Pa}a=1, χ) from N for the homomorphism game for ((S, M,Mn),G) with respect to some quantum edge basis. c (3) There is a PVM {Pa}a=1 in M ⊗ N satisfying the following: and if 1 ≤ a, b ≤ c and a 6∼ b in G, then 0 ⊥ Pa((S ∩ (M ) ) ⊗ 1Mn(N ))Pb = 0. Pm ∗ (4) There is a CPTP map Φ: M ⊗ N∗ → Dc of the form Φ(·) = i=1 Fi(·)Fi such that 0 ⊥ ∗ ⊥ Fi((S ∩ (M ) ) ⊗ I)Fj ⊆ SG ∩ (Dc) for all i,j. Proof. Clearly (1) implies (2). We will show that (2) =⇒ (3) =⇒ (4) =⇒ (1). Let n {v1, ..., vn} be an orthonormal basis for C . Let U be the unitary such that Uei = vi for ∗ c all i. Suppose that we can establish (3) for the PVM {U PaU}a=1, the quantum graph ∗ ∗ ∗ 0 ∗ 0 (U SU, U MU, Mn). Then (U MU) = U M U, so the condition in (3) can be written as ∗ ∗ ∗ 0 ⊥ ∗ (U PaU)((U SU) ∩ (U M U) ⊗ 1N )(U PbU) = 0 if a 6∼ b. It is not hard to see that (U ∗M0U)⊥ = U ∗(M0)⊥U, so that the above reduces to ∗ 0 ⊥ U Pa((S ∩ (M ) ) ⊗ 1N )PbU = 0. c Since U is a unitary, we obtain the desired condition for {Pa}a=1 with respect to the quantum graph (S, M,Mn). Hence, we may assume without loss of generality that vi = ei for all i. P ∗ P Then, given a matrix Y = p,q yα,pqvpvq with associated unit vector y = p,q yα,pqvp ⊗ vq, the probability of Alice and Bob outputting a and b respectively, given y and using the c synchronous strategy ({Pa}a=1, χ), is ! X ∗ X X p(a, b| yp,qvp ⊗ vq) = h(Pa,ijPb,k`)(i,j),(k,`) ypqvp ⊗ χ ⊗ vq , yrsvr ⊗ χ ⊗ vsi p,q p,q r,s X ∗ X = h vi ⊗ Pa,ikyk`Pb,j`χ ⊗ vj, yrsvr ⊗ χ ⊗ vsi i,j,k,` r,s X ∗ = hPa,ikyk`Pb,j`yijχ, χi X = τ(Pa,ikyk`Pb,`jyij)

X ∗ = Tr( Pa,ikyk`Pb,`j (Y ⊗ I)) ∗ = Tr ⊗ τ(Pa(Y ⊗ I)Pb(Y ⊗ I)) ∗ (4.1) = Tr ⊗ τ(Pa(Y ⊗ I)Pb(Y ⊗ I)Pa).

Now, suppose that F = {Yα}α is a quantum edge basis for (S, M,Mn), and suppose that c 0 ({Pa}a=1, χ) is a winning strategy with respect to this quantum edge basis. If Yα ∈ M , then Equation 4.1 and faithfulness of the trace gives Pa(Yα ⊗ I)Pb = 0 whenever a 6= b. Then c c ! X X Pa(Yα ⊗ I)Pa = Pa(Yα ⊗ I)Pb = Pa(Yα ⊗ I) Pb = Pa(Yα ⊗ I). b=1 b=1 0 Similarly, Pa(Yα ⊗ I)Pa = (Yα ⊗ I)Pa. Hence, Pa commutes with Yα ⊗ I whenever Yα ∈ M . 0 0 This shows that Pa ∈ (M ⊗ I) ∩ (Mn ⊗ N ) = M ⊗ B(H) ∩ (Mn ⊗ N ) = M ⊗ N . THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 27

0 Similarly, if Yα ⊥ M , then the rules of the game and the faithfulness of the trace force Pa(Y ⊗ I)Pb = 0 whenever a 6∼ b, which shows that (3) holds. Now we show that (3) implies (4). If (3) holds, then there is a projection-valued c 0 ⊥ measure {Pa}a=1 in M ⊗ N such that Pa(Y ⊗ I)Pb = 0 for all X ∈ S ∩ (M ) and a 6∼ b. Then the map Ψ : Dc → M ⊗ N given by Ψ(Ekk) = Pk is a unital ∗-homomorphism. Since Dc is finite-dimensional, Ψ is normal. Hence, we may find Kraus operators F1,F2, ... in B(Cn ⊗ H, Cc) such that m X ∗ Ψ(·) = Fi (·)Fi, i=1 ∗ where m is either finite or ℵ0. In the infinite case, these sums converge in the SOT -topology. ∗ Then Ψ = Θ for a CPTP map Θ : M∗ ⊗ N∗ = M ⊗ N∗ → Dc given by m X ∗ Θ(·) = Fi(·)Fi . i=1 We claim that, if Y ∈ S ∩ (M0)⊥, 1 ≤ i, j ≤ n and 1 ≤ a, b ≤ c are such that a 6∼ b, then ∗ ∗ EaaFi(Y ⊗ I)Fj Ebb = 0 for all i, j. Indeed, let us set Za,b,i,j = EaaFi(Y ⊗ I)Fj Ebb. Then we have ∗ ∗ ∗ ∗ Tr(Za,b,i,jZa,b,i,j) = Tr(EaaFi(Y ⊗ I)Fj EbbFj(Y ⊗ I)Fi Eaa) ∗ ∗ ∗ = Tr(EaaFi(Y ⊗ I)Fj EbbFj(Y ⊗ I)Fi ) ∗ ∗ ∗ = Tr(Fi EaaFi(Y ⊗ I)Fj EbbFj(Y ⊗ I)) ∗ ∗ = hFi EaaFi(Y ⊗ I)Fj EbbFj, (Y ⊗ I)i. Summing over all i, j, we obtain m * m ! m ! + X ∗ X ∗ X ∗ Tr(Za,b,i,jZa,b,i,j) = Fi EaaFi (Y ⊗ I) Fj EbbFj ,Y ⊗ I i,j=1 i=1 i=1

= hΨ(Eaa)(Y ⊗ I)Ψ(Ebb), (Y ⊗ I)i

= hPa(Y ⊗ I)Pb,Y ⊗ Ii = 0. ∗ Since each term in the sum is positive, Tr(Za,b,i,jZa,b,i,j) = 0. By faithfulness of the trace, we ∗ ∗ obtain EaaFi(Y ⊗ I)Fj Ebb = 0 for all i, j and a 6∼ b. Since Fi(Y ⊗ I)Fj ∈ Mc, we can write c ∗ X ∗ X ∗ Fi(Y ⊗ I)Fj = EaaFi(Y ⊗ I)Fj Ebb = EaaFi(Y ⊗ I)Fj Ebb. a,b=1 a∼b ⊥ P P ∗ Note that SG ∩ (Dc) = a∼b CEab = a∼b EaaMcEbb, so we see that Fi(Y ⊗ I)Fj ∈ ⊥ 0 ⊥ SG ∩ (Dc) for all i, j and X ∈ (S ∩ (M ) ). Thus, (4) holds. Lastly, suppose that (4) holds; we will obtain a winning strategy for the game. Suppose Pm ∗ that Φ : M∗ ⊗ N∗ → Dc is a CPTP map of the form Φ(·) = i=1 Ri(·)Ri , such that ∗ ⊥ 0 ⊥ ∗ Pm ∗ Ri(Y ⊗I)Rj ∈ SG ∩Dc for all 1 ≤ i, j ≤ m and Y ∈ S ∩(M ) . Then Φ (·) = i=1 Ri (·)Ri ∗ Pm ∗ deﬁnes a normal ucp map from Dc to M ⊗ N . Let Pa = Φ (Eaa) = i=1 Ri EaaRi for each ∗ c 1 ≤ a ≤ c. Since Φ is UCP, {Pa}a=1 is a POVM in M ⊗ N . As in the proof of (2) =⇒ (3), ∗ c ∗ ∗ by considering the POVM {U PaU}a=1 and the quantum graph (U SU, U MU, Mn), we (a,b) may assume without loss of generality that vi = ei for all i. We will show that X(i,j),(k,`) = ∗ (τ(Pa,ijPb,k`)) deﬁnes a winning t-strategy for the quantum graph homomorphism game for 28 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

0 ⊥ ((S, M,Mn),G). Let Y ∈ S ∩ (M ) . Then for each a 6∼ b and 1 ≤ i, j ≤ m, we have by ∗ ∗ assumption that Ri(Y ⊗ I)Rj is zero in the (a, b)-entry. Hence, EaaRi(Y ⊗ I)Rj Ebb = 0, so that m X ∗ ∗ ∗ 0 = Tr(Eaa(Ri(Y ⊗ I)Rj EbbRj(Y ⊗ I)Ri Eaa) i,j=1 m X ∗ ∗ = hRi EaaRi(Y ⊗ I)Rj EbbRj,Y ⊗ Ii i,j=1 ∗ ∗ = hΦ (Eaa)(Y ⊗ I)Φ (Ebb),Y ⊗ Ii

= hPa(Y ⊗ I)Pb,Y ⊗ Ii. P ∗ Therefore, if {Yα}α is a quantum edge basis for (S, M,Mn), Yα = p,q yα,pqvpvq and Yα ⊥ M0, then X p(a, b | yα,pqvp ⊗ vq) = hPaYαPb,Yαi = 0. if a 6∼ b. p,q 0 P Similarly, if Yα belongs to M and a 6= b, then the above calculation shows that p(a, b| p,q yα,pqvp⊗ s vq) = 0 as well. Hence, X ∈ Qt (n, c) defines a winning loc-strategy for the homomorphism t game for ((S, M,Mn),G), so (S, M,Mn) −→ G. In the loc model, condition (3) of the above theorem is essentially the condition of B. Musto, D.J. Reutter and D. Verdon for graph homomorphisms from a quantum graph to a classical graph [26]. In the loc model, condition (4) is a direct generalization of Stahlke’s notion of graph homomorphism from a non-commutative graph (i.e. a quantum graph with M = Mn) into a classical graph [33]. Hence, Theorem 4.7 shows that these two notions are equivalent in this context. Moreover, these definitions are equivalent in the q, qa and qc models. In fact, we have proved something stronger: if we start with a projection-valued c measure {Pa}a=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 of the four conditions are satisfied. Notice that we needed to start with a PVM and a faithful normal trace for this to happen. Using Theorem 4.7 and the characterizations of synchronous correlations, we obtain the following theorem:

Theorem 4.8. Let (S, M,Mn) be a quantum graph and let G be a classical graph on c vertices. loc (1)( S, M,Mn) −→ G if and only if there is a CPTP map Φ: M → Dc of the form Pm ∗ Φ(·) = i=1 Fi(·)Fi such that 0 ⊥ ∗ ⊥ Fi(S ∩ (M ) )Fj ⊆ SG ∩ (Dc) for all i, j. q (2)( S, M,Mn) −→ G if and only if there is d ∈ N and a CPTP map Φ: M ⊗ Md → Dc Pm ∗ of the form Φ(·) = i=1 Fi(·)Fi such that 0 ⊥ ∗ ⊥ Fi((S ∩ (M ) ) ⊗ Id)Fj ⊆ SG ∩ (Dc) for all i, j. qa U (3)( S, M,Mn) −→ G if and only if there is a CPTP map Φ: M ⊗ (R )∗ → Dc of the Pm ∗ form Φ(·) = i=1 Fi(·)Fi such that 0 ⊥ ∗ ⊥ Fi((S ∩ (M ) ) ⊗ 1RU )Fj ⊆ SG ∩ (Dc) for all i, j. THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 29

qc (4)( S, M,Mn) −→ G if and only if there is a von Neumann algebra N , a faithful normal Pm ∗ trace τ on N , and a CPTP map Φ: M ⊗ N∗ → Dc of the form Φ(·) = i=1 Fi(·)Fi such that 0 ⊥ ∗ ⊥ Fi((S ∩ (M ) ) ⊗ 1N )Fj ⊆ SG ∩ (Dc) for all i, j.

loc Proof. We consider the case t = loc first. If (S, M,Mn) −→ G, then there is a winning loc-strategy for the homomorphism game from (S, M,Mn) into G. Since Qloc(n, c) is convex and non-empty, one may obtain an extreme point in Qloc(n, c) that wins the game with probability 1. Applying Corollary 2.8, there is a realization of this correlation using a PVM c {Pa}a=1 in Mn = Mn ⊗ C. Then the result follows by condition (4) of Theorem 4.7 with N = C. The converse of (1) is immediate since C is an abelian C∗-algebra. The argument is similar for t = q. Indeed, if there is a winning strategy for the homomorphism game in the q-model, then an application of Corollary 2.9 shows that there s is a winning q-strategy using an extreme point in Qq(n, c), which can be realized using projections whose entries are in Md, for some d. Then condition (4) of Theorem 4.7 with N = Md yields the desired CPTP map. The converse is trivial. 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 00 πτ : A → B(Hτ ) is the GNS representation of τ with cyclic vector ξ, then πτ (A) is a finite 00 von Neumann algebra and h(·)ξ, ξi is a faithful normal trace on πτ (A) . We leave the details to the reader. 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 idempotents whose products were 0 when the related pair of outputs was not allowed [18]. One can define a game algebra in our context as follows.

Deﬁnition 4.9. Let (S, M,Mn) be a quantum graph and let G be a classical graph on c vertices. The game algebra for the homomorphism game for ((S, M,Mn),G), denoted A(Hom((S, M,Mn),G)), is the universal ∗-algebra generated by entries {pa,ij : 1 ≤ a ≤ c, 1 ≤ i, j ≤ n} subject to the relations ∗ • pa,ij = pa,ji, 2 Pc • pa = (pa,ij)i,j satisﬁes pa = pa and a=1 pa = In, where In is the n × n identity matrix; and 0 ⊥ • pa((S ∩ (M ) ) ⊗ 1)pb = 0 for each a 6∼ b. 0 • pa(M ⊗ 1)pb = 0 for each a 6= b. We say that the algebra exists if 1 6= 0 in the algebra.

t In general, we will write (S, M,Mn) −→ G provided that there is a winning t-strategy for the graph homomorphism game for ((S, M,Mn),G). We obtain the following characterizations of homomorphisms:

Theorem 4.10. Let (S, M,Mn) be a quantum graph and G a classical graph. loc (1)( S, M,Mn) −→ G ⇐⇒ there is a unital ∗-homomorphism A(Hom((S, M,Mn),G)) → . C q (2)( S, M,Mn) −→ G if and only if there is a unital ∗-homomorphism A(Hom((S, M,Mn),G)) → Md for some d ∈ N. 30 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

qa (3)( S, M,Mn) −→ G if and only if there is a unital ∗-homomorphism A(Hom((S, M,Mn),G)) → RU . qc (4)( S, M,Mn) −→ G if and only if there is a unital ∗-homomorphism A(Hom((S, M,Mn),G)) → C, where C is a tracial C∗-algebra. alg (5)( S, M,Mn) −→ G if and only if A(Hom((S, M,Mn),G)) 6= 0. One can also define C∗-homomorphisms and hereditary homomorphisms of quantum C∗ graphs into classical graphs. We write (S, M,Mn) −→ G provided that there is a unital ∗-homomorphism π : A(Hom((S, M,Mn),G)) → B(H), for some Hilbert space H. (Equiva- lently, by the Gelfand-Naimark theorem, one may simply require that the game algebra has a representation into some unital C∗-algebra.) For hereditary homomorphisms, we recall the concept of a hereditary (unital) ∗-algebra. Recall that a unital ∗-algebra A is said to be hereditary if, whenever x1, ..., xn ∈ A are ∗ ∗ such that x1x1 + ··· + xnxn = 0, then x1 = x2 = ··· = xn = 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+) = {0}. Every unital C -algebra is hereditary as a unital ∗-algebra. hered With this background in hand, we will write (S, M,Mn) −−−→ G provided that there is a unital ∗-homomorphism from A(Hom((S, M,Mn),G)) into a (non-zero) hereditary unital ∗-algebra. One has the following sequence of implications for these types of homomorphism: q qa qc (S, M,Mn) → G =⇒ (S, M,Mn) −→ G =⇒ (S, M,Mn) −→ G =⇒ (S, M,Mn) −→ G C∗ hered alg =⇒ (S, M,Mn) −→ G =⇒ (S, M,Mn) −−−→ G =⇒ (S, M,Mn) −→ G. Our notions of homomorphisms reduce to the analogous types of homomorphisms for classical graphs in the case when (S, M,Mn) is a classical graph. Recall that, for a classical graph G on n vertices, the graph operator system SG is defined as

SG = span ({Eii : 1 ≤ i ≤ n} ∪ {Eij : i ∼ j in G}). (REFERENCE FOR this operator system?) Corollary 4.11. Let G and H be classical graphs on n and c vertices, respectively. Let ∗ t t t ∈ {loc, q, qa, qc, C , hered, alg}. Then G −→ H if and only if (SG,Dn,Mn) −→ H. Proof. It suﬃces to show that the algebra A(Hom(G, H)) from [18] is ∗-isomorphic to the algebra A(Hom((SG,Dn,Mn),H)). The former algebra is the universal unital ∗-algebra Pn generated by self-adjoint idempotents ex,a such that x=1 ex,a = 1, ex,aex,b = 0 if a 6= b, 0 and ex,aey,b = 0 if x ∼ y in G but a 6∼ b in H. Since Dn = Dn, the latter algebra is the universal unital ∗-algebra generated by elements pa,ij such that pa = (pa,ij) ∈ Mn(A) is a Pc ⊥ self-adjoint idempotent with a=1 pa = In, pa((SG ∩(Dn) )⊗1)pb = 0 whenever a 6∼ b in H, and pa(Dn ⊗ 1)pb = 0 whenever a 6= b. Since Eii ∈ Dn, using the fact that pa(Dn ⊗ 1)pb = 0 for a 6= b, we obtain c X pa(Eii ⊗ 1)pa = pa(Eii ⊗ 1)pb = pa(Eii ⊗ 1). b=1

Similarly, pa(Eii ⊗ 1)pa = (Eii ⊗ 1)pa, so that Eii ⊗ 1 commutes with pa. It follows that 2 ∗ 2 ∗ pa,ij = 0 whenever i 6= j. Since pa = pa = pa, we see that pa,ii = pa,ii = pa,ii. For THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 31

1 ≤ a ≤ c and 1 ≤ x ≤ n, we deﬁne qx,a = pa,xx. Then qx,a is a self-adjoint idempotent and Pc a=1 qx,a = 1 for all 1 ≤ x ≤ n. Note that, if x ∼ y in G but a 6∼ b in H, then

qx,aqy,b = pa,xxpb,yy = pa(Exy ⊗ 1)pb = 0, ⊥ since Exy ∈ SG ∩ (Dn) and a 6∼ b in H. Similarly, if a 6= b, then qx,aqx,b = pa(Exx ⊗ 1)pb = 0 since Exx ∈ Dn. By the universal property of A(Hom(G, H)), there is a unital ∗-homomorphism π : A(Hom(G, H)) → A(Hom((SG,Dn,Mn),H)) such that π(ex,a) = qx,a for all x, a. Conversely, in A(Hom(G, H)), one can construct the n × n matrices fa = (fa,ij) with 2 ∗ Pc fa,ij = 0 for i 6= j and fa,ii = ea,i. Then evidently fa = fa = fa and a=1 fa = In. Since ex,aex,b = 0 for a 6= b, we see that fa(Exx ⊗ 1)fb = 0 if a 6= b. Since Dn = span {Exx : 1 ≤ x ≤ n}, it follows that fa(Dn ⊗ 1)fb = 0 for a 6= b. Similarly, it is not hard to see that fa(Exy ⊗ 1)fb = 0 whenever x ∼ y in G but a 6∼ b in H. By the universal property, there is a unital ∗-homomorphism ρ : A(Hom((SG,Dn,Mn),H)) → A(Hom(G, H)) such that ρ(pa,ij) = fa,ij. Evidently ρ and π are mutual inverses on the generators, so we conclude that the algebras are ∗-isomorphic. The result follows. Might be nice to give examples known of the separations from the classical graphs case. Separations are known for the loc/q, the q/qa, and the hered/alg. The first two come from synchronous BCS games and the latter comes from [18] and our final theorem. A special case of the homomorphism game is when the target graph is the classical complete graph Kc on c vertices. In this case, the resulting game is a generalization of the coloring game for classical graphs. ∗ Definition 4.12. Let t ∈ {loc, q, qs, qa, qc, C , hered, alg}. Let (S, M,Mn) be a quantum graph. We define t χt((S, M,Mn)) = min{c ∈ N :(S, M,Mn) −→ Kc}, t and we define χt((S, M,Mn)) = ∞ if (S, M,Mn) 6−→ Kc for all c ∈ N. Due the inclusions of the models, we always have

χloc((S, M,Mn)) ≥ χq((S, M,Mn)) ≥ χqa((S, M,Mn)) ≥ χqc((S, M,Mn))

≥ χC∗ ((S, M,Mn)) ≥ χhered((S, M,Mn)) ≥ χalg((S, M,Mn)).

As a consequence of Corollary 4.11, whenever G is a classical graph, we have χt(G) = χt((SG,Dn,Mn)). Example 4.13. Let S = span {I,Eij : i 6= j} ⊆ Mn, which is a quantum graph on Mn. It is known [22] that χ((S,Mn)) = n. Here, we will show that χqc((S,Mn)) = n as well, which shows that χt((S,Mn)) = n for any t ∈ {loc, q, qa, qc}. Evidently the basis F = {I,Eij : i 6= j} is a quantum edge basis for (S,Mn). Now, suppose that P1, ..., Pc are projections in Mn(B(H)) with Pa(Ek` ⊗ I)Pa = 0 for all 1 ≤ a ≤ c and 1 ≤ k 6= ` ≤ n. A winning strategy in the qc-model for coloring (S,Mn) with c colors would mean that there is a trace τ on the algebra generated by the Pa,ij’s and that

p(a, a|ei ⊗ ej) = 0 if i 6= j. This implies that ∗ τ(Pa,iiPa,jj) = 0 for all i 6= j. 32 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

By faithfulness of τ and positivity of Pa,jj, we have Pa,iiPa,jj = 0 for all i 6= j. Now, choose i 6= j. Notice that, for each i, the set {Pa,ii} is a POVM on H. Moreover, for any a, b ∈ {1, ..., c}, ∗ p(a, b|ei ⊗ ej) = τ(Pa,iiPb,jj) = τ(Pa,iiPb,jj). Thus, the only information relevant to Alice and Bob winning the game is the correlation s (τ(Pa,iiPb,jj))a,b,i,j ∈ Cqc(n, c). By taking a quotient by the kernel of the GNS representation for τ, we may assume that τ is faithful. It is well-known that this forces each Pa,ii to be a projection. By the synchronous condition, the previous equation and faithfulness of the trace, we obtain Pa,iiPa,jj = 0 = Pa,iiPb,ii bs whenever a 6= b and i 6= j. Therefore, (τ(Pa,iiPb,jj))a,b,i,j ∈ Cqc (n, c); that is, the correlation is bisynchronous in the sense of [28]. By [28], we must have c ≥ n. Therefore, χqc(S,Mn) ≥ n. It follows that χt(S,Mn) = n for every t ∈ {loc, q, qa, qc}.

5. Quantum Complete Graphs and Algebraic Colorings In this section, we consider quantum complete graphs; that is, graphs of the form (Mn, M,Mn), where M ⊆ Mn is a non-degenerate von Neumann algebra. We show that ∗ χt((Mn, M,Mn)) = dim(M) for all t ∈ {q, qa, qc, C , hered}. In contrast, we will see that χloc((Mn, M,Mn)) 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. In contrast, the algebraic model for colorings is known to be very wild. At the end of this section, we will extend a surprising result of [18]: 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 5.1. Let M ⊆ Mn be a non-degenerate von Neumann algebra. Then there ∗ Lm is a unitary U ∈ Mn such that UMU = r=1 CInr ⊗ Mkr . Moreover, for any t ∈ {loc, q, qa, qs, qc, C∗, hered, alg}, we have m ! M χt((Mn, M,Mn)) = χt Mn, CInr ⊗ Mkr ,Mn . r=1 Proof. The existence of the unitary U is a consequence of the theory of ﬁnite-dimensional ∗ ∗ 0 0 ∗ C -algebras. It is not hard to see that (UMU ) = UM U . Now, an element X ∈ Mn belongs to M0 if and only if Tr(XY ) = 0 for all Y ∈ M0. This statement is equivalent to having Tr((UXU ∗)(UYU ∗)) = 0 for all Y ∈ M0, since U is unitary. It follows that U(M0)⊥U ∗ = (UM0U ∗)⊥. c Now, suppose that {Pa}a=1 ⊆ Mn ⊗ A is a collection of self-adjoint idempotents sum- 0 ⊥ ming to In ⊗1A, where A is a unital ∗-algebra. Then it is evident that Pa((M ) ⊗1A)Pa = 0 0 ∗ ⊥ ∗ if and only if Pea((UM U ) ⊗ 1A)Pea = 0, where Pea = (U ⊗ 1A)Pa(U ⊗ 1A). Similarly, if 0 0 ∗ ⊥ a 6= b, then Pa(M ⊗ 1A)Pb = 0 if and only if Pea((UM U ) ⊗ 1A)Peb = 0. Thus, there is a bijective correspondence between algebraic c-colorings of (Mn, M,Mn) and algebraic Lm c-colorings of (Mn, r=1 CInr ⊗ Mkr ,Mn). This yields the equality of chromatic numbers for t = alg; the other cases are similar. The diﬀerent chromatic numbers satisfy a certain monotonicity as well. THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 33

Proposition 5.2. If (S, M,Mn) and (T , M,Mn) are quantum graphs with S ⊆ T , then

χt((S, M,Mn)) ≤ χt((T , M,Mn)).

Proof. We deal with the t = alg case; all the other cases are similar. If (T , M,Mn) has no algebraic coloring, then χalg((T , M,Mn)) = ∞, so the desired result holds. Otherwise, let A c be a (non-zero) unital ∗-algebra. Suppose that {Pa}a=1 are self-adjoint idempotents in Mn(A) Pc 0 ⊥ 0 such that a=1 Pa = In, Pa((T ∩ (M ) ) ⊗ 1A)Pa = 0 for all a, and Pa(M ⊗ 1A)Pb = 0 0 ⊥ for all a 6= b. Then evidently Pa((S ∩ (M ) ) ⊗ 1A)Pa = 0 as well, so the self-adjoint idempotents form an algebraic c-coloring of (S, M,Mn). This shows that χalg((S, M,Mn)) ≤ χalg((T , M,Mn)). The proof for the other models is the same. By Propositon 5.2, to establish that every quantum graph has a ﬁnite quantum coloring, it suﬃces to consider quantum complete graphs. First, we look at (Mn,Mn,Mn), the quantum complete graph. While we will have an alternative quantum coloring of this quantum graph from Theorem 5.4, the protocol given in Theorem 5.3 is minimal for (Mn,Mn,Mn) 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 (Mn, M,Mn) when M is not isomorphic to a matrix algebra.

Theorem 5.3. Let d, k ∈ N, and let n = dk. Let M = CId ⊗Mk. Then χq((Mn, M,Mn)) ≤ k2.

Proof. We construct our projections from the canonical orthonormal basis for Ck ⊗ Ck that consists of maximally entangled vectors; that is, the basis of the form k−1 1 X 2πia(p + b) ϕ = √ exp e ⊗ e , a,b k b+p p k p=0 where addition in the indices of the vectors is done modulo k. (REFERENCE for these?) We deﬁne projections in M ⊗ M, for all 1 ≤ a, b ≤ n, by k−1 1 X 2πia(p − q) P = exp I ⊗ E ⊗ I ⊗ E . (a,b) k k d b+p,b+q d pq p,q=0 n n Since the set {ϕ(a,b)}a,b=1 is orthonormal, it is not hard to see that {P(a,b)}a,b=1 is a family of Pn mutually orthogonal projections. Moreover, a,b=1 P(a,b) = Id ⊗ Ik ⊗ Id ⊗ Ik. With respect 0 ⊥ to Mn,(M ) is spanned by elements of the form Exy ⊗ Evw and Exy ⊗ (Evv − Eww) for 1 ≤ x, y ≤ d and 1 ≤ v, w ≤ k with v 6= w. For Y = Exy ⊗ Evw ⊗ (Id ⊗ Ik), one computes P(a,b)YP(a,b) and obtains k−1 1 X 2πia(p + p0 − q − q0) exp E ⊗ E E E 0 0 ⊗ I ⊗ E E 0 0 . k2 k xy b+p,b+q vw b+p ,b+q d pq p q p,q,p0,q0=0 For a term in the above sum to be non-zero, one requires that b + q = v, w = b + p0, and q = p0. Equivalently, a term in the sum is non-zero only when q = p0 and b + q = v = w. Hence, if v 6= w, then the above sum is 0. In the case when v = w, one obtains k−1 1 X 2πia(p − q0) exp E ⊗ E 0 ⊗ I ⊗ E 0 . k2 k xy b+p,b+q d pq p,q0=0 34 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

The above expression does not depend on v, so we conclude that, for all 1 ≤ v, w ≤ k,

P(a,b)(Exy ⊗ Evv ⊗ Id ⊗ Ik)P(a,b) = P(a,b)(Exy ⊗ Eww ⊗ Id ⊗ Ik)P(a,b).

This shows that P(a,b)(X⊗Id⊗Ik)P(a,b) = 0 whenever X = Exy⊗Evw or X = Exy⊗(Evv−Eww) for v 6= w. As such elements span (M0)⊥, we see that 0 ⊥ P(a,b)(X ⊗ Id ⊗ Ik)P(a,b) = 0 ∀X ∈ (M ) . 0 0 0 0 Finally, we show that P(a,b)(M ⊗ Id ⊗ Ik)P(a0,b0) = 0 whenever (a, b) 6= (a , b ). If Y ∈ M , then Y ⊗ (Id ⊗ Ik) commutes with each P(a,b), since P(a,b) ∈ M ⊗ (Id ⊗ Mk). Therefore, if (a, b) 6= (a0, b0), we have

P(a,b)(Y ⊗ (Id ⊗ Ik))P(a0,b0) = P(a,b)P(a0,b0)(Y ⊗ (Id ⊗ Ik)) = 0. Putting all of these equations together, we see that there is a representation of the game 2 algebra A(Hom((Mn, M,Mn),Kk2 )) into CId ⊗ Mk. Therefore, χq((Mn, M,Mn)) ≤ k , which yields the claimed result.

For a general complete quantum graph (Mn, M,Mn), we require a slightly diﬀerent approach. The protocol in the previous proof is used in the context of quantum teleportation, and essentially arises from the use of a “shift and multiply” unitary error basis for Mn (REFERENCE). To give a dim(M)-coloring for (Mn, M,Mn) in the q-model, we will use what we refer to as a “global shift and local multiply” framework.

Theorem 5.4. Let M be a non-degenerate von Neumann algebra in Mn. For the quantum complete graph (Mn, M,Mn), we have χq((Mn, M,Mn)) ≤ dim(M). Lm Proof. Up to unitary equivalence in Mn, we may write M = r=1(CInr ⊗ Mkr ), where Pm Pm 2 n = r=1 nrkr. We will exhibit a PVM in M⊗Md, with d = dim(M) = r=1 kr , satisfying the properties of a quantum coloring for (Mn, M,Mn). For notational convenience, we set k0 = 0, and index our colors a by the set {0, 1, ..., d − 1}. For 1 ≤ r ≤ m and 1 ≤ i ≤ kr, we set

r−1 X 2 γ(r, i) = kp + (i − 1)kr. p=1

Lm (r) (r) (r) kr For 0 ≤ a ≤ d − 1, we deﬁne Pa = r=1 Inr ⊗ Pa , where Pa = (Pa,ij)i,j=1 ∈ Mkr (Md) is given by kr−1 (r) 1 (i−j)a X Pa,ij = ωr Eγ(r,i)+`+a,γ(r,j)+`+a. kr `=0 Here, the addition in the indices is done modulo d, while ωr is a primitive kr-th root of unity. By our choice of the operators Pa, it is immediate that Pa belongs to M ⊗ Md for each 0 ≤ a ≤ d − 1. Pd First, we show that a=1 Pa = In ⊗ Id. For each 1 ≤ r ≤ d and 1 ≤ i, j ≤ kr,

d−1 d−1 kr−1 X (r) 1 X X (i−j)a Pa,ij = ωr Eγ(r,i)+`+a,γ(r,j)+`+a. kr a=0 a=0 `=0 Pd (r) If 1 ≤ p, q ≤ d and p 6= q, then Epp a=1 Pa,ij Eqq will either be 0, or it will be a sum of terms where γ(r, i) + ` + a = p and γ(r, j) + ` + a = q, which yields ` + a = p − γ(r, i) and THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 35

`+a = q−γ(r, j). To each of the kr values of `, there is a unique a for which `+a = p−γ(r, i) and `+a = q−γ(r, j); moreover, the possible values of a are all distinct modulo kr. Therefore, 1 Pkr−1 (i−j)f the coefficient on Epq will be of the form k f=0 ωr . This sum is 0 if i 6= j. Hence, r Pd P (r) = 0 if i 6= j. By a similar argument, E (Pd P (r))E = k 1 E = E . a=1 a,ij pp a=1 a,ii pp r kr pp pp Pd (r) Pd Hence, a=1 Pa,ii = Ikr . Putting these facts together, we see that a=1 Pa = In ⊗ Id. (r) Next, we check that each Pa is an orthogonal projection. By definition of Pa,ij, it is ∗ 2 easy to see that Pa = Pa for all a. To compute Pa , we note that m 2 M (r) 2 Pa = Inr ⊗ (Pa ) , r=1 (r) so it suffices to show that each Pa is an idempotent in Mkr . We compute

kr−1 (r) (r) 1 X P P = ω(v−w)a E E a,vj a,jw k2 r γ(r,v)+`+a,γ(r,j)+`+a γ(r,j)+p+a,γ(r,w)+p+a r `,p=0 k −1 1 Xr = ω(v−w)a E k2 r γ(r,v)+`+a,γ(r,w)+`+a r `=0

1 (r) = Pa,vw. kr

(r) Pkr (r) (r) Since this happens for all 1 ≤ v, w ≤ kr, it follows that Pa,vw = j=1 Pa,vjPa,jw. Therefore, (r) Pa is an idempotent, so Pa is an orthogonal projection. 0 ⊥ 0 Lm Now, we show that Pa((M ) ⊗ Id)Pa = 0 for all a. We note that M = r=1 Mnr ⊗ 0 ⊥ Lm CIkr . Hence, (M ) is spanned by all elements of the form Eij, where Eij 6∈ r=1 Mnr ⊗Mkr ,

and elements from each Mnr ⊗ Mkr of the form Eij ⊗ Evw and Eij ⊗ (Evv − Eww), where Lm v 6= w. By a consideration of blocks, if Eij does not belong to r=1 Mnr ⊗ Mkr , then for any 1 ≤ k, ` ≤ n, either Pa,ki is 0 or Pa,j` is 0, which yields Pa,kiPa,j` = 0. Since the (k, `) entry of Pa(Eij ⊗ Id)Pa is Pa,kiPa,j`, we see that Pa(Eij ⊗ Id)Pa = 0 in this case. Next, we suppose that 1 ≤ v, w ≤ kr with v 6= w, and consider an element Eij ⊗ Evw ∈

Mnr ⊗ Mkr . Note that |γ(r, v) − γ(r, w)| ≥ kr. Since the projection Pa acts as the identity

on Mnr , we obtain

kr X (r) (r) Pa(Eij ⊗ Evw ⊗ Id)Pa = Eij ⊗ Ek` ⊗ Pa,kvPa,w`. k,`=1

We observe that, if 1 ≤ v, w, k, ` ≤ kr and v 6= w, then

kr−1 (r) (r) 1 X P P = ω(k−v+w−`)a E E = 0, a,kv a,w` k2 r γ(r,k)+`+a,γ(r,v)+`+a γ(r,w)+p+a,γ(r,`)+p+a r `,p=0

since 0 ≤ ` ≤ kr − 1 and |γ(r, v) − γ(r, w)| ≥ kr. Thus, Pa(Eij ⊗ Evw ⊗ Id)Pa = 0 in this

case. Lastly, we deal with Eij ⊗ (Evv − Eww) ∈ Mnr ⊗ Mkr . We can write

kr X (r) (r) (r) (r) Pa(Eij ⊗ (Evv − Eww) ⊗ Id)Pa = Eij ⊗ (Pa,kvPa,v` − Pa,kwPa,w`) = 0, k,`=1 36 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

since P (r) P (r) = 1 P (r) for any r and 1 ≤ k, v, ` ≤ k . a,kv a,v` kr a,k` r d Putting all of these facts together, we conclude that {Pea}a=1 ⊆ M ⊗ Md is a quantum d-coloring of (Mn, M,Mn), as desired. Remark 5.5. In general, the above ancillary algebra is not the minimal choice. Indeed, in the case of the quantum complete graph (Mn,Mn,Mn), one can verify that the smallest possible ancillary algebra is Mn, whereas the above construction would have used Mn2 when M = Mn. In the abelian case, one can always use C as the ancilla, using our results on correlations in the loc model. In general, the ancillary algebra can always be taken to be Md for some d; however, we are not sure what the minimal choice for d is in general.

Next, we will show that χhered((Mn, M,Mn)) ≥ dim(M), which will show that, for ∗ every t ∈ {q, qa, qc, C , hered}, we have χt((Mn, M,Mn)) = dim(M). Moreover, we will show that dim(M)-colorings of (Mn, M,Mn) must arise from ∗-isomorphisms on Ddim(M) that preserves a certain tracial δ-form for Ddim(M) (REFERENCE). In this context, we will 1 consider Ddim(M), equipped with its tracial δ-form ψDdim(M) satisfying ψDdim(M) (ea) = dim(M) Lm for all 1 ≤ a ≤ dim(M). The von Neumann algebra M' r=1 CInr ⊗ Mkr is equipped with its tracial δ-form given by m M kr ψ = Tr (·). M n dim(M) nrkr r=1 r In fact, we will show the preservation of trace in a much more general model: the hereditary ∗-algebra model. Our goal is to establish the hereditary coloring number for a complete quantum graph (Mn, M,Mn). In addition, we will show that any minimal hereditary coloring induces a quantum version of isomorphism between (Mn, M,Mn) and the complete graph Kdim(M) on dim(M) vertices. Here, the notion of a “quantum isomorphism” means an isomorphism in the sense of REFERENCE, when using an ancilliary hereditary unital ∗-algebra A. We consider the case when M' CId ⊗ Mk ﬁrst.

Lemma 5.6. Let d, k ∈ N and let n = dk. Consider the quantum graph (Mn, M,Mn) with M = CId ⊗ Mk. Let A be a unital ∗-algebra, and let {P1, ..., Pc} ∈ M ⊗ A be a family of Pc mutually orthogonal projections such that a=1 Pa = Idk ⊗ 1A and 0 ⊥ Pa(X ⊗ 1A)Pa = 0 for all X ∈ (M ) . k Then for each a, the element Ra = d dim(M) (Trdk ⊗ idA)(Pa) is a self-adjoint idempotent in Pc 2 A, and a=1 Ra = k 1A. 0 Proof. Since M = Id ⊗ Mk, we have M = Md ⊗ Ik and n = dk. Now, let 1 ≤ v, w ≤ k with 0 ⊥ x 6= y, and let 1 ≤ i, j ≤ d. Then Eij ⊗ (Evv − Eww) belongs to (M ) , so we must have

Pa(Eij ⊗ (Evv − Eww) ⊗ 1A)Pa = 0 ∀1 ≤ a ≤ c. 0 ⊥ Similarly, Eij ⊗ Evw is in (M ) , so

Pa(Eij ⊗ Evw ⊗ 1A)Pa = 0 ∀1 ≤ a ≤ c. Pk Pd Note that Pa ∈ M ⊗ A = Id ⊗ Mk ⊗ A, so Pa = p,q=1 x=1 Exx ⊗ Epq ⊗ Pa,x,pq, with the property that Pa,x,pq = Pa,y,pq for any 1 ≤ x, y ≤ d. For simplicity, we set Pa,pq = Pa,x,pq for THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 37

any 1 ≤ x ≤ d. The quantity on the left of the above is exactly k X Eij ⊗ Epq ⊗ Pa,pvPa,wq p,q=1 so this says that Pa,pvPa,wq = 0 and Pa,pvPa,vq = Pa,pwPa,wq. Now, since Pa is a projection, Pk 2 we have Pa,pq = v=1 Pa,pvPa,vq = kPa,pvPa,vq for all p, q. In particular, Pa,vv = kPa,vv. By scaling, we see that kPa,vv is a self-adjoint idempotent. Similarly, since Pa,pvPa,wq = 0 n if v 6= w, we see that Pa,vvPa,ww = 0. Therefore, {kPa,vv}v=1 is a collection of mutually orthogonal projections in A. Pk Next, we set Ra = v=1 kPa,vv for each 1 ≤ a ≤ c. Then Ra is a self-adjoint idempotent, so 1A − Ra is a self-adjoint idempotent. We see that c c k k X X X X 2 Ra = kPa,vv = k1A = k 1A, a=1 a=1 v=1 v=1 which completes the proof. Now, we deal with the case of a general quantum complete graph.

Theorem 5.7. Let (Mn, M,Mn) be a quantum complete graph. Let A be a hereditary c ∗-algebra, and let {Pa}a=1 ⊆ M ⊗ A be a hereditary c-coloring of (Mn, M,Mn). Then c ≥ dim(M). Moreover, if c = dim(M), then for each 1 ≤ a ≤ dim(M) we have 1 (ψ ⊗ id )(P ) = 1 . M A a dim(M) A Lm Proof. Up to unitary equivalence, we may write M = r=1 CInr ⊗ Mkr . Then m 0 M M = Mnr ⊗ CIkr . r=1 0 Deﬁne Er = 0 ⊕ · · · ⊕ Inr ⊗ Ikr ⊕ 0 ⊕ · · · ⊕ 0, which belongs to M ∩ M. Then deﬁning Pea = (Er ⊗ 1A)Pa(Er ⊗ 1A) ∈ (ErMEr) ⊗ A, we obtain a family of mutually orthogonal 0 0 projections whose sum is Er. Since Er is central in M, we see that (ErMEr) = ErM Er, n 0 ⊥ while ErMnEr = Mnrkr . It is evident that X ∈ B(ErC )∩(ErM Er) if and only if X = ErXEr 0 n 0 ⊥ and X ⊥ M in Mn. Therefore, for X ∈ B(ErC ) ∩ (ErM Er) and 1 ≤ a ≤ c, one has

Pea(X ⊗ 1A)Pea = (Er ⊗ 1A)Pa(ErXEr ⊗ 1A)Pa(Er ⊗ 1A) = 0, 0 c using the fact that ErXEr = X and X belongs to M . Therefore, {Pea}a=1 is a hereditary

coloring of the quantum complete graph (Mnrkr , ErMEr,Mnrkr ). (r) kr Since E ME = I ⊗M , by Lemma 5.6, we see that Ra := (Tr ⊗id )(P ) is a r r C nr kr nr nrkr A ea Pc (r) 2 self-adjoint idempotent in A for each 1 ≤ a ≤ c and 1 ≤ r ≤ m. Moreover, a=1 Ra = kr 1A. (r) (s) Next, we claim that Ra Ra = 0 if r 6= s. To show this orthogonality relation, it suﬃces to show that Pa,xxPa,yy = 0 whenever Pa,xx is a block from (ErMEr) ⊗ A and Pa,yy is a block from (EsMEs) ⊗ A. If x and y are chosen in this way, then the matrix unit Exy in Mn satisﬁes Er(Exy)Es = Exy and EpExyEq = 0 for all other pairs (p, q). It is not hard to see 0 that Exy belongs to M , so that Pa(Exy ⊗ 1A)Pa = 0. Considering the (x, y)-block of this (r) (s) equation gives Pa,xxPa,yy = 0. It follows that Ra Ra = 0 for r 6= s. 38 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

(r) m Since {Ra }r=1 is a collection of mutually orthogonal projections in A, the element Pm (r) Ra := r=1 Ra is a self-adjoint idempotent in A for each a. Considering blocks, it is not hard to see that c c m m X X X (r) X 2 Ra = Ra = kr 1A = dim(M)1A. a=1 a=1 r=1 r=1 Since Ra is a self-adjoint idempotent, so is 1A − Ra. Their sum is given by c c X X (1A − Ra) = c1A − Ra = (c − dim(M))1A. a=1 a=1 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 1A−Ra = 0 Pm (r) (r) kr for all a. Hence, R = 1 . Since R = Ra and Ra = (Tr ⊗ id )(P ), we see a A a r=1 nr nrkr A a that m X kr (Tr ⊗ id )(P ) = 1 . n nrkr A a A r=1 r Therefore, m X kr 1 (ψ ⊗ id )(P ) = (Tr ⊗ id )(P ) = 1 . M A a dim(M)n nrkr A a dim(M) A r=1 r

Remark 5.8. In essence, Theorem 5.7 proves that any q-coloring of (Mn, M,Mn) with dim(M) colors induces a quantum isomorphim between the quantum graph (Mn, M,Mn) and the classical graph Kdim(M). This isomorphism occurs because any such coloring with ancillary algebra A yields a (necessarily injective) unital ∗-isomorphism π : Ddim(M) → M ⊗ A satisfying the properties of a quantum graph homomorphism, with the additional property that (ψM ⊗ idA) ◦ π = π ◦ ψDdim(M) . In contrast to the case of q-colorings, the existence of a loc-coloring for a complete quantum graph is equivalent to the von Neumann algebra being abelian.

Theorem 5.9. Let M ⊆ Mn be a non-degenerate von Neumann algebra. Then χloc((Mn, M,Mn)) is ﬁnite if and only if M is abelian. In particular, if M is non-abelian, then χ((Mn, M,Mn)) 6= χq((Mn, M,Mn)).

Proof. Suppose that there is a c-coloring of (Mn, M,Mn) in the loc-model. Up to unitary Lm equivalence, we write M = r=1 CInr ⊗ Mkr . We may choose projections Pa ∈ M such that Pc P = I and P ((M0)⊥)P = 0 for all a. Let R = Pm kr Tr (P ) as in the a=1 a n a a a r=1 nr nrkr a proof of the last theorem. Each Ra is an idempotent in C; hence, either Ra = 0 or Ra = 1. Pc We know that a=1 Ra = dim(M), so exactly dim(M) of the Ra’s are non-zero. Since Ra is given by a trace on M which is faithful, having Ra = 0 implies that Pa = 0. Hence, by discarding any projections Pa for which Ra = 0, we may assume without loss of generality that Ra = 1 for all a, and that c = dim(M).

Let Er be the orthogonal projection onto the copy of CInr ⊗ Mkr inside of M = Lm dim(M) r=1 CInr ⊗ Mkr . Then, as before, the PVM {ErPaEr}a=1 yields a classical dim(M)- coloring for (Mnrkr , CInr ⊗Mkr ,Mnrkr ). We will show that kr = 1. By the same argument as above, by discarding values of a for which kr Tr (E P E ) = 0, we may assume that there nr nrkr r a r THE QUANTUM-TO-CLASSICAL GRAPH HOMOMORPHISM GAME 39

2 2 are exactly kr non-zero projections ErPaEr that yield a kr -classical coloring for (Mnrkr , CInr ⊗ M ,M ). Set P = E P E . By THEOREM, for each a, we have kr Tr (P ) = 1. kr nrkr ea r a r nr nrkr ea

Notice that krPea = Inr ⊗ krQa for some projection krQa ∈ Mkr . Hence, Trkr (krQa) = 1. Pkr Let λ1, ..., λkr be the eigenvalues of krQa in Mkr . Since each λi ∈ {0, 1} and i=1 λi =

Trkr (krQa) = 1, there is exactly one λi that is non-zero. Hence, Qa is rank one. The sum over all non-zero Qa gives Ikr , and each Qa is rank one. Hence, the number of a for which 2 2 Qa is non-zero must be kr. Since we assumed that this number is kr , we must have kr = kr . Lm Since kr > 0, we have kr = 1. Since r was arbitrary, we see that M = r=1 CInr ⊗ Mkr = Lm r=1 CInr is abelian. Conversely, suppose that M is abelian. Then the proof of THEOREM yields projec- Pd tions Pa ∈ M ⊗ Md, where d = dim(M), such that a=1 Pa = In ⊗ Id and Pa(X ⊗ Id)Pa = 0 0 ⊥ whenever X ∈ (M ) . Moreover, the projections obtained in this case satisfy Pa,ijPb,k` = Pb,k`Pa,ij for all 1 ≤ a, b ≤ d and 1 ≤ i, j, k, ` ≤ n. Thus, the entries of the projections Pa must ∗-commute with each other, so the C∗-algebra they generate is abelian. Since there is a d-coloring for (Mn, M,Mn) with an abelian ancilla, this implies that χloc((Mn, M,Mn)) ≤ d.

Using the monotonicity of colorings and the results above on quantum complete graphs, we see that every quantum graph has a ﬁnite quantum coloring. As a result, we obtain the following generalization of a theorem from [18].

Theorem 5.10. Let (S, M,Mn) be a quantum graph. Then χalg((S, M,Mn)) ≤ 4.

Proof. Suppose that χalg(S, M,Mn) ≤ c for some c < ∞. Then A(Hom((S, M,Mn),Kc)) exists. We will let p1, ..., pc be the canonical self-adjoint idempotents in the matrix algebra Mn(A(Hom((S, M,Mn),Kc)). By [18], there is an algebraic homomorphism Kc → K4. Thus, there are self-adjoint idempotents fa,v in A(Hom(Kc,K4)) for 1 ≤ a ≤ c and 1 ≤ v ≤ 4 P4 such that v=1 fa,v = 1 for all a and fa,vfb,v = 0 whenever a 6= b. Deﬁne

c X qv,ij = pa,ij ⊗ fa,v ∈ A(Hom((S, M,Mn),Kc)) ⊗ A(Hom(Kc,K4)). a=1 Then

n n c ! c ! X X X X qv,ikqv,kj = pa,ik ⊗ fa,v pb,kj ⊗ fb,v k=1 k=1 a=1 b=1 n c X X = pa,ikpb,kj ⊗ fa,vfb,v k=1 a,b=1 n X 2 = pa,ikpa,kj ⊗ fa,v k=1 = pa,ij ⊗ fa,v = qv,ij.

∗ Therefore, qv = (qv,ij) is an idempotent for each v. Similarly, one can see that qv = qv (that ∗ P4 is, qv,ij = qv,ji) and v=1 qv,ij is 0 if i 6= j and 1 if i = j. Therefore, letting rv,ij be the 40 MICHAEL BRANNAN, PRIYANGA GANESAN AND SAMUEL J. HARRIS

canonical generators of A(Hom((S, M,Mn),K4)), we obtain a unital ∗-homomorphism

π : A(Hom((S, M,Mn),K4)) → A(Hom((S, M,Mn),Kc)) ⊗ A(Hom(Kc,Kr)),

rv,ij 7→ qv,ij.

As the range algebra is non-zero, A(Hom((S, M,Mn),Kr)) is also non-zero. We conclude that χalg((S, M,Mn)) ≤ 4.

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