Here we revisit some known Lasserre-type hierarchies for the classical stability num-ber α(G) and chromatic numnum-ber χ(G) and we show that their tracial noncommu-tative analogues can be used to recover known parameters such as the projective
8.3. Bounding quantum graph parameters 143 packing number αp(G), the projective rank ξf(G), and the tracial rank ξtr(G). Both the commutative hierarchies and the tracial noncommutative hierarchies can be viewed as strengthenings of the Lov´asz theta number towards either the (quantum) stability number or the (quantum) chromatic number: the first level corresponds to the theta number. 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 HG=xi− x2i : i ∈ V ∪ xixj: {i, j} ∈ E
in the variables x = (xi : i ∈ V ) (which are commutative or noncommutative depending on the context). Note that 1 − x2i ∈ M2(∅) + I2(HG) for all i ∈ V , so that M(∅) + I(HG) 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
α(G) = supn X
i∈V
xi: x ∈ Rn, h(x) = 0 for all h ∈ HG
o
, (8.7)
the r-th level of the Lasserre hierarchy for α(G) (introduced in [Las01, Lau03]) is defined by
lasstabr (G) = supn
L X
i∈V
xi : L ∈ R[x]∗2r positive, L(1) = 1, L = 0 on I2r(HG)o .
Then we have lasstabr+1(G) ≤ lasstabr (G) and the first bound is Lov´asz’ theta number:
lasstab1 (G) = ϑ(G). Finite convergence to α(G) is shown in [Lau03]:
lasstabα(G)(G) = α(G).
Roberson [Rob13] introduces the projective packing number αp(G) = supn1
d X
i∈V
rank Xi: d ∈ N, X ∈ (Sd)n projectors, (8.8) XiXj= 0 for {i, j} ∈ Eo
= supn1
dTr X
i∈V
Xi
: d ∈ N, X ∈ (Sd)n, h(X) = 0 for h ∈ HG
o
(8.9)
as an upper bound for the quantum stability number αq(G). Note that the in-equality αq(G) ≤ αp(G) also follows from Proposition 8.7 below. Comparing (8.7) and (8.9) we see that the parameter αp(G) can be viewed as a noncommutative analogue of α(G).
144 Chapter 8. Quantum graph parameters For r ∈ N ∪ {∞} we define the noncommutative analogue of lasstabr (G) by
ξrstab(G) = supn L X
i∈V
xi
: L ∈ Rhxi∗2r tracial, symmetric, and positive, L(1) = 1, L = 0 on I2r(HG)o
,
and ξ∗stab(G) by adding the constraint rank(M (L)) < ∞ to the definition of ξ∞stab(G).
In view of Theorems 4.5 and 4.6, both ξ∞stab(G) and ξ∗stab(G) can be reformu-lated in terms of C∗-algebras: ξ∞stab(G) (resp., ξ∗stab(G)) is the largest value of τ (P
i∈V Xi), where A is a (resp., finite-dimensional) C∗-algebra with tracial state τ and Xi ∈ A (i ∈ [n]) are projectors satisfying XiXj= 0 for all {i, j} ∈ E. Moreover, as we now see, the parameter ξ∗stab(G) coincides with the projective packing num-ber and the parameters ξ∗stab(G) and ξ∞stab(G) upper bound the quantum stability numbers.
Proposition 8.7. For every graph G we have
ξ∗stab(G) = αp(G) ≥ αq(G) and ξ∞stab(G) ≥ αqc(G).
Proof. By (8.9), αp(G) is the largest value of L(P
i∈V xi) taken over all linear functionals L that are normalized trace evaluations at projectors X ∈ (Sd)n (for some d ∈ N) with XiXj = 0 for {i, j} ∈ E. By convexity the optimum remains unchanged when considering a convex combination of such trace evaluations. In view of Theorem 4.6 (the equivalence between (1) and (3)), we can conclude that this optimum value is precisely the parameter ξ∗stab(G). This shows equality αp(G) = ξ∗stab(G).
Consider a C∗-algebra A with tracial state τ and a set of projectors Xci∈ A (for i ∈ V, c ∈ [k]) satisfying (8.4)-(8.5). Then, setting Xi=P
c∈[k]Xcifor i ∈ V , we ob-tain projectors Xi∈ A that satisfy XiXj= 0 if {i, j} ∈ E. Moreover, the following holds: τ (P
i∈V Xi) =P
c∈[k]τ (P
i∈V Xci) = k. This shows that ξstab∞ (G) ≥ αqc(G) and, when restricting A to be finite-dimensional, ξstab∗ (G) ≥ αq(G).
Using Lemma 4.15 one can verify that ξrstab(G) converges to ξstab∞ (G) as r → ∞, and for r ∈ N ∪ {∞} the infimum in ξrstab(G) is attained. Moreover, by Theo-rem 4.7, if ξstabr (G) admits a flat optimal solution, then equality ξrstab(G) = ξstab∗ (G) holds. The first bound ξstab1 (G) coincides with the theta number, since ξ1stab(G) = lasstab1 (G) = ϑ(G). Summarizing, we have αqc(G) ≤ ξstab∞ (G) and the following chain of inequalities
αq(G) ≤ αp(G) = ξstab∗ (G) ≤ ξ∞stab(G) ≤ ξrstab(G) ≤ ξ1stab(G) = ϑ(G), where the bounds ξrstab(G) (r ∈ N) are semidefinite programs, and αq(G) is NP-hard to compute.
8.3. Bounding quantum graph parameters 145 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:
χf(G) := minn X
S∈SG
λS : λ ∈ RS+G, X
S∈SG:i∈S
λS = 1 for all i ∈ Vo ,
where SG is the set of stable sets of G. Clearly, χf(G) ≤ χ(G). The following Lasserre type lower bounds for the classical chromatic number χ(G) are defined in [GL08b]:
lascolr (G) = infL(1) : L ∈ R[x]∗2rpositive, L(xi) = 1 (i ∈ V ), L = 0 on I2r(HG) . Note that we may view χf(G) as minimizing L(1) over all linear functionals L ∈ R[x]∗ that are conic combinations of evaluations at characteristic vectors of stable sets. From this we see that lascolr (G) ≤ χf(G) for all r ≥ 1. In [GL08b] it is shown that finite convergence to χf(G) holds:
lascolα(G)(G) = χf(G).
The bound of order r = 1 coincides with the theta number: lascol1 (G) = ϑ(G).
The following parameter ξf(G), called the projective rank of G, was introduced in [MR16b] as a lower bound on the quantum chromatic number χq(G):
ξf(G) := inf d
r : d, r ∈ N, X1, . . . , Xn ∈ Sd, Tr(Xi) = r (i ∈ V ), Xi2= Xi (i ∈ V ), XiXj = 0 ({i, j} ∈ E) . Proposition 8.8 ([MR16b]). For every graph G we have ξf(G) ≤ χq(G).
Proof. Set k = χq(G). It is shown in [CMN+07] that in the definition of χq(G) from (8.2)-(8.3), one may assume w.l.o.g. that Xic are projectors that all have the same rank, say, r. Then, for any given color c ∈ [k], the matrices Xic(i ∈ V ) provide a feasible solution to ξf(G) with value d/r. This shows ξf(G) ≤ d/r. Finally, d/r = k holds since by (8.2)-(8.3) we have d = rank(I) =Pk
c=1rank(Xic) = kr.
In [PSS+16, Prop. 5.11] it is shown that the projective rank can equivalently be defined as
ξf(G) = infλ : ∃ finite-dimensional C∗-algebra A with tracial state τ, Xi∈ A projector with τ (Xi) = 1
λ for i ∈ V, XiXj = 0 for {i, j} ∈ E .
Paulsen et al. [PSS+16] also define the tracial rank ξtr(G) of G as the parameter obtained by omitting in the above definition of ξf(G) the restriction that A has
146 Chapter 8. Quantum graph parameters to be finite-dimensional. The motivation for the parameter ξtr(G) is that it lower bounds the commuting quantum chromatic number [PSS+16, Thm. 5.11]:
ξtr(G) ≤ χqc(G).
Using Theorems 4.5 and 4.6 (which we apply to L/L(1) when L is not normal-ized), we obtain the following reformulations:
ξf(G) = infL(1) : L ∈ Rhxi∗ tracial, symmetric, positive, rank(M (L)) < ∞, L(xi) = 1 (i ∈ V ), L = 0 on I(HG) ,
and ξtr(G) is obtained by the same program without the restriction rank(M (L)) <
∞. In addition, we obtain that in this formulation of ξf(G) we can equivalently optimize over all L that are conic combinations of trace evaluations at projectors Xi ∈ Sd (for some d ∈ N) satisfying XiXj = 0 for all {i, j} ∈ E. If we restrict the optimization to conic combinations of 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 [MR16b, PSS+16].
The above formulations of the parameters ξtr(G) and ξf(G) in terms of linear functionals also show that they fit within the following hierarchy {ξcolr (G)}r∈N∪{∞}, defined as the noncommutative tracial analogue of the hierarchy {lascolr (G)}r:
ξrcol(G) = infL(1) : L ∈ Rhxi∗2r tracial, symmetric, and positive, L(xi) = 1 (i ∈ V ), L = 0 on I2r(HG) .
Again, ξcol∗ (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 8.9. For every graph G we have
ξ∗col(G) = ξf(G) ≤ χq(G) and ξcol∞(G) = ξtr(G) ≤ χqc(G).
Using Lemma 4.15 one can verify that the parameters ξrcol(G) converge to ξ∞col(G). Moreover, by Theorem 4.7, if ξcolr (G) admits a flat optimal solution, then we have ξrcol= ξ∗col(G). Also, the parameter ξ1col(G) coincides with lascol1 (G) = ϑ(G).
Summarizing we have ξcol∞(G) = ξtr(G) ≤ χqc(G) and the following chain of inequal-ities
ϑ(G) = ξ1col(G) ≤ ξrcol(G) ≤ ξ∞col(G) = ξtr(G) ≤ ξcol∗ (G) = ξf(G) ≤ χq(G).
Observe that the bounds lascolr (G) and ξrcol(G) remain below the fractional chro-matic number χf(G), since ξf(G) = ξcol∗ (G) ≤ lascol∗ (G) = χf(G). Hence, these bounds are weak if χf(G) is close to ϑ(G) and far from χ(G) or χq(G). In the clas-sical 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 Lov´asz [Lov78, Lov79], the fractional chromatic number is n/r, which is known to be equal to ϑ(K(n, r)), while the chromatic number is
8.3. Bounding quantum graph parameters 147 n − 2r + 2. In [GL08b] this was used as a motivation to define a new hierarchy of lower bounds {Λr(G)} on the chromatic number that can go beyond the frac-tional chromatic number. In Section 8.3.3 we recall this approach and show that its extension to the tracial setting recovers the hierarchy {γrcol(G)} introduced in Section 8.3.1. We also show how a similar technique can be used to recover the hierarchy {γstabr (G)}.
A link between ξrstab(G) and ξrcol(G)
In [GL08b, Thm. 3.1] it is shown that the bounds lasstabr (G) and lascolr (G) satisfy lasstabr (G)lascolr (G) ≥ |V | for any r ≥ 1,
with equality if G is vertex-transitive. This extends a well-known property of the theta number (i.e., the case r = 1). The same property holds for the noncommuta-tive analogues ξrstab(G) and ξrcol(G).
Lemma 8.10. For a graph G = (V, E) and r ∈ N ∪ {∞, ∗} we have ξstabr (G)ξrcol(G) ≥ |V |,
with equality if G is vertex-transitive.
Proof. Let L be feasible for ξrcol(G). Then ˜L = L/L(1) provides a solution to ξstabr (G) with value ˜L P
i∈V xi = |V |/L(1), implying that ξstabr (G) ≥ |V |/L(1) and therefore ξrstab(G)ξcolr (G) ≥ |V |.
Assume G is vertex-transitive. Let L be a feasible solution for ξrstab(G). As G is vertex-transitive we may assume (after symmetrization) that L(xi) takes a constant value. Set L(xi) =: 1/λ for all i ∈ V , so that the objective value of L for ξrstab(G) is
|V |/λ. Then ˜L = λL provides a feasible solution for ξrcol(G) with value λ, implying ξcolr (G) ≤ λ. This shows ξrcol(G)ξrstab(G) ≤ |V |.
For a vertex-transitive graph G, the inequality ξf(G)αq(G) ≤ |V | is shown in [MR16b, Lem. 6.5]; it can be recovered from the r = ∗ case of Lemma 8.10 and αq(G) ≤ αp(G).
Comparison to existing semidefinite programming bounds
By adding the constraints L(xixj) ≥ 0, for all i, j ∈ V , to the program defining ξcol1 (G), we obtain the strengthened theta number ϑ+(G) (from [Sze94]). Moreover, if we add the constraints
L(xixj) ≥ 0 for i 6= j ∈ V, (8.10)
X
j∈C
L(xixj) ≤ 1 for i ∈ V, (8.11)
L(1) + X
i∈C,j∈C0
L(xixj) ≥ |C| + |C0| for C, C0 distinct cliques in G (8.12)
148 Chapter 8. Quantum graph parameters to the program defining the parameter ξcol1 (G), then we obtain the parameter ξSDP(G), which is introduced in [PSS+16, Thm. 7.3] as a lower bound on ξtr(G).
We will now show that the inequalities (8.10)–(8.12) are in fact valid for ξcol2 (G), which implies
ξcol2 (G) ≥ ξSDP(G) ≥ ϑ+(G).
For this, given a clique C in G, we define the polynomial gC:= 1 −X
i∈C
xi∈ Rhxi.
Then (8.11) and (8.12) can be reformulated as L(xigC) ≥ 0 and L(gCgC0) ≥ 0, respectively, using the fact that L(xi) = L(x2i) = 1 for all i ∈ V . Hence, to show that any feasible L for ξcol2 (G) satisfies (8.10)-(8.12), it suffices to show Lemma 8.11 below. Recall that a commutator is a polynomial of the form [p, q] = pq − qp with p, q ∈ Rhxi. We denote by Θr the set of linear combinations of commutators [p, q]
with deg(pq) ≤ r.
Lemma 8.11. Let C and C0 be cliques in a graph G and let i, j ∈ V . Then we have
gC∈ M2(∅) + I2(HG), and xixj, xigC, gCgC0 ∈ M4(∅) + I4(HG) + Θ4. Proof. The claim gC∈ M2(∅) + I2(HG) follows from the identity
gC =
1 −X
i∈C
xi
| {z }
gC
2
+X
i∈C
(xi− x2i) + X
i6=j∈C
xixj
| {z }
h
= gC2 + h, (8.13)
where h ∈ I2(HG). We also have
xixj = xix2jxi+ xj(xi− x2i) + x2i(xj− x2j) + [xi, xix2j] + [xi− x2i, xj], xigC= xigC2xi+ gC2(xi− x2i) + [xi− x2i, gC2] + [xi, xigC2],
and, writing analogously gC0 = gC20+ h0 with h0∈ I2(HG), we have gCgC0 = gCgC20gC+ [gC, gCg2C0] + [h, g2C0] + gC2h0+ hh0+ gC20h.
Example 8.12. Using the bound ξSDP(G) it is shown in [PSS+16, Thm. 7.4] that the tracial rank of the odd cycle C2n+1 on 2n + 1 vertices equals (2n + 1)/n. That is, ξtr(C2n+1) = ξcol∞(C2n+1) = (2n + 1)/n. Combining this with Lemma 8.10 gives the inequality n = ξstab∞ (C2n+1) ≥ αqc(C2n+1). In fact, equality holds since
αqc(C2n+1) ≥ α(C2n+1) = n. 4
8.3.3 Links between γ
rcol(G), ξ
rcol(G), γ
rstab(G), and ξ
rstab(G)
In this last section, we make the link between the two hierarchies {ξstabr (G)} (resp.
{ξrcol(G)}) and {γrstab(G)} (resp. {γrcol(G)}). The key tool is the interpretation of the coloring and stability numbers in terms of certain graph products.
8.3. Bounding quantum graph parameters 149 We start with the (quantum) coloring number. For an integer k, recall that the Cartesian product GKk of G and the complete graph Kk is the graph with vertex set V × [k], where two vertices (i, c) and (j, c0) are adjacent if ({i, j} ∈ E and c = c0) or (i = j and c 6= c0). The following is a well-known reduction of the chromatic number χ(G) to the stability number of the Cartesian product GKk:
χ(G) = mink ∈ N : α(GKk) = |V | .
It was used in [GL08b] to define the following lower bounds on the chromatic num-ber:
Λr(G) = mink ∈ N : lasstabr (GKk) = |V | ,
where it was also shown that lascolr (G) ≤ Λr(G) ≤ χ(G) for all r ≥ 1, with equality Λ|V |(G) = χ(G). Hence the bounds Λr(G) may go beyond the fractional chromatic number. This is the case for the above-mentioned Kneser graphs; see [GL08a] for other graph instances.
The above reduction from coloring to stability number has been extended to the quantum setting in [MR16b], where it is shown that
χq(G) = min{k ∈ N : αq(GKk) = |V |}.
It is therefore natural to use the upper bounds ξrstab(GKk) on αq(GKk) in order to get the following lower bounds on the quantum coloring number:
min{k : ξstabr (GKk) = |V |}, (8.14) which are thus the noncommutative analogues of the bounds Λr(G).
Observe that, for any k ∈ N and r ∈ N ∪ {∞, ∗}, we have ξstabr (GKk) ≤ |V |, which follows from Lemma 8.11 and the fact that the cliques Ci = {(i, c) : c ∈ [k]}, for i ∈ V , cover all vertices in GKk. Let
CGKk=gCi : i ∈ V , where gCi= 1 − X
c∈[k]
xci,
denote the set of polynomials corresponding to these cliques. We now show that the parameter (8.14) in fact coincides with the parameter γrcol(G) for all r ∈ N ∪ {∞}.
For this observe first that the quadratic polynomials in the set HcolG,k correspond precisely to the edges of GKk, and that the projector constraints are included in I2(HcolG,k) (see (8.6)). Hence we have
I2r(HcolG,k) = I2r(HGKk∪ CGKk). (8.15) We will also use the following result.
Lemma 8.13. Let r ∈ N∪{∞, ∗} and assume L is feasible for ξrstab(GKk). Then, we have L(P
i∈V,c∈[k]xci) = |V | if and only if L = 0 on I2r(CGKk).
Proof. Assume L = 0 on I2r(CGKk). Then 0 =P
i∈V L(gCi) = |V | − L(P
i,cxci).
Conversely assume that 0 = L P
i∈V,c∈[k]xci − |V | = Pi∈V L(gCi). We will show L = 0 on I2r(CGKk). For this we first observe that gCi−(gCi)2∈ I2(HGKk)
150 Chapter 8. Quantum graph parameters by (8.13). Hence L(gCi) = L(gC2
i) ≥ 0, which, combined with P
iL(gCi) = 0, im-plies L(gCi) = 0 for all i ∈ V . Next we show L(wgCi) = 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 positiv-ity of L, the Cauchy-Schwarz inequalpositiv-ity gives |L(uvgCi)| ≤ L(u∗u)1/2L(v∗g2C
iv)1/2. Note that it suffices to show L(v∗gCiv) = 0 since, using again (8.13), this implies L(v∗gC2
iv) = 0 and thus L(uvgCi) = 0. We have deg(vv∗) < deg(w), and there-fore, using the tracial property of L and the induction assumption, we see that L(v∗gCiv) = L(vv∗gCi) = 0.
Proposition 8.14. For every graph G and r ∈ N ∪ {∞} we have γrcol(G) = min{k : ξstabr (GKk) = |V |}.
Proof. Let L be a linear functional certifying γcolr (G) ≤ k. Then, using (8.15) we see that L is feasible for ξrstab(GKk) and Lemma 8.13 shows that L(P
i,cxci) = |V |.
This shows ξrstab(GKk) ≥ |V | and thus equality holds (since the reverse inequality always holds). Therefore, min{k : ξrstab(GKk) = |V |} ≤ k.
Conversely, assume ξrstab(GKk) = |V |. Since the optimum is attained, there exists a linear functional L feasible for ξstabr (GKk) with L(P
i,cxci) = |V |. Using Lemma 8.13 we can conclude that L is zero on I2r(CGKk). Hence, in view of (8.15), L is zero on I2r(HcolG,k). This shows γcolr (G) ≤ k.
Note that the proof of Proposition 8.14 also works in the commutative setting;
this shows that the sequence Λr(G) corresponds to the usual Lasserre hierarchy for the feasibility problem defined by the equations (8.2)–(8.3), 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 Kk ? G, with vertex set [k] × G, and with an edge between two vertices (c, i) and (c0, j) when (c 6= c0, i = j) or (c = c0, i 6= j) or (c 6= c0, {i, j} ∈ E). The product Kk? G coincides with the homomorphic product Kkn G used in [MR16b, Sec. 4.2], where it is shown that
αq(G) = maxk ∈ N : αq(Kk? G) = k .
This suggests using the upper bounds ξrstab(Kk? G) on αq(Kk? G) to define the following upper bounds on αq(G):
maxk ∈ N : ξstabr (Kk? G) = k . (8.16) For each c ∈ [k], the set Cc= {(c, i) : i ∈ V } is a clique in Kk? G, and we let
CKk?G =gCc: c ∈ [k] , where gCc= 1 −X
i∈V
xic,
denote the set of polynomials corresponding to these cliques. As these k cliques cover the vertex set of Kk? G, we can use Lemma 8.11 to conclude that ξrstab(Kk? G) ≤ k for all r ∈ N ∪ {∞, ∗}.
8.4. Discussion 151