Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

In this paper we study bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. For synchronous correlations, we show a correspondence between the minimal entanglement dimension and the completely positive semidefinite rank of an associated matrix. We then study optimization over the set of synchronous correlations by investigating quantum graph parameters. We unify existing bounds on the quantum chromatic number and the quantum stability number by placing them in the framework of tracial optimization. In particular, we show that the projective packing number, the projective rank, and the tracial rank arise naturally when considering tracial analogues of the Lasserre hierarchy for the stability and chromatic number of a graph. We also introduce semidefinite programming hierarchies converging to the commuting quantum chromatic number and commuting quantum stability number.


Introduction 1.Bipartite quantum correlations
One of the distinguishing features of quantum mechanics is quantum entanglement, which allows for nonclassical correlations between spatially separated parties.In this paper we consider the problems of quantifying the advantage entanglement can bring (first investigated through Bell inequalities in the seminal work [3]) and quantifying the minimal amount of entanglement necessary for generating a given correlation (initiated in [5] and continued, e.g., in [38,54,47]).
Quantum entanglement has been widely studied in the bipartite correlation setting (for a survey, see, e.g., [39]).Here we have two parties, Alice and Bob, where Alice receives a question s taken from a finite set S and Bob receives a question t taken from a finite set T .The parties do not know each other's questions, and after receiving the questions they do not communicate.Then, according to some predetermined protocol, Alice returns an answer a from a finite set A and Bob returns an answer b from a finite set B. The probability that the parties answer (a, b) to questions (s, t) is given by a bipartite correlation P (a, b|s, t), which satisfies P (a, b|s, t) ≥ 0 for all (a, b, s, t) ∈ Γ and a,b P (a, b|s, t) = 1 for all (s, t) ∈ S × T .Throughout we set Γ = A × B × S × T .Which bipartite correlations P = (P (a, b|s, t)) ∈ R Γ are possible depends on the additional resources available to the two parties Alice and Bob.
If the parties do not have access to additional resources, then the correlation is deterministic, which means it is of the form P (a, b|s, t) = P A (a|s) P B (b|t), with P A (a|s) and P B (b|t) taking values in {0, 1} and a P A (a|s) = b P B (b|t) = 1 for all s, t.If the parties have access to local randomness, then P A and P B take values in [0, 1].If the parties have access to shared randomness, then the resulting correlation is a convex combination of deterministic correlations and is said to be a classical correlation.The classical correlations form a polytope, denoted C loc (Γ), whose valid inequalities are known as Bell inequalities [3].
We are interested in the quantum setting, where the parties have access to a shared quantum state on which they can perform measurements.The quantum setting can be modeled in different ways, leading to the so-called tensor and commuting models; see the discussion, e.g., in [52,31,11].
In the tensor model, Alice and Bob each have access to "one half" of a finite dimensional quantum state, which is modeled by a unit vector ψ ∈ C d ⊗ C d .Alice and Bob determine their answers by performing a measurement on their part of the state.Such a measurement is modeled by a positive operator valued measure (POVM), which consists of a set of d × d Hermitian positive semidefinite matrices labeled by the possible answers and summing to the identity matrix.If Alice uses the POVM {E a s } a∈A when she gets question s ∈ S and Bob uses the POVM {F b t } b∈B when he gets question t ∈ T , then the probability of obtaining the answers (a, b) is given by P (a, b|s, t) = Tr(( If the state ψ cannot be written as a single tensor product ψ A ⊗ ψ B , then ψ is entangled, which means it can be used to produce a nonclassical correlation P .A correlation of the above form ( 1) is a quantum correlation, realizable in the tensor model in local dimension d (or in dimension d 2 ).Let C d q (Γ) be the set of such correlations and define Denote the smallest dimension needed to realize P ∈ C q (Γ) in the tensor model by The set C 1 q (Γ) contains the deterministic correlations.Hence, by Carathéodory's theorem, C loc (Γ) ⊆ C c q (Γ) holds for c = |Γ| + 1 − |S||T |; that is, quantum entanglement can be used as an alternative to shared randomness.If A, B, S, and T all contain at least two elements, then Bell [3] shows the inclusion C loc (Γ) ⊆ C q (Γ) is strict; that is, quantum entanglement can be used to obtain nonclassical correlations.
The second commonly used model to define quantum correlations is the commuting model (or relativistic field theory model).Here a correlation P ∈ R Γ is called a commuting quantum correlation if it is of the form where {X a s } a and {Y b t } b are POVMs consisting of bounded operators on a separable Hilbert space H, satisfying [X a s , Y b t ] = X a s Y b t − Y b t X a s = 0 for all (a, b, s, t) ∈ Γ, and where ψ is a unit vector in H.Such a correlation is said to be realizable in dimension d = dim(H) in the commuting model.Denote the set of such correlations by C d qc (Γ) and set C qc (Γ) = C ∞ qc (Γ).The smallest dimension needed to realize a quantum correlation P ∈ C qc (Γ) is given by We have C d q (Γ) ⊆ C d 2 qc (Γ), which follows by setting X a s = E a s ⊗ I and Y b t = I ⊗ F b t .This shows D qc (P ) ≤ D q (P ) for all P ∈ C q (Γ).
The minimum Hilbert space dimension in which a given quantum correlation P can be realized quantifies the minimal amount of entanglement needed to represent P .Computing D q (P ) is NP-hard [49], so a natural question is to find good lower bounds for the parameters D q (P ) and D qc (P ).A main contribution of this paper is proposing a hierarchy of semidefinite programming lower bounds for these parameters.
Further variations on the above definitions are possible.For instance, we can consider a mixed state ρ (a Hermitian positive semidefinite matrix ρ with Tr(ρ) = 1) instead of a pure state ψ, where we replace the rank 1 matrix ψψ * by ρ in the above definitions.By convexity this does not change the sets C q (Γ) and C qc (Γ).It is shown in [47] that this also does not change the parameter D q (P ), but it is unclear whether or not D qc (P ) might decrease.Another variation would be to use projection valued measures (PVMs) instead of POVMs, where the operators are projectors instead of positive semidefinite matrices.This again does not change the sets C q (Γ) and C qc (Γ) [35], but the dimension parameters can be larger when restricting to PVMs.
When the two parties have the same question sets (S = T ) and the same answer sets (A = B), a bipartite correlation P ∈ R Γ is called synchronous if P (a, b|s, s) = 0 for all s and a = b.The sets of synchronous (commuting) quantum correlations, denoted C q,s (Γ) and C qc,s (Γ), are rich enough, so that Connes' embedding conjecture still holds if and only if cl(C q,s (Γ)) = C qc,s (Γ) for all Γ [12,Thm. 3.7].The quantum graph parameters discussed in Section 1.3 will be defined through optimization problems over these sets.
A matrix M ∈ R n×n is completely positive semidefinite if there exist d ∈ N and Hermitian positive semidefinite matrices X 1 , . . ., X n ∈ C d×d with M = (Tr(X i X j )).The minimal such d is its completely positive semidefinite rank, denoted cpsd-rank(M ).Completely positive semidefinite matrices are used in [25] to model quantum graph parameters and the completely positive semidefinite rank is investigated in [43,16,44,15].By combining the proofs from [46] (see also [28]) and [41] one can show the following link between synchronous correlations and completely positive semidefinite matrices. 1roposition A.1.The smallest local dimension in which a synchronous quantum correlation P can be realized is given by the completely positive semidefinite rank of the matrix M P indexed by S × A with entries (M P ) (s,a),(t,b) = P (a, b|s, t).
In [15] we use techniques from tracial polynomial optimization to define a semidefinite programming hierarchy {ξ cpsd r (M )} of lower bounds on cpsd-rank(M ).By the above result this hierarchy gives lower bounds on the smallest local dimension in which a synchronous correlation can be realized in the tensor model.However, in [15] we show that the hierarchy typically does not converge to cpsd-rank(M ) but instead (under a certain flatness condition) to a parameter ξ cpsd * (M ), which can be seen as a block-diagonal version of the completely positive semidefinite rank.
Here we use similar techniques, now exploiting the special structure of quantum correlations, to construct a hierarchy {ξ q r (P )} of lower bounds on the minimal dimension D q (P ) of anynot necessarily synchronous -quantum correlation P .The hierarchy converges (under flatness) to a parameter ξ q * (P ), and using the additional structure we can show that ξ q * (P ) is equal to an interesting parameter A q (P ) ≤ D q (P ).This parameter describes the minimal average entanglement dimension of a correlation when the parties have free access to shared randomness; see Section 1.2.
In the rest of the introduction we give a road map through the contents of the paper and state the main results.We will introduce the necessary background along the way.

A hierarchy for the average entanglement dimension
We are interested in the minimal entanglement dimension needed to realize a given correlation P ∈ C q (Γ).If P is deterministic or only uses local randomness, then D q (P ) = D qc (P ) = 1.But other classical correlations (which use shared randomness) have D q (P ) ≥ D qc (P ) > 1, which means the shared quantum state is used as a shared randomness resource.In [5] the concept of dimension witness is introduced, where a d-dimensional witness is defined as a halfspace containing conv(C d q (Γ)), but not the full set C q (Γ).As a measure of entanglement this suggests the parameter inf max i∈[I] D q (P i ) : Observe that, for a bipartite correlation P , this parameter is equal to 1 if and only if P is classical.Hence, it more closely measures the minimal entanglement dimension when the parties have free access to shared randomness.From an operational point of view, (5) can be interpreted as follows.Before the game starts the parties select a finite number of pure states ψ i (i ∈ I) (instead of a single one), in possibly different dimensions d i , and POVMs {E a s (i)} a , {F b t (i)} b for each i ∈ I and (s, t) ∈ S × T .As before, we assume that the parties cannot communicate after receiving their questions (s, t), but now they do have access to shared randomness, which they use to decide on which state ψ i to use.The parties proceed to measure state ψ i using POVMs {E a s (i)} a , {F b t (i)} b , so that the probability of answers (a, b) is given by the quantum correlation P i .Equation (5) then asks for the largest dimension needed in order to generate P when access to shared randomness is free.
It is not clear how to compute (5).Here we propose a variation of (5), and we provide a hierarchy of semidefinite programs that converges to it under flatness.Instead of considering the largest dimension needed to generate P , we consider the average dimension.That is, we minimize i∈I λ i D q (P i ) over all convex combinations P = i∈I λ i P i .Hence, the minimal average entanglement dimension is given by in the tensor model.In the commuting model, A qc (P ) is given by the same expression with D q (P i ) replaced by D qc (P i ).Observe that we need not replace C q (Γ) by C qc (Γ) since D qc (P ) = ∞ for any P ∈ C qc (Γ) \ C q (Γ).It follows by convexity that for the above definitions it does not matter whether we use pure or mixed states.We show that for the average minimal entanglement dimension it also does not matter whether we use the tensor or commuting model.Proposition 2.1.For any P ∈ C q (Γ) we have A q (P ) = A qc (P ).
We have A q (P ) ≤ D q (P ) and A qc (P ) ≤ D qc (P ) for P ∈ C q (Γ), with equality if P is an extreme point of C q (Γ).Hence, we have D q (P ) = D qc (P ) if P is an extreme point of C q (Γ).We show that the parameter A q (P ) can be used to distinguish between classical and nonclassical correlations.
Proposition 2.2.For a correlation P ∈ R Γ we have A q (P ) = 1 if and only if P ∈ C loc (Γ).
As mentioned before, there exist Γ for which C q (Γ) is not closed [48,13], which implies the existence of a sequence {P i } ⊆ C q (Γ) such that D q (P ) → ∞.We show this also implies the existence of such a sequence with A q (P i ) → ∞.
Using tracial polynomial optimization we construct a hierarchy {ξ q r (P )} of lower bounds on A qc (P ).For each r ∈ N this is a semidefinite program, and for r = ∞ it is an infinite dimensional semidefinite program.We further define a (hyperfinite) variation ξ q * (P ) of ξ q ∞ (P ) by adding a finite rank constraint, so that ξ q 1 (P ) ≤ ξ q 2 (P ) ≤ . . .≤ ξ q ∞ (P ) ≤ ξ q * (P ) ≤ A qc (P ).
We do not know whether ξ q ∞ (P ) = ξ q * (P ) always holds; this question is related to Connes' embedding conjecture [22].First we show that we imposed enough constraints in the bounds ξ q r (P ) so that ξ q * (P ) = A qc (P ).
Then we show that the infinite dimensional semidefinite program ξ q ∞ (P ) is the limit of the finite dimensional semidefinite programs.Proposition 2.9.For any P ∈ C q (Γ) we have ξ q r (P ) → ξ q ∞ (P ) as r → ∞.
Finally we give a criterion under which finite convergence ξ q r (P ) = ξ q * (P ) holds.The definition of flatness follows later in the paper; here we only note that it is an easy to check criterion given the output of the semidefinite programming solver.Proposition 2.10.If ξ q r (P ) admits a (⌈r/3⌉ + 1)-flat optimal solution, ξ q r (P ) = ξ q * (P ).

Quantum graph parameters
Nonlocal games have been introduced in quantum information theory as abstract models to quantify the power of entanglement, in particular, in how much the sets C q (Γ) and C qc (Γ) differ from C loc (Γ).A nonlocal game is defined by a probability distribution π : S × T → [0, 1] and a predicate f : A × B × S × T → {0, 1}.Alice and Bob receive a question pair (s, t) ∈ S × T with probability π(s, t).They know the game parameters π and f , but they do not know each other's questions, and they cannot communicate after they receive their questions.Their answers (a, b) are determined according to some correlation P ∈ R Γ , called their strategy, on which they may agree before the start of the game, and which can be classical or quantum depending on whether P belongs to C loc (Γ), C q (Γ), or C qc (Γ).Then their corresponding winning probability is given by A strategy P is called perfect if the above winning probability is equal to one, that is, if for all (a, b, s, t) ∈ Γ we have π(s, t) > 0 and f (a, b, s, t) = 0 =⇒ P (a, b|s, t) = 0.
Computing the maximum winning probability of a nonlocal game is an instance of linear optimization over C loc (Γ) in the classical setting, and over C q (Γ) or C qc (Γ) in the quantum setting.Since the inclusion C loc (Γ) ⊆ C q (Γ) can be strict, the winning probability can be higher when the parties have access to entanglement.In fact there are nonlocal games that can be won with probability 1 by using entanglement, but with probability strictly less than 1 in the classical setting.
The quantum graph parameters are analogues of the classical parameters defined through the coloring and stability number games as described below.These nonlocal games use the set [k] (whose elements are denoted as a, b) and the set V of vertices of G (whose elements are denoted as i, j) as question and answer sets.
In the quantum coloring game, introduced in [1, 9], 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 = b) or ({i, j} ∈ E and a = b).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 corresponds to a 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.Since such a strategy is necessarily synchronous we get the following definition.
Definition 1.1.The (commuting) quantum chromatic number χ q (G) (resp., χ qc (G)) is the smallest integer k ∈ N for which there exists a synchronous correlation P = (P (a, b|i, j) In the quantum stability number game, introduced in [28,45], we again have a graph G = (V, E) and k ∈ N, but now 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 = j) or (a = b and (i = j or {i, j} ∈ E)).This expresses the fact that if 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 = 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 1.2.The (commuting) stability number α q (G) (resp., α qc (G)) is the largest integer k ∈ N for which there exists a synchronous correlation The classical parameters χ(G) and α(G) are NP-hard.The same holds for the quantum coloring number χ q (G) [19] and also for the quantum stability number α q (G) in view of the following reduction to coloring shown in [28]: Here G K k is the Cartesian product of the graph G = (V, E) and the complete graph K k .By construction we have χ qc (G) ≤ χ q (G) ≤ χ(G) and α(G) ≤ α q (G) ≤ α qc (G).The separations between χ q (G) and χ(G), and between α q (G) and α(G), can be exponentially large in the number of vertices; this is the case for the graphs with vertex set {±1} n for n a multiple of 4, where two vertices are adjacent if they are orthogonal [1,28,29].While it was recently shown that the sets C q,s (Γ) and C qc,s (Γ) can be different, it is not known whether there is a separation between the parameters χ q (G) and χ qc (G), and between α q (G) and α qc (G).
We now give an overview of the results of Section 3 and refer to that section for formal definitions.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 {γ col r (G)} of lower bounds on the commuting quantum chromatic number and a hierarchy {γ stab r (G)} of upper bounds on the commuting quantum stability number.We show the following convergence results for these hierarchies.

Proposition 3.2. There is an r
Then we define tracial analogues {ξ stab r (G)} and {ξ col r (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 γ col r (G) and γ stab r (G) (since they use less variables) and also permit to recover some known bounds for the quantum parameters.We show that ξ stab * (G), which is the parameter ξ stab ∞ (G) with an additional rank constraint on the matrix variable, coincides with the projective packing number α p (G) from [45] and that ξ stab ∞ (G) upper bounds α qc (G).
Next, we consider the chromatic number.The tracial hierarchy {ξ col r (G)} unifies two known bounds: the projective rank ξ f (G), a lower bound on the quantum chromatic number from [28], and the tracial rank ξ tr (G), a lower bound on the commuting quantum chromatic number from [41].In [12,Cor. 3.10] it is shown that the projective rank and the tracial rank coincide if Connes' embedding conjecture is true.Proposition 3.6.We have We compare the hierarchies ξ col r (G) and γ col r (G), and the hierarchies ξ stab r (G) and γ stab r (G).For the coloring parameters, we show the analogue of reduction (7).
We show an analogous statement for the stability parameters, when using the homomorphic graph product of K k with the complement of G, denoted here as K k ⋆ G, and the following reduction shown in [28]:

Techniques from noncommutative polynomial optimization
To derive our bounds we use techniques from tracial polynomial optimization.This is a noncommutative extension of the widely used moment and sum-of-squares techniques from Lasserre [23] and Parrilo [40] in polynomial optimization, dealing with the problem of minimizing a multivariate polynomial over a feasible region defined by polynomial inequalities.These techniques have been adapted to the noncommutative setting in [31] and [11] for approximating the set C qc (Γ) of commuting quantum correlations and the winning probability of nonlocal games over C qc (Γ) (and, more generally, computing Bell inequality violations).In [42,32] this approach has been extended to the general eigenvalue optimization problem, of the form Here, the matrix variables X i have free dimension d ∈ N and {f } ∪ G ⊆ R x 1 , . . ., x n is a set of symmetric polynomials in noncommutative variables.In tracial optimization, instead of minimizing the smallest eigenvalue of f (X 1 , . . ., X n ), we minimize its normalized trace Tr(f (X 1 , . . ., X n ))/d (so that the identity matrix has trace one) [7,6,8,21].The moment approach for these problems relies on minimizing L(f ), where L is a linear functional on the space of noncommutative polynomials satisfying some necessary conditions, and L(f ) models By truncating the degrees one gets hierarchies of lower bounds for the original problem.The asymptotic limit of these bounds involves operators X i on a Hilbert space (possibly of infinite dimension).In tracial optimization this leads to allowing solutions X i in a C * -algebra A equipped with a tracial state τ , where τ (f (X 1 , . . ., X n )) is minimized.An important feature in noncommutative optimization is the dimension independence: the optimization is over all possible matrix sizes d ∈ N. In some applications one may want to restrict to optimizing over matrices with restricted size d.In [33,30] techniques are developed that allow to incorporate this dimension restriction by suitably selecting the linear functionals L in a specified space; this is used to give bounds on the maximum violation of a Bell inequality in a fixed dimension.A related natural problem is to decide what is the minimum dimension d needed to realize a given algebraically defined object, such as a (commuting) quantum correlation P .We propose an approach based on tracial optimization: starting from the observation that the trace of the d × d identity matrix gives its size d, we consider the problem of minimizing L(1) where L is a linear functional modeling the non-normalized matrix trace.This approach has been used in several recent works [51,34,15] for lower bounding factorization ranks of matrices and tensors.

A hierarchy for the minimal entanglement dimension 2.1 The minimal average entanglement dimension
We start by showing that it does not matter whether we use the tensor or the commuting model when defining the average entanglement dimension.Proposition 2.1.For any P ∈ C q (Γ) we have A q (P ) = A qc (P ).

Proof. The easy inequality
For the other inequality we suppose P = I i=1 λ i P i is feasible for A qc (P ).This means we have POVMs {X a s (i)} a and We will construct a feasible solution to A q (P ) with value at most i λ i d i .
Fix some index i ∈ [I].By Artin-Wedderburn theory applied to C {X a s (i)} a,s , the * -algebra generated by the matrices X a s (i) for (a, s) ∈ A × S, there exists a unitary matrix U i and integers By the commutation relations each matrix Y b t (i) commutes with all the matrices in C {X a s (i)} a,s , and thus . Hence, we may assume , , where e l is the lth unit vector in R min{m k ,n k } .Then we have We now show the parameter A q (•) permits to characterize classical correlations.
Proposition 2.2.For a correlation P ∈ R Γ we have A q (P ) = 1 if and only if P ∈ C loc (Γ).
Proof.If P ∈ C loc (Γ), then P can be written as a convex combination of deterministic correlations (which are contained in C 1 q (Γ)), hence A q (P ) = 1.On the other hand, if A q (P ) = 1, then there exist convex decompositions indexed by l ∈ N: P = i∈I l λ l i P l i with {P l i } ⊆ C q (Γ) and lim l→∞ i∈I l λ l D q (P l i ) = 1.Decompose I l as the disjoint union I l − ∪ I l + so that D q (P i ) is equal to 1 for i ∈ I l − and strictly greater than 1 for i ∈ I l + .Let ε > 0. For all l sufficiently large we have This shows that P is the limit of convex combinations of deterministic correlations, which implies that P ∈ C loc (Γ).
Proof.Assume for contradiction there exists an integer K such that A q (P ) ≤ K for all P ∈ C q (Γ); we show this results in a uniform upper bound K ′ on D qc (P ), which implies C q (Γ) = C K ′ qc (Γ) is closed.For this, we will first show that P ∈ conv(C K qc (Γ)).In a first step observe that any P ∈ C q (Γ) \ conv(C K qc (Γ)) can be decomposed as where ), and 0 < µ 1 ≤ K/(K + 1).Indeed, by assumption and using Proposition 2.1, A qc (P ) = A q (P ) ≤ K, so P can be written as a convex combination , we may repeat the same argument for R 1 .By iterating we obtain for each integer k ∈ N a decomposition Hence the sequence ( Qk ) k has a limit Q and P = Q holds.As all Qk lie in the compact set conv(C K qc (Γ)), we also have , where c = |Γ| + 1 − |S||T |.By using a direct sum construction one can obtain D qc (P ) ≤ cK, which shows K ′ := cK is a uniform upper bound on D qc (P ) for all P ∈ C q (Γ).

Setup of the hierarchy
We will now construct a hierarchy of lower bounds on the minimal entanglement dimension, using its formulation via A qc (P ).Our approach is based on noncommutative polynomial optimization, thus similar to the approach in [15] for bounding matrix factorization ranks.We first need some notation.Set x = x a s : (a, s) ∈ A×S and y = y b t : (b, t) ∈ B ×T , and let x, y, z r be the set of all words of length at most r in the n = |S||A| + |T ||B| + 1 symbols x a s , y b t , and z.Moreover, set x, y, z = x, y, z ∞ .We equip x, y, z r with an involution w → w * that reverses the order of the symbols in the words and leaves the symbols x a s , y b t , z invariant; e.g., (x a s z) * = zx a s .Let R x, y, z r be the vector space of all real linear combinations of the words of length (aka degree) at most r.The space R x, y, z = R x, y, z ∞ is the *algebra with Hermitian generators {x a s }, {y b t }, and z, and the elements in this algebra are called noncommutative polynomials in the variables {x a s }, {y b t }, z.The hierarchy is based on the following idea: For any feasible solution to A qc (P ), its objective value can be modeled as L(1) for a certain tracial linear form L on the space of noncommutative polynomials (truncated to degree 2r).
Indeed, assume {(P i , λ i ) i } is a feasible solution to the program A qc (P ) defined in Section 1.2, where , and d i = D qc (P i ).For r ∈ N ∪ {∞}, consider the linear functional L ∈ R x, y, z * 2r defined by Here, for each index i, we set , and replace the variables x a s , y b t , z by X a s (i), Y b t (i), and is the objective value of the feasible solution {(P i , λ i ) i } to A qc (P ).We will identify several computationally tractable properties that this L satisfies.Then the hierarchy of lower bounds on A qc (P ) consists of optimization problems where we minimize L(1) over the set of linear functionals that satisfy these properties.First note that L is symmetric, that is, L(w) = L(w * ) for all w ∈ x, y, z 2r , and tracial, that is, L(ww ′ ) = L(w ′ w) for all w, w ′ ∈ x, y, z with deg(ww ′ ) ≤ 2r.
For all p ∈ R x, y, z r−1 we have then L is nonnegative (denoted as L ≥ 0) on the truncated quadratic module Similarly, setting we have L = 0 on the truncated ideal Moreover, we have L(z) = i λ i Re(Tr(ψ i ψ * i )) = 1.In addition, for any matrices U, V ∈ C d i ×d i we have therefore, in particular, L(wzuzvz) = L(wzvzuz) for all u, v, w ∈ x, y, z with deg(wzuzvz) ≤ 2r.
That is, we have L = 0 on I 2r (R r ), where We get the idea of adding these last constraints from [32], where this is used to study the mutually unbiased bases problem.
Note that for order r = 1 we get the trivial lower bound ξ q 1 (P ) = 1.For each finite r ∈ N the parameter ξ q r (P ) can be computed by semidefinite programming.Indeed, the condition L ≥ 0 on M 2r (G) means that L(p * gp) ≥ 0 for all g ∈ G ∪ {1} and all polynomials p ∈ R x, y, z with degree at most r − ⌈deg(g)/2⌉.This is equivalent to requiring that the matrices (L(w * gw ′ )), indexed by all words w, w ′ with degree at most r − ⌈deg(g)/2⌉, are positive semidefinite.To see this, write p = w p w w and let p = (p w ) denote the vector of coefficients, then L(p * gp) ≥ 0 is equivalent to pT (L(w * gw ′ ))p ≥ 0. When g = 1, the matrix (L(w * w ′ )) is indexed by the words of degree at most r, it is called the moment matrix of L and denoted by M r (L) (or M (L) when r = ∞).The entries of the matrices (L(w * gw ′ )) are linear combinations of the entries of M r (L), and the constraint L = 0 on I 2r (H ∪ R r ) can be written as a set of linear constraints on the entries of M r (L).It follows that for finite r ∈ N, the parameter ξ q r (P ) is indeed computable by a semidefinite program.

Background on positive tracial linear forms
Before we show the convergence results we give some background on positive tracial linear forms, which we use again in Section 3. We state these results using the variables x 1 , . . ., x n , where we use the notation x = x 1 , . . ., x n .The results stated below do not always appear in this way in the sources cited; we follow the presentation of [15], where full proofs for these results are also provided.
First we need a few more definitions.A polynomial p ∈ R x is called symmetric if p * = p, and we denote the set of symmetric polynomials by Sym R x .Given G ⊆ Sym R x and for some R > 0. We will use the concept of a C * -algebra, which for our purposes can be defined as a norm closed * -subalgebra of the space B(H) of bounded operators on a complex Hilbert space H.We say that A is unital if it contains the identity operator (denoted 1).An element a ∈ A is called positive if a = b * b for some b ∈ A. A linear form τ on a unital C * -algebra A is said to be a state if τ (1) = 1 and τ is positive; that is, τ (a) ≥ 0 for all positive elements a ∈ A.
We say that a state τ is tracial if τ (ab) = τ (ba) for all a, b ∈ A. See, for example, [4] for more information on C * -algebras.The first result relates positive tracial linear forms to C * -algebras; see [32] for the noncommutative (eigenvalue) setting and [8] for the tracial setting.
Theorem 2.4.Let G ⊆ Sym R x and H ⊆ R x and assume that M(G)+I(H) is Archimedean.For a linear form L ∈ R x * , the following are equivalent: (1) L is symmetric, tracial, nonnegative on M(G), zero on I(H), and L(1) = 1; (2) there is a unital C * -algebra A with tracial state τ and X ∈ A n such that g(X) is positive in A for all g ∈ G, and h(X) = 0 for all h ∈ H, with The following can be seen as the finite dimensional analogue of the above result.The proof of the unconstrained case (G = H = ∅) can be found in [7], and for the constrained case in [8].Given a linear form L ∈ R x * , recall that the moment matrix M (L) is given by Theorem 2.5.Let G ⊆ Sym R x and H ⊆ R x .For L ∈ R x * , the following are equivalent: (1) L is a symmetric, tracial, linear form with L(1) = 1 that is nonnegative on M(G), zero on I(H), and has rank(M (L)) < ∞; (2) there is a finite dimensional C * -algebra A with a tracial state τ and X ∈ A n satisfying (11), with g(X) positive in A for all g ∈ G and h(X) = 0 for all h ∈ H; (3) L is a convex combination of normalized trace evaluations at tuples X of Hermitian matrices that satisfy g(X) 0 for all g ∈ G and h(X) = 0 for all h ∈ H.
A truncated linear functional L ∈ R x 2r is called δ-flat if the principal submatrix M r−δ (L) of M r (L) indexed by monomials up to degree r − δ has the same rank as M r (L); L is flat if it is δ-flat for some δ ≥ 1.The following result claims that any flat linear functional on a truncated polynomial space can be extended to a linear functional L on the full algebra of polynomials.It is due to Curto and Fialkow [10] in the commutative case and extensions to the noncommutative case can be found in [42] (for eigenvalue optimization) and [7,21] (for trace optimization).
2r is symmetric, tracial, δ-flat, nonnegative on M 2r (G), and zero on I 2r (H), then L extends to a symmetric, tracial, linear form on R x that is nonnegative on M(G), zero on I(H), and whose moment matrix has finite rank.
The following technical lemma, based on the Banach-Alaoglu theorem, is a well-known tool to show asymptotic convergence results in polynomial optimization.
Lemma 2.7.Let G ⊆ Sym R x , H ⊆ R x , and assume that for some d ∈ N and R > 0 2r is tracial, nonnegative on M 2r (G) and zero on I 2r (H).Then |L r (w)| ≤ R |w|/2 L r (1) for all w ∈ x 2r−2d+2 .In addition, if sup r L r (1) < ∞, then {L r } r has a pointwise converging subsequence in R x * .

Convergence results
We first show equality ξ q * (P ) = A qc (P ), and then we consider convergence properties of the bounds ξ q r (P ) to the parameters ξ q ∞ (P ) and ξ q * (P ).
Proof.We already know that ξ q * (P ) ≤ A qc (P ).To show ξ q * (P ) ≥ A qc (P ) we let L be feasible for ξ q * (P ), so that L ≥ 0 on M(G) and L = 0 on I(H ∪ R ∞ ).By Theorem 2.5, there exist finitely many scalars λ i ≥ 0, Hermitian matrix tuples X(i) = (X a s (i)) a,s and Y(i) = (Y b t (i)) b,t , and Hermitian matrices Z i , so that g(X(i), Y(i), Z i ) 0 for all g ∈ G, h(X(i), Y(i), Z i ) = 0 for all h ∈ H ∪ R ∞ , and By Artin-Wedderburn theory we know that for each i there is a unitary matrix U i such that Hence, after applying this further block diagonalization we may assume that in the decomposition (12), for each i, C X(i), Y(i), Z i is a full matrix algebra Since Z i is a projector, there exists a unitary matrix U i such that U i Z i U * i = Diag(1, . . ., 1, 0, . . ., 0).The above then implies that for all T 1 and T 2 , the leading principal submatrices of size rank(Z i ) of U i T 1 U * i and U i T 2 U * i commute.This implies rank(Z i ) ≤ 1 and thus Tr(Z i ) ∈ {0, 1}.Let I be the set of indices with Tr(Z i ) = 1.
For each i ∈ I define and [X a s (i), Y b t (i)] = 0 by the ideal conditions.We have P = i∈I λ i P i , so that (P i , λ i ) i∈I forms a feasible solution to A qc (P ) with objective value i∈I λ i D qc The problem ξ q r (P ) differs in two ways from a standard tracial optimization problem.It does not have the normalization L(1) = 1 (and instead minimizes L(1)), and it has ideal constraints L = 0 on I 2r (R r ) where R r depends on r.We show asymptotic convergence still holds.Proposition 2.9.For any P ∈ C q (Γ) we have ξ q r (P ) → ξ q ∞ (P ) as r → ∞.
, where H 0 contains the symmetric polynomials in H; i.e., omitting the commutators [x a s , y b t ].Indeed, we have 1 x a s , and the same for y b t .Hence R − z 2 − a,s (x a s ) 2 − b,t (y b t ) 2 ∈ M 4 (G ∪ H 0 ) for some R > 0. Fix ε > 0 and for each r ∈ N let L r be feasible for ξ q r (P ) with value L r (1) ≤ ξ q r (P )+ε.As L r is tracial and zero on I 2r (H 0 ), it follows (using the identity p * gp = pp * g+[p * g, p]) that L = 0 on M 2r (H 0 ).Hence, L r ≥ 0 on M 2r (G ∪ H 0 ).Since sup r L r (1) ≤ A q (P ) + ε, we can apply Lemma 2.7 and conclude that {L r } r has a converging subsequence; denote its limit by L ε ∈ R x * .One can verify that L ε is feasible for ξ q ∞ (P ), and ξ q ∞ (P ) ≤ L ε (1) ≤ lim r→∞ ξ q r (P ) + ε ≤ ξ q ∞ (P ) + ε.Letting ε → 0 we obtain that ξ q ∞ (P ) = lim r→∞ ξ q r (P ).
Next we show that if ξ q r (P ) admits a δ-flat optimal solution with δ = ⌈r/3⌉ + 1, then we have ξ q r (P ) = ξ q * (P ).This result is a variation of the flat extension result from Theorem 2.6, where δ now depends on the order r because the ideal constraints in ξ q r (P ) depend on r.
Proof.Let δ = ⌈r/3⌉ + 1 and let L be a δ-flat optimal solution to ξ q r (P ).We have to show ξ q r (P ) ≥ ξ q * (P ), which we do by constructing a feasible solution to ξ q * (P ) with the same objective value.The main step in the proof of Theorem 2.6 consists of extending the linear form L to a tracial symmetric linear form L on R x, y, z that is nonnegative on M(G), zero on I(H), and satisfies rank(M ( L)) < ∞ (see the proof of [15, Thm.2.3] for a detailed exposition).To do this a subset W of x, y, z r−δ is found such that we have the vector space direct sum R x, y, z = span(W ) ⊕ I(N r (L)), where N r (L) is the vector space N r (L) = p ∈ R x, y, z r : L(qp) = 0 for all q ∈ R x, y, z r .
It is moreover shown that I(N r (L)) ⊆ N ( L).For p ∈ R x, y, z we denote by r p the unique element in span(W ) such that p − r p ∈ I(N r (L)).We now show that L is zero on I(R ∞ ).For this fix u, v, w ∈ R x, y, z .Then we have L(w(zuzvz − zvzuz)) = L(wzuzvz) − L(wzvzuz).
Since L is tracial and u − r u , v − r v , w − r w ∈ I(N r (L)) ⊆ N ( L), we have L(wzuzvz) = L(r w zr u zr v z) and L(wzvzuz) = L(r w zr v zr u z).
Since deg(r u zr v zr w z) = deg(r v zr u zr w z) ≤ 2r we have L(r w zr u zr v z) = L(r w zr u zr v z) and L(r w zr v zr u z) = L(r w zr v zr u z).
Since L extends L we have L(z) = L(z) = 1 and L(x a s y b t z) = L(x a s y b t z) = P (a, b|s, t) for all a, b, s, t.So, L is feasible for ξ q * (P ) and has the same objective value L(1) = L(1).
3 Bounding quantum graph parameters 3.1 Hierarchies γ col r (G) and γ stab r

(G) based on synchronous correlations
In Section 1.3 we introduced quantum chromatic numbers (Definition 1.1) and quantum stability numbers (Definition 1.2) in terms of synchronous quantum correlations satisfying certain linear constraints.We first give (known) reformulations in terms of C * -algebras, and then we reformulate those in terms of tracial optimization, which leads to the hierarchies γ col r (G) and γ stab r (G).The following result from [41] allows us to write a synchronous quantum correlation in terms of C * -algebras admitting a tracial state.

Theorem 3.1 ([41]
).Let Γ = A 2 × S 2 and P ∈ R Γ .We have P ∈ C qc,s (Γ) (resp., P ∈ C q,s (Γ)) if and only if there exists a unital (resp., finite dimensional) C * -algebra A with a faithful tracial state τ and a set of projectors {X a s : s ∈ S, a ∈ A} ⊆ A satisfying a∈A X a s = 1 for all s ∈ S and P (a, b|s, t) = τ (X a s X b t ) for all s, t ∈ S, a, b ∈ A.
Here we add the condition that τ is faithful, that is, τ (X * X) = 0 implies X = 0, since it follows from the GNS construction in the proof of [41].This means that It follows that χ qc (G) is equal to the smallest k ∈ N for which there exists a C * -algebra A, a tracial state τ on A, and a family of projectors The quantum chromatic number χ q (G) is equal to the smallest k ∈ N for which there exists a finite dimensional C * -algebra A with the above properties.Analogously, α qc (G) is equal to the largest k ∈ N for which there is a C * -algebra A, a tracial state τ on A, and a set of projectors X i c X j c ′ = 0 if (i = j and c = c ′ ) or ((i = j or {i, j} ∈ E) and c = c ′ ), (16) and α q (G) is equal to the largest k ∈ N for which A can be taken finite dimensional.These reformulations of χ q (G), χ qc (G), α q (G) and α qc (G) also follow from [36,Thm. 4.7], where general quantum graph homomorphisms are considered; the formulations of χ q (G) and χ qc (G) are also made explicit in [36,Thm. 4.12].
By Artin-Wedderburn theory [53,2], a finite dimensional C * -algebra is isomorphic to a matrix algebra.So the above reformulations of χ q (G) and α q (G) can be seen as feasibility problems of systems of equations in matrix variables of unspecified (but finite) dimension; such formulations are given in [9,28,46].Restricting to scalar solutions (1 × 1 matrices) in these feasibility problems recovers the classical graph parameters χ(G) and α(G).
We now reinterpret the above formulations in terms of tracial optimization.Given a graph G = (V, E), let i ≃ j denote {i, j} ∈ E or i = j.For k ∈ N, let H col G,k and H stab G,k denote the sets of polynomials corresponding to equations ( 13)-( 14) and ( 15)-( 16): ), and the analogous statements hold for H stab G,k .Hence, both M(∅) + I(H col k ) and M(∅) + I(H stab k ) are Archimedean and we can apply Theorems 2.4 and 2.5 to express the quantum graph parameters in terms of positive tracial linear functionals.Namely, and χ q (G) is obtained by adding the constraint rank(M (L)) < ∞.Likewise, and α q (G) is given by this program with the additional constraint rank(M (L)) < ∞.
Starting from these formulations it is natural to define a hierarchy {γ col r (G)} of lower bounds on χ qc (G) and a hierarchy {γ stab r (G)} of upper bounds on α qc (G), where the bounds of order r ∈ N are obtained by truncating L to polynomials of degree at most 2r and truncating the ideal to degree 2r.Then, by defining γ col * (G) and The optimization problems γ col r (G), for r ∈ N, can be computed by semidefinite programming and binary search on k, since the positivity condition on L can be expressed by requiring that its truncated moment matrix M r (L) = (L(w * w ′ )) (indexed by words with degree at most r) is positive semidefinite.If there is an optimal solution (k, L) to γ col r (G) with L flat, then, by Theorem 2.6, we have equality γ col r (G) = χ q (G).Since {γ col r (G)} r∈N is a monotone nondecreasing sequence of lower bounds on χ q (G), there exists an r 0 such that for all r ≥ r 0 we have The analogous statements hold for the parameters γ stab r (G).Hence, we have shown the following result.

Proposition 3.2.
There is an r 0 ∈ N such that γ col r (G) = χ qc (G) and γ stab r (G) = α qc (G) for all r ≥ r 0 .Moreover, if γ col r (G) admits a flat optimal solution, then γ col r (G) = χ q (G), and if γ stab r (G) admits a flat optimal solution, then γ stab r (G) = α q (G).Remark 3.3.A hierarchy {Q r (Γ)} of outer semidefinite approximations for the set C qc (Γ) of commuting quantum correlations was constructed in [41], revisiting the approach in [31,42].This hierarchy is converging, that is, The approximations Q r (Γ) are based on the eigenvalue optimization approach, applied to the formulation (3) of commuting quantum correlations, and thus they use linear functionals on polynomials involving two sets of variables x a s , y b t for (a, b, s, t) ∈ Γ.The authors of [41] use these outer approximations of C qc (Γ) to define a converging hierarchy of lower bounds on χ qc (G) in terms of feasibility problems over the sets Q r (Γ).
For synchronous correlations we can use the result of Theorem 3.1 and the tracial optimization approach used here to define directly a converging hierarchy {Q r,s (Γ)} of outer semidefinite approximations for the set C qc,s (Γ) of synchronous commuting quantum correlations.These approximations now use linear functionals on polynomials involving only one set of variables x a s .Namely, define Q r,s (Γ) as the set of P ∈ R Γ for which there exists a symmetric, tracial, positive linear functional L ∈ R {x a s : (a, s) ∈ A × S} * 2r such that L(1) = 1 and L = 0 on the ideal generated by the polynomials x a s − (x a s ) 2 ((a, s) ∈ A × S) and 1 − a∈A x a s (s ∈ S), truncated at degree 2r.Then we have The synchronous value of a nonlocal game is defined in [12] as the maximum value of the objective function (6) over the set C qc,s (Γ).By maximizing the objective (6) over the relaxations Q r,s (Γ) we get a hierarchy of semidefinite programming upper bounds that converges to the synchronous value of the game.Finally note that one can also view the parameters γ col r (G) as solving feasibility problems over the sets Q r,s (Γ).Here we revisit some known Lasserre type hierarchies for the classical stability number α(G) and chromatic number χ(G) and we show that their tracial noncommutative analogues can be used to recover known parameters such as the projective packing number α p (G), the projective rank ξ f (G), and the tracial rank ξ tr (G).Compared to the hierarchies defined in the previous section, these Lasserre type hierarchies use less variables (they only use variables indexed by the vertices of the graph G), but they also do not converge to the (commuting) quantum chromatic or stability number.Given a graph G = (V, E), define the set of polynomials in the variables x = (x i : i ∈ V ) (which are commutative or noncommutative depending on the context).Note that 1 − x 2 i ∈ M 2 (∅) + I 2 (H G ) for all i ∈ V , so M(∅) + I(H G ) is Archimedean.

Semidefinite programming bounds on the projective packing number
We first recall the Lasserre hierarchy of bounds for the classical stability number α(G).Starting from the formulation of α(G) via the optimization problem the r-th level of the Lasserre hierarchy for α(G) (introduced in [23,24]) is defined by Then las stab r+1 (G) ≤ las stab r (G), the first bound is Lovász' theta number: las stab 1 (G) = ϑ(G), and finite convergence to α(G) is shown in [24]: las stab α(G) (G) = α(G).Roberson [45] introduces the projective packing number as an upper bound for the quantum stability number α q (G); the inequality α q (G) ≤ α p (G) also follows from Proposition 3.4 below.In view of ( 17), the parameter α p (G) can be seen as a noncommutative analogue of α(G).
For r ∈ N ∪ {∞} we define the noncommutative analogue of las stab r (G) by (G) can be reformulated in terms of C * -algebras: where A is a (resp., finite-dimensional) C * -algebra with tracial state τ and X i ∈ A (i ∈ [n]) are projectors satisfying X i X j = 0 for all {i, j} ∈ E.Moreover, as we now see, the parameter ξ stab * (G) coincides with the projective packing number and the parameters ξ stab * (G) and ξ stab ∞ (G) upper bound the quantum stability numbers.Proposition 3.4.We have Proof.By (17), α p (G) is the largest value of L( i∈V x i ) over linear functionals L that are normalized trace evaluations at projectors X ∈ (S d ) n (for some d ∈ N) with X i X j = 0 for {i, j} ∈ E. By convexity the optimum remains unchanged when considering a convex combination of such trace evaluations.In view of Theorem 2.5(3), this optimum is precisely the parameter ξ stab * (G).This shows equality α p (G) = ξ stab * (G).Consider a C * -algebra A with tracial state τ and projectors 15)-( 16).Then, setting

Semidefinite programming bounds on the projective rank and tracial rank
We now turn to the (quantum) chromatic numbers.First recall the definition of the fractional chromatic number: where S is the set of stable sets of G. Clearly, χ f (G) ≤ χ(G).The following Lasserre type lower bounds for the classical chromatic number χ(G) are defined in [18]: By viewing χ f (G) as minimizing L(1) over linear functionals L ∈ R[x] * that are conic combinations of evaluations at characteristic vectors of stable sets, we see that las col r (G) ≤ χ f (G) for all r ≥ 1.In [18] it is shown that las col α(G) (G) = χ f (G).Moreover, the order 1 bound coincides with the theta number: las col 1 (G) = ϑ(G).
The following parameter ξ f (G), called the projective rank of G, was introduced in [28] as a lower bound on the quantum chromatic number χ q (G): Proposition 3.5 ([28]).For any graph G we have ξ f (G) ≤ χ q (G).
Proof.Set k = χ q (G).It is shown in [9] that in the definition of χ q (G) from ( 13)-( 14), one may assume w.l.o.g. that all matrices X c i have the same rank, say, r.Then, for any given color c ∈ [k], the matrices X c i (i ∈ V ) provide a feasible solution to ξ f (G) with value d/r.Finally, d/r = k holds since by ( 13)-( 14) we have d = rank(I) = k c=1 rank(X c i ) = kr.
In [41,Prop. 5.11] it is shown that the projective rank can equivalently be defined as They also define the tracial rank ξ tr (G) of G as the parameter obtained by omitting in the above definition of ξ f (G) the restriction that A has to be finite dimensional.The motivation for the parameter ξ tr (G) is that it lower bounds the commuting quantum chromatic number [41,Thm. 5.11]: In view of Theorems 2.4 and 2.5, we obtain the following reformulations: and ξ tr (G) is obtained by the same program without the restriction rank(M (L)) < ∞.In addition, using Theorem 2.5(3), we see that in this formulation of ξ f (G) we can equivalently optimize over all L that are conic combinations of trace evaluations at projectors X i ∈ S d (for some d ∈ N) satisfying X i X j = 0 for all {i, j} ∈ E. If we restrict the optimization to scalar evaluations (d = 1) we obtain the fractional chromatic number.This shows that the projective rank can be seen as the noncommutative analogue of the fractional chromatic number, as was already observed in [28,41].The above formulations of the parameters ξ tr (G) and ξ f (G) in terms of linear functionals also show that they fit within the following hierarchy {ξ col r (G)} r∈N∪{∞} , defined as the noncommutative tracial analogue of the hierarchy {las col r (G)} r : Again, ξ col * (G) is the parameter obtained by adding the constraint rank(M (L)) < ∞ to the program defining ξ col ∞ (G).By the above discussion the following holds.
Proposition 3.6.We have Using Lemma 2.7 one can verify that the parameters ξ col r (G) converge to ξ col ∞ (G).Moreover, by Theorem 2.6, if ξ col r (G) admits a flat optimal solution, then we have ξ col r = ξ col * (G).Also, the parameter ξ col 1 (G) coincides with las col 1 (G) = ϑ(G).Summarizing we have ξ col ∞ (G) = ξ tr (G) ≤ χ qc (G) and the following chain of inequalities Observe that the bounds las col r (G) and ξ col r (G) remain below the fractional chromatic number Hence, these bounds are weak if χ f (G) is close to ϑ(G) and far from χ(G) or χ q (G).In the classical setting this is the case, e.g., for the class of Kneser graphs G = K(n, r), with vertex set the set of all r-subsets of [n] and having an edge between any two disjoint r-subsets.By results of Lovász [26,27], the fractional chromatic number is n/r, which is known to be equal to ϑ(K(n, r)), while the chromatic number is n − 2r + 2. In [18] this was used as a motivation to define a new hierarchy of lower bounds {Λ r (G)} on the chromatic number that can go beyond the fractional chromatic number.In Section 3.3 we recall this approach and show that its extension to the tracial setting recovers the hierarchy {γ col r (G)} introduced in Section 3.1.We also show how a similar technique can be used to recover the hierarchy {γ stab r (G)}.

A link between ξ stab
Assume G is vertex-transitive.Let L be a feasible solution for ξ stab r (G).As G is vertextransitive we may assume (after symmetrization) that L(x i ) is constant, set L(x i ) =: 1/λ for all i ∈ V , so that the objective value of L for ξ stab For vertex-transitive G, the inequality ξ f (G)α q (G) ≤ |V | is shown in [28, Lem.6.5]; it can be recovered from the r = * case of Lemma 3.7 and α q (G) ≤ α p (G).

Comparison to existing semidefinite programming bounds
By adding the inequalities L(x i x j ) ≥ 0, for all i, j ∈ V , to ξ col 1 (G), we obtain the strengthened theta number ϑ + (G) (from [50]).Moreover, if we add the constraints to the program defining the parameter ξ col 1 (G), then we obtain the parameter ξ SDP (G), which is introduced in [41, Thm.7.3] as a lower bound on ξ tr (G).We will now show that the inequalities ( 18)-( 20) are in fact valid for ξ col 2 (G), which implies For this, given a clique C in G, we define the polynomial g C := 1 − i∈C x i ∈ R x .Then (19) and ( 20) can be reformulated as L(x i g C ) ≥ 0 and L(g C g C ′ ) ≥ 0, respectively, using the fact that L(x i ) = L(x 2 i ) = 1 for all i ∈ V .Hence, to show that any feasible L for ξ col 2 (G) satisfies ( 18)- (20), it suffices to show Lemma 3.8 below.Recall that a commutator is a polynomial of the form [p, q] = pq − qp with p, q ∈ R x .We denote the set of linear combinations of commutators [p, q] with deg(pq) ≤ r by Θ r .Lemma 3.8.Let C and C ′ be cliques in a graph G and let i, j ∈ V .Then we have where h ∈ I 2 (H G ).We also have , and, writing analogously g  Here, in this last section, we make the link between the hierarchies {ξ stab r (G)} (resp.{ξ col r (G)}) and {γ stab r (G)} (resp.{γ col r (G)}).The key fact is the interpretation of the coloring and stability numbers in terms of certain graph products.
We start with the (quantum) coloring number.For an integer k, recall that the Cartesian product G K k is the graph with vertex set V × [k], where the vertices (i, c) and (j, c ′ ) are adjacent if ({i, j} ∈ E and c = c ′ ) or (i = j and c = c ′ ).The following is a well-known reduction of the chromatic number χ(G) to the stability number of the Cartesian product G K k : χ(G) = min k ∈ N : α(G K k ) = |V | .It was used in [18] to define the following lower bounds on the chromatic number: where it was also shown that las col r (G) ≤ Λ r (G) ≤ χ(G) for all r ≥ 1, with equality Λ |V | (G) = χ(G).Hence the bounds Λ r (G) may go beyond the fractional chromatic number.This is the case for the above mentioned Kneser graphs; see [17] for other graph instances.
The above reduction from coloring to stability number has been extended to the quantum setting by [28], where it is shown that χ q (G) = min{k ∈ N : α q (G K k ) = |V |}.It is therefore natural to use the upper bounds ξ stab r (G K k ) on α q (G K k ) in order to get the following lower bounds on the quantum coloring number: which are thus the noncommutative analogues of the bounds Λ r (G).Observe that, for any integer k ∈ N and r ∈ N ∪ {∞, * }, we have ξ stab r (G K k ) ≤ |V |, which follows from Lemma 3.8 and the fact that the cliques i , denote the set of polynomials corresponding to these cliques.We now show that the parameters (22) coincide in fact with γ col r (G) for all r ∈ N ∪ {∞}.For this observe first that the quadratic polynomials in the set H col G,k correspond precisely to the edges of G K k , and the projector constraints are included in I 2 (H col G,k ) (see Section 3.1), so that We will also use the following result.
Lemma 3.9.Let r ∈ N ∪ {∞, * } and assume L is feasible for ξ . We will show L = 0 on I 2r (C G K k ).For this we first observe that g C i − (g C i ) 2 ∈ I 2 (H G K k ) by (21).Hence L(g C i ) = L(g 2 C i ) ≥ 0, which, combined with i L(g C i ) = 0, implies L(g C i ) = 0 for all i ∈ V .Next we show L(wg C i ) = 0 for all words w with degree at most 2r − 1, using induction on deg(w).The base case w = 1 holds by the above.Assume now w = uv, where deg(v) < deg(u) ≤ r.Using the positivity of L, the Cauchy-Schwarz inequality gives |L(uvg Note that it suffices to show L(v * g C i v) = 0 since, using again (21), this implies L(v * g 2 C i v) = 0 and thus L(uvg C i ) = 0. Using the tracial property of L and the induction assumption, we see that L(v * g C i v) = L(vv * g C i ) = 0 since deg(vv * ) < deg(w).Note that the proof of Proposition 3.10 also works in the commutative setting; this shows that the sequence Λ r (G) corresponds to the usual Lasserre hierarchy for the feasibility problem defined by the equations ( 13)-( 14), which is another way of showing Λ ∞ (G) = χ(G).
We now turn to the (quantum) stability number.For k ∈ N, consider the graph product K k ⋆ G, with vertex set [k] × G, and with an edge between (c, i) and (c ′ , j) when (c = c ′ , i = j) or (c = c ′ , i = j) or (c = c ′ , {i, j} ∈ E).The product K k ⋆ G coincides with the homomorphic product K k ⋉G used in [28,Sec. 4.2], where it is shown that α q (G) = max k ∈ N : α q (K k ⋆G) = k .This suggests using the upper bounds ξ stab r (K k ⋆ G) on α q (K k ⋆ G) to define the following upper bounds on α q (G): max k ∈ N : Proof.We may restrict to r ∈ N since we have seen earlier that the inequalities hold for r ∈ {∞, * }.The proof for the coloring parameters is similar to the proof of [18,Prop. 3.3] in the classical case and thus omitted.We show the inequality ξ stab ).Also, we set L(1) = L(1) = 1.Then, we have L( i∈V x i ) = k.Moreover, one can easily check that L is indeed tracial, symmetric, positive, and vanishes on I 2r (H G ).

A Synchronous quantum correlations
We prove the following result by combining proofs from [46] (see also [28]) and [41].
Proposition A.1.The smallest local dimension in which a synchronous quantum correlation P can be realized is given by the completely positive semidefinite rank of the matrix M P indexed by S × A with entries (M P ) (s,a),(t,b) = P (a, b|s, t).
Proof.Suppose first that (ψ, E By using the identities vec(K) = ψ and vec(K) * (E a s ⊗ F b t )vec(K) = Tr(KE a s KF b t ) = Tr(K 1/2 E a s K 1/2 K 1/2 F b t K 1/2 ), we see that P (a, b|s, t) = X a s , Y b
is positive semidefinite.In the same way we have L(p * y b t p) ≥ 0 and L(p * zp) ≥ 0. That is, if we set

r
(G) and ξ col r (G) In[18, Thm.3.1]  it is shown that, for any r ≥ 1, the bounds las stab r (G) and las col r (G) satisfy las stab r (G)las col r (G) ≥ |V |, with equality if G is vertex-transitive, which extends a well-known property of the theta number (case r = 1).The same holds for the noncommutative analogues ξ stab r (G) and ξ col r (G).Lemma 3.7.For any graph G = (V, E) and r ∈ N ∪ {∞, * } we have ξ stab r (G)ξ col r (G) ≥ |V |, with equality if G is vertex-transitive.Proof.Let L be feasible for ξ col r (G).Then L = L/L(1) provides a solution to ξ stab r (G) with value L i∈V x i = |V |/L(1), implying that ξ stab r (G) ≥ |V |/L(1) and therefore ξ stab r

Proposition 3 . 10 .
For r ∈ N ∪ {∞} we have γ col r (G) = min{k :ξ stab r (G K k ) = |V |}.Proof.Let L be a linear functional certifying γ col r (G) ≤ k.Then L is feasible for ξ stab r (G K k ) and, as L = 0 on I 2r (C G K k ), Lemma 3.9 shows that L( i,c x c i ) = |V |.This shows that ξ stab r (G K k ) = |V | and thus min{k : ξ stab r (G K k ) = |V |} ≤ k.Conversely, assume ξ stab r (G K k ) = |V |.Since the optimum is attained, there exists a linear functional L feasible for ξ stab r (G K k ) with L( i,c x c i ) = |V |.Using Lemma 3.9 we can conclude that L is zero on I 2r (C G K k ) and thus also on I 2r (H col G,k ).This shows γ col r (G) ≤ k.

ForProposition 3 . 12 .
each c ∈ [k], the set C c = {(c, i) : i ∈ V } is a clique in K k ⋆ G and we let C K k ⋆G = g C c : c ∈ [k], where g C c = 1 − i∈V x i c , denote the set of polynomials corresponding to these cliques.Since these k cliques cover the vertex set of K k ⋆ G, we can use Lemma 3.8 to conclude ξ stab r (K k ⋆ G) ≤ k for all r ∈ N ∪ {∞, * }.Again, observe that the quadratic polynomials in the set H stab G,k correspond precisely to the edges of K k ⋆ G and that we haveI 2r (H stab G,k ) = I 2r (H K k ⋆G ∪ C K k ⋆G ).Based on this, one can show the analogue of Lemma 3.9:If L is feasible for the program ξ stab r (K k ⋆ G), then we have L( i,c x i c ) = k if and only if L = 0 on I 2r (C K k ⋆G ), which implies the following result.Proposition 3.11.For r ∈ N ∪ {∞} we have γ stab r (G) = max{k : ξ stab r (K k ⋆ G) = k}.We do not know whether the results of Propositions 3.10 and 3.11 hold for r = * , since we do not know whether the supremum is attained in the parameter ξ stab * (•) = α p (•) (as was already observed in[45, p. 120]).Hence we can only claim the inequalitiesγ col * (G) ≥ min{k : ξ stab * (G K k ) = |V |} and γ stab * (G) ≤ max{k : ξ stab * (K k ⋆ G) = k}.As mentioned above, we have las col r (G) ≤ Λ r (G) for any r ∈ N[18, Prop.3.3].This result extends to the noncommutative setting and the analogous result holds for the stability parameters.In other words the hierarchies {γ col r (G)} and {γ stab r (G)} refine the hierarchies {ξ col r (G)} and ξ stab r (G)}.For r ∈ N ∪ {∞, * }, ξ col r (G) ≤ γ col r (G) and ξ stab r (G) ≥ γ stab r (G).
and, using Proposition 3.11, letL ∈ R x i c : i ∈ V, c ∈ [k] * 2r be optimal for ξ stab r (K k ⋆G) = k.That is, L is tracial, symmetric, positive, and satisfies L(1) = 1, L( i,c x i c ) = k, and L = 0 on I(H K k ⋆G ).It suffices now to construct a tracial symmetric positive linear form L ∈ R x i : i ∈ V * 2r such that L(1) = 1, L( i∈V x i ) = k, and L = 0 on I 2r (H G ), since this will imply ξ stab r (G) ≥ k.For this, for any wordx i 1 • • • x it with degree 1 ≤ t ≤ 2r, we define L(x i 1 • • • x it ) := c∈[k] L(x i 1 c • • • x it c a s , F b t ) is a realization of P in local dimension d.That is, ψ is a unit vector in C d ⊗ C d , E a s , F b t are d × d Hermitian positive semidefinite matrices such that a E a s = b F b t = I for all s, t and P (a, b|s, t) = ψ * (E a s ⊗ F b t )ψ for all (a, b, s, t) ∈ Γ.We will show cpsd-rank C (A P ) ≤ d.The Schmidt decomposition of the unit vector ψ gives nonnegative scalars {λ i } and orthonormal bases {u i } and{v i } of C d such that ψ = d i=1 √ λ i u i ⊗ v i .If we replace ψ by d i=1 √ λ i v i ⊗ v i and E a s by U * E a s U, where U is the unitary matrix for which u i = U v i for all i, then we obtain a new realization (d i=1 √ λ i v i ⊗ v i , U * E a s U, F b t ) of P still in local dimension d.For the simplicity of notation we rename U * E a s U as E a s .Then we define the matrices