On Quantum Contextuality and

NonlocalityMarcelo Terra Cunha

Unicamp

Keep in Mind 1• A game with 5 boxes where QT contrasts with

usual probability theory

• Concepts of probability theory (to be generalized)

• Compatibility (Bell) Scenarios

• Important sets of behaviors

• Exercise: PR-behaviors are non signaling

Keep in Mind 2• Description of Local and No Signaling CHSH polytopes

• Definition and Properties of Quantum Set

• Bell Inequalities, NC Inequalities, Graph Approach, KCBS Scenario

• Independence Number

• Lovász Number

• Beautiful history

• Beautiful geometry

Keep in Mind 2.2• Global Sections (Topology)

• For the 5-box (Hardy) game

• For a more stringent situation

• For assignalations and for probabilities

• A. Fine Theorem

Exercises• PR is no signaling

• PR is nonlocal

• PR is not quantum (Tsirelson bound)

• Bell Theorem

• Quantum Set is no signaling, convex, and not a polytope

E-PrincipleThe sum of the probabilities of

pairwise exclusive events can notbe more than 1.

01|01

01|12

01|2301|34

01|40

p =1

2

p =1

2

p =1

2p =

1

2

p =1

2

4X

i=0

p(0, 1|i, i+ 1)E 5

2

↵⇤(C5)

FractionalPackingNumber

The General Picture4X

i=0

p(0, 1|i, i+ 1)NC 2

QT

p5

E 5

24X

i=0

p(0, 1|i, i+ 1)NC ↵(C5)

QT #(C5)

E ↵⇤(C5)

S(G)NC ↵(G)

QT #(G)

E ↵⇤(G)

S(G,w)NC ↵(G,w)

QT #(G,w)

E ↵⇤(G,w)

S(G,w) =X

e

wep(e)where ~w · ~p ↵ (G,w)

Questions?

Can you have a little more?

Some SurprisesState Independent (Quantum) Contextuality

�x

⌦ I I ⌦ �x

�x

⌦ �x

I ⌦ �z �z ⌦ I �z ⌦ �z

�x

⌦ �z

�z

⌦ �x

�y ⌦ �y

Peres, Phys. Lett. A (1990); Mermin, Phys. Rev. Lett. (1990)

Some Surprises3

0,1,0,1|0 ,1 ,0��1V V S S

,

0,1,0,1|0 ,1 ,1��2V V S S

,0,1,0,1|0 ,1 ,4V V S S

,0

0,1,0,1|0 ,1 ,2V V S S

,3

0,1,0,1|1 ,2 ,4V V S S

,0

0,1,0,1|1 ,2 ,0��1V V S S

,

0,1,0,1|1 ,2 ,1V V S S

,2

0,1,0,1|1 ,2 ,2V V S S

,3

0,1,0,1|1 ,2 ,3V V S S

,4

0,1,0,1|2 ,3 ,0��1V V S S

,

0,1,0,1|2 ,3 ,1V V S S

,2

0,1,0,1|2 ,3 ,2V V S S

,3

0,1,0,1|2 ,3 ,4V V S S

,0

0,1,0,1|2 ,3 ,3V V S S

,4

0,1,0,1|3 ,4 ,1V V S S

,2

0,1,0,1|3 ,4 ,2V V S S

,3

0,1,0,1|3 ,4 ,3V V S S

,4

0,1,0,1|3 ,4 ,4V V S S

,0

0,1,0,1|3 ,4 ,0��1V V S S

,

0,1,0,1|4 ,0 ,2V V S S

,3

0,1,0,1|4 ,0 ,3V V S S

,4

0,1,0,1|4 ,0 ,4V V S S

,0

0,1,0,1|4 ,0 ,0��1V V S S

,

0,1,0,1|4 ,0 ,1V V S S

,2

0,1,0,1|0 ,1 ,3V V S S

,4

FIG. 2: (Color online) Graph of the relationships of exclu-sivity between the 25 events obtained from considering twoexperiments testing the KCBS inequality. Each vertex repre-sents an event a, b, c, d|xV , yV , zS , kS denoting “the results aand b are respectively obtained when compatible tests x andy are performed in Vienna, and the results c and d are respec-tively obtained when compatible tests z and k are performedin Stockholm.” Exclusive events are represented by adjacentvertices. Any event belongs to a set of 5 pairwise exclusiveevents. One of these sets is indicated in red.

in both cases, by the same graph G, corresponds to theOR product (also called co-normal product, disjunctiveproduct, or disjunction product) of two copies of G [43].The OR product of two graphs G and H is a new graphG ∗ H whose vertex set is V (G) × V (H) and in whichtwo vertices (g, h) and (g′, h′) in G ∗ H are adjacent ifg and g′ were adjacent vertices in G or h and h′ wereadjacent vertices in H . The graph in Fig. 2 is the ORproduct of two copies of the graph in Fig. 1. Similarly,the graph of the relationships of exclusivity for n copiesof the experiment is given by the OR product of n copiesof G, denoted as G∗n.The CHSH inequality.—The Clauser-Horne-Shimony-

Holt (CHSH) inequality [44] is the tight Bell inequalitycorresponding to the bipartite scenario in which Alicechooses between two tests x ∈ {0, 1} and Bob choosesbetween two tests y ∈ {0, 1}. Each test has two pos-sible results: Alice’s are denoted a ∈ {0, 1} and Bob’sb ∈ {0, 1}. If we complete the CHSH inequality withits maximum violation in QM (Tsirelson’s bound [45])and the upper bound imposed by nonsignaling [8] (whichequals the one imposed by E), we obtain the followingexpression:

!

P (a, b|x, y)NCHV,LHV

≤ 3QM

≤ 2 +√2

E,NS

≤ 4, (3)

where the sum is extended to all x, y ∈ {0, 1} anda, b ∈ {0, 1} such that a ⊕ b = xy, where ⊕ denotessum modulo 2, LHV means local hidden variables, andNS means nonsignaling.

0,0|0,0

1,1|0,0

0,0|0,1

0,1|1,1 1,0|1,1

0,0|1,0

1,1|1,0 1,1|0,1

1,1|0,1 0,0|1,0

1,1|0,00,0|0,0

1,0|1,1

(a) (b)

FIG. 3: (a) Graph of the relationships of exclusivity between5 events to which a PR box assigns a probability 1

2to each

event. (b) Graph of the relationships of exclusivity betweenthe 8 events involved in the CHSH inequality. This graphis the 8-vertex (1,4)-circulant graph Ci8(1, 4). It contains 8induced pentagons. The pentagon emphasized corresponds tothe one in (a). Vertex ab|xy represents the event “the results aand b are respectively obtained when spacelike separated testsx and y are performed,” where x is performed in Alice’s sideand y in Bob’s. Exclusive events are represented by adjacentvertices.

A PR box [8] is a two-party nonsignaling device whichachieves the maximum algebraic violation of the CHSHinequality, which is equal to the maximum violation sat-isfying NS and E. A PR box produces joint probabilitiesP (a, b|x, y) = 1

2 , if a⊕ b = xy, and 0 otherwise.Now the question is, why are PR boxes not allowed in

nature despite that they do not violate nonsignaling [8].Many reasons have been given for this [9–12, 46]. Here

we show that a recent proof [36] can be simplified by thefollowing observation.Observation 1: A PR box assigns probability 1

2 to 5joint probabilities of events whose relationships of ex-clusivity are exactly the ones of the KCBS inequality.Therefore, the proof of Result 1 does not only excludesWright’s assignment to the KCBS inequality, but alsoexcludes PR’s assignment to the CHSH inequality.Proof.—A PR box assigns probability 1

2 to the 5 jointprobabilities of events whose relationships of exclusivityare represented in Fig. 3 (a).The proof of Result 1 does not single out Tsirelson’s

bound 2 +√2 ≈ 3.4142, but states that the maximum

quantum nonlocality should be less than or equal to8√5≈ 3.5778. Interestingly, this is the same bound ob-

tained by considering all the restrictions that E imposes,after assuming the principle of local orthogonality, to allpossible combinations of the 64 events resulting from twoPR boxes [36]. This emphasizes the fundamental role ofthe elementary Bell inequalities introduced in [38] to un-derstand the quantum violation of Bell inequalities.The next question is, why does the quantum violation

of the CHSH inequality stop at Tsirelson’s bound [8].Here we show that, if we only use GE applied to mul-

tiple copies of the CHSH experiment, then the answer isnot known and is related to an open problem in graphtheory. However, a curious observation can be made.Observation 2: If the Shannon capacity [47] of the 8-

vertex (1,2)-circulant graph Ci8(1, 2) were equal to its

01|01

01|12

01|2301|34

01|40

01|01

01|12

01|2301|34

01|40

StockholmVienna

Global

Collective Bounds3

0,1,0,1|0 ,1 ,0��1V V S S

,

0,1,0,1|0 ,1 ,1��2V V S S

,0,1,0,1|0 ,1 ,4V V S S

,0

0,1,0,1|0 ,1 ,2V V S S

,3

0,1,0,1|1 ,2 ,4V V S S

,0

0,1,0,1|1 ,2 ,0��1V V S S

,

0,1,0,1|1 ,2 ,1V V S S

,2

0,1,0,1|1 ,2 ,2V V S S

,3

0,1,0,1|1 ,2 ,3V V S S

,4

0,1,0,1|2 ,3 ,0��1V V S S

,

0,1,0,1|2 ,3 ,1V V S S

,2

0,1,0,1|2 ,3 ,2V V S S

,3

0,1,0,1|2 ,3 ,4V V S S

,0

0,1,0,1|2 ,3 ,3V V S S

,4

0,1,0,1|3 ,4 ,1V V S S

,2

0,1,0,1|3 ,4 ,2V V S S

,3

0,1,0,1|3 ,4 ,3V V S S

,4

0,1,0,1|3 ,4 ,4V V S S

,0

0,1,0,1|3 ,4 ,0��1V V S S

,

0,1,0,1|4 ,0 ,2V V S S

,3

0,1,0,1|4 ,0 ,3V V S S

,4

0,1,0,1|4 ,0 ,4V V S S

,0

0,1,0,1|4 ,0 ,0��1V V S S

,

0,1,0,1|4 ,0 ,1V V S S

,2

0,1,0,1|0 ,1 ,3V V S S

,4

FIG. 2: (Color online) Graph of the relationships of exclu-sivity between the 25 events obtained from considering twoexperiments testing the KCBS inequality. Each vertex repre-sents an event a, b, c, d|xV , yV , zS , kS denoting “the results aand b are respectively obtained when compatible tests x andy are performed in Vienna, and the results c and d are respec-tively obtained when compatible tests z and k are performedin Stockholm.” Exclusive events are represented by adjacentvertices. Any event belongs to a set of 5 pairwise exclusiveevents. One of these sets is indicated in red.

in both cases, by the same graph G, corresponds to theOR product (also called co-normal product, disjunctiveproduct, or disjunction product) of two copies of G [43].The OR product of two graphs G and H is a new graphG ∗ H whose vertex set is V (G) × V (H) and in whichtwo vertices (g, h) and (g′, h′) in G ∗ H are adjacent ifg and g′ were adjacent vertices in G or h and h′ wereadjacent vertices in H . The graph in Fig. 2 is the ORproduct of two copies of the graph in Fig. 1. Similarly,the graph of the relationships of exclusivity for n copiesof the experiment is given by the OR product of n copiesof G, denoted as G∗n.The CHSH inequality.—The Clauser-Horne-Shimony-

Holt (CHSH) inequality [44] is the tight Bell inequalitycorresponding to the bipartite scenario in which Alicechooses between two tests x ∈ {0, 1} and Bob choosesbetween two tests y ∈ {0, 1}. Each test has two pos-sible results: Alice’s are denoted a ∈ {0, 1} and Bob’sb ∈ {0, 1}. If we complete the CHSH inequality withits maximum violation in QM (Tsirelson’s bound [45])and the upper bound imposed by nonsignaling [8] (whichequals the one imposed by E), we obtain the followingexpression:

!

P (a, b|x, y)NCHV,LHV

≤ 3QM

≤ 2 +√2

E,NS

≤ 4, (3)

where the sum is extended to all x, y ∈ {0, 1} anda, b ∈ {0, 1} such that a ⊕ b = xy, where ⊕ denotessum modulo 2, LHV means local hidden variables, andNS means nonsignaling.

0,0|0,0

1,1|0,0

0,0|0,1

0,1|1,1 1,0|1,1

0,0|1,0

1,1|1,0 1,1|0,1

1,1|0,1 0,0|1,0

1,1|0,00,0|0,0

1,0|1,1

(a) (b)

FIG. 3: (a) Graph of the relationships of exclusivity between5 events to which a PR box assigns a probability 1

2to each

event. (b) Graph of the relationships of exclusivity betweenthe 8 events involved in the CHSH inequality. This graphis the 8-vertex (1,4)-circulant graph Ci8(1, 4). It contains 8induced pentagons. The pentagon emphasized corresponds tothe one in (a). Vertex ab|xy represents the event “the results aand b are respectively obtained when spacelike separated testsx and y are performed,” where x is performed in Alice’s sideand y in Bob’s. Exclusive events are represented by adjacentvertices.

A PR box [8] is a two-party nonsignaling device whichachieves the maximum algebraic violation of the CHSHinequality, which is equal to the maximum violation sat-isfying NS and E. A PR box produces joint probabilitiesP (a, b|x, y) = 1

2 , if a⊕ b = xy, and 0 otherwise.Now the question is, why are PR boxes not allowed in

nature despite that they do not violate nonsignaling [8].Many reasons have been given for this [9–12, 46]. Here

we show that a recent proof [36] can be simplified by thefollowing observation.Observation 1: A PR box assigns probability 1

2 to 5joint probabilities of events whose relationships of ex-clusivity are exactly the ones of the KCBS inequality.Therefore, the proof of Result 1 does not only excludesWright’s assignment to the KCBS inequality, but alsoexcludes PR’s assignment to the CHSH inequality.Proof.—A PR box assigns probability 1

2 to the 5 jointprobabilities of events whose relationships of exclusivityare represented in Fig. 3 (a).The proof of Result 1 does not single out Tsirelson’s

bound 2 +√2 ≈ 3.4142, but states that the maximum

quantum nonlocality should be less than or equal to8√5≈ 3.5778. Interestingly, this is the same bound ob-

tained by considering all the restrictions that E imposes,after assuming the principle of local orthogonality, to allpossible combinations of the 64 events resulting from twoPR boxes [36]. This emphasizes the fundamental role ofthe elementary Bell inequalities introduced in [38] to un-derstand the quantum violation of Bell inequalities.The next question is, why does the quantum violation

of the CHSH inequality stop at Tsirelson’s bound [8].Here we show that, if we only use GE applied to mul-

tiple copies of the CHSH experiment, then the answer isnot known and is related to an open problem in graphtheory. However, a curious observation can be made.Observation 2: If the Shannon capacity [47] of the 8-

vertex (1,2)-circulant graph Ci8(1, 2) were equal to its

Symmetry p(0, 1|i, i+ 1) = p

Independencyp(0, 1, 0, 1|iV , i+ 1V , jS , j + 1S) = pV pS = p2

E Principle 5p2 1 pMax

1p5

KCBS4X

i=0

p(0, 1|i, i+ 1) 5pMax

p5

We did not use Quantum Theory!A. Cabello, PRL (2013)

More Results

• Sets in correlation space also have graph-theoretical analogues:

• NC = STAB

• Q = THETA

• E = QSTAB

Cabello, Severini, Winter, PRL (2014)

More Results

• E-Principle gives the set of quantum correlations if this set is known for the complementary graph

• E-Principle forbids larger-than-quantum sets of correlations (for self-complementary G)

Barbara Amaral, Marcelo Terra Cunha, and Adán Cabello,Phys. Rev. A (2014)

• E-Principle gives the set of quantum correlations! To be checked, written, and published…

Complete Exclusiveness

• Not only the events obey E-Principle for its own graph

• If jointly and independently considered with any other exclusivity structure, E-Principle must be valid

Compare with the definition of Completely Positive Maps, and its role as quantum channels…

The Holy Grail

The Holy Grail

Can we obtain Quantum Theory from Informational Principles?

Can we obtain Quantum Correlations from Informational Principles?

Envy of Special Relativity…

We have seen• A game with 5 boxes where QT contrasts with usual

probability theory (better understood today)

• Concepts of probability theory (to be generalized)

• Compatibility (Bell) Scenarios

• Important sets of behaviors

• Two polytopes and a convex set

• Bell Inequalities, NC Inequalities, and relation to graphs

We have seen• Graph Invariants

• Graph Sets

• Global Sections

• A Fine Theorem

• Exclusivity and Complete Exclusivity Principles

• A possible explanation for Quantum Correlations

Tools and ArgumentsComplementary Graphs G, G

“Two-city pairing”4.2 The Pentagon 75

2

1

5

4 3

40

10

30

50 20

Fig. 4.1: Exclusivity graphs of the sets of events ei and e0i.

4^40

1^10

3^30

5^50 2^20

Fig. 4.2: The exclusivity graph for the set of events { fi}.

The E-principle implies

5

Âi=1

p( fi) =5

Âi=1

p(ei)p(e0i) 1. (4.10)

Using the symmetry of the pentagon, we can assume (see lemma 4.1 below) that themaximum is reached when all the probabilities are equal, that is

p(ei) = p(e0i) = p, 8 i 2 V(G) (4.11)

Hence, we have5

Âi=1

p2 = 5p2 1 (4.12)

which implies that

p 1p5. (4.13)

Now, if we substitute this value into equation (4.6) for S we have

Dual interpretation of

~w · ~p 1

No larger than quantum…

Unicity!

We did not cover• A huge collection of Scenarios and Inequalities

• Many interesting formulations of Contextuality

• A colored graph approach, where parts come into Contextuality

• All but two approaches…

• Many other Informational Principles related to Quantum Correlations

Some of this…Barbara Amaral and Marcelo Terra Cunha

Graph approach to Contextuality

and its role in Quantum Theory

August 8, 2017

Springer

Soon, in the best libraries and bookshops

Keep in Mind• Quantum Theory is about (Generalized) Probabilities

• Incompatibility is a central issue

• We cannot have answers to all the questions!

• Possibly we can obtain Quantum Correlations from informational postulates (CE-Principle)

• Can help in merging Quantum Theory with Gravity