Quantum Mechanics As a Sandwich Theory Simone Severini University - - PowerPoint PPT Presentation

Quantum Mechanics As a Sandwich Theory

Simone Severini University College London with Adán Cabello (Sevilla), Runyao Duan (UTS), Laura Mancinska (IQC/NUS), Giannicola Scarpa (CWI), Andreas Winter (ICREA)

Quantum Mechanics As a Sandwich Theory

Simone Severini University College London with Adán Cabello (Sevilla), Runyao Duan (UTS), Laura Mancinska (IQC), Giannicola Scarpa (CWI), Andreas Winter (ICREA)

Combinatorial optimization techniques, like semidef. programming (see the Parillo-Lasserre SDP hierarchy) are traditionally important in quantum theory. We propose a framework based on optimization on graphs to unify

• non-contextuality,
• non-locality (via non-local games),
• and certain information transmission tasks that

use quantum resources.

Overview

Plan

• 1. Introduction:

non-contextuality

• 2. Results:

a general framework to study non contextuality; non-contextuality and non-locality

• 3. Open problems:

theoretical; applied; complexity perspective

• 4. Information theory:

zero-error capacities [if there is time]

Plan:

• 1. Introduction:

non-contextuality for

• 1. classical theories
• 2. non-signaling theories
• 3. quantum theory
• 1. Introduction: non-contextuality

• 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Question 4 Question 5

• 1. Introduction: non-contextuality

Question 1 Question 2 Question 3

context:

a set of mutually compatible questions

• 1. Introduction: non-contextuality

Question 1: Is the colour red? Answer: Yes Question 2: What is the suit? Answer: Diamonds

compatibility:

• 1. Introduction: non-contextuality

non-contextuality:

answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2: Is Y true? Answer: Yes Question 1: Is X true? Answer: Yes

Context 1

non-contextuality:

answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2: Is Y true? Answer: Yes Question 3: Is Z true? Answer: No

Context 2

non-contextuality:

answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2: Is Y true? Answer: Yes Question 3: Is Z true? Answer: No Question 1: Is X true? Answer: Yes

Context 1 Context 2

non-contextuality:

answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2: Is Y true? Answer:

Yes

Question 3: Is Z true? Answer: No Question 1: Is X true? Answer: Yes

Context 1 Context 2

non-contextuality:

answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2 Question 3 Question 4 Question 5 Question 1

compatibility structure

• 1. Introduction: non-contextuality

Question 2 Question 3 Question 4 Question 5 Question 1

compatibility structure

• 1. Introduction: non-contextuality

Question 2 Question 3 Question 4 Question 5 Question 1

compatibility structure

• 1. Introduction: non-contextuality

Question 2 Question 3 Question 4 Question 5 Question 1

compatibility structure

• 1. Introduction: non-contextuality

Question 2 Question 3 Question 4 Question 5 Question 1

compatibility structure

• 1. Introduction: non-contextuality

Question 2 Question 3 Question 4 Question 5 Question 1

compatibility structure

• 1. Introduction: non-contextuality

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality

Question 2 Answer: 0/1 Question 3 Answer: 0/1 Question 4 Answer: 0/1 Question 5 Answer: 0/1 Question 1 Answer: 0/1

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality

Question 2 Answer: 0/1 Question 3 Answer: 0/1 Question 4 Answer: 0/1 Question 5 Answer: 0/1 Question 1 Answer: 1

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality

Question 2 Answer: Question 3 Answer: 0/1 Question 4 Answer: 0/1 Question 5 Answer: 0/1 Question 1 Answer: 1

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality

Question 2 Answer: Question 3 Answer: 0/1 Question 4 Answer: 0/1 Question 5 Answer: Question 1 Answer: 1

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality

the answers 1 have expectation ≤ 2 if non-contextual and exclusive

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• 1. Introduction: non-contextuality

classical theories

give expectation ≤ 2

• 1. Introduction: non-contextuality

non-contextuality:

probabilities of answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2: Is Y true? Pr[answer Yes] = b Question 1: Is X true? Pr[answer Yes] = a

Context 1

non-contextuality:

probabilities of answers do not depend

• n contexts

• 1. Introduction: non-contextuality

Question 2: Is Y true? Pr[answer Yes] = b Question 3: Is Z true? Pr[answer Yes] = c

non-contextuality:

probabilities of answers do not depend

• n contexts

Context 2

• 1. Introduction: non-contextuality

Question 2: Is Y true? Pr[answer Yes] = b Question 3: Is Z true? Pr[answer Yes] = c

non-contextuality:

probabilities of answers do not depend

• n contexts

Context 2

Question 1: Is X true? Pr[answer Yes] = a

Context 1

• 1. Introduction: non-contextuality

Question 3: Is Z true? Pr[answer Yes] = c

non-contextuality:

probabilities of answers do not depend

• n contexts

Context 2

Question 2: Is Y true?

Pr[answer Yes] = b

Question 1: Is X true? Pr[answer Yes] = a

Context 1

• 1. Introduction: non-contextuality

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

Question 2 Answer: 0/1 Question 3 Answer: 0/1 Question 4 Answer: 0/1 Question 5 Answer: 0/1 Question 1 Answer: 0/1

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

Question 2 Answer: Pr[1] = Question 3 Answer: Pr[1] = Question 4 Answer: Pr[1] = Question 5 Answer: Pr[1] = Question 1 Answer: Pr[1] =

exclusiveness:

answers to adjacent questions are not both 1 p1 p2 p3 p5 p4

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

Question 2 Answer: Pr[1] = Question 1 Answer: Pr[1] =

exclusiveness:

answers to adjacent questions are not both 1 p1 p2

Context 1

p1p2=1

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

Question 2 Answer: Pr[1] = 1/2 Question 3 Answer: Pr[1] = 1/2 Question 4 Answer: Pr[1] = 1/2 Question 5 Answer: Pr[1] = 1/2 Question 1 Answer: Pr[1] = 1/2

exclusiveness:

answers to adjacent questions are not both 1

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

non-signaling theories*

give expectation ≤ 2.5

*also called “general probabilistic theories”

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

axiomatically:

non-signaling theories

give expectation ≤ 2.5

classical theories

give expectation ≤ 2

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

axiomatically:

non-signaling theories

give expectation ≤ 2.5

classical theories

give expectation ≤ 2

quantum theory

give expectation ≤ ?

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

∣ 〉

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

Question 2 Answer: 0/1 Question 3 Answer: 0/1 Question 4 Answer: 0/1 Question 5 Answer: 0/1 Question 1 Answer: 0/1

exclusiveness:

answers to adjacent questions are not both 1

∣ 〉

• 1. Introduction: non-contextuality
• 1. Introduction: non-contextuality

extra axiom: Born rule

∣ 〉

Question 2 Answer: Pr[1] = Question 3 Answer: Pr[1] = Question 4 Answer: Pr[1] = Question 5 Answer: Pr[1] = Question 1 Answer: Pr[1] = ∣〈∣v 1〉∣

2

∣〈∣v 5〉∣

2

∣〈∣v 2〉∣

2

∣〈∣v 3〉∣

2

∣〈∣v 4〉∣

2

expectation:

∑i=1

5 ∣〈∣v i〉∣ 2

Context i

〈v i∣v i1mod5〉=0 Question i + mod(5) Answer: Pr[1] = Question i Answer: Pr[1] = ∣〈∣v i〉∣

2

∣〈∣v i1mod5〉∣

2

compatibility

• 1. Introduction: non-contextuality

• 1. Introduction: non-contextuality

∣v1〉

∣ 〉∈ℝ

3

∣v 2〉 ∣v 3〉 ∣v 4〉 ∣v5〉 ∣〈∣v i〉∣

2= 1

5

• 1. Introduction: non-contextuality

∣v1〉

∣ 〉∈ℝ

3

∣v 2〉 ∣v 3〉 ∣v 4〉 ∣v5〉 ∣〈∣v i〉∣

2= 1

5

∑i=1

5 ∣〈∣v i〉∣ 2=5

• 1. Introduction: non-contextuality

∣v1〉

∣ 〉∈ℝ

3

∣v 2〉 ∣v 3〉 ∣v 4〉 ∣v5〉

quantum theory

gives expectation ≤ 5

∑i=1

5 ∣〈∣v i〉∣ 2=5

∣〈∣v i〉∣

2= 1

5

• 1. Introduction: non-contextuality

non-signaling theories* give expectation ≤ 2.5 classical theories* give expectation ≤ 2

quantum theory**

gives expectation ≤ 5≈2.23

*Wright (1978); **Klyachko et al. (2008)

• 1. Introduction: non-contextuality
• 2. Results:

a general framework to study non-contextuality:

• 1. general compatibility structures
• 2. perfectness
• 3. non-locality

Question 2 Question 3 Question 4 Question 5 Question 1

• 2. Results

every graph/hypergraph is a compatibility structure

Question 2 Question 3 Question 4 Question 5 Question 1

• 2. Results

every graph/hypergraph is a compatibility structure

Question 2 Question 3 Question 4 Question 5 Question 1

every graph/hypergraph is a compatibility structure

• 2. Results

• 2. Results

Let be a compatibility structure, seen as a hypergraph. Let be the graph obtained by connecting contextual

• questions. The maximum expectation values for

exclusive answers are for classical, quantum, and non-signaling theories, respectively; where is the independence number, is the Lovász -function, and is the fractional packing number. Let be a compatibility structure, seen as a hypergraph. Let be the graph obtained by connecting contextual

• questions. The maximum expectation values for

exclusive answers are for classical, quantum, and non-signaling theories, respectively; where is the independence number, is the Lovász -function, and is the fractional packing number. Classification theorem  G

GG

FP

G G 

FP



• 2. Results

Is the maximum number of mutually non-adjacent vertices in a graph G. Is the maximum number of mutually non-adjacent vertices in a graph G. Independence number G

C5=2

NP-complete; hard to approximate

• 2. Results

Is the maximum number of mutually non-adjacent vertices in a graph G. Is the maximum number of mutually non-adjacent vertices in a graph G. Independence number G

C5=2

NP-complete; hard to approximate classical theories give expectation ≤ 2

• 2. Results

An orthogonal representation of G is a set of unit vectors associated to the vertices such that two vectors are orthogonal if the corresponding vertices are adjacent: An orthogonal representation of G is a set of unit vectors associated to the vertices such that two vectors are orthogonal if the corresponding vertices are adjacent: Lovász -function

C5=5  G G:= max

• rth. repr.∑i=1

n ∣〈∣v i〉∣ 2

∣v5〉 ∣v1〉 ∣v 2〉 ∣v 3〉 ∣v 4〉

semidefinite program*

*Lovász (1978)

• 2. Results

An orthogonal representation of G is a set of unit vectors associated to the vertices such that two vectors are orthogonal if the corresponding vertices are adjacent: An orthogonal representation of G is a set of unit vectors associated to the vertices such that two vectors are orthogonal if the corresponding vertices are adjacent: Lovász -function

C5=5  G G:= max

• rth. repr.∑i=1

n ∣〈∣v i〉∣ 2

∣v1〉 ∣v 2〉 ∣v 3〉 ∣v 4〉

semidefinite program* quantum theory gives expectation ≤ 5

*Lovász (1978)

∣v5〉

• 2. Results

Let be a compatibility structure, seen as a hypergraph: Let be a compatibility structure, seen as a hypergraph: Fractional packing number 

FP

 

FPC5=5/2

linear program



FP=max∑ i

w i s.t.∀ i 0w i1 and ∀ context C∈ , ∑

i ∈C

w i1 1/2 1/2 1/2 1/2 1/2

• 2. Results

Let be a compatibility structure, seen as a hypergraph: Let be a compatibility structure, seen as a hypergraph: Fractional packing number



FP=max∑ i

w i s.t.∀ i 0w i1 and ∀ context C∈ , ∑

i ∈C

w i1 

FP

 

FPC5=5/2

linear program non-signaling theories give expectation ≤ 2.5

1/2 1/2 1/2 1/2 1/2

*Klyachko et al. (2008)

is the max. violation of the Klyachko-Can-Biniciouglu-Shumovsky (KCBS) inequality*. The inequality can be used to detect genuine quantum effects and it is the simplest non-contextual inequality violated by a qutrit (because the orthogonal representation has dimension 3). is the max. violation of the Klyachko-Can-Biniciouglu-Shumovsky (KCBS) inequality*. The inequality can be used to detect genuine quantum effects and it is the simplest non-contextual inequality violated by a qutrit (because the orthogonal representation has dimension 3). Remark.

C5

• 2. Results

Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Quantum mechanics as a “sandwich theory”*

• 2. Results

*cf. Knuth (1994)

EC⊂EQ⊂E NS

Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Quantum mechanics as a “sandwich theory”*

EC⊂EQ⊂E NS

• 2. Results

*cf. Knuth (1994)

EC EQ E NS 

FP

G G

Maxima

Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Quantum mechanics as a “sandwich theory”*

EC⊂EQ⊂E NS

• 2. Results

*cf. Knuth (1994)

EC EQ E NS

membership in of a vector can be tested with a semidefinite program!

EQ

*Liang-Spekkens-Wiseman (2010)

Standard result about the Lovász function can be then used to give the max. violation of known inequalities. For example, the max. violation for the n-cycle generalization

• f the KCBS inequality, recently computed in * is

Standard result about the Lovász function can be then used to give the max. violation of known inequalities. For example, the max. violation for the n-cycle generalization

• f the KCBS inequality, recently computed in * is

Remark.

Cn= ncos/n 1cos/n

• 2. Results

A graph G is perfect* if for every induced subgraph H. So, perfect graphs are “the most classical ones”. For a perfect graph Whenever we have a difference between classical theories and quantum mechanics and a “state dependent” proof of the Kochen-Specker theorem**. A graph G is perfect* if for every induced subgraph H. So, perfect graphs are “the most classical ones”. For a perfect graph Whenever we have a difference between classical theories and quantum mechanics and a “state dependent” proof of the Kochen-Specker theorem**. Classicality and perfectness

• 2. Results

*Berge (1961); **where effects sum to unity

H=H=   H EC=EQ=E NS GG

The KCBS inequality is based on which is the smallest non-perfect graph.

C5

• 1. Many intractable problems are tractable for perfect

graphs (i.e., when classical and quantum theories coincide).

• 2. There are graphs s.t. and

(i.e., classical and quantum theories can have arbitrarily large separation)*.

• 1. Many intractable problems are tractable for perfect

graphs (i.e., when classical and quantum theories coincide).

• 2. There are graphs s.t. and

(i.e., classical and quantum theories can have arbitrarily large separation)*. Two remarks

• 2. Results

*Koniagin (1981)

G=2 G=n

1/3

• 2. Results

non-locality non-contextuality

Non-local experiments give compatibility structures: compatible questions are the local measurement. Non-local experiments give compatibility structures: compatible questions are the local measurement. Observation

• 2. Results

Non-local experiments give compatibility structures: compatible questions are the local measurement. Non-local experiments give compatibility structures: compatible questions are the local measurement. Observation

• 2. Results
• 2. Results

Alice Bob

x ∈X

settings

• utcomes a∈A

x ∈X

settings

• utcomes a∈A

settings

• utcomes

y ∈Y

settings

• utcomes b∈B

Compatibility graph for a non-local experiment

• 2. Results

*complete subgraphs

V =A×B×X ×Y G=V ,E  {abxy ,a' b' x ' y }∈E iff x=x '∧a≠a' ∨y=y '∧b≠b' 

 [ is the hypergraph of all cliques* in G]

Compatibility graph for a non-local experiment

• 2. Results

V =A×B×X ×Y {abxy ,a' b' x ' y }∈E iff x=x '∧a≠a' ∨y=y '∧b≠b' 

exclusiveness; compatibility:

the observables of Alice and Bob commute. V =A×B×X ×Y G=V ,E 

• 2. Results

Let be the compatibility hypergraph for a non-local experiment: are the sets of correlations obtainable by local hidden variables, local quantum measurements on a bipartite state, idem but without completeness relation for the measurement, and non-signaling theories, respectively: Let be the compatibility hypergraph for a non-local experiment: are the sets of correlations obtainable by local hidden variables, local quantum measurements on a bipartite state, idem but without completeness relation for the measurement, and non-signaling theories, respectively: A classification theorem for correlations

EC

1 ⊂EQ id⊂E Q 1 ⊂E NS 1 

 E X=C ,Q, NS

1

:=E X∩{ w : ∀ xy ∑

w ab∣xy

w ab∣xy=1} EQ

id:={w ab∣xyabxy : ∀ xy ∑ ab

P ab∣xy=id}

• 2. Results

EC EQ

1

E NS

1

EQ

id

EC

1

• 2. Results

EC EQ

1

E NS

1

EQ

id

EC

1

maximization over and is equivalent to maximization over and

E NS

1

E NS

1

E NS

1

EC

1

E NS EC Fact

• 2. Results

EC EQ

1

EQ

id

maxima via SDP are efficient upper bounds to maximum quantum violations

Fact

• 2. Results

EC EQ

1

EQ

id

there is no efficient algorithm, unless the polynomial hierarchy collapses*

Fact

*Ito-Kobayashi-Matsumoto (2009)

*Ito-Kobayashi-Matsumoto (2009)

• 2. Results

EC EQ

1

EQ

id

maxima via SDP are efficient upper bounds to maximum quantum violations

Fact EQ

id

Problem: how well approximates ? EQ

1

• 2. Results

*Clause-Horne-Shimony-Holt (1969)

Clause-Horne-Shimony-Holt (CHSH) inequality*

• 2. Results

CHSH inequality

settings and outcomes: A=B=X =Y ={0,1}

∑

w ab∣xy : x⋅y=a XOR b

w ab∣xy

constraint:

• 1. Introduction: non-contextuality
• 2. Results

settings and outcomes: A=B=X =Y ={0,1}

∑

w ab∣xy : x⋅y=a XOR b

w ab∣xy

constraint:

CHSH inequality

• 1. Introduction: non-contextuality
• 2. Results

settings and outcomes: A=B=X =Y ={0,1}

∑

w ab∣xy : x⋅y=a XOR b

w ab∣xy

constraint:

CHSH inequality

00∣00 01∣11 11∣10 10∣00 00∣11 01∣00 11∣01 10∣11

• 1. Introduction: non-contextuality
• 2. Results

settings and outcomes: A=B=X =Y ={0,1}

∑

w ab∣xy : x⋅y=a XOR b

w ab∣xy

constraint:

CHSH inequality

00∣00 01∣11 11∣10 10∣00 00∣11 01∣00 11∣01 10∣11 G=3

classical max.

• 1. Introduction: non-contextuality
• 2. Results

settings and outcomes: A=B=X =Y ={0,1}

∑

w ab∣xy : x⋅y=a XOR b

w ab∣xy

constraint:

CHSH inequality

1/2

classical max. non-signaling max.

G=3 1/2 1/2 1/2 1/2 1/2 1/2 1/2 

FP=4

• 1. Introduction: non-contextuality
• 2. Results

settings and outcomes: A=B=X =Y ={0,1}

∑

w ab∣xy : x⋅y=a XOR b

w ab∣xy

constraint:

CHSH inequality

G=3

classical max.



FP=4

non-signaling max. quantum max.

G=22≈3.4

∣〈∣v 1〉∣

2

∣〈∣v 2〉∣

2

∣〈∣v 3〉∣

2

∣〈∣v 4〉∣

2

∣〈∣v 5〉∣

2

∣〈∣v 6〉∣

2

∣〈∣v 7〉∣

2

∣〈∣v 8〉∣

2

• 1. Introduction: non-contextuality
• 2. Results

CHSH inequality

G=3

classical max.



FPG=4

non-signaling max.

G=22≈3.4

quantum max.

it attains the Tsirelson bound

• 1. Introduction: non-contextuality
• 2. Results

*Collins-Gisin (2004); **Navascués-Acín-Pironio (2008)

EC EQ

1

EQ

id

Collins-Gisin inequality (I3322)*

• max. 6.2514
• max. 6.2508**

EQ

id⊂EQ 1

• 3. Open problems:
• 1. theoretical:

relations to Bell inequalities

• 2. applied:

loophole-free experiments

• 3. a complexity perspective:

degree of perfectness

• 3. Open problems

• 3. Open problems

Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality? Is your entropy 5 bits?

“theoretical open problem”

• 3. Open problems

Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality?

Is your entropy 5 bits?

“applied open problem”

So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors?

• 3. Open problems

Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality?

Is your entropy 5 bits?

“complexity open problem”

So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors?

Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum description. Can we define a notion of parametric complexity according to the classical-quantum gap?

• 3. Open problems

Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality?

Is your entropy 5 bits?

• pen problems

So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum description. Can we define a notion of parametric complexity according to the classical-quantum gap?

The Lovász function is fundamental in zero-error classical and quantum information theory*. Can we recast the non-contextuality framework into an information theoretic one?

• 3. Open problems

Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality?

Is your entropy 5 bits?

• pen problems

So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum description. Can we define a notion of parametric complexity according to the classical-quantum gap? The Lovász function is fundamental in zero-error classical and quantum information theory. Can we recast the non-contextuality framework into an information theoretic one?

• 3. Open problems

Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality?

Is your entropy 5 bits?

• pen problems

So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum description. Can we define a notion of parametric complexity according to the classical-quantum gap? The Lovász function is fundamental in zero-error classical and quantum information theory. Can we recast the non-contextuality framework into an information theoretic one?

• 3. Open problems

Is your entropy 5 bits?

• pen problems

The Lovász function is fundamental in zero-error classical and quantum information theory. Can we recast the non-contextuality framework into an information theoretic one?

Yes! In fact, we have gain in quantum coloring if and only if the union of the measurements is always a projective KS set (not defined in this talk). Also, the (one- shot) entanglement-assisted zero-error capacity of the associated channel results to be larger than its classical analogue.

References

Is your entropy 5 bits?

references

• L. Mancinska, G. Scarpa, S. Severini, Generalized

Kochen-Specker sets relate Quantum Coloring to Entanglement-Assisted Channel Capacity, IEEE Trans.

• Inf. Theory, arXiv:1207.1111v1 [quant-ph] (2013)
• R. Duan, S. Severini, A. Winter, Zero-error

communication via quantum channels, noncommutative graphs and a quantum Lovász ϑ- function, IEEE Trans. Inf. Theory., arXiv:1002.2514 (2010).

• A. Cabello, S. Severini, A. Winter, Graph approach

to physical correlations, Phys. Rev. Lett. arXiv:1010.2163v1 (2010).

• 3. Open problems

Is your entropy 5 bits?

• pen problems

The Lovász function is fundamental in zero-error classical and quantum information theory. Can we recast the non-contextuality framework into an information theoretic one?

Yes! In fact, we have gain in quantum coloring if and only if the union of the measurements is always a projective KS set (not defined in this talk). Also, the (one- shot) entanglement-assisted zero-error capacity of the associated channel results to be larger than its classical analogue.