Spectral Lower Bounds for the Quantum Chromatic Number of a Graph -- Part II

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \ge \ldots \ge \mu_n$ and chromatic number $\chi(G)$ satisfies: \[ \chi \ge 1 + \kappa \] where $\kappa$ is the smallest integer such that \[ \mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i} \le 0. \] We strengthen this well known result by proving that $\chi(G)$ can be replaced by the quantum chromatic number, $\chi_q(G)$, where for all graphs $\chi_q(G) \le \chi(G)$ and for some graphs $\chi_q(G) \ll \chi(G)$. We also prove a similar result, and investigate implications of these inequalities for the quantum chromatic number of various classes of graphs, which improves many known results. For example we demonstrate that the Kneser graph $KG_{p,2}$ has $\chi_q = \chi = p - 2$.


Introduction
For any connected graph G let V denote the set of vertices where |V | = n, A denote the adjacency matrix, µ max = µ 1 ≥ µ 2 ≥ . . .≥ µ n denote the eigenvalues of A, χ(G) denote the chromatic number and ω(G) the clique number.Let χ q (G) denote the quantum chromatic number, as defined by Cameron et al [2].It is evident that χ q (G) ≤ χ(G), and Mancinska and Roberson [10] found a graph on 14 vertices with χ(G) > χ q (G), which they suspect is the smallest possible example.There exist graphs for which χ q (G) ≪ χ(G).
Elphick and Wocjan [3] proved that many spectral lower bounds for χ(G) are also lower bounds for χ q (G), using linear algebra techniques called pinching and twirling.In this paper, we prove that stronger lower bounds on the chromatic number are also lower bound on the quantum chromatic number.
The following purely combinatorial definition of the quantum chromatic number is due to [10, Definition 1].For d > 0, let I d and 0 d denote the identity and zero matrices in C d×d .

Definition 1 (Quantum c coloring). A quantum c-coloring of the graph
• for all edges vw ∈ E and for all k ∈ [c] The quantum chromatic number χ q (G) is the smallest c for which the graph G admits a quantum c-coloring for some dimension d > 0.
According to the above definition, any classical c-coloring can be viewed as a 1-dimensional quantum coloring, where we set P v,k = 1 if vertex v has color k and we set P v,k = 0, otherwise.Therefore, quantum coloring is a relaxation of classical coloring.As noted in [10], it is surprising that the quantum chromatic number can be strictly and even exponentially smaller than the chromatic number for certain families of graphs.
We use the following alternative characterization of the quantum chromatic number due to [3].Before stating this characterization, we briefly review the definition of pinching.
Definition 2 (Pinching).Let P k ∈ C m×m for k ∈ [c] be orthogonal projectors such that they form a resolution of the identity, that is, ( The operation D : C m×m → C m×m defined by is called pinching.We say that it annihilates X if D(X) = 0.
For d > 0, recall that I d and 0 d denote the identity and zero matrices in C d×d respectively.
Then, the following block-diagonal projectors define a pinching operation that annihilates A ⊗ I d , that is, Remark 1.The classical case corresponds simply to the special case d = 1.
In this case, the projectors P k are a diagonal in the standard basis of C n and each projector corresponds to a color class, that is, each projector P k projects onto the subspace spanned by the standard basis vectors corresponding to the vertices that have been colored with the kth color.

New bounds for the quantum chromatic number
We use ↑ to indicate that the eigenvalues are sorted in increasing order.The ith smallest eigenvalue of a hermitian matrix X ∈ C n×n is denoted by µ ↑ i (X) so that Similarly, we use ↓ to indicate that the eigenvalues are sorted in decreasing order.The ith largest eigenvalue of X is denoted by µ ↓ i (X) so that When the eigenvalues are sorted in decreasing order, we often omit ↓ so that We have µ ↓ i (X) = µ n+1−i (X).It would be somewhat inconvenient to always have to write n + i − 1, so this is why the ↓ notation simplifies the indices.
The following lemma is a standard result in matrix analysis.We describe it in detail since it is the main result that we rely on to prove the new stronger bounds on the quantum chromatic number.
Lemma 1.Let X ∈ C n×n be a hermitian matrix and let S ∈ C n×m be a matrix such that S † S = I m , that is, its m column vectors s 1 , . . ., s m are orthonormal vectors.Then, Proof.This follows from the Courant-Weyl-Fisher theorem stating where the minimum is taken over subspaces M i of dimension i and the maximum is taken over unit vectors x ∈ M i .By multiplying X by S † and S from the left and right, respectively, we effectively restrict the subspaces M i to have the form where We obtain the following corollary by using the identity µ ↑ i (−X) = −µ i (X) and applying the lemma to the matrix −X.
In the following, we will only use this corollary with i = 1, that is, 2.1 First new bound for χ q (G) Theorem 2 (First bound on quantum chromatic number).Let χ q (G) be the quantum chromatic number of the graph G with adjacency matrix A. Let κ be the smallest integer such that holds.Then the quantum chromatic number is bounded from below by Proof.Assume that the quantum chromatic number c = χ q is attained by a quantum coloring P k in dimension d.Let z ∈ C n denote the unique1 eigenvector of A corresponding to the largest eigenvalue µ max (A).Let f j ∈ C d for j ∈ [d] denote the standard basis vectors.Let s 1 , . . ., s m be an orthonormal basis of the subspace For the lower bound, observe that the d orthogonal vectors z ⊗ f j are contained in S since For the upper bound, observe that there are exactly cd vectors in eq. ( 17) and, thus, m cannot be larger than cd.
Let S ∈ C nd×m be the matrix with s 1 , . . ., s m as column vectors.The following two arguments show that the largest eigenvalue of the matrix S † (A ⊗ I d )S is equal to µ max (A) and its multiplicity is equal to d.First, there exist d orthogonal vectors y 1 , . . ., y d ∈ C m such that Sy j = z ⊗ f j since the latter vectors are contained in the subspace S, or equivalently, the column space of S. We have Second, using Corollary 1, the largest eigenvalue of S † (A ⊗ I d )S cannot be greater than the largest eigenvalue of A ⊗ I d .We can always choose the orthonormal basis vectors s 1 , . . ., s m such that for each i ∈ [m] there exists a unique k i ∈ [c] with This is because S = k∈[c] S k , where S k = span{(P k (z ⊗ f j ) : j ∈ [d]} since the projectors P k form a resolution of the identity.We now see that the diagonal entries of the matrix S † (A ⊗ I)S must all be zero since For the last equality we used that P k (A ⊗ I d )P k = 0 for all k ∈ [c].So using Lemma 1, we obtain Using 18, we have Note that K d ≥ (κ − 1)d + 1 must hold because otherwise the condition that κ = K 1 is minimal would be violated.This implies c − 1 A weaker version of Theorem 2, with χ(G) replacing χ q (G), was proved by Hoffman [9] in 1970.It is a trivial corollary that as proved in [3].

Second new bound for χ q (G)
Theorem 3 (Second bound on quantum chromatic number).For any graph G with µ 2 > 0: where g is the multiplicity of µ n (A).
Proof.Consider an arbitrary quantum c-coloring in dimension d.Assume that c ≤ g.Let S be defined as in the proof of the previous theorem in (17).Let T be the subspace spanned by the eigenvectors corresponding to the cd smallest eigenvalues µ ↑ 1 (A ⊗ I), . . ., µ ↑ cd (A ⊗ I).There exists a non-zero unit vector y with y ∈ S ⊥ ∩ T .
The intersection is non-trivial because both S ⊥ and T are contained in the subspace R ⊥ , where Consider y i = P i y for i ∈ [c].Let m be the number of nonzero y i .We now show that at least two of them (w.l.o.g.y 1 and y 2 ) must be nonzero, that is, m ≥ 2. First of all, at least one must be nonzero because otherwise we would have 0 = y = Iy = k∈[c] P k y = k∈[c] y k = 0. Now assume that only y 1 were nonzero, or equivalent, y 1 = y.But this leads to the contradiction where the first inequality holds because c ≤ g, the second inequality holds because y ∈ T and P 1 (A ⊗ I d )P 1 = 0 because the projectors P k form a pinching that annihilates A ⊗ I d .Define s i = y i / y i for i ∈ [m] and S to be the matrix with s i as columns.Consider the matrix X = A ⊗ I d − ∆ • zz † ⊗ I d , where ∆ = µ max − µ min .The diagonal entries of S † XS are all zero because we have = 0 (33) because P i (A ⊗ I d )P i = 0 and y ⊥ z ⊗ f j as y ∈ S ⊥ and z ⊗ f j ∈ S. Since y is in the column space of S, the smallest eigenvalue of S † XS is at most y † Xy, which in turn is at most µ ↑ cd (A ⊗ I d ) = µ ↑ c (A) = µ n (A) as y ∈ T and c ≤ g.Putting everything together, we obtain A weaker version of this bound, with χ(G) replacing χ q (G), is already known, for example in Corollary 3.6.4 in [1].
We note that both Theorems are also valid for weighted adjacency matrices of the form W • A, where W is an arbitrary Hermitian matrix and • denotes the Hadamard product (also called the Schur product).

Non-SRGs
The orthogonality graph, Ω(n), has vertex set the set of ±1−vectors of length n, with two vertices adjacent if they are orthogonal.With 4|n (see [11]), it is known that χ q (Ω(n)) = n but χ(Ω(n)) is exponential in n.A proof that χ q (Ω(n)) = n is as follows.It is immediate from the definition of Ω(n) that ξ ′ (Ω(n)) = n, and it is known that χ q (G) ≤ ξ ′ (G), where ξ ′ (G) is the normalized orthogonal rank of G [12].However, using Theorem 2 and results in section 4.3 of [5] on the eigenvalues of orthogonality graphs we have that: We can construct a quantum coloring of Ω(n) using n colors as follows.Let dimension d = n, let U = diag(1, ω, . . ., ω n−1 ) be a unitary matrix where ω = e 2πi/n , and let z v denote the ±1 vector of length n assigned to vertex v. Then let It is straightforward that this collection of orthogonal projectors satisfy the completeness and orthogonality conditions in Definition 1, so this completes the quantum coloring.We note in passing that with 4|n, a proof that ω(Ω(n)) = n would provide a proof of the Hadamard Conjecture, which dates from 1867.

Open questions
The pentagon (C 5 ) demonstrates that both of the bounds in this paper are not lower bounds for the vector chromatic number or the fractional chromatic number.The orthogonal rank, ξ(G), is incomparable to χ q (G).We do not know whether ξ(G) can replace χ q (G) in Theorems 2 and 3.
We have shown that the Kneser graph K p,2 has χ q = χ, but is this true for all Kneser graphs?Are there any strongly regular graphs with χ q < χ?
In Definition 1 the dimension d is any finite positive integer.Let χ d (G) denote the smallest c for which graph G admits a quantum c-coloring in dimension d.From Section 3.2, we know that χ 1 (Ω(n)) = χ(Ω(n)) which is exponential in n, but χ n (Ω(n)) = n.This raises the question of what is the value of χ d (Ω(n)) for 2 ≤ d ≤ n − 1?