Graph Approach for Quantum Contextuality

Marcelo Terra CunhaUniversidade Federal de Minas Gerais - UFMG

Brazil

IPQI - 2014Bhubaneswar, 28th February

arXiv:1010.2163 arXiv: 1306.6289

Adán CabelloSimone Severini

Andreas Winter

Barbara Amaral

The Result

There is a translation from Non-Contextuality Inequalities into

Graph-Theoretical concepts

Personal Motivation

•To clearly understand/explain the novelties that Quantum Theory brings

•Try to “derive” Quantum Theory (or something like it) from other Principles Envy about Relativity Theory

The Surprise

•The E-Principle can be such foundational stone

E = Exclusivity

Ernest Specker

The Most Famous Case

Bell InequalitiesBell Scenario

Bell InequalitiesLocal Correlations

Quantum Correlations

The CHSH Scenario

J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt,

Phys. Rev. Lett. 23, 880 (1969).

The CHSH Graph

CompatibilityGraph

ExclusivityGraph

Beware: Lines are Cliques!

CHSH Inequality

CHSH Ineq. Graph

What is the relation?

Independencenumber

KCBS Graphs

Exclusivity InequalityCompatibility

Quantum Theory?

WhereOrthogonal Representation of

Handle

Lovász Theta Function

Small piece of History

•Zero-error information theory includes the concept of Shannon Capacity of a Graph

•Conjectured as intractable

•For the pentagon, a lower bound is very simple

•Lovász defined his function to be a computable upper bound

•And obtained the exact value for the pentagon!

Beautiful Geometry

Umbrela

Handle

E-PrincipleThe sum of the probabilities of

pairwise exclusive events can notbe more than

FractionalPackingNumber

The General Picture

where

Questions?

Can you have a little more?

Some Surprises

StockholmVienna

Global

Collective BoundsSymmetry

Independency

E Principle

KCBS

We did not use Quantum Theory!

More Results

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

•NC = STAB

•Q = THETA

•E = QSTAB

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 correlationsBarbara Amaral, Marcelo Terra Cunha, and

Adán Cabello,Phys. Rev. A (2014)

GeneralizationsGraph Approach to Nonlocality

GeneralizationsGraph Approach to Nonlocality

GeneralizationsGraph Approach to Nonlocality

Pool: Next Friday•Bit-

Commitment

arXiv:1307.0156

A protocol, using Bell Inequality violation, that avoids the no-go theorem of Lo-Chau-Mayers

Security is conjectural

• Graph-Approach to Contextuality

Cabello, Severini & Winter idea for encoding NC Inequalities as graph-theoretical invariants;

Extensions, E-Principle,Quantum

CorrelationsRaise hands Raise hands

08/02/14

Take home messages

•Graph approach encodes all contextuality inequality and some generalizations

•Extensions to Nonlocality under construction

•E Principle is a good candidate for explaining quantum correlations

•No larger-than-quantum set!

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