Graph-Theoretic Approach to Quantum Correlations

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by  Adan Cabello

Graph-Theoretic Approach to Quantum Correlations
Ad´an Cabello,1,∗ Simone Severini,2,† and Andreas Winter3,‡
1Departamento de F´ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain
2Department of Computer Science, and Department of Physics and Astronomy,
University College London, WC1E 6BT London, United Kingdom
3ICREA and F´ısica Teo`rica: Informacio´ i Fenomens Qua`ntics,
Universitat Auto`noma de Barcelona, E-08193 Bellaterra (Barcelona), Spain
(Dated: January 30, 2014)
CorrelationsinBellandnoncontextualityinequalitiescanbeexpressedasapositivelinearcombi-
nationofprobabilitiesofevents. Exclusiveeventscanberepresentedasadjacentverticesofagraph,
so correlations can be associated to a subgraph. We show that the maximum value of the corre-
lations for classical, quantum, and more general theories is the independence number, the Lova´sz
number, and the fractional packing number of this subgraph, respectively. We also show that, for
4 any graph, there is always a correlation experiment such that the set of quantum probabilities is
1 exactly the Gro¨tschel-Lova´sz-Schrijver theta body. This identifies these combinatorial notions as
0
fundamental physical objects and provides a method for singling out experiments with quantum
2
correlations on demand.

n
a PACSnumbers: 03.65.Ta,03.65.Ud,02.10.Ox
J

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Introduction.—Quantum theory (QT) is the basis of abilities to “events” defined as follows. We assume that
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our current description of nature and is arguably the preparationsofphysicalsystemsarereproducibleandcan

] most successful theory in the history of science. How- be compared through their statistics with respect to the
h
ever, we still do not understand QT in the sense that we available experiments. Preparations that yield the same
p
- cannot single out QT as the only theory that satisfies a probabilities for each of the experiments are considered
t set of principles similar to the two principles from which equivalent and define the same state. Reciprocally, ex-
n
a special relativity is derived [1]. periments that yield the same statistics for all states are
u In the search for similar principles for QT, much ef- considered equivalent and define the same test. Sets of
q
fort has been devoted to seeking principles singling out tests and their corresponding outcomes that occur with
[
quantumnonlocalcorrelations(i.e.,thosethatcannotbe thesameprobabilityineitherstateareconsideredequiv-

1 explained with local theories) [2–5]. However, this em- alent and define the same event: outcome a for test x
v
phasis on nonlocality assumes that experiments involv- and outcomes b,...,c for tests y,...,z are equivalent if
1
ing spacelike separated tests are more fundamental than probabilities P(a|x) and P(b,...,c|y,...,z) are equal in
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0 other types of correlation experiments, while nothing in either state. Then, a|x and b,...,c|y,...,z will denote
7 the rules of QT supports this assumption. two possible realizations of the same event. Two events
1. A more general approach to quantum correlations fol- ei and ej are exclusive if there exist two jointly measur-
0 lows from the question of whether there exists a joint able observables µi, defined by ei, and µj, defined by ej,
4 probability distribution that gives the marginals pre- that distinguish between them.
1 dictedbyQT.Thisquestionisequivalenttothequestion Exclusivity graphs.—To any correlation experiment,
:
v of whether a specific set of linear correlation inequalities we can associate a graph G in which events are repre-
i is satisfied [6–10]. For experiments with spacelike sepa- sented by vertices and pairs of exclusive events are rep-
X
rated tests, these inequalities are called Bell inequalities resented by adjacent vertices. We will refer to G as the
r
a [11, 12] and for more general scenarios are called non- exclusivity graph of the experiment.
contextuality (NC) inequalities [9, 13–25]. The exper- For example, in the Clauser-Horne-Shimony-Holt
imental violation of NC inequalities reveals contextual (CHSH)Bellexperiment[12]therearefourtests0,1,2,3,
correlations that cannot be explained with theories in eachofthemwithtwopossibleoutcomes: 0and1. Tests
which outcomes are predefined and do not depend on 0 and 2 can be performed by Alice, and tests 1 and 3
which combination of jointly measurable observables is byBob. Theexperimentconsistsofperformingthepairs
considered (i.e., noncontextual theories [26, 27]). of tests (0,1), (1,2), (2,3), and (3,0) on systems in the
In this Letter we present a novel approach to quan- samequantumstate. TheexclusivitygraphoftheCHSH
tum correlations within this more general framework of experiment G is represented in Fig. 1 (a). It has
CHSH
contextual correlations. We use graphs to characterize 16 vertices and 12 cliques of size 4 (sets of four pairwise
correlations and we show that three different classes of adjacent vertices).
theories, namely, noncontextual theories, QT, and more Similarly, in the Klyachko-Can-Binicio˘glu-Shumovsky
general probabilistic theories, allow for different sets of (KCBS) contextuality experiment [9, 25] there are five
probabilities. Wewillconsidertheoriesthatassignprob- tests i = 0,...,4 with two possible outcomes 0 and 1,
2
and the experiment consists of performing the five pairs P(e )isinS,adjacentverticesrepresentexclusiveevents,
i
of tests (i,i+1), with the sum modulo 5, on systems in and the vertex weights represent the weights w of the
i
the same quantum state. The exclusivity graph of the probabilities P(e ). We will call (G,w) the exclusivity
i
KCBS experiment G is shown in Fig. 1 (b). It has graph of S. The exclusivity graphs of S and S
KCBS CHSH KCBS
20 vertices and 15 cliques of size 4. are represented in Figs. 1 (a) and (b), respectively.
The correlations in any Bell or NC inequality are ex- In order to define a general class of theories assigning
pressed as a linear combination of probabilities of a sub- probabilities to events, we will consider theories satis-
set of events of the corresponding experiment. The fact fying the following principle: The sum of probabilities
thatthesumofprobabilitiesofoutcomesofatestis1can of any set of pairwise exclusive events cannot be higher
be used to express these correlations as a positive linear than 1. This class has been previously considered in
(cid:80)
combination of probabilities of events, S = w P(e ), [29, 30]. Specker noticed that classical and QT satisfy
i i i
withw >0. Forexample,theCHSHandKCBSinequal- this principle, but that there are theories that do not
i
ities can be expressed [28], respectively, as [29, 31]. Following [32, 33], we will refer to this princi-
ple as the exclusivity principle. We will denote by E1
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(cid:88)(cid:88) LHV those theories satisfying the exclusivity principle applied
S = P(a,b|i,i+1) ≤ 3, (1a)
CHSH
to G alone. The index 1 in E1 is used to distinguish
i=0 a,b
these theories from those satisfying the exclusivity prin-
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(cid:88) NCHV ciple applied jointly to G and other independent graphs
S = P(0,1|i,i+1) ≤ 2, (1b)
KCBS
[32, 33].
i=0
We first show that the exclusivity graph of S can be
where the second sum in Eq. (1a) is extended to a,b ∈ usedtocalculatethelimitsofthecorrelationsinclassical,
{0,1} with a = b if i (cid:54)= 2 and a (cid:54)= b if i = 2, the sum quantum, and theories satisfying E1.
in i+1 is taken modulo 4 in Eq. (1a) and modulo 5 in Result 1: Given S corresponding to a Bell or NC in-
Eq.(1b), andLHVandNCHVdenotelocalandnoncon- equality,themaximumvalueofS forclassical(LHVand
textual hidden variables, respectively. Although in these NCHV)theories, QT,andtheoriessatisfyingE1isgiven
examplesallprobabilitieshaveweight1,eachprobability by
P(e ) may have a different weight w . A vertex-weighted
i i
graph (G,w) is a graph G with vertex set V and weight S LHV≤,NCHVα(G,w)≤Q ϑ(G,w) ≤E1α∗(G,w), (2)
assignment w :V →R .
+
WecanassociatetoS avertex-weightedgraph(G,w),
where α(G,w) is the independence number of (G,w)
where G ⊆ G and i ∈ V represents event e such that
i [34], ϑ(G,w) is the Lov´asz number of (G,w) [34–36],
and α∗(G,w) is the fractional packing number of (G,w)
[34, 36, 37]. ϑ(G,w) might be only an upper bound to
(a) (b)
0,1|0,1 the maximum quantum value of S in cases in which the
1,1|0,1 0,0|1,2
0,1|0,1 0,1|1,2 0,0|0,1 1,1|0,1 particular physical settings of the experiment testing S
1,1|4,0 0,0|1,2 add further constraints.
0,0|0,1 1,0|0,1 1,0|1,2 1,1|1,2 0,1|4,0 1,0|0,1 0,1|1,2 Proof: The maximum value of S for classical theories
1,0|4,0 1,0|1,2
is always reached by a model in which each of the tests
0,0|4,0 1,1|1,2
1,1|3,0 1,0|3,0 1,0|2,3 0,0|2,3 1,1|3,4 1,0|3,41,0|2,3 0,0|2,3 has a predefined outcome, i.e., a model in which each of
theeventsinS haseitherprobability0or1. Sincemutu-
0,1|3,0 0,0|3,0 1,1|2,3 0,1|2,3 0,1|3,4 0,0|3,4 1,1|2,3 0,1|2,3 ally exclusive events cannot both have probability 1, the
maximum value of S for classical theories occurs when
probability 1 is assigned to each vertex in a set of non-
FIG.1: (a)Simplifiedrepresentationoftheexclusivitygraph
adjacent vertices in G. A set of nonadjacent vertices is
of the CHSH experiment, G . (b) Idem of the KCBS ex-
CHSH
periment, G . Events are represented by vertices. Here, called an independent or stable set of vertices of G.
KCBS
forsimplicity,setsofpairwiseexclusiveeventsarerepresented Therefore, the maximum of S for classical theories is
(cid:80)
by vertices in the same straight line or circumference rather themaximumof w wherethemaximumistakenover
i i
thanbycliques. (a)TheexclusivitygraphofSCHSH,denoted all stable sets of vertices of G. This is exactly the inde-
as G , is the induced subgraph of G obtained by
CHSH CHSH pendence number of (G,w), denoted as α(G,w) [34].
removing all but the eight black vertices. An induced sub- An orthonormal representation (OR) in Rd of a graph
graph is obtained by selecting a subset of vertices and their
G with vertex set V assigns a nonzero unit vector |v (cid:105)∈
incident edges. We use G instead of (G,w) whenever vertex i
weights are all 1. GCHSH is isomorphic to the eight-vertex Rd to each i ∈ V such that (cid:104)vi|vj(cid:105) = 0 for all pairs i,j
circulant (1,4) graph Ci (1,4). (b) The exclusivity graph of of nonadjacent vertices. A further unit vector |ψ(cid:105) ∈ Rd,
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SKCBS,denotedasGKCBS,istheinducedsubgraphofGKCBS called handle [35], is sometimes specified together with
obtained by removing all but the five black vertices. GKCBS the OR. The definition of OR does not require that dif-
is isomorphic to a five-cycle C (i.e., a pentagon).
5 ferent vertices be mapped onto different vectors nor that
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adjacentverticesbemappedontononorthogonalvectors. Examples: When we apply Result 1 to S , we
C√HSH
ThecomplementGofagraphGisthegraphwithvertex obtain α(G ) = 3, ϑ(G ) = 2 + 2, and
CHSH CHSH
set V such that two vertices i,j are adjacent in G if and α∗(G ) = 4, which correspond to the maximum for
CHSH
only if i,j are not adjacent in G. The Lov´asz number of local [12], quantum [39], and no-signaling theories [2],
(G,w) can be defined [34] as respectively.
When we apply Result 1 to S , we obtain
(cid:88) √ KCBS
ϑ(G,w):=max w |(cid:104)ψ|v (cid:105)|2, (3) α(G ) = 2, ϑ(G ) = 5, and α∗(G ) = 5,
i i KCBS KCBS KCBS 2
i∈V whichcorrespondtothemaximumfornoncontextual[9],
quantum [28], and no-disturbance theories [40], respec-
where the maximum is taken over all ORs of G and han- tively.
dles in any dimension. TheKCBSinequalitycanbeextendedtoanyoddnum-
Taken into account that the maximum value of S in ber of settings n≥5, and then Result 1 leads to
QT is always obtained for a quantum pure state |ψ(cid:105) and
a set of projectors Π in a real Hilbert space of suitable n−1
i (cid:88) NCHV n−1 Q ncos(π/n) E1 n
dimension, the fact that ϑ(G,w) equals the maximum P(0,1|i,i+1) ≤ ≤ ≤ ,
2 1+cos(π/n) 2
valueofS inQTisevidentbynoticingthat(cid:104)ψ|Π |ψ(cid:105)can i=0
i
(cid:112) (5)
bewrittenas|(cid:104)ψ|v (cid:105)|2,where|v (cid:105)=Π |ψ(cid:105)/ (cid:104)ψ|Π |ψ(cid:105)for
i i i i
wherethesumini+1istakenmodulon. Thisinequality
alli∈V isanORofGandthehandle|ψ(cid:105)isthequantum
was also obtained in [31].
state leading to the maximum value of S in QT.
For a given Bell or NC inequality (i.e., for a given S),
The maximum value of S for theories satisfying E1
ϑ(G,w) only provides an upper bound to its quantum
is proven by recalling that the fractional packing num-
maximum [41]. Then, a natural question is whether,
ber (or fractional stability number) of (G,w) [34, 37] is
given G, there is a NC inequality that reaches ϑ(G).
defined as
Result 2: For any graph G, there is always a NC
(cid:88)
α∗(G,w):=max w p , (4) inequality such that the quantum maximum is exactly
i i
ϑ(G) and the set of quantum probabilities is exactly the
i∈V
Gr¨otschel-Lov´asz-Schrijver theta body TH(G) [34].
where the maximum is taken over all p ≥ 0 and for all
i Proof: Given G, by Eq. (3), there is always an OR
(cid:80)
cliques C of G, under the restriction i∈Cpi ≤ 1. This of G in Rd, {|vi(cid:105)}, and a handle |ψ(cid:105) such that ϑ(G) =
last restriction imposes that the sum of the probabili- max(cid:80) |(cid:104)ψ|v (cid:105)|2. Let D be the minimum dimension
i∈V i
ties of any set of pairwise exclusive events in S cannot
d in which this OR exists. Then, consider the following
exceed 1.
positive linear combination of probabilities of events:
Comments: Computing α(G,w) for arbitrary graphs
is NP (nondeterministic polynomial time) complete even (cid:88)
S = P(1,0,...,0|i,i ,...,i ), (6)
1 n(i)
when all vertex weights are 1 [34, 38], which is in agree-
i∈V
ment with the well-known result that computing the up-
per bound for correlation inequalities for classical theo- where test i is defined as Π = |v (cid:105)(cid:104)v |, with |v (cid:105) in the
i i i i
riesisNP-hard. TheLov´asznumberofGwasintroduced OR, test i is Π = |v (cid:105)(cid:104)v |, with |v (cid:105) in the OR, and
j ij ij ij ij
[35] as an upper bound to the independence number and {i ,...,i } is theset of testscorresponding tovertices
1 n(i)
theShannoncapacityofG[37](whichisnotevenknown adjacent to i. This S reaches ϑ(G) when a quantum
to be computable). The extension of the Lov´asz number system is prepared in the quantum state |ψ(cid:105). Notice
to weighted graphs was introduced in [34, 36]. For any that 1,0,...,0|i,i ,...,i , which belongs to the same
1 n(i)
vertex weights, the Lov´asz number can be computed to equivalenceclassas1|i,isarepeatableeventwithawell-
any desired precision in polynomial time [36, 38]. Com- defined probability, even though i,i ,...,i need not
1 n(i)
puting α∗(G,w) for arbitrary graphs is NP-hard [36, 38] be jointly measurable.
becauseweneedthelistofcliquesofG. Ifthesearegiven, In QT, the set of probabilities that can be assigned
thenα∗(G,w)isalinearprogram,andassuchefficiently to the vertices of G by performing tests on a quantum
computable. system is
Q(G):=(cid:8)(|(cid:104)ψ|v (cid:105)|2 :i∈V):(|v (cid:105):i∈V) is an OR of G and |ψ(cid:105) a handle(cid:9). (7)
i i
If the quantum system has dimension greater than or equal to D, then Q(G) is exactly the theta body TH(G)
4
introduced in [34] [see Theorem 3.5 in [34] and Corollary (ii)Q(G)=E1(G)ifandonlyifGhasnooddcycleC ,
n
9.3.22 (c) in [38]]. with n≥5, or its complement C as induced subgraphs.
n
Comments: Result 2 identifies ϑ(G) as a fundamental (iii) Q(G) is a polytope if and only if G has no odd
physical limit for quantum correlations and TH(G) as cycle C , with n ≥ 5, or its complement C as induced
n n
the set of physical correlations for a given G. Result 2 subgraphs.
suggests that an important question for understanding Proof: C(G) is exactly the stable set polytope of G,
quantumcorrelationsiswhichistheprinciplethatsingles STAB(G) (also called vertex packing polytope) [34, 38].
out TH(G) among all possible sets of probabilities that E1(G) is exactly the fractional stable set polytope of G,
can be assigned to the vertices of G. QSTAB(G) (or clique-constrained stable set polytope)
Although here we have shown that, for quantum introduced in [37].
physics, TH(G) is a fundamental set, TH(G) was orig- In[34]itisproventhatSTAB(G)=TH(G)ifandonly
inally introduced to bound the size of independent sets if G is perfect. Perfect graphs were introduced in [45] in
[34]. connection to the problem of the zero-error capacity of a
Notice that Result 2 is not longer true if we replace graph [37]. The strong perfect graph theorem [46] states
“NC inequality” by “Bell inequality.” For example, if G that G is perfect if and only if G has no odd cycle Cn,
is a pentagon, there is no Bell inequality reaching ϑ(G) with n≥5, or its complement Cn as induced subgraphs.
[41]. This is due to the extra constraints imposed by This proves (i).
the Bell scenario which enforce a specific labeling of the In [34] it is proven that TH(G) = QSTAB(G) if and
events (see [41] for details). This shows the advantage only if G is perfect. This proves (ii).
of discussing quantum correlations in the framework of Finally, in [34] it is proven that TH(G) is a polytope
NCinequalitiesnotreferringtoanyspecificexperimental if and only if G is perfect [34]. This proves (iii).
scenario. Applications.—Results 1 and 2 provide a general
Notice that this strategy of focusing on graphs with- method to construct NC inequalities. Specifically, they
outreferringtoanyspecificexperimentalscenario(whose show that, for every graph G such that α(G) < ϑ(G),
existenceisguaranteedbyResult2)substantiallysimpli- there is a NC inequality with classical limit given by
fies the problem of characterizing the quantum set with α(G) and quantum violation given by ϑ(G) and that,
respecttothecaseinwhichthelabelingisgiven[39,42– for every G such that α(G) < ϑ(G) = α∗(G), there is a
44]. NCinequalityinwhichthemaximumquantumviolation
In a similar way that we have defined the quantum cannotbehigherwithoutviolatingtheexclusivityprinci-
set Q(G), we can define the corresponding sets for clas- ple. This allows us to design experiments with quantum
sical and more general theories satisfying E1. They are, contextuality on demand by selecting graphs with the
respectively, desired relationships between these three numbers.
The fact that these three numbers have been studied
C(G):=convex hull(cid:8)xS :xS is a stable labeling of G(cid:9), for a long time and that there is an extensive literature
on the subject also opens the possibility of identifying
(8a) novel interesting quantum correlations. For example, it
(cid:40) (cid:41) is known that for arbitrarily large n there are graphs
E1(G):= p∈R|V| :(cid:88)p ≤1 for all cliques C , for which α(G) ≈ 2logn and ϑ(G) ≈ √n or for which
+ i √
i∈C α(G) = 3 and ϑ(G) ≈ 4n [47]. This shows that the
(8b) quantum violation of NC inequalities can be arbitrarily
large. In addition, a question such as which are the cor-
where a stable labeling of G is a labeling xS ∈ {0,1}|V| relationswithmaximumquantumcontextualitycannow
for a stable set S of G such that xS = 1 if i ∈ S and be addressed, since now it can be related to the ques-
i
xS = 0 if i ∈/ S. Clearly, the quantum set is sandwiched tion of which graphs have the maximum ϑ(G)/α(G) for
i
between the classical and the E1 set, a given number of vertices. Similarly, all possible forms
of quantum contextuality can be classified by classifying
C(G)⊆Q(G)⊆E1(G). (9) graphs according to their combinatorial numbers.
Conclusions.—Herewehaveintroducedagraph-based
A natural question is which graph properties distin- approachtothestudyofquantumcorrelations. First,we
guishquantumfromclassicalcorrelations. Anotherinter- have shown that we can associate a graph G to any cor-
esting question is when quantum correlations are singled relation experiment such that the possible correlations
out by E1. Notice that C(G) and E1(G) are polytopes, are given by the possible probability distributions that
but Q(G), in general, is not. can be assigned to the vertices of the graph. Hence, the
Result 3: (i) C(G)=Q(G) if and only if G has no odd correlations considered in any Bell or NC inequality can
cycle C , with n ≥ 5, or its complement C as induced be associated to a weighted subgraph (G,w) such that
n n
subgraphs. G ⊆ G. We have shown that the limits imposed to the
5
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O. Gu¨hne, M. Kleinmann, J.-˚A. Larsson, A. J. L´opez- and A. Cabello, Phys. Rev. Lett. 109, 250402 (2012).
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FIS2011-29400 with FEDER funds (MINECO, Spain), 070401 (2013).
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the FQXi large grant project “The Nature of Informa-
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tion in Sequential Quantum Measurements,” and the
3, 011012 (2013).
Brazilian program Science without Borders. S.S.’s work
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is supported by the Royal Society and EPSRC. A.W.’s Sci. Rep. 3, 2170 (2013).
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derMartin-LutherUniversita¨tHalle-Wittenberg10,114