Let G = (V, E) be a simple, undirected graph with vertex set V and edge set E ⊆ V × V . Assume |V | = n ∈ N. The stability number of G, denoted α(G), is defined as the size of the largest stable set in the graph G:
α(G) = max{|S| : S ⊆ V, (S × S) ∩ E = ∅}.
The chromatic number, also called the coloring number, of G is defined as the smallest number of colors needed to color the vertices of G in such a way that no two adjacent vertices receive the same color:
χ(G) = min{k ∈ N : ∃ c : V → [k] s.t. (i, j) ∈ E ⇒ c(i) 6= c(j)}.
Alternatively, one can define each of these parameters in terms of a nonlocal game in which two parties try to convince a referee that the graph has a certain stability number/chromatic number. For instance, in the graph coloring game the two players Alice and Bob try to convince the referee that they know a coloring c
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136 Chapter 8. Quantum graph parameters using only k colors. The referee attempts to verify this by selecting a pair of vertices (i, j) ∈ V × V uniformly at random, and asking Alice how she colors i and Bob how he colors j. Alice and Bob are allowed to decide on a strategy before the game starts, but during the game they are not allowed to communicate (in particular they don’t know each other’s questions). Alice and Bob ‘win’ if their answers are consistent with the same k-coloring. The referee becomes convinced that they know a valid k-coloring if they win with probability 1, i.e., if they have a perfect strategy (see Equation (3.9)). Below we describe these nonlocal games formally and we show how they can be used to define the quantum analogues of the classical parameters.
These nonlocal games use the set [k] (whose elements are denoted as a, b) and the set V of vertices of a graph G (whose elements are denoted as i, j) as question and answer sets.
The quantum coloring number. In the quantum coloring game, introduced in [AHKS06, CMN+07], we have a graph G = (V, E) and an integer k. Here we have question sets S = T = V and answer sets A = B = [k], and the distribution π is strictly positive on V × V . The predicate f is such that the players’ answers have to be consistent with having a k-coloring of G; that is, f (a, b, i, j) = 0 precisely when (i = j and a 6= b) or ({i, j} ∈ E and a = b), and f (a, b, i, j) = 1 otherwise. This expresses the fact that if Alice and Bob receive the same vertex, they should return the same color and if they receive adjacent vertices, they should return distinct colors. A perfect classical strategy exists if and only if a perfect deterministic strategy exists, and a perfect deterministic strategy means that the players agree on a fixed k-coloring of G. Hence the smallest number k of colors for which there exists a perfect classical strategy is equal to the classical chromatic number χ(G).
It is therefore natural to define the quantum chromatic number as the smallest k for which there exists a perfect quantum strategy. Recall that a strategy is perfect if the probability of giving a wrong answer is zero (see Equation (3.9)). In this case, a strategy P is perfect if P (a, b|i, j) = 0 whenever (i = j and a 6= b) or ({i, j} ∈ E and a = b). Here the first condition says precisely that a perfect strategy P needs to be synchronous (see Equation (3.19)); when Alice and Bob receive the same question, they should provide the same answer.1 We therefore have the following definition of the quantum chromatic number:
Definition 8.1. The quantum chromatic number χq(G) is the smallest k ∈ N for which there exists a synchronous correlation P = (P (a, b|i, j)) in Cq,s([k]2× V2) such that
P (a, a|i, j) = 0 for all a ∈ [k], {i, j} ∈ E.
The commuting quantum chromatic number χqc(G) is defined analogously by taking P ∈ Cqc,s([k]2× V2).
The quantum stability number. In the quantum stability number game, intro-duced in [MR16b, Rob13], we again have a graph G = (V, E) and k ∈ N, but now
1Recall that the set of synchronous quantum correlations is denoted by Cq,s(Γ) in the tensor model and Cqc,s(Γ) in the commuting operator model.
8.2. Our results 137 we use the question set [k] × [k] and the answer set V × V . The distribution π is again strictly positive on the question set and now the predicate f of the game is such that the players’ answers have to be consistent with having a stable set of size k, that is, f (i, j, a, b) = 0 precisely when (a = b and i 6= j) or (a 6= b and (i = j or {i, j} ∈ E)). This expresses the fact that when Alice and Bob receive the same index a = b ∈ [k], they should answer with the same vertex i = j of G, and if they receive distinct indices a 6= b from [k], they should answer with distinct nonadjacent vertices i and j of G. There is a perfect classical strategy precisely when there exists a stable set of size k, so that the largest integer k for which there exists a perfect classical strategy is equal to the stability number α(G). Again, such a strategy is necessarily synchronous, so we get the following definition.
Definition 8.2. The quantum stability number αq(G) is the largest integer k ∈ N for which there exists a synchronous correlation P = (P (i, j|a, b)) in Cq,s(V2× [k]2) such that
P (i, j|a, b) = 0 whenever (i = j or {i, j} ∈ E) and a 6= b ∈ [k].
The commuting quantum stability number αqc(G) is defined analogously by taking P ∈ Cqc,s(V2× [k]2).
The classical parameters χ(G) and α(G) are NP-hard. The same holds for the quantum coloring number χq(G) [Ji13], and also for the quantum stability number αq(G) in view of the following reduction to coloring shown in [MR16b]:
χq(G) = min{k ∈ N : αq(GKk) = |V |}. (8.1) Here GKk is the Cartesian product of the graph G = (V, E) and the complete graph Kk. By construction we have
χqc(G) ≤ χq(G) ≤ χ(G) and α(G) ≤ αq(G) ≤ αqc(G).
A natural question is whether or not the above inequalities can be strict. We revisit this topic in Section 8.4. In short, the quantum parameters can indeed be strictly separated from their classical analogues, but we do not know how to separate the quantum parameter and its commuting operator model analogue. Such a separation would require infinite-dimensional entanglement to be useful for either of the nonlocal games. Finding such a separation is a motivation for the work in this chapter: new bounds on these parameters could potentially lead to a separation there as well.
8.2 Our results
We now give an overview of the results of Section 8.3 and refer to that section for for-mal definitions. In Section 8.3.1 we first reformulate the quantum graph parameters in terms of C∗-algebras, which allows us to use techniques from tracial polynomial optimization to formulate bounds on the quantum graph parameters. We define a hierarchy {γrcol(G)} of lower bounds on the commuting quantum chromatic number and a hierarchy {γrstab(G)} of upper bounds on the commuting quantum stability number. We show the following convergence results for these hierarchies.
138 Chapter 8. Quantum graph parameters Proposition 8.5. There is an integer r0 ∈ N such that γrcol(G) = χqc(G) and γrstab(G) = αqc(G) for all r ≥ r0. Moreover, if γcolr (G) admits a flat optimal solution, then γrcol(G) = χq(G), and if γrstab(G) admits a flat optimal solution, then γrstab(G) = αq(G).
Then in Section 8.3.2 we define tracial analogues {ξstabr (G)} and {ξrcol(G)} of Lasserre-type bounds on α(G) and χ(G) that provide hierarchies of bounds for their quantum analogues. These bounds are more economical than the bounds γrcol(G) and γrstab(G) (since they use less variables) and they also permit to recover some known bounds for the quantum parameters. We show that ξ∗stab(G), which is the parameter ξstab∞ (G) with an additional rank constraint on the matrix variable, coincides with the projective packing number αp(G) from [Rob13] and that ξstab∞ (G) upper bounds αqc(G).
Proposition 8.7. For every graph G we have
ξ∗stab(G) = αp(G) ≥ αq(G) and ξ∞stab(G) ≥ αqc(G).
Next, we consider the chromatic number. The tracial hierarchy {ξcolr (G)} unifies two known bounds: the projective rank ξf(G), a lower bound on the quantum chromatic number from [MR16b], and the tracial rank ξtr(G), a lower bound on the commuting quantum chromatic number from [PSS+16].
Proposition 8.9. For every graph G we have
ξ∗col(G) = ξf(G) ≤ χq(G) and ξcol∞(G) = ξtr(G) ≤ χqc(G).
Let us put the result in perspective. For each graph G we have the inequality ξtr(G) ≤ ξf(G). In [DP16, Cor. 3.10] it is shown that the projective rank and the tracial rank coincide if Connes’ embedding conjecture is true. That is, if Connes’
embedding conjecture is true, then ξ∗col(G) = ξ∞col(G) for every graph G.
Next, we establish some relations between the four hierarchies ξrcol(G), γcolr (G), ξrstab(G), and γrstab(G). For the coloring parameters, we show the analogue of reduction (8.1).
Proposition 8.14. For every graph G and r ∈ N ∪ {∞} we have γrcol(G) = min{k : ξstabr (GKk) = |V |}.
We show an analogous statement for the stability parameters, when using the homomorphic graph product of Kk with the complement of G, denoted here as Kk? G, and the following reduction shown in [MR16b]:
αq(G) = max{k ∈ N : αq(Kk? G) = k}.
Proposition 8.15. For every graph G and r ∈ N ∪ {∞} we have γrstab(G) = max{k : ξrstab(Kk? G) = k}.
Finally, we show that the hierarchies {γrcol(G)} and {γrstab(G)} refine the hier-archies {ξrcol(G)} and {ξstabr (G)}.
Proposition 8.16. For every graph G and r ∈ N ∪ {∞, ∗} we have ξrcol(G) ≤ γrcol(G) and ξstabr (G) ≥ γrstab(G).