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Bounds on entanglement dimensions and quantum graph
parameters via noncommutative polynomial optimization
Sander Gribling∗ David de Laat† Monique Laurent‡
Sept ember 1, 2017
A bst r act
In t his paper we st udy bipart it e quant um correlat ions using t echniques from t racial poly-nomial opt imizat ion. We const ruct a hierarchy of semidefinit e programming lower bounds on t he minimal ent anglement dimension of a bipart it e correlat ion. T his hierarchy converges t o a new paramet er: t he minimal average ent anglement dimension, which measures t he amount of ent anglement needed t o reproduce a quant um correlat ion when access t o shared random-ness is free. For synchronous correlat ions, we show a correspondence between t he minimal ent anglement dimension and t he complet ely posit ive semidefinit e rank of an associat ed ma-t rix. We ma-t hen sma-t udy opma-t imizama-t ion over ma-t he sema-t of synchronous correlama-t ions by invesma-t igama-t ing quant um graph paramet ers. We unify exist ing bounds on t he quant um chromat ic number and t he quant um st ability number by placing t hem in t he framework of t racial opt imiza-t ion. In parimiza-t icular, we show imiza-t haimiza-t imiza-t he projecimiza-t ive packing number, imiza-t he projecimiza-t ive rank, and t he t racial rank arise nat urally when considering t racial analogues of t he Lasserre hierarchy for t he st ability and chromat ic number of a graph. We also int roduce semidefinit e program-ming hierarchies converging t o t he commut ing quant um chromat ic number and commut ing quant um st ability number.
1
Int r od u ct ion
1.1 B ip ar t it e qu ant u m cor r elat ion s
One of t he dist inguishing feat ures of quant um mechanics is quant um ent anglement , which allows for nonclassical correlat ions between spat ially separat ed part ies. By performing a measurement on t heir part of an ent angled syst em, t he part ies – who cannot communicat e – can use such correlat ions t o complet e t asks t hat are impossible wit hin classical mechanics. In t his paper we consider t he problems of quant ifying t he advant age ent anglement can bring and quant ifying t he minimal amount of ent anglement necessary for generat ing a given correlat ion. For t his we use t echniques from t racial polynomial opt imizat ion.
Quant um ent anglement has been widely st udied in t he bipart it e correlat ion set t ing. Here we have two part ies, Alice and Bob, where Alice receives a quest ion s from a finit e set S and Bob receives a quest ion t from a finit e set T . T he part ies do not know each ot her’s quest ions, and aft er receiving t he quest ions t hey do not communicat e. T hen, according t o some predet ermined prot ocol, Alice ret urns an answer a from a finit e set A and Bob ret urns an answer b from a finit e set B . T he probability t hat t he part ies answer (a, b) t o quest ions (s, t) is given by
∗CW I and QuSoft , Amst erdam, t he Net herlands. Supp ort ed by t he Net herlands Organizat ion for Scient ific
Research, grant numb er 617.001.351. [email protected]
†CW I and QuSoft , Amst erdam, t he Net herlands. Supp ort ed by t he Net herlands Organizat ion for
Sci-ent ific Research, grant numb er 617.001.351, and by t he ERC Consolidat or Grant QP ROGRESS 615307. [email protected]
a bipartite correlation P (a, b|s, t), which sat isfies P (a, b|s, t) ≥ 0 for all (a, b, s, t) Γ and ∑
a,bP (a, b|s, t) = 1 for all (s, t) S × T . T hroughout we set Γ = A × B × S × T .
T he bipart it e correlat ions P = (P (a, b|s, t)) RΓ depend on t he addit ional resources t hat
are available t o t he two part ies Aice and Bob. As we discuss below, it is of fundament al import ance in quant um informat ion t heory t hat quant um ent anglement allows for correlat ions t hat are not possible in a classical set t ing.
If t he part ies do not have access t o any addit ional resources, t hen t he correlat ion will be
deterministic, which means it is of t he form P (a, b|s, t) = PA(a|s) PB(b|t), wit h PA(a|s) and
PB(b|t) t aking values in { 0, 1} and
∑
aPA(a|s) =
∑
bPB(b|t) = 1 for all s, t. If t he part ies
have access t o local randomness, t hen PA and PB t ake values in [0, 1]. If t he part ies have
access t o shared randomness (t hey can draw from a shared random variable), t hen t he result ing correlat ion will be a convex combinat ion of det erminist ic correlat ions, and is said t o be a classical
correlation. T he classical correlat ions form a polyt ope, denot ed by Cl oc(Γ), and valid inequalit ies
for it are known as Bell inequalit ies [Bel64].
We are int erest ed in t he quant um set t ing, where t he part ies have access t o a shared quant um st at e upon which t hey can perform measurement s. T he quant um set t ing can be modeled in different ways, leading t o t he so-called t ensor model and commut ing model; see t he discussion, e.g., in [T si06, NPA08, DLT W08].
In t he tensor model, Alice and Bob each have access t o one half of a finit e dimensional
quantum state, which is modeled by a unit vect or ψ Cd Cd. Alice and Bob det ermine
t heir answers by performing a measurement on t heir part of t he st at e. Such a measurement is modeled by a posit ive operat or valued measure (POVM), which consist s of a set of d × d Hermit ian posit ive semidefinit e mat rices labeled by t he possible answers and summing t o t he
ident ity mat rix. If Alice uses t he POVM { Ea
s}a A when she get s quest ion s S and Bob uses
t he POVM { Fb
t}b B when he get s quest ion t T , t hen t he probability of obt aining t he answers
(a, b) is given by
P (a, b|s, t) = Tr((Esa Ftb)ψψ∗) = ψ∗(Ea
s Ftb)ψ. (1)
If t he st at e ψ cannot be writ t en as a single t ensor product ψA ψB, t hen ψ is said t o be
entangled, and t his can lead t o t he above correlat ion P t o be nonclassical.
A correlat ion of t he above form (1) is called a (tensor) quantum correlation, and we say it
is realizable in t he t ensor model in local dimension d or in dimension d2. Let Cqd(Γ) be t he set
of quant um correlat ions realizable in local dimension d, denot e t he smallest dimension needed
t o realize t he correlat ion P Cq(Γ) in t he t ensor model by
Dq(P ) = min
{
d2: d N, P Cqd(Γ)}, (2)
and define t he set
Cq(Γ) =
⋃
d N
Cqd(Γ).
T he set Cq(Γ) is convex, for if P1, P2 Cq(Γ) wit h Pi(a, b|s, t) = ψ∗i(Esa(i ) Ftb(i ))ψi for i = 1, 2,
and if λ [0, 1], t hen, wit h ψ = √λψ1
√
1 − λψ2, Esa= Esa(1) Esa(2), and Ftb= Ftb(1) Ftb(2),
we have (λP1+ (1 − λ)P2)(a, b|s, t) = ψ∗(Esa Ftb)ψ, which shows λP1+ (1 − λ)P2 Cq(Γ).
T he set Cq1(Γ) cont ains t he det erminist ic correlat ions, so by Carat h´eodory’s t heorem Cl oc(Γ)
is cont ained in Cc
q(Γ), where c is at most |A||S| + |B ||T | + 1; t hat is, quant um ent anglement
can be used as an alt ernat ive t o shared randomness. If A, B , S, and T all cont ain at least
two element s, t hen Bell’s t heorem says t he inclusion Cl oc(Γ) Cq(Γ) is st rict ; t hat is, quant um
ent anglement can be used t o obt ain nonclassical correlat ions [Bel64].
T he second model commonly used in quant um informat ion t heory t o define quant um corre-lat ions is t he commuting model (or recorre-lativistic field theory model). In t his model a correcorre-lat ion
P RΓ is called a commuting quantum correlation if it is of t he form
P (a, b|s, t) = Tr(XsaYtbψψ∗) = ψ∗(Xa
where { Xsa}a and { Ytb}b are POVMs consist ing of bounded operat ors on a separable Hilbert
space H , sat isfying [Xa
s, Ytb] = XsaYtb− YtbXsa = 0 for all (a, b, s, t) Γ, and where ψ is a
unit vect or in H . Such a correlat ion is said t o be realizable in dimension d = dim(H ) in t he
commut ing model, and we denot e t he set of such correlat ions by Cqcd(Γ) and set Cqc(Γ) = Cqc∞(Γ).
We denot e t he smallest dimension needed t o realize a quant um correlat ion P Cqc(Γ) by
Dqc(P ) = min
{
d N { ∞ } : P Cqcd(Γ)}. (4)
We have Cqd(Γ) Cqcd2(Γ), which follows by set t ing Xsa= Esa I and Ytb= I Ftb. T his shows
Dqc(P ) ≤ Dq(P ) for all P Cq(Γ).
T he minimum Hilbert space dimension in which a given quant um correlat ion P can be realized in t he t ensor or commut ing model quant ifies t he minimal amount of ent anglement
needed t o represent P . Comput ing t he paramet er Dq(P ) is in fact an NP-hard problem [St a15].
Hence a nat ural quest ion is t o find good lower bounds for t he paramet ers Dq(P ) and Dqc(P ),
and a main cont ribut ion of t his paper is proposing a hierarchy of semidefinit e programming
lower bounds for t hese paramet ers. A lower bound for Dq(P ) based on t he not ion of fidelity is
given in [SVW16].
As said above we have Cqd(Γ) Cqcd2(Γ). Conversely, each finit e dimensional commut ing
quant um correlat ion can be realized in t he t ensor model, alt hough not necessarily in t he same dimension [T si06] (see, e.g., [DLT W08] for a det ailed proof ). T his shows
Cq(Γ) =
⋃
d N
Cqcd(Γ) Cqc(Γ).
Whet her t he two set s Cq(Γ) and Cqc(Γ) coincide is known as T sirelson’s problem. In a
recent breakt hrough result Slofst ra [Slo17] shows t hat if |S| = 184, |T | = 235, |A| = 8, and
|B | = 2, t hen Cq(Γ) is not closed. T his implies t he exist ence of a sequence { Pi} Cq(Γ)
wit h Dq(Pi) → ∞ . Since Cqc(Γ) is closed [Fri12, Prop. 3.4], t his also implies t he inclusion
Cq(Γ) Cqc(Γ) is st rict , t hus set t ling T sirelson’s problem. Whet her t he closure of Cq(Γ) equals
Cqc(Γ) is an open problem t hat is relat ed t o an import ant conject ure in operat or t heory: We have
cl(Cq(Γ)) = Cqc(Γ) for all Γ if and only if Connes’ embedding conject ure holds [JNP+11, Oza12].
Furt her variat ions on t he above definit ions are possible. For inst ance, we can consider a mixed st at e ρ (a Hermit ian posit ive semidefinit e mat rix ρ wit h Tr(ρ) = 1) inst ead of a pure
st at e ψ, where we replace t he rank 1 mat rix ψψ∗ by ρ in t he above definit ions. By convexity t his
does not change t he set s Cq(Γ) and Cqc(Γ), but t he dimension paramet ers Dq(P ) and Dqc(P ) can
be smaller when allowing mixed st at es. Anot her variat ion would be t o use project ion valued measures (PVMs) inst ead of POVMs, where t he operat ors are project ors inst ead of posit ive
semidefinit e mat rices. T his again does not change t he set s Cq(Γ) and Cqc(Γ) [NC00], but t he
dimension paramet ers can be larger when rest rict ing t o PVMs.
In t he rest of t he int roduct ion we give a road map t hrough t he cont ent s of t he paper. We st at e t he main result s, which we number according t o t he sect ion where t hey will be proved, and we will int roduce t he necessary background along t he way.
1.2 Fr om syn ch r on ou s cor r elat ion s t o h ier ar ch ies
When t he two part ies have t he same quest ion set s (S = T ) and t he same answer set s (A = B ), a
bipart it e correlat ion P RΓis called synchronous if P (a, b|s, s) = 0 for all s and a = b. T he set s
Cq,s(Γ) and Cqc,s(Γ) of synchronous correlat ions form part icularly int erest ing subset s of bipart it e
correlat ions; T he quant um graph paramet ers discussed in Sect ion 1.4 will be defined t hrough opt imizat ion problems over t hese set s. T he set s of synchronous correlat ions are rich enough, so t hat t he above ment ioned result about Connes’ embedding conject ure st ill holds when we rest rict
t o synchronous correlat ions; t hat is, t he conject ure holds if and only if cl(Cq,s(Γ)) = Cqc,s(Γ)
for all Γ [DP16, T hm. 3.7].
We show t hat t he minimal local dimension in which a synchronous quant um correlat ion P can be realized is given by t he complet ely posit ive semidefinit e rank of an associat ed mat rix
MP, indexed by A × S and defined by
(MP)(s,a),(t,b) = P (a, b|s, t) for all (a, b, s, t) Γ.
A mat rix M Rn× n is said t o be completely positive semidefinite if t here exist d N and
Hermit ian posit ive semidefinit e mat rices X1, . . . , Xn Cd× d such t hat Mi j = Tr(XiXj) for
all i , j [n]. T he minimal such d is called t he completely positive semidefinite rank of M
and denot ed by cpsd-rankC(M ). Complet ely posit ive semidefinit e mat rices are invest igat ed
in [LP15], mot ivat ed by t heir use t o model quant um graph paramet ers, and t he complet ely posit ive semidefinit e rank in [PSVW16, GdLL17b, PV17, GdLL17a]. To show t he following
result we combine proofs from [SV17] (see also [MR16]) and [PSS+16]; t he proof can be found
in t he Appendix.
P r op osit ion A .1. T he smallest local dimension in which a synchronous quantum correlation
P can be realized is given by cpsd-rankC(MP).
In [GdLL17a] we use t echniques from t racial polynomial opt imizat ion t o define a semidefinit e
programming hierarchy of lower bounds { ξrcpsd(M )}r ≥ 1on cpsd-rankC(M ). By t he above result
t his hierarchy can be used t o obt ain lower bounds on t he smallest local dimension in which a synchronous correlat ion can be realized in t he t ensor model. However, in [GdLL17a] we show
t hat t he hierarchy typically does not converge t o cpsd-rankC(M ) but inst ead (under a cert ain
flat ness condit ion) t o a paramet er ξ∗cpsd(M ), which can be seen as a block-diagonal version of
t he complet ely posit ive semidefinit e rank.
We will use similar t echniques t o const ruct a hierarchy { ξrq(P )}r ≥ 1 of lower bounds on
t he minimal dimension Dq(P ) of a quant um correlat ion P Cq(Γ). T his new hierarchy will
have t hree advant ages over t he above approach. 1) It works for all correlat ions and not just for synchronous correlat ions. 2) T he special st ruct ure of a quant um correlat ion allows us t o add const raint s t hat st rengt hen t he lower bounds. 3) T he hierarchy converges (under flat ness)
t o ξ∗q(P ), and by using t he ext ra const raint s ment ioned above we can show ξ∗q(P ) is equal
t o an int erest ing paramet er Aq(P ) ≤ Dq(P ). T his paramet er describes t he minimal average
ent anglement dimension of a correlat ion when t he part ies have free access t o shared randomness; see t he next sect ion.
1.3 A h ier ar chy for t h e aver age ent an glem ent d im en sion
We are int erest ed in t he minimal ent anglement dimension needed t o realize a given quant um
correlat ion P Cq(Γ). If P is det erminist ic or only uses local randomness, t hen Dq(P ) =
Dqc(P ) = 1, but ot herwise we have Dq(P ) ≥ Dqc(P ) > 1. T hat is, t he shared quant um st at e
is used as a shared randomness resource. We define a new paramet er Aq(P ) ≤ Dq(P ) t hat
more closely measures t he minimal ent anglement dimension when t he part ies have free access
t o shared randomness, so t hat Aq(P ) = 1 if and only if P is classical.
For t his we assume t hat before t he game st art s t he part ies select a finit e number of pure
st at es ψi (i I ) (inst ead of a single one), in possibly different dimensions di, and POVMs
{ Esa(i )}a, { Ftb(i )}b for each i I and (s, t) S × T . As before, we assume t hat t he part ies
cannot communicat e aft er receiving t heir quest ions (s, t), but now t hey do have access t o shared
randomness, which t hey use t o decide on which st at e ψi t o use. T he part ies proceed t o measure
st at e ψi using POVMs { Esa(i )}a, { Ftb(i )}b, so t hat t he probability of answers (a, b) is given
by t he quant um correlat ion Pi. We want t o know what t he minimal average dimension of
t he average dimension ∑ i I λidi over all convex combinat ions P =
∑
i I λiPi. Hence, in t he
t ensor model t he minimal average entanglement dimension is given by
Aq(P ) = inf I i = 1 λiDq(Pi) : I N, λ RI+, I i = 1 λi = 1, P = I i = 1 λiPi, Pi Cq(Γ) ,
and, in t he commut ing model, Aqc(P ) is given by t he same expression wit h Dq(Pi) replaced
by Dqc(Pi). Observe t hat we need not replace Cq(Γ) by Cqc(Γ) since Dqc(P ) = ∞ for any
P Cqc(Γ) \ Cq(Γ).
It follows by convexity t hat for t he above definit ions it does not mat t er whet her we use pure or mixed st at es. In t he following proposit ion we show t hat for t he average minimal ent anglement dimension it also does not mat t er whet her we use t he t ensor or commut ing model.
P r op osit ion 2.1. For any P Cq(Γ) we have Aq(P ) = Aqc(P ).
We have Aq(P ) ≤ Dq(P ) and Aqc(P ) ≤ Dqc(P ) for P Cq(Γ), wit h equality if P is an
ext reme point of Cq(Γ). Hence, we have Dq(P ) = Dqc(P ) if P is an ext reme point of Cq(Γ).
We show t hat t he paramet er Aq(·) can be used t o dist inguish between classical and nonclassical
correlat ions.
P r op osit ion 2.2. For a correlation P RΓ we have Aq(P ) = 1 if and only if P Cl oc(Γ).
As ment ioned before, Slofst ra showed t he exist ence of Γ for which Cq(Γ) is not closed, which
implies t he exist ence of a sequence { Pi} Cq(Γ) such t hat Dq(P ) → ∞ . By t he following
proposit ion t his also implies t he exist ence of such a sequence wit h Aq(Pi) → ∞ .
P r op osit ion 2.3. If Cq(Γ) is not closed, then there exists { Pi} Cq(Γ) with Aq(Pi) → ∞ .
Using t racial polynomial opt imizat ion and building on t he t echniques from [GdLL17a] we const ruct a hierarchy of increasingly large opt imizat ion problems whose opt imal values give
increasingly good lower bounds { ξrq(P )}r ≥ 1 on Aqc(P ). For each r N t his is a semidefinit e
program, and for r = ∞ it is an infinit e dimensional semidefinit e program. We furt her define a
(hyperfinit e) variat ion ξ∗q(P ) of ξ∞q(P ) by adding a finit e rank const raint , so t hat
ξ1q(P ) ≤ ξq2(P ) ≤ . . . ≤ ξ∞q (P ) ≤ ξ∗q(P ) ≤ Aqc(P ).
We do not know whet her ξq∞(P ) = ξ∗q(P ) always holds; t his quest ion is relat ed t o Connes’
embedding conject ure [K S08].
First we show t hat we imposed enough const raint s in t he bounds ξrq(P ) so t hat ξ∗q(P ) =
Aqc(P ).
P r op osit ion 2.8. For any P Cq(Γ) we have ξq∗(P ) = Aqc(P ).
T hen we show t hat t he infinit e dimensional semidefinit e program ξq∞(P ) is t he limit of t he
finit e dimensional semidefinit e programs.
P r op osit ion 2.9. For any P Cq(Γ) we have ξqr(P ) → ξ∞q(P ) as r → ∞ .
Finally we give a crit erion under which finit e convergence ξrq(P ) = ξ∗q(P ) holds. T he
defini-t ion of fladefini-t ness follows ladefini-t er in defini-t he paper; here we only nodefini-t e defini-t hadefini-t idefini-t is an easy defini-t o check cridefini-t erion given t he out put of t he semidefinit e programming solver.
1.4 Q u ant u m gr ap h p ar am et er s
Nonlocal games have been int roduced in quant um informat ion t heory as abst ract models t o
quant ify t he power of ent anglement , in part icular, in how much t he set s Cq(Γ) and Cqc(Γ) differ
from Cl oc(Γ). A nonlocal game is defined by a probability dist ribut ion π : S × T → [0, 1] and a
funct ion f : A × B × S × T → { 0, 1} , known as t he predicate of t he game, where f (a, b, s, t) = 0 means t hat t he answer pair (a, b) is wrong for t he quest ion pair (s, t). Alice and Bob receive a
quest ion pair (s, t) S × T wit h probability π(s, t). T hey know t he game paramet ers π and f ,
but t hey do not know each ot her’s quest ions, and t hey cannot communicat e aft er t hey receive
t heir quest ions. T heir answers (a, b) are det ermined according t o some correlat ion P RΓ,
called t heir strategy, on which t hey may agree before t he st art of t he game, and which can be
classical or quant um depending on whet her P belongs t o Cl oc(Γ), Cq(Γ), or Cqc(Γ). T hen t heir
corresponding winning probability is given by
(s,t) S× T
π(s, t)
(a,b) A × B
P (a, b|s, t)f (a, b, s, t). (5)
A st rat egy P is called perfect if t he above winning probability is equal t o one, t hat is, if t he
probability of giving a wrong answer is zero: 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.
Comput ing t he maximum winning probability of a nonlocal game is an inst ance of linear
opt imizat ion over Cl oc(Γ) in t he classical set t ing, and over Cq(Γ) or Cqc(Γ) in t he quant um
set t ing. Since t he inclusion Cl oc(Γ) Cq(Γ) can be st rict , it is not surprising t hat t he winning
probability can be higher when t he part ies have access t o ent anglement . Perhaps more surprising is t he exist ence of nonlocal games t hat can be won wit h probability 1 when using ent anglement , but wit h opt imal winning probability st rict ly less t han 1 in t he classical set t ing.
T he quant um graph paramet ers αq(G) and χq(G) (and t he variant s αqc(G) and χqc(G)) are
quant um analogues of t he classical stability number α(G), which is t he size of a largest st able set in a graph G, and t he chromatic number χ(G), which is t he minimal number of colors needed t o color t he vert ices of G such t hat no two adjacent vert ices have t he same color. T hese quant um graph paramet ers are defined t hrough t he coloring st ability number games as described below. T hese nonlocal games use t he set [k] (whose element s are denot ed as a, b) and t he set V of vert ices of G (whose element s are denot ed as i , j ) as quest ion and answer set s.
In t he quantum coloring game, int roduced in [AHK S06, CMN+07], we are given a graph
G = (V, E ) and an int eger k N. We select S = T = V as quest ion set s and A = B = [k] as
answer set s. T he dist ribut ion π is st rict ly posit ive for all element s of V × V (e.g., it is uniform) and t he predicat e f of t he game is such t hat t he players’ answers have t o be consist ent wit h
having a k-coloring of G, t hat is, f (a, b, i , j ) = 0 precisely when (i = j and a = b) or ({ i , j } E
and a = b). T his expresses t he fact t hat if Alice and Bob receive t he same vert ex t hey should ret urn t he same color and if t hey receive adjacent vert ices t hey should ret urn dist inct colors. A perfect classical st rat egy exist s if and only if a perfect det erminist ic st rat egy exist s, and a perfect det erminist ic st rat egy corresponds t o a k-coloring of G. Hence t he smallest number k
of colors for which t here exist s a perfect classical st rat egy P Cl oc(Γ) is equal t o t he classical
chromat ic number χ(G). It is t herefore nat ural t o define t he quantum chromatic number χq(G)
(resp., t he commuting quantum chromatic number χqc(G)) as t he smallest k for which t here
exist s a perfect (resp., commut ing) quant um st rat egy P Cq(Γ) (resp., P Cqc(Γ)), where
Γ = [k]2× V2. Not e t hat such a st rat egy P is necessarily synchronous. In ot her words:
D efi nit ion 1.1. T he (commuting) quantum chromatic number χq(G) (resp., χqc(G)) is the
smallest integer k N for which there exists a synchronous correlation P = (P (a, b|i , j )) in
Cq,s([k]2× V2) (resp., Cqc,s([k]2× V2)) such that
In t he quantum stability number game, int roduced in [MR16, Rob13], we again have a graph
G = (V, E ) and k N, but now we use t he quest ion set [k] × [k] and t he answer set V × V . T he
dist ribut ion π is again st rict ly posit ive on t he quest ion set and now t he predicat e f of t he game is such t hat t he players’ answers have t o be consist ent wit h having a st able set of size k, t hat
is, f (i , j , a, b) = 0 precisely when (a = b and i = j ) or (a = b and (i = j or { i , j } E )). T his
expresses t he fact t hat if Alice and Bob receive t he same index a = b [k] t hey should answer
wit h t he same vert ex i = j of G and if t hey receive dist inct indices a = b from [k] t hey should answer wit h dist inct nonadjacent vert ices i and j of G. T here is a perfect classical st rat egy precisely when t here exist s a st able set of size k, so t hat t he largest int eger k for which t here exist s a perfect classical st rat egy is equal t o t he st ability number α(G). T he largest int eger k
for which t here exist s a perfect quant um st rat egy P Cq(Γ) (resp., Cqc(Γ)) is t he (commuting)
quantum stability number αq(G) (resp., αqc(G)), where we now have Γ = V2× [k]2. Again, a
perfect st rat egy P must be synchronous. In ot her words:
D efi nit ion 1.2. T he (commuting) stability number αq(G) (resp., αqc(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)
(resp., Cqc,s(V2× [k]2)) such that
P (i , j |a, b) = 0 whenever (i = j or { i , j } E ) and a = b [k].
As is well known, t he classical paramet ers χ(G) and α(G) are NP-hard t o comput e. T he
same holds for t he quant um coloring number χq(G) [Ji13] and also for t he quant um st ability
number αq(G), in view of t he following reduct ion t o coloring shown in [MR16]:
χq(G) = min{ k N : αq(G Kk) = |V |} . (6)
Here G Kk is t he Cart esian product of t he graph G = (V, E ) and t he complet e graph Kk. Not e
t hat (6) ext ends t o t he quant um set t ing t he analogous well-known reduct ion for t he classical
paramet ers. By const ruct ion we have χqc(G) ≤ χq(G) ≤ χ(G) and α(G) ≤ αq(G) ≤ αqc(G).
Int erest ingly, t he separat ion between χq(G) and χ(G), and between αq(G) and α(G), can be
exponent ially large in t he number of vert ices; T his is t he case for t he graphs Gn wit h vert ex
set V = { ± 1}n for n a mult iple of 4, where two vert ices x, y V are adjacent if t hey are
ort hogonal [AHK S06, MR16, MSS13].
By definit ion, t he paramet ers αq(G) and χq(G) involve synchronous quant um correlat ions,
while t he paramet ers αqc(G) and χqc(G) involve synchronous commut ing quant um correlat ions.
It is not known whet her t here is a separat ion between t he paramet ers χq(G) and χqc(G), and
between αq(G) and αqc(G). A mot ivat ion for st udying bot h versions of t he games lies in t he fact
t hat it it is not known whet her t he two set s Cq,s(Γ) and Cqc,s(Γ) coincide, where Γ = A2× S2for
finit e set s A and S. In t he asynchronous set t ing, as already ment ioned earlier, t his has recent ly
been set t led by Slofst ra [Slo17]: t here exist s a Γ = A × B × S × T for which Cq(Γ) = Cqc(Γ).
A second mot ivat ion is t he st udy of t he following lower bounds on t he (commut ing) quant um
chromat ic number: t he project ive rank ξf(G) [MR16] and t he t racial rank ξtr(G) [PSS+16].
Re-cent ly it has been shown in [DP16, Cor. 3.10] t hat t he project ive rank and t racial rank coincide if Connes’ embedding conject ure is t rue. In Sect ion 3 we provide a hierarchy of semidefinit e
pro-gramming bounds { ξcol
r (G)}r t hat asympt ot ically converges t o t he t racial rank, and has finit e
convergence t o t he project ive rank if a cert ain ‘flat ness’ condit ion holds.
We now give an overview of t he result s of Sect ion 3 and refer t o t hat sect ion for formal
definit ions. In Sect ion 3.1.1 we reformulat e t he quant um graph paramet ers in t erms of C∗
-algebras, using a reformulat ion from [PSS+16] for quant um synchronous correlat ions in t erms
of C∗-algebras. We t hen use t his in Sect ion 3.1.2 t o express t he quant um graph paramet ers in
t erms of posit ive t racial linear forms, which allows us t o use t echniques from t racial polynomial opt imizat ion t o formulat e bounds on t he quant um graph paramet ers. In part icular, we define a
hierarchy { γrcol(G)}r N { ∞ } of semidefinit e programming lower bounds on t he commut ing
quan-t um chromaquan-t ic number. We moreover define quan-t he paramequan-t er γ∗col(G) as γ∞col(G) wit h an addit ional
rank const raint on t he mat rix variable. Similarly, we define a hierarchy { γst ab
r (G)}r N { ∞ } of
upper bounds on t he commut ing quant um st ability number, and t he corresponding paramet er γ∗st ab(G). We show t he following convergence result s for t hese hierarchies.
L em m a 3.2. Let G be a graph. T here exists an r0 N such that γrcol(G) = χqc(G) and
γst ab
r (G) = αqc(G) for all r ≥ r0. Moreover, if γrcol(G) admits a flat optimal solution, then
γrcol(G) = χq(G), and similarly if γrst ab(G) admits a flat optimal solution, then γst abr (G) = αq(G).
T hen, in Sect ion 3.2, we use t racial analogues of Lasserre type bounds on α(G) and χ(G) t o obt ain hierarchies of semidefinit e programming bounds for t heir quant um analogues, which
are more economical t han t he bounds γrcol(G) and γrst ab(G) (since t hey use less variables) and
also permit t o recover some known bounds for t he quant um paramet ers. T he classical st ability number α(G) has a nat ural formulat ion as a polynomial opt imizat ion problem. Applying t he
st andard Lasserre hierarchy [Las01] t o t hat problem gives a hierarchy { lasst abr (G)}r N { ∞ } of
upper bounds on t he st ability number. We define t he t racial analogue ξst ab
r (G) of lasst abr (G) for
r N { ∞ } and t he corresponding paramet er ξst ab∗ (G). We show t hat ξ∗st ab(G) coincides wit h
t he project ive packing number αp(G) and t hat ξ∞st ab(G) upper bounds αqc(G).
P r op osit ion 3.3. We have ξ∗st ab(G) = αp(G) ≥ αq(G) and ξ∞st ab(G) ≥ αqc(G).
Next , we consider t he chromat ic number. A Lasserre-type hierarchy { lascolr (G)}r N { ∞ } of
semidefinit e programming lower bounds on t he chromat ic number χ(G) is defined in [GL08b].
We again consider t he t racial analogue ξrcol(G) of lascolr (G) for r N { ∞ } and t he corresponding
paramet er ξ∗col(G). T he t racial hierarchy { ξrcol(G)} unifies two known bounds: t he project ive
rank ξf (G), a lower bound on t he quant um chromat ic number [MR16]; and t he t racial rank
ξtr(G), a lower bound on t he commut ing chromat ic number [PSS+16].
P r op osit ion 3.5. We have ξ∗col(G) = ξf(G) ≤ χq(G) and ξcol∞ (G) = ξtr(G) ≤ χqc(G).
Aft er t hat we show ξrst ab(G)ξcolr (G) ≥ |V |, wit h equality if G is vert ex-t ransit ive; t his ext ends
t he corresponding known result for t he commut at ive paramet ers (cf. Sect ion 3.2.3). T he bounds
of order 1 correspond t o t he well-known t het a number: ξst ab1 (G) = ϑ(G) and ξ1col(G) = ϑ(G),
and we point out t he relat ion between ξcol2 (G) and t he semidefinit e programming bound ξSDP(G)
from [PSS+16] (cf. Sect ion 3.2.4).
In Sect ion 3.3, we compare t he hierarchies ξrcol(G) and γrcol(G), and t he hierarchies ξst abr (G)
and γrst ab(G). For t he coloring paramet ers, t he analogue of reduct ion (6) applies t o t he
semidef-init e programming bounds.
P r op osit ion 3.9. For r N { ∞ } we have γcol
r (G) = min{ k : ξst abr (G Kk) = |V |} .
An analogous st at ement holds for t he st ability paramet ers, when using t he homomorphic
graph product of Kk wit h t he complement of G, denot ed here as Kk ⋆ G, and t he following
reduct ion shown in [MR16]:
αq(G) = max{ k N : αq(Kk⋆ G) = k} .
We show t he following result for t he corresponding semidefinit e programming bounds.
P r op osit ion 3.10. For r N { ∞ } we have γst ab
r (G) = max{ k : ξrst ab(Kk ⋆ G) = k} .
Finally, we show t hat t he hierarchies { γcol
r (G)} and { γrst ab(G)} refine t he hierarchies { ξrcol(G)}
and { ξrst ab(G)} .
1.5 Tech n iqu es fr om n on com m u t at ive p olyn om ial op t im izat ion
To derive our bounds we use t echniques from t racial polynomial opt imizat ion. T his is a
noncommut at ive ext ension of t he widely used moment and sum-of-squares t echniques from Lasserre [Las01] and Parrilo [Par00] in polynomial opt imizat ion, dealing wit h t he problem of
minimizing a mult ivariat e polynomial funct ion f (x1, . . . , xn) over a feasible region defined by
polynomial inequalit ies g(x1, . . . , xn) ≥ 0 (for g G R[x1, . . . , xn]). T hese t echniques have
been adapt ed t o t he noncommut at ive set t ing in [NPA08] and [DLT W08] for approximat ing t he
set Cqc(Γ) of commut ing quant um correlat ions and t he winning probability of nonlocal games
over Cqc(Γ) (and, more generally, comput ing Bell inequality violat ions). In [PNA10, NPA12]
t his approach has been ext ended t o t he general eigenvalue opt imizat ion problem, of t he form inf{ψ∗f (X
1, . . . , Xn)ψ : d N, ψ Cd unit vect or, X1, . . . , Xn Cd× d,
g(X1, . . . , Xn) 0 for g G
} .
Here, t he mat rix variables Xi have free dimension d N and { f } G R x1, . . . , xn is
a set of symmet ric polynomials in noncommut at ive variables. In t racial opt imizat ion,
in-st ead of minimizing t he smallein-st eigenvalue of f (X1, . . . , Xn), we minimize it s normalized t race
Tr(f (X1, . . . , Xn))/ d (so t hat t he ident ity mat rix has t race one) [BK 12, BCK P13, BK P16,
K P16]. T he moment approach for t hese problems relies on minimizing L (f ), where L is a linear funct ional on t he space of noncommut at ive polynomials sat isfying some necessary condit ions,
so t hat L (f ) models eit her ψ∗f (X
1, . . . , Xn)ψ or Tr(f (X1, . . . , Xn))/ d. By t runcat ing t he
de-grees one get s hierarchies of lower bounds for t he original problem. By t he GNS const ruct ion,
t he asympt ot ic limit of t hese bounds involves operat ors Xi on a Hilbert space (possibly wit h
infinit e dimension). In t racial opt imizat ion t his leads t o allowing solut ions Xi in a C∗-algebra
A equipped wit h a t racial st at e τ, so t hat τ(f (X1, . . . , Xn)) is minimized.
In [PSS+16] hierarchies of out er approximat ions { Q
r(Γ)} for t he set Cqc(Γ) of commut ing
quant um correlat ions are const ruct ed and used t o derive semidefinit e programming bounds
con-verging t o t he commut ing quant um coloring number χqc(G). T hey are based on t he eigenvalue
opt imizat ion approach, applied t o t he formulat ion (3) of commut ing quant um correlat ions. In
t his paper we const ruct new hierarchies of semidefinit e programming bounds for χqc(G) and
αqc(G), exploit ing t he fact t hat t hese paramet ers are defined in t erms of synchronous
corre-lat ions and t he fact (from [PSS+16]) t hat such correlat ions admit a reformulat ion in t erms of
C∗-algebras wit h a t racial st at e. So our bounds are based on t racial opt imizat ion and t hey use
less variables, roughly speaking t hey involve only t he variables { xas} while t he previous bounds
of [PSS+16] use t he larger set of variables { xas, ybt} .
An import ant feat ure in noncommut at ive opt imizat ion is t he dimension independence: t he
opt imizat ion is over all possible mat rix sizes d N. In some applicat ions one may want t o
rest rict t o opt imizing over mat rices wit h rest rict ed size d. In [NV15, NFAV15] t echniques are developed t hat allow t o incorporat e t his dimension rest rict ion by suit ably select ing t he linear funct ionals L in a specified space; t his is used t o give bounds on t he maximum violat ion of a Bell inequality t hat can be achieved in a fixed dimension. A relat ed nat ural problem is t o decide what is t he minimum dimension d needed t o realize a given algebraically defined object , like a (commut ing) quant um correlat ion P . We propose an approach based on t racial opt imizat ion: st art ing from t he observat ion t hat t he t race of t he d × d ident ity mat rix gives it s size d, we consider t he problem of minimizing L (1) where L is a linear funct ional modeling t he non-normalized mat rix t race. T his approach has been developed in t he recent work [GdLL17a] for t he
problem of finding smallest mat rix fact orizat ion ranks: Given a nonnegat ive mat rix M Rm × n,
t he smallest dimension d for which t here exist Hermit ian posit ive semidefinit e mat rices Xi, Yj
so t hat M = (Tr(XiYj))i [m ],j [n]is called t he posit ive semidefinit e rank of M ; when m = n and
we rest rict t o using t he same fact ors Xi = Yi t he analogous paramet er is called t he complet ely
t hese mat rix fact orizat ion ranks (and for t heir commut at ive analogues, where all fact ors are diagonal mat rices: t he nonnegat ive rank and t he complet ely posit ive rank). Similar ideas are used here t o derive semidefinit e programming bounds for t he minimum dimension paramet ers
Dq(P ), Dqc(P ) considered in t his paper.
2
A h ier ar chy for t h e m in im al ent an glem ent d im en sion
2.1 T h e m in im al aver age ent an glem ent d im en sion
We st art by showing t hat it does not mat t er whet her we use t he t ensor or t he commut ing model when defining t he average ent anglement dimension.
P r op osit ion 2.1. For any P Cq(Γ) we have Aq(P ) = Aqc(P ).
P roof. T he easy inequality Aqc(P ) ≤ Aq(P ) follows from t he ident ity Esa Ftb= (Esa I )(I Ftb).
For t he ot her inequality we suppose P = ∑Ii = 1λiPi is feasible for Aqc(P ). T his means we
have POVMs { Xsa(i )}aand { Ytb(i )}bin Cdi× di wit h [Xsa(i ), Ytb(i )] = 0 and unit vect ors ψi Cdi
such t hat Pi(a, b|s, t) = ψ∗iXsa(i )Ytb(i )ψi for all (a, b, s, t) Γ and i [I ]. We will const ruct a
feasible solut ion t o Aq(P ) wit h value at most
∑
iλidi, t hus showing Aq(P ) ≤ Aqc(P ).
Fix some index i [I ]. By Art in-Wedderburn t heory applied t o C { Xsa(i )}a,s , t he ∗-algebra
generat ed by t he mat rices Xa
s(i ) wit h (a, s) A × S, t here exist s a unit ary mat rix Ui and int egers
Ki, mk, nk such t hat UiC { Xsa(i )}a,s Ui∗= Ki k= 1 (Cnk× nk I mk) and di = Ki k= 1 mknk.
By t he commut at ion relat ions each mat rix Ytb(i ) commut es wit h all mat rices in C { Xsa(i )}a,s ,
and t hus UiYtb(i )Ui∗ lies in t he algebra k(Ink C
mk× mk). Hence, we may assume
Xsa(i ) = Ki k= 1 Esa(i , k) Imk, Y b t (i ) = Ki k= 1 Ink F b t(i , k), ψi = Ki k= 1 ψi ,k, wit h Ea
s(i , k) Cnk× nk, Ftb(i , k) Cmk× mk, and ψi ,k Cmknk. T hen we have
Pi(a, b|s, t) = Tr(Xsa(i )Ytb(i )ψiψi∗) = k ψi ,k 2 Tr Esa(i , k) Ftb(i , k)ψi ,kψ ∗ i ,k ψi ,k 2 ︸ ︷ Qi k(a,b|s,t) , where Qi ,k Cq(Γ). As ∑ k ψi ,k 2= ψi 2= 1, Pi = ∑
k ψi ,k 2Qi ,k is a convex combinat ion.
We now show t hat Qi ,k Cqmin{ mk,nk}(Γ). For t his consider t he Schmidt decomposit ion
ψi ,k/ ψi ,k =
min{ mk,nk}
l = 1
λi ,k,lvi ,k,l wi ,k,l,
where { vi ,k,l}nl = 1k Cnk and { wi ,k,l}ml = 1k Cmk are ort honormal bases, and λi ,k,l ≥ 0. Define
unit ary mat rices Vk Cnk× nk and Wk Cmk× mk such t hat Vkvi ,k,l is t he lt h unit vect or in Rnk
and Wkwi ,k,l is t he lt h unit vect or in Rmk for l ≤ min{ mk, nk} . Let Esa(i , k)′(resp., Ftb(i , k)′) be
t he leading principal submat rices of VkEsa(i , k)Vk∗ (resp., WkFtb(i , k)Wk∗) of size min{ mk, nk} .
Moreover, set φi ,k =
∑min{ mk,nk}
T hen Qi ,k(a, b|s, t) = Tr Esa(i , k) Ftb(i , k) ψi ,kψi ,k∗ ψi ,k 2 = min{ mk,nk} l ,l′= 1 λi ,k,lλi ,k,l′v∗i ,k,lEsa(i , k)vi ,k,l′w∗i ,k,lFtb(i , k)wi ,k,l′ = min{ mk,nk} l ,l′= 1 λi ,k,lλi ,k,l′e∗lEsa(i , k)′el′e∗lFtb(i , k)′el′ = Tr((Esa(i , k)′ Ftb(i , k)′)φi ,kφ∗i ,k),
t hus showing Qi ,k Cmin{ mk,nk}
q (Γ). From t he convex decomposit ion P =
∑ i ,kλi ψi ,k 2Qi ,k, we obt ain Aq(P ) ≤ i ,k λi ψi ,k 2min{ mk, nk}2≤ i ,k λimin{ mk, nk}2≤ i ,k λimknk = i λidi,
which complet es t he proof.
We now show t hat t he paramet er Aq(·) permit s t o charact erize classical correlat ions.
P r op osit ion 2.2. For a correlation P RΓ we have A
q(P ) = 1 if and only if P Cl oc(Γ).
P roof. If P Cl oc(Γ), t hen P can be writ t en as a convex combinat ion of det erminist ic
correla-t ions (which are concorrela-t ained in C1
q(Γ)), hence Aq(P ) = 1.
On t he ot her hand, if Aq(P ) = 1, t hen t here exist convex decomposit ions indexed by l N:
P =
i Il
λliPil wit h { Pil} Cq(Γ) and lim
l →∞ i Il
λlDq(Pil) = 1.
Decompose Il as t he disjoint union Il
− I+l so t hat Dq(Pi) is equal t o 1 for i I−l and st rict ly
great er t han 1 for i I+l . Let ε > 0. For all l sufficient ly large we have
( 1 − i Il + λli + 2 i Il + λli ≤ i Il − λli + i Il + λliDq(Pil) ≤ 1 + ε,
which shows t hat ∑ i Il
+λ
l
i ≤ ε. T his shows t hat P is t he limit of convex combinat ions of
det erminist ic correlat ions, which implies t hat P Cl oc(Γ).
P r op osit ion 2.3. If Cq(Γ) is not closed, then there exists { Pi} Cq(Γ) with Aq(Pi) → ∞ .
P roof. Assume for cont radict ion t hat t here exist s an int eger K such t hat Aq(P ) < K for all
P Cq(Γ). We will show t his result s in a uniform upper bound on Dq(P ) for P Cq(Γ), which
implies Cq(Γ) is closed. For t his we first observe t hat any P Cq(Γ) can be decomposed as
P = µ1R1+ (1 − µ1)Q1, (7)
where R1 Cq(Γ), Q1 conv(CqK(Γ)), and µ1≤ K / (K + 1). Indeed, by assumpt ion, P can be
writ t en as a convex combinat ion P =
i I
λiPi wit h { Pi} Cq(Γ) and
i I
We can decompose I as t he disjoint union I− I+ so t hat Dq(Pi) is at most K for i I− and
at least K + 1 for i I+. T hen,
(K + 1) i I+ λi ≤ i I+ λiDq(Pi) ≤ K , and t hus µ1:= ∑
i I+ λi ≤ K / (K + 1). Hence (7) holds aft er set t ing R1= (
∑
i I+ λiPi)/ µ1and
Q1= (
∑
i I− λiPi)/ (1 − µ1).
By repeat ing t he same argument for R1 and it erat ing we obt ain for each int eger k N a
decomposit ion
P = µ1µ2· · · µkRk + (1 − µ︸ 1)Q1+ µ1(1 − µ2)Q2+ . . . + µ︷ 1µ2· · · µk− 1(1 − µk)Qk = (1− µ1µ2···µk) ˆQk
,
where Rk Cq(Γ), ˆQk conv(CqK(Γ)) and µ1µ2· · · µk ≤ (K / (K + 1))k. As t he ent ries of Rk
lie in [0, 1] we can conclude t hat µ1µ2· · · µkRk t ends t o 0 as k → ∞ . Hence t he sequence ( ˆQk)k
has a limit ˆQ and P = ˆQ holds. As all ˆQk lie in t he compact set conv(CqK(Γ)), we also have
P conv(CqK(Γ)). T he ext reme point s of t he compact convex set conv(CqK(Γ)) lie in CqK(Γ),
so, by t he Carat h´eodory t heorem, P conv(CqK(Γ)) is a convex combinat ion of at most c
element s from CK
q (Γ) where c is at most |A||S| + |B ||T | + 1. By a direct sum const ruct ion (see
Sect ion 1.1) we t hen obt ain Dq(P ) ≤ cK .
2.2 Set u p of t h e h ier ar chy
We will now const ruct a hierarchy of lower bounds on t he minimal ent anglement dimension,
using it s formulat ion via Aqc(P ). Our approach is based on noncommut at ive polynomial opt
i-mizat ion, t hus similar t o t he approach in [GdLL17a] for bounding mat rix fact orizat ion ranks. We first need some not at ion. Set
x = {xas : (a, s) A × S} and y = {ytb: (b, t) B × T},
and let x , y , z r be t he set of all words of lengt h at most r in t he n = |S||A| + |T ||B | + 1 symbols
xas, ytb, and z. Moreover, set x , y , z = x , y , z ∞. We equip x , y , z r wit h an involut ion
w → w∗ t hat reverses t he order of t he symbols in t he words and leaves t he symbols xa
s, ybt, z
invariant ; e.g., (xasz)∗= zxa
s. Let R x , y , z r be t he vect or space of all real linear combinat ions
of t he words of lengt h (aka degree) at most r . T he space R x , y , z = R x , y , z ∞ is t he
∗-algebra wit h Hermit ian generat ors { xa
s} , { ybt} , and z, and t he element s in t his algebra are called
noncommutative polynomials in t he variables { xas} , { ytb} , z.
T he hierarchy is based on t he following idea: For any feasible solut ion t o Aqc(P ), it s object ive
value can be modeled as L (1) for a cert ain t racial linear form L on t he space of noncommut at ive polynomials (t runcat ed t o degree 2r ).
Indeed, assume { (Pi, λi)i} is a feasible solut ion t o t he program Aqc(P ) defined in
Sec-t ion 1.3, where Pi(a, b|s, t) = Tr
( Xa s(i )Ytb(i )ψiψ∗i ) wit h Xa s(i ), Ytb(i ) Cdi× di, ψi Cdi, and
di = Dqc(Pi). Fix r N { ∞ } , and consider t he linear funct ional L R x , y , z ∗2r defined by
L (p) =
i
λiRe(Tr(p(X (i ), Y (i ), ψiψi∗))) for p R x , y , z 2r.
Here, for each index i , we set X (i ) = (Xa
s(i ) : (a, s) A × S), Y (i ) = (Ytb(i ) : (b, t) B × T ), and
replace t he variables xas, ybt, z by Xsa(i ), Ytb(i ), and ψiψ∗i. T hen L (1) =
∑
iλidi. T hat is, L (1)
is t he object ive value of t he feasible solut ion { (Pi, λi)i} t o Aqc(P ). We will now ident ify several
comput at ionally t ract able propert ies t hat t his linear funct ional L sat isfies. T hen t he hierarchy consist s of opt imizat ion problems where we minimize L (1) over t he set of linear funct ionals t hat
First not e t hat L is symmetric, t hat is, L (w) = L (w∗) for all w x , y , z
2r, and tracial,
t hat is, L (ww′) = L (w′w) for all w, w′ x , y , z wit h deg(ww′) ≤ 2r .
For all p R x , y , z r − 1 we have
L (p∗xasp) =
i
λiRe(Tr(M (i )∗Xsa(i )M (i )), where M (i ) = p(X (i ), Y (i ), ψiψi∗).
Since Xa
s(i ) is a posit ive semidefinit e mat rix, M (i )∗Xsa(i )M (i ) is posit ive semidefinit e t oo, and
t hus we have L (p∗xa
sp) ≥ 0. In t he same way we have L (p∗ytbp) ≥ 0 and L (p∗zp) ≥ 0. T hat is,
if we set
G = {xas : s S, a A} {ybt : t T, b B} { z} ,
t hen L is nonnegat ive (denot ed as L ≥ 0) on t he truncated quadratic module
M 2r(G) = cone p∗gp : p R x , y , z , g G { 1} , deg(p∗gp) ≤ 2r . (8)
Similarly, set t ing
H = {z − z2} { 1 −
a A
xas : s S} { 1 −
b B
ybt : t T} {[xas, ytb] : (s, t, a, b) Γ},
we have L = 0 on t he truncated ideal
I2r(H ) = ph : p R x , y , z , h H , deg(ph) ≤ 2r . (9)
Moreover, we have L (z) = ∑ iλiRe(Tr(ψiψi∗)) = 1. In addit ion, for any mat rices U, V Cdi× di
we have
ψiψi∗Uψiψi∗V ψiψ∗i = ψiψ∗iV ψiψ∗iUψiψ∗i,
and t herefore, in part icular,
L (wzuzvz) = L (wzvzuz) for all u, v, w x , y , z wit h deg(wzuzvz) ≤ 2r.
T hat is, we have L = 0 on I2r(Rr), where
Rr =
{
zuzvz − zvzuz : u, v u, v x , y , z wit h deg(zuzvz) ≤ 2r}.
We get t he idea of adding t hese last const raint s from [NPA12], where t his is used t o st udy t he mut ually unbiased bases problem.
We call M (G) = M ∞(G) t he quadrat ic module generat ed by G, and we call I (H R∞) =
I∞(H R∞) t he ideal generat ed by H R∞.
For r N { ∞ } we can now define t he paramet er:
ξqr(P ) = min L (1) : L R x , y , z ∗
2r t racial and symmet ric,
L (z) = 1, L (xasybtz) = P (a, b|s, t) for all (a, b, s, t) Γ,
L ≥ 0 on M 2r(G), L = 0 on I2r(H Rr) .
Addit ionally, we define ξ∗q(P ) by adding t he const raint rank(M (L )) < ∞ t o ξ∞q(P ). By
con-st ruct ion t his gives a hierarchy of lower bounds for Aqc(P ):
ξ1q(P ) ≤ . . . ≤ ξrq(P ) ≤ ξ∞q (P ) ≤ ξ∗q(P ) ≤ Aqc(P ).
Not e t hat for order r = 1 we get t he t rivial lower bound ξ1q(P ) = 1.
For each finit e r N t he paramet er ξrq(P ) can be comput ed by semidefinit e programming.
polynomials p R x , y , z wit h degree at most r − ⌈ deg(g)/ 2⌉ . T his is equivalent t o requiring
t hat t he mat rices (L (w∗w)), indexed by all words w, w′ wit h degree at most r − ⌈ deg(g)/ 2⌉ ,
are posit ive semidefinit e. To see t his, writ e p = ∑wpww and let ˆp = (pw) denot e t he vect or
of coefficient s, t hen L (p∗gp) ≥ 0 is equivalent t o ˆpT(L (w∗gw′)) ˆp ≥ 0. When g = 1, t he mat rix
(L (w∗w′)) is indexed by t he words of degree at most r , it is called t he moment matrix of L
and denot ed by Mr(L ) (or M (L ) when r = ∞ ). T he ent ries of t he mat rices (L (w∗gw′)) are
linear combinat ions of t he ent ries of Mr(L ), and t he const raint L = 0 on I2r(H Rr) can be
writ t en as a set of linear const raint s on t he ent ries of Mr(L ). It follows t hat for finit e r N,
t he paramet er ξqr(P ) is indeed comput able by a semidefinit e program.
2.3 B ackgr ou n d on p osit ive t r acial lin ear for m s
Before we show t he convergence result s we give some background on posit ive t racial linear forms,
which we use again in Sect ion 3. We st at e t hese result s using t he variables x1, . . . , xn, where
we use t he not at ion x = x1, . . . , xn . T he result s st at ed below do not always appear in t his
way in t he sources cit ed; we follow t he present at ion of [GdLL17a], where full proofs for t hese result s are also provided.
First we need a few more definit ions. A polynomial p R x is called symmet ric if p∗ = p,
and we denot e t he set of symmet ric polynomials by Sym R x . Given G Sym R x and
H R x , t he set M (G) + I (H ) is called Archimedean if it cont ains t he polynomial R −∑ ni = 1x2
i
for some R > 0. We will use t he concept of a C∗-algebra, which for our purposes can be defined
as a norm closed ∗-subalgebra of t he space B(H ) of bounded operat ors on a complex Hilbert space H . We say t hat A is unital if it cont ains t he ident ity operat or (denot ed 1). An element
a A is called positive if a = b∗b for some b A . A linear form τ on a unit al C∗-algebra A is
said t o be a state if τ(1) = 1 and τ is posit ive; t hat is, τ(a) ≥ 0 for all posit ive element s a A .
We say t hat a st at e τ is t racial if τ(ab) = τ(ba) for all a, b A . See, for example, [Bla06] for
more informat ion on C∗-algebras.
T he first result , which relat es posit ive t racial linear forms t o C∗-algebras, is due t o [NPA12]
in t he noncommut at ive set t ing, and due t o [BK P16] in t he t racial set t ing.
T heor em 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 An such that g(X ) is positive
in A for all g G, and h(X ) = 0 for all h H , with
L (p) = τ(p(X )) for all p R x . (10)
T he following can be seen as t he finit e dimensional analogue of t he above result . T he proof
of t he unconst rained case (G = H = ) can be found in [BK 12], and for t he const rained case
in [BK P16]. Given a linear form L R x ∗, recall t hat t he moment mat rix M (L ) is given by
M (L )u,v = L (u∗v) for u, v x .
T heor em 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 An
satisfy-ing (10), 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 = (X1, . . . , Xn) of
A t runcat ed linear funct ional L R x 2r is δ-flat if t he principal submat rix Mr − δ(L ) of
Mr(L ) indexed by monomials up t o degree r − δ has t he same rank as Mr(L ). We call a
t runcat ed linear funct ional flat if it is δ-flat for some δ ≥ 1. T he following result claims t hat any flat linear funct ional on a t runcat ed polynomial space can be ext ended t o a linear funct ional L on t he full algebra of polynomials. It is due t o Curt o and Fialkow [CF96] in t he commut at ive case and ext ensions t o t he noncommut at ive case can be found in [PNA10] (for eigenvalue opt imizat ion) and [BK 12] (for t race opt imizat ion).
T heor em 2.6. Let 1 ≤ δ ≤ t < ∞ , G Sym R x 2r, and H R x 2r. If L R x ∗2r is
symmetric, tracial, δ-flat, nonnegative on M 2r(G), and zero on I2r(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.
T he following t echnical lemma, based on t he Banach-Alaoglu t heorem, is a well-known t ool t o show asympt ot ic convergence result s in (t racial) polynomial opt imizat ion.
L em m a 2.7. Let G Sym R x , H R x , and assume R − (x2
1+ · · · + x2n) M2d(G) + I2d(H )
for some d N and R > 0. For r N assume Lr R x ∗2r is tracial, nonnegative on M2r(G)
and zero on I2r(H ). T hen we have |Lr(w)| ≤ R|w|/ 2Lr(1) for all w x 2r − 2d+ 2. In addition,
if suprLr(1) < ∞ , then { Lr}r has a pointwise converging subsequence in R x ∗.
2.4 C onver gen ce r esu lt s
We first show equality ξ∗q(P ) = Aqc(P ), and t hen we consider convergence propert ies of t he
bounds ξrq(P ) t o t he paramet ers ξ∞q (P ) and ξq∗(P ).
P r op osit ion 2.8. For any P Cq(Γ) we have ξq∗(P ) = Aqc(P ).
P roof. Since we know ξq∗(P ) ≤ Aqc(P ), it remains t o show ξ∗q(P ) ≥ Aqc(P ). For t his let L be
feasible for ξ∗q(P ), so t hat L ≥ 0 on M (G) and L = 0 on I (H R∞). By T heorem 2.5, t here exist
finit ely many scalars λi ≥ 0, Hermit ian mat rix t uples X (i ) = (Xsa(i ))a,s and Y (i ) = (Ytb(i ))b,t,
and Hermit ian mat rices Zi, so t hat g(X (i ), Y (i ), Zi) 0 for all g G, h(X (i ), Y (i ), Zi) = 0 for
all h H R∞, and
L (p) =
i
λiTr(p(X (i ), Y (i ), Zi)) for all p R x , y , z . (11)
Here we may assume wit hout loss of generality t hat , for each i , t he algebra C X (i ), Y (i ), Zi
is a full mat rix algebra Cdi× di. Indeed, if t his is not t he case, by t he Art in–Wedderburn
t heorem t here exist s a unit ary mat rix U for which t he algebra U∗C X (i ), Y (i ), Z
i U can be
block diagonalized int o smaller blocks and t hus we obt ain anot her conic decomposit ion of L involving only full mat rix algebras.
Since h(E(i ), F (i ), Zi) = 0 for all h R∞ { z − z2} , t he commut at or
[
ZiuZi, ZivZi
]
vanishes for all u, v E(i ), F (i ), Zi , and hence for all u, v C E(i ), F (i ), Zi . T his means
t hat [ZiT1Zi, ZiT2Zi] = 0 for all T1, T2 Cdi× di. Since Zi is a project or, t here exist s a unit ary
mat rix Ui such t hat
UiZiUi∗= Diag(1, . . . , 1, 0, . . . , 0).
T he above t hen implies t hat for all T1and T2, t he leading principal submat rices of size rank(Zi)
of UiT1Ui∗ and UiT2Ui∗ commut e. T his implies rank(Zi) = 1 and t herefore Tr(Zi) = 1. T hus we
have 1 = L (z) = ∑iλiTr(Zi) =
∑
iλi.
For each index i define t he correlat ion Pi Cq(Γ) by
Pi(a, b|s, t) = Tr
(
Esa(i )Ftb(i )Zi
)
T hen, P = ∑iλiPi, so t hat (Pi, λi) forms a feasible solut ion t o Aqc(P ) wit h object ive value i λidi = i λiTr(Idi) = L (1). T his shows ξq∗(P ) ≥ Aqc(P ).
T he problem ξqr(P ) differs in two ways from a st andard t racial opt imizat ion problem. It
does not have t he normalizat ion condit ion L (1) = 1 (and inst ead minimizes L (1)), and it has
t he ext ra ideal const raint s L = 0 on I2r(Rr), where Rr depends on r . T he following proof
is a st raight forward adapt at ion of a similar proof for general t racial opt imizat ion problems from [K P16] and it relies on Lemma 2.7.
P r op osit ion 2.9. For any P Cq(Γ) we have ξqr(P ) → ξ∞q(P ) as r → ∞ .
P roof. First we observe t hat t he polynomials 1− z2, 1− (xas)2, and 1− (ytb)2lie in M 4(G H0),
where H0 cont ains t he symmet ric polynomials in H (i.e., omit t ing t he polynomials [xas, ybt]).
Indeed, we have 1 − z2= (1 − z)2+ 2(z − z2), 1 − (xas)2= (1 − xas)2+ 2(1 − xsa)xas(1 − xas) + 2xas( (1 − a′ xas′) + a′= a xas′)xas,
and analogously for yb
t. Hence R − z2−
∑
a,s(xas)2−
∑
b,t(ytb)2 M 4(G H0) for some R > 0.
Fix ε > 0 and for each r N let Lr be feasible for ξrq(P ) wit h value Lr(1) ≤ ξrq(P ) + ε. As
Lr is t racial and zero on I2r(H0) it follows (using t he ident ity p∗gp = pp∗g + [p∗g, p]) t hat
L = 0 on M 2r(H0). Hence, Lr ≥ 0 on M2r(G H0). Since suprLr(1) ≤ Aq(P ) + ε, we can
apply Lemma 2.7 and conclude t hat { Lr}r has a converging subsequence; denot e it s limit by
Lε R x ∗. T hen one can verify t hat Lεis feasible for ξ∞q (P ), and we have
ξ∞q (P ) ≤ Lε(1) ≤ lim
r →∞ ξ q
r(P ) + ε ≤ ξ∞q(P ) + ε.
Let t ing ε → 0 we obt ain t hat ξ∞q(P ) = limr →∞ ξqr(P ).
Recall t hat a feasible solut ion L of ξrq(P ) is said t o be δ-flat if rank(Mr(L )) = rank(Mr − δ(L )),
where Mr − δ(L ) is t he principal submat rix of Mr(L ) whose rows and columns are indexed by
e, f , z r − δ. Since comput ing t he rank of a mat rix is easy, it is easy t o check whet her t he
solut ion given by t he semidefinit e programming solver is flat . In t he following proposit ion we
show t hat if ξrq(P ) admit s a δ-flat opt imal solut ion wit h δ = ⌈ r / 3⌉ + 1, t hen ξrq(P ) = ξq∗(P ).
T his proposit ion and it s proof are a small ext ension of t he flat ext ension result from [K P16]
for t racial opt imizat ion, where now δ depends on r because t he set Rr for t he ideal const raint
depends on r .
P r op osit ion 2.10. If ξrq(P ) admits a (⌈ r / 3⌉ + 1)-flat optimal solution, then ξrq(P ) = ξ∗q(P ).
P roof. Let δ = ⌈ r / 3⌉ + 1 and let L be a δ-flat opt imal solut ion t o ξqr(P ). We have t o show
ξrq(P ) ≥ ξ∗q(P ), which we do by const ruct ing a feasible solut ion t o ξ∗q(P ) wit h t he same object ive
value. In t he proof of T heorem 2.6 (see [GdLL17a, T hm. 2.3], and also [K P16, Prop. 6.1] for t he original proof of t his t heorem), t he linear form L is ext ended t o a t racial symmet ric linear form
ˆL on R x, y, z that is nonnegative on M 2r(G), zero on I (H ), and sat isfies rank(M ( ˆL )) < ∞ .
To do t his a subset W of x , y , z t− δ can be found such t hat we have t he vect or space direct
sum
R x , y , z = span(W ) I (Nr(L )),
where Nr(L ) is t he vect or space
Nr(L ) =
{
p R x , y , z r : L (qp) = 0 for all q R x , y , z r
} .
It is moreover shown t hat I (Nr(L )) N ( ˆL ). For p R x , y , z we denot e by rp t he unique
element in span(W ) such t hat p − rp I (Nr(L )).
Fix u, v, w R x , y , z . T hen we have
ˆL (w(zuzvz − zvzuz)) = ˆL(wzuzvz) − ˆL (wzvzuz).
Since ˆL is t racial and u − ru, v − rv, w − rw I (Nr(L )) N ( ˆL ), we have
ˆL (wzuzvz) = ˆL(rwzruzrvz) and ˆL (wzvzuz) = ˆL (rwzrvzruz).
Since deg(ruzrvzrwz) = deg(rvzruzrwz) ≤ 2r we have
ˆL (rwzruzrvz) = L (rwzruzrvz) and ˆL (rwzrvzruz) = L (rwzrvzruz).
So L I2r(Rr) implies ˆL I (R∞).
Since ˆL ext ends L we have ˆL (z) = L (z) = 1 and ˆL (xa
sybtz) = L (xasytbz) = P (a, b|s, t) for all
a, b, s, t. So, ˆL is feasible for ξ∗q(P ) and has t he same object ive value ˆL (1) = L (1).
3
B ou n d in g qu ant u m gr ap h p ar am et er s
3.1 H ier ar ch ies γcol
r (G) an d γrst ab(G) b ased on syn ch r on ou s cor r elat ion s
In Sect ion 1.4 we int roduced quant um chromat ic numbers (Definit ion 1.1) and quant um st a-bility numbers (Definit ion 1.2) in t erms of t he exist ence of synchronous quant um correlat ions sat isfying cert ain linear const raint s. We use t his in Sect ion 3.1.1 t o reformulat e t hese
prob-lems in t erms of C∗-algebras, and t hen in Sect ion 3.1.2 t o reformulat e t his in t erms of t racial
opt imizat ion, which leads t o t he hierarchies γcol
r (G) and γst abr (G).
3.1.1 G r aph par am et er s in t er m s of C∗-algebr as
T he following result from [PSS+16] allows us t o writ e a synchronous quant um correlat ion in
t erms of C∗-algebras admit t ing a t racial st at e.
T heor em 3.1 ([PSS+16]). Let Γ = A2 × S2 and P RΓ. We have P Cqc,s(Γ) (resp.,
P Cq,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 { Xa
s : s S, a A} A satisfying
∑
a A Xsa= 1
for all s S and
P (a, b|s, t) = τ(XsaXtb) for all s, t S, a, b A.
Here we add t he condit ion t hat τ is fait hful, t hat is, τ(X∗X ) = 0 implies X = 0, since it
follows from t he GNS const ruct ion in t he proof of [PSS+16]. T his means t hat
0 = P (a, b|s, t) = τ(XsaXtb) = τ((Xsa)2(Xtb)2) = τ((XsaXtb)∗XsaXtb)
implies Xa
sXtb= 0.
It follows t hat χqc(G) is equal t o t he smallest k N for which t here exist s a C∗-algebra A ,
a t racial st at e τ on A , and a family of project ors { Xic: i V, c [k]} A sat isfying
c [k]
Xic− 1 = 0 for all i V, (12)
XicXjc′ = 0 if (c = c′ and i = j ) or (c = c′ and { i , j } E ). (13)
T he quant um chromat ic number χq(G) is equal t o t he smallest k N for which t here exist s a
Analogously, αqc(G) is equal t o t he largest int eger k N for which t here exist s a C∗-algebra
A , a t racial st at e τ on A , and a family of project ors { Xi
c: c [k], i V } A sat isfying
i V
Xci − 1 = 0 for all c [k], (14)
XciXcj′ = 0 if (i = j and c = c′) or ((i = j or { i , j } E ) and c = c′), (15)
and t he quant um st ability number αq(G) is equal t o t he largest k N for which t here exist s a
finit e dimensional C∗-algebra A wit h t he above propert ies.
T hese reformulat ions of t he paramet ers χq(G), χqc(G), αq(G) and αqc(G) can be obt ained
from [OP16, T hm. 4.7], where general quant um graph homomorphisms are considered; t he
formulat ions of χq(G) and χqc(G) are also made explicit in [OP16, T hm. 4.12].
By Art in-Wedderburn t heory [Wed64, BEK 78], a finit e dimensional C∗-algebra is isomorphic
t o a mat rix algebra. So t he above reformulat ions of χq(G) and αq(G) can be seen as feasibility
problems of syst ems of equat ions in mat rix variables of unspecified (but finit e) dimension;
such formulat ions are given in [CMN+07, MR16, SV17] and t hey also follow from t he proof of
Proposit ion A.1. If we rest rict t o scalar solut ions (1 × 1 mat rices) in t hese feasibility problems, t hen we recover t he classical graph paramet ers χ(G) and α(G).
In [OP16] variat ions on t he above paramet ers are considered where t he C∗-algebras are not
required t o admit a t racial st at e.
3.1.2 G r aph par am et er s in t er m s of p osit ive t r acial linear for m s
Given a graph G = (V, E ) and an int eger k N, we let Hcol
G,k and Hst abG,k denot e t he set of
polynomials corresponding t o equat ions (12)–(13) and (14)–(15): HcolG,k = {1 −
c [k]
xic: i V} {xcixic′ : (c = c′ and i = j ) or (c = c′and { i , j } E )},
Hst abG,k = {1−
i V
xic: c [k]} {xicxjc′ : (i = j and c = c′) or ((i = j or { i , j } E ) and c = c′)
} .
We have 1 − (xci)2 M2( ) + I2(HcolG,k), since 1 − (xci)2= (1 − xci)2+ 2(xci − (xci)2) and
xci − (xci)2= xci 1 −
c′
xci′ +
c′:c′= c
xcixci′ I2(HcolG,k),
and t he analogous st at ement s hold for Hst abG,k. Hence, M ( ) + I (Hcolk ) and M ( ) + I (Hkst ab) are
Archimedean and we can apply T heorems 2.4 and 2.5 t o express t he quant um graph paramet ers in t erms of posit ive t racial linear funct ionals. Namely,
χqc(G) = min
{
k N : L R { xci : i V, c [k]} ∗ symmet ric, t racial, posit ive,
L (1) = 1, L = 0 on I (HcolG,k)},
and χq(G) is obt ained by adding t he const raint rank(M (L )) < ∞ . Likewise,
αqc(G) = min
{
k N : L R { xic: c [k], i V } ∗ symmet ric, t racial, posit ive,
L (1) = 1, L = 0 on I (Hst abG,k)},
and αq(G) is given by t he same program wit h t he addit ional const raint rank(M (L )) < ∞ .
St art ing from t hese formulat ions it is nat ural t o define a hierarchy { γcol
r (G)} of lower bounds
on χqc(G) and a hierarchy { γrst ab(G)} of upper bounds on αqc(G), where t he bounds of order
t o degree 2r . T hen, if we define γ∗col(G) and γ∗st ab(G) by adding t he const raint rank(M (L )) < ∞ t o γcol
∞ (G) and γ∞st ab(G), it follows by definit ion t hat
γ∞col(G) = χqc(G), γ∞st ab(G) = αqc(G), γcol∗ (G) = χq(G), and γ∗st ab(G) = αq(G).
T he opt imizat ion problems γcol
r (G), for r N, can be comput ed by semidefinit e
program-ming and binary search on k, since t he posit ivity condit ion on L can be expressed by requiring
t hat it s t runcat ed moment mat rix Mr(L ) = (L (w∗w′)) (indexed by words wit h degree at most
r ) is posit ive semidefinit e. If t here is an opt imal solut ion (k, L ) t o γrcol(G) wit h L flat , t hen,
by T heorem 2.6, we have equality γcol
r (G) = χq(G). Since { γrcol(G)}r N is a monot one
nonde-creasing sequence of lower bounds on χq(G), t here exist s an r0such t hat for all r ≥ r0 we have
γrcol(G) = γcolr0 (G), which is equal t o γ∞col(G) = χqc(G) by Lemma 2.7. T he analogous st at ement s
hold for t he paramet ers γst ab
r (G). Hence, we have shown t he following result .
L em m a 3.2. Let G be a graph. T here exists an r0 N such that γrcol(G) = χqc(G) and
γst ab
r (G) = αqc(G) for all r ≥ r0. Moreover, if γrcol(G) admits a flat optimal solution, then
γcol
r (G) = χq(G), and similarly if γrst ab(G) admits a flat optimal solution, then γst abr (G) = αq(G).
Going back t o t he reformulat ion of synchronous commut ing quant um correlat ions in T he-orem 3.1 we can obt ain in t he same way a hierarchy of semidefinit e programming based out er
approximat ions for t he set Cqc,s(Γ): Define Qr ,s(Γ) as t he set of P RΓ for which t here exist s
a symmet ric, t racial, posit ive linear form L R { xas : (a, s) A × S} ∗
2r such t hat L (1) = 1 and
L = 0 on t he ideal generat ed by t he polynomials xa
s− (xas)2 ((a, s) A × S) and 1 −
∑
a Axas
(s S), t runcat ed at degree 2r . T hen we have
Cqc,s(Γ) = Q∞ ,s(Γ) =
⋂
r N
Qr ,s(Γ).
Compared t o t he approximat ion Qr(Γ) from [PSS+16], only one set of variables { xas} is used
t o define Qr ,s in t he synchronous case while two set s of variables { xas, ybt} are used t o define
Qr(Γ). T he synchronous value of a nonlocal game is defined in [DP16] as t he maximum value
of t he object ive funct ion (5) over t he set Cqc,s(Γ). By maximizing t he object ive (5) over t he
relaxat ions Qr ,s(Γ) we get a hierarchy of semidefinit e programming upper bounds t hat converges
t o t he synchronous value.
We will now present ot her hierarchies of bounds for t he quant um paramet ers, inspired by exist ing result s on t he classical paramet ers α(G) and χ(G), and more economical since t hey involve variables indexed only by t he vert ices of G. T hese hierarchies capt ure exist ing bounds like project ive packing, project ive rank and t racial rank and are in fact t ight ly linked t o t he
bounds γcol
r (·) and γrst ab(·) via suit able graph product s.
3.2 H ier ar ch ies ξcol
r (G) an d ξst abr (G) b ased on Lasser r e t yp e b ou n d s
Here we revisit some known Lasserre type hierarchies for t he classical st ability number α(G) and chromat ic number χ(G) and we show t hat t heir t racial noncommut at ive analogues can be
used t o recover known paramet ers such as t he project ive packing number αp(G), t he project ive
rank ξf(G), and t he t racial rank ξt r(G). Compared t o t he hierarchies defined in t he previous
sect ion, t hese Lasserre type hierarchies use less variables (t hey only use variables indexed by t he vert ices of t he graph G), but t hey also do not converge t o t he (commut ing) quant um chromat ic or st ability number.
Given a graph G = (V, E ), define t he set of polynomials
HG = { xi− x2i : i V } { xixj : { i , j } E }
in t he variables x = (xi : i V ) (which are commut at ive or noncommut at ive depending on t he
