[PPT] - Wiring of No-Signaling Boxes Expands the Hypercontractivity Ribbon PowerPoint Presentation

SLIDE 1

Wiring of No-Signaling Boxes Expands the Hypercontractivity Ribbon

Salman A. Beigi

Institute for Research in Fundamental Sciences (IPM) Tehran, Iran

January 12, 2015 Joint work with Amin Gohari arXiv:1409.3665

SLIDE 2

Closed sets of nonlocal correlations

SLIDE 3

Closed sets of nonlocal correlations

Foundations of Physics, Vol. 24, No. 3, 1994

Q u a n t u m N

n

l

c

a l i t y a s a n A x i

m

S a n d u P

p

e s c u t a n d D a n i e l R

h

r l i c h 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum mechanics, &determinism is an axiom and nonlocality is a theorem. We consider inverting the logical order, mak#1g nonlocality an axiom and indeterminism a theorem. Nonlocal "superquantum" correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

What is the quantum principle? J. Wheeler named it the "Merlin principle" after the legendary magician who, when pursued, could change his form again and again. The more we pursue the quantum principle, the more it changes: from discreteness, to indeterminism, to sums over paths, to many worlds, and so on. By comparison, the relativity principle is easy to grasp. Relativity theory and quantum theory underlie all of physics, but we do not always know how to reconcile them. Here, we take nonlocality as the quantum principle, and we ask what nonlocality and relativistic causality together imply. It is a pleasure to dedicate this paper to Professor Fritz Rohrlich, who has contributed much to the juncture of quantum theory and relativity theory, including its most spectacular success, quantum electrodynamics, and who has written both on quantum paradoxes tll and the logical structure of physical theory, t2~ Bell t31 proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables. Although Bell worked within nonrelativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its back- ward light cone, not by events outside, and can influence events in its

i U n i v e r s i t 6 L i b r e d e B r u x e l l e s , C a m p u s P l a i n e , C . P . 2 2 5 , B

u

l e v a r d d u T r i

m

p h e , B

1

5 B r u x e l l e s , B e l g i u m . 2 S c h

l
f

P h y s i c s a n d A s t r

n
m

y , T e I

A

v i v U n i v e r s i t y , R a m a t

A

v i v , T e l

A

v i v 6 9 9 7 8 I s r a e l . 3 7 9 8 2 5 / 2 4 / 3

4

1 5

9

1 8 / 9 4 / 3

3

7 9 5 7 . / ~

,

1 9 9 4 P l e n u m P u b l i s h i n g C

r

p

r

a t i

n

SLIDE 4

Closed sets of nonlocal correlations

Foundations of Physics, Vol. 24, No. 3, 1994

Q u a n t u m N

n

l

c

a l i t y a s a n A x i

m

S a n d u P

p

e s c u t a n d D a n i e l R

h

r l i c h 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum mechanics, &determinism is an axiom and nonlocality is a theorem. We consider inverting the logical order, mak#1g nonlocality an axiom and indeterminism a theorem. Nonlocal "superquantum" correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

What is the quantum principle? J. Wheeler named it the "Merlin principle" after the legendary magician who, when pursued, could change his form again and again. The more we pursue the quantum principle, the more it changes: from discreteness, to indeterminism, to sums over paths, to many worlds, and so on. By comparison, the relativity principle is easy to grasp. Relativity theory and quantum theory underlie all of physics, but we do not always know how to reconcile them. Here, we take nonlocality as the quantum principle, and we ask what nonlocality and relativistic causality together imply. It is a pleasure to dedicate this paper to Professor Fritz Rohrlich, who has contributed much to the juncture of quantum theory and relativity theory, including its most spectacular success, quantum electrodynamics, and who has written both on quantum paradoxes tll and the logical structure of physical theory, t2~ Bell t31 proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables. Although Bell worked within nonrelativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its back- ward light cone, not by events outside, and can influence events in its

i U n i v e r s i t 6 L i b r e d e B r u x e l l e s , C a m p u s P l a i n e , C . P . 2 2 5 , B

u

l e v a r d d u T r i

m

p h e , B

1

5 B r u x e l l e s , B e l g i u m . 2 S c h

l
f

P h y s i c s a n d A s t r

n
m

y , T e I

A

v i v U n i v e r s i t y , R a m a t

A

v i v , T e l

A

v i v 6 9 9 7 8 I s r a e l . 3 7 9 8 2 5 / 2 4 / 3

4

1 5

9

1 8 / 9 4 / 3

3

7 9 5 7 . / ~

,

1 9 9 4 P l e n u m P u b l i s h i n g C

r

p

r

a t i

n

L i m i t

n

N

n

l

c

a l i t y i n A n y W

r

l d i n W h i c h C

m

m u n i c a t i

n

C

m

p l e x i t y I s N

t

T r i v i a l

Gilles Brassard,1 Harry Buhrman,2,3 Noah Linden,4 Andre ´ Allan Me ´thot,1 Alain Tapp,1 and Falk Unger 3

1

D e ´ p a r t e m e n t I R O , U n i v e r s i t e ´ d e M

n

t r e ´ a l , C . P . 6 1 2 8 , S u c c u r s a l e C e n t r e

V

i l l e , M

n

t r e ´ a l , Q u e ´ b e c H 3 C 3 J 7 , C a n a d a

2

I L L C , U n i v e r s i t e i t v a n A m s t e r d a m , P l a n t a g e M u i d e r g r a c h t 2 4 , 1 1 8 T V A m s t e r d a m , T h e N e t h e r l a n d s

3

C e n t r u m v

r

W i s k u n d e e n I n f

r

m a t i c a ( C W I ) , P

s

t O f fi c e B

x

9 4 7 9 , 1 9 G B A m s t e r d a m , T h e N e t h e r l a n d s

4

D e p a r t m e n t

f

M a t h e m a t i c s , U n i v e r s i t y

f

B r i s t

l

, U n i v e r s i t y W a l k , B r i s t

l

, B S 8 1 T W , U n i t e d K i n g d

m

( R e c e i v e d 2 M a r c h 2 6 ; p u b l i s h e d 2 7 J u n e 2 6 ) B e l l p r

v

e d t h a t q u a n t u m e n t a n g l e m e n t e n a b l e s t w

s

p a c e l i k e s e p a r a t e d p a r t i e s t

e

x h i b i t c l a s s i c a l l y e n t h

u

g h t h e s e c

r

r e l a t i

n

s a r e s t r

n

g e r t h a n a n y t h i n g c l a s s i c a l l y a c h i e v a b l e , l i g h t ) c

m

m u n i c a t i

n

p

s

s i b l e . Y e t , P

p

e s c u i n s t a n t a n e

u

s

PRL 96, 250401 (2006) P H Y S I C A L R E V I E W

SLIDE 5

Closed sets of nonlocal correlations

Foundations of Physics, Vol. 24, No. 3, 1994

Q u a n t u m N

n

l

c

a l i t y a s a n A x i

m

S a n d u P

p

e s c u t a n d D a n i e l R

h

r l i c h 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum mechanics, &determinism is an axiom and nonlocality is a theorem. We consider inverting the logical order, mak#1g nonlocality an axiom and indeterminism a theorem. Nonlocal "superquantum" correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

What is the quantum principle? J. Wheeler named it the "Merlin principle" after the legendary magician who, when pursued, could change his form again and again. The more we pursue the quantum principle, the more it changes: from discreteness, to indeterminism, to sums over paths, to many worlds, and so on. By comparison, the relativity principle is easy to grasp. Relativity theory and quantum theory underlie all of physics, but we do not always know how to reconcile them. Here, we take nonlocality as the quantum principle, and we ask what nonlocality and relativistic causality together imply. It is a pleasure to dedicate this paper to Professor Fritz Rohrlich, who has contributed much to the juncture of quantum theory and relativity theory, including its most spectacular success, quantum electrodynamics, and who has written both on quantum paradoxes tll and the logical structure of physical theory, t2~ Bell t31 proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables. Although Bell worked within nonrelativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its back- ward light cone, not by events outside, and can influence events in its

i U n i v e r s i t 6 L i b r e d e B r u x e l l e s , C a m p u s P l a i n e , C . P . 2 2 5 , B

u

l e v a r d d u T r i

m

p h e , B

1

5 B r u x e l l e s , B e l g i u m . 2 S c h

l
f

P h y s i c s a n d A s t r

n
m

y , T e I

A

v i v U n i v e r s i t y , R a m a t

A

v i v , T e l

A

v i v 6 9 9 7 8 I s r a e l . 3 7 9 8 2 5 / 2 4 / 3

4

1 5

9

1 8 / 9 4 / 3

3

7 9 5 7 . / ~

,

1 9 9 4 P l e n u m P u b l i s h i n g C

r

p

r

a t i

n

L i m i t

n

N

n

l

c

a l i t y i n A n y W

r

l d i n W h i c h C

m

m u n i c a t i

n

C

m

p l e x i t y I s N

t

T r i v i a l

Gilles Brassard,1 Harry Buhrman,2,3 Noah Linden,4 Andre ´ Allan Me ´thot,1 Alain Tapp,1 and Falk Unger 3

1

D e ´ p a r t e m e n t I R O , U n i v e r s i t e ´ d e M

n

t r e ´ a l , C . P . 6 1 2 8 , S u c c u r s a l e C e n t r e

V

i l l e , M

n

t r e ´ a l , Q u e ´ b e c H 3 C 3 J 7 , C a n a d a

2

I L L C , U n i v e r s i t e i t v a n A m s t e r d a m , P l a n t a g e M u i d e r g r a c h t 2 4 , 1 1 8 T V A m s t e r d a m , T h e N e t h e r l a n d s

3

C e n t r u m v

r

W i s k u n d e e n I n f

r

m a t i c a ( C W I ) , P

s

t O f fi c e B

x

9 4 7 9 , 1 9 G B A m s t e r d a m , T h e N e t h e r l a n d s

4

D e p a r t m e n t

f

M a t h e m a t i c s , U n i v e r s i t y

f

B r i s t

l

, U n i v e r s i t y W a l k , B r i s t

l

, B S 8 1 T W , U n i t e d K i n g d

m

( R e c e i v e d 2 M a r c h 2 6 ; p u b l i s h e d 2 7 J u n e 2 6 ) B e l l p r

v

e d t h a t q u a n t u m e n t a n g l e m e n t e n a b l e s t w

s

p a c e l i k e s e p a r a t e d p a r t i e s t

e

x h i b i t c l a s s i c a l l y e n t h

u

g h t h e s e c

r

r e l a t i

n

s a r e s t r

n

g e r t h a n a n y t h i n g c l a s s i c a l l y a c h i e v a b l e , l i g h t ) c

m

m u n i c a t i

n

p

s

s i b l e . Y e t , P

p

e s c u i n s t a n t a n e

u

s

PRL 96, 250401 (2006) P H Y S I C A L R E V I E W

461, 1101-1104 (22 October 2009) | doi:10.1038/nature08400; Received 8 May 2009; Accepted 13 August 2009

I n f

r

m a t i

n

c a u s a l i t y a s a p h y s i c a l p r i n c i p l e

M a r c i n P a w

w

s k i

1

, T

m

a s z P a t e r e k

2

, D a g

m

i r K a s z l i k

w

s k i

2

, V a l e r i

S

c a r a n i

2

, A n d r e a s W i n t e r

2 , 3

& M a r e k u k

w

s k i

1 Institute of Theoretical Physics and Astrophysics, University of Gda sk, 80-952 Gda sk, Poland 1. Centre for Quantum Technologies and Department of Physics, National University of 117543 Singapore, Singapore 2. Department of Mathematics, University of B 3. Correspond

SLIDE 6

Closed sets of nonlocal correlations

Foundations of Physics, Vol. 24, No. 3, 1994

Q u a n t u m N

n

l

c

a l i t y a s a n A x i

m

S a n d u P

p

e s c u t a n d D a n i e l R

h

r l i c h 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum mechanics, &determinism is an axiom and nonlocality is a theorem. We consider inverting the logical order, mak#1g nonlocality an axiom and indeterminism a theorem. Nonlocal "superquantum" correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

What is the quantum principle? J. Wheeler named it the "Merlin principle" after the legendary magician who, when pursued, could change his form again and again. The more we pursue the quantum principle, the more it changes: from discreteness, to indeterminism, to sums over paths, to many worlds, and so on. By comparison, the relativity principle is easy to grasp. Relativity theory and quantum theory underlie all of physics, but we do not always know how to reconcile them. Here, we take nonlocality as the quantum principle, and we ask what nonlocality and relativistic causality together imply. It is a pleasure to dedicate this paper to Professor Fritz Rohrlich, who has contributed much to the juncture of quantum theory and relativity theory, including its most spectacular success, quantum electrodynamics, and who has written both on quantum paradoxes tll and the logical structure of physical theory, t2~ Bell t31 proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables. Although Bell worked within nonrelativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its back- ward light cone, not by events outside, and can influence events in its

i U n i v e r s i t 6 L i b r e d e B r u x e l l e s , C a m p u s P l a i n e , C . P . 2 2 5 , B

u

l e v a r d d u T r i

m

p h e , B

1

5 B r u x e l l e s , B e l g i u m . 2 S c h

l
f

P h y s i c s a n d A s t r

n
m

y , T e I

A

v i v U n i v e r s i t y , R a m a t

A

v i v , T e l

A

v i v 6 9 9 7 8 I s r a e l . 3 7 9 8 2 5 / 2 4 / 3

4

1 5

9

1 8 / 9 4 / 3

3

7 9 5 7 . / ~

,

1 9 9 4 P l e n u m P u b l i s h i n g C

r

p

r

a t i

n

L i m i t

n

N

n

l

c

a l i t y i n A n y W

r

l d i n W h i c h C

m

m u n i c a t i

n

C

m

p l e x i t y I s N

t

T r i v i a l

Gilles Brassard,1 Harry Buhrman,2,3 Noah Linden,4 Andre ´ Allan Me ´thot,1 Alain Tapp,1 and Falk Unger 3

1

D e ´ p a r t e m e n t I R O , U n i v e r s i t e ´ d e M

n

t r e ´ a l , C . P . 6 1 2 8 , S u c c u r s a l e C e n t r e

V

i l l e , M

n

t r e ´ a l , Q u e ´ b e c H 3 C 3 J 7 , C a n a d a

2

I L L C , U n i v e r s i t e i t v a n A m s t e r d a m , P l a n t a g e M u i d e r g r a c h t 2 4 , 1 1 8 T V A m s t e r d a m , T h e N e t h e r l a n d s

3

C e n t r u m v

r

W i s k u n d e e n I n f

r

m a t i c a ( C W I ) , P

s

t O f fi c e B

x

9 4 7 9 , 1 9 G B A m s t e r d a m , T h e N e t h e r l a n d s

4

D e p a r t m e n t

f

M a t h e m a t i c s , U n i v e r s i t y

f

B r i s t

l

, U n i v e r s i t y W a l k , B r i s t

l

, B S 8 1 T W , U n i t e d K i n g d

m

( R e c e i v e d 2 M a r c h 2 6 ; p u b l i s h e d 2 7 J u n e 2 6 ) B e l l p r

v

e d t h a t q u a n t u m e n t a n g l e m e n t e n a b l e s t w

s

p a c e l i k e s e p a r a t e d p a r t i e s t

e

x h i b i t c l a s s i c a l l y e n t h

u

g h t h e s e c

r

r e l a t i

n

s a r e s t r

n

g e r t h a n a n y t h i n g c l a s s i c a l l y a c h i e v a b l e , l i g h t ) c

m

m u n i c a t i

n

p

s

s i b l e . Y e t , P

p

e s c u i n s t a n t a n e

u

s

PRL 96, 250401 (2006) P H Y S I C A L R E V I E W

461, 1101-1104 (22 October 2009) | doi:10.1038/nature08400; Received 8 May 2009; Accepted 13 August 2009

I n f

r

m a t i

n

c a u s a l i t y a s a p h y s i c a l p r i n c i p l e

M a r c i n P a w

w

s k i

1

, T

m

a s z P a t e r e k

2

, D a g

m

i r K a s z l i k

w

s k i

2

, V a l e r i

S

c a r a n i

2

, A n d r e a s W i n t e r

2 , 3

& M a r e k u k

w

s k i

1 Institute of Theoretical Physics and Astrophysics, University of Gda sk, 80-952 Gda sk, Poland 1. Centre for Quantum Technologies and Department of Physics, National University of 117543 Singapore, Singapore 2. Department of Mathematics, University of B 3. Correspond

A glance beyond the quantum model

BY MIGUEL NAVASCUÉS

1 , 2 ,

* AND HARALD WUNDERLICH

1 , 2 , 3 1Institute for Mathematical Sciences, Imperial College London,

S W 7 2 P G , U K

SLIDE 7

Closed sets of nonlocal correlations

Foundations of Physics, Vol. 24, No. 3, 1994

Q u a n t u m N

n

l

c

a l i t y a s a n A x i

m

S a n d u P

p

e s c u t a n d D a n i e l R

h

r l i c h 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum mechanics, &determinism is an axiom and nonlocality is a theorem. We consider inverting the logical order, mak#1g nonlocality an axiom and indeterminism a theorem. Nonlocal "superquantum" correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

What is the quantum principle? J. Wheeler named it the "Merlin principle" after the legendary magician who, when pursued, could change his form again and again. The more we pursue the quantum principle, the more it changes: from discreteness, to indeterminism, to sums over paths, to many worlds, and so on. By comparison, the relativity principle is easy to grasp. Relativity theory and quantum theory underlie all of physics, but we do not always know how to reconcile them. Here, we take nonlocality as the quantum principle, and we ask what nonlocality and relativistic causality together imply. It is a pleasure to dedicate this paper to Professor Fritz Rohrlich, who has contributed much to the juncture of quantum theory and relativity theory, including its most spectacular success, quantum electrodynamics, and who has written both on quantum paradoxes tll and the logical structure of physical theory, t2~ Bell t31 proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables. Although Bell worked within nonrelativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its back- ward light cone, not by events outside, and can influence events in its

i U n i v e r s i t 6 L i b r e d e B r u x e l l e s , C a m p u s P l a i n e , C . P . 2 2 5 , B

u

l e v a r d d u T r i

m

p h e , B

1

5 B r u x e l l e s , B e l g i u m . 2 S c h

l
f

P h y s i c s a n d A s t r

n
m

y , T e I

A

v i v U n i v e r s i t y , R a m a t

A

v i v , T e l

A

v i v 6 9 9 7 8 I s r a e l . 3 7 9 8 2 5 / 2 4 / 3

4

1 5

9

1 8 / 9 4 / 3

3

7 9 5 7 . / ~

,

1 9 9 4 P l e n u m P u b l i s h i n g C

r

p

r

a t i

n

L i m i t

n

N

n

l

c

a l i t y i n A n y W

r

l d i n W h i c h C

m

m u n i c a t i

n

C

m

p l e x i t y I s N

t

T r i v i a l

Gilles Brassard,1 Harry Buhrman,2,3 Noah Linden,4 Andre ´ Allan Me ´thot,1 Alain Tapp,1 and Falk Unger 3

1

D e ´ p a r t e m e n t I R O , U n i v e r s i t e ´ d e M

n

t r e ´ a l , C . P . 6 1 2 8 , S u c c u r s a l e C e n t r e

V

i l l e , M

n

t r e ´ a l , Q u e ´ b e c H 3 C 3 J 7 , C a n a d a

2

I L L C , U n i v e r s i t e i t v a n A m s t e r d a m , P l a n t a g e M u i d e r g r a c h t 2 4 , 1 1 8 T V A m s t e r d a m , T h e N e t h e r l a n d s

3

C e n t r u m v

r

W i s k u n d e e n I n f

r

m a t i c a ( C W I ) , P

s

t O f fi c e B

x

9 4 7 9 , 1 9 G B A m s t e r d a m , T h e N e t h e r l a n d s

4

D e p a r t m e n t

f

M a t h e m a t i c s , U n i v e r s i t y

f

B r i s t

l

, U n i v e r s i t y W a l k , B r i s t

l

, B S 8 1 T W , U n i t e d K i n g d

m

( R e c e i v e d 2 M a r c h 2 6 ; p u b l i s h e d 2 7 J u n e 2 6 ) B e l l p r

v

e d t h a t q u a n t u m e n t a n g l e m e n t e n a b l e s t w

s

p a c e l i k e s e p a r a t e d p a r t i e s t

e

x h i b i t c l a s s i c a l l y e n t h

u

g h t h e s e c

r

r e l a t i

n

s a r e s t r

n

g e r t h a n a n y t h i n g c l a s s i c a l l y a c h i e v a b l e , l i g h t ) c

m

m u n i c a t i

n

p

s

s i b l e . Y e t , P

p

e s c u i n s t a n t a n e

u

s

PRL 96, 250401 (2006) P H Y S I C A L R E V I E W

461, 1101-1104 (22 October 2009) | doi:10.1038/nature08400; Received 8 May 2009; Accepted 13 August 2009

I n f

r

m a t i

n

c a u s a l i t y a s a p h y s i c a l p r i n c i p l e

M a r c i n P a w

w

s k i

1

, T

m

a s z P a t e r e k

2

, D a g

m

i r K a s z l i k

w

s k i

2

, V a l e r i

S

c a r a n i

2

, A n d r e a s W i n t e r

2 , 3

& M a r e k u k

w

s k i

1 Institute of Theoretical Physics and Astrophysics, University of Gda sk, 80-952 Gda sk, Poland 1. Centre for Quantum Technologies and Department of Physics, National University of 117543 Singapore, Singapore 2. Department of Mathematics, University of B 3. Correspond

A glance beyond the quantum model

BY MIGUEL NAVASCUÉS

1 , 2 ,

* AND HARALD WUNDERLICH

1 , 2 , 3 1Institute for Mathematical Sciences, Imperial College London,

S W 7 2 P G , U K

2012 | Accepted 8 Jul 2013 | Published 16 Aug 2013

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T. Fritz1,2, A.B. Sainz1, R. Augusiak1, J. Bohr Brask1, R. Chaves1,3, A. Leverrier1,4,5 & A. Acı

´n1,6

I n r e c e n t y e a r s , t h e

DOI: 10.1038/ncomms3263

SLIDE 8

Closed sets of nonlocal correlations

Foundations of Physics, Vol. 24, No. 3, 1994

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r l i c h 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum mechanics, &determinism is an axiom and nonlocality is a theorem. We consider inverting the logical order, mak#1g nonlocality an axiom and indeterminism a theorem. Nonlocal "superquantum" correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

What is the quantum principle? J. Wheeler named it the "Merlin principle" after the legendary magician who, when pursued, could change his form again and again. The more we pursue the quantum principle, the more it changes: from discreteness, to indeterminism, to sums over paths, to many worlds, and so on. By comparison, the relativity principle is easy to grasp. Relativity theory and quantum theory underlie all of physics, but we do not always know how to reconcile them. Here, we take nonlocality as the quantum principle, and we ask what nonlocality and relativistic causality together imply. It is a pleasure to dedicate this paper to Professor Fritz Rohrlich, who has contributed much to the juncture of quantum theory and relativity theory, including its most spectacular success, quantum electrodynamics, and who has written both on quantum paradoxes tll and the logical structure of physical theory, t2~ Bell t31 proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables. Although Bell worked within nonrelativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its back- ward light cone, not by events outside, and can influence events in its

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PRL 96, 250401 (2006) P H Y S I C A L R E V I E W

461, 1101-1104 (22 October 2009) | doi:10.1038/nature08400; Received 8 May 2009; Accepted 13 August 2009

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1 Institute of Theoretical Physics and Astrophysics, University of Gda sk, 80-952 Gda sk, Poland 1. Centre for Quantum Technologies and Department of Physics, National University of 117543 Singapore, Singapore 2. Department of Mathematics, University of B 3. Correspond

A glance beyond the quantum model

BY MIGUEL NAVASCUÉS

1 , 2 ,

* AND HARALD WUNDERLICH

1 , 2 , 3 1Institute for Mathematical Sciences, Imperial College London,

S W 7 2 P G , U K

2012 | Accepted 8 Jul 2013 | Published 16 Aug 2013

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T. Fritz1,2, A.B. Sainz1, R. Augusiak1, J. Bohr Brask1, R. Chaves1,3, A. Leverrier1,4,5 & A. Acı

´n1,6

I n r e c e n t y e a r s , t h e

DOI: 10.1038/ncomms3263

Closed sets of nonlocal correlations

Jonathan Allcock,1 Nicolas Brunner,2 Noah Linden,1 Sandu Popescu,2 Paul Skrzypczyk,2 and Tamás Vértesi3

1Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom 2H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom 3Institute of Nuclear Research, Hungarian Academy of Sciences, P.O. Box 51, H-4001 Debrecen, Hungary

Received 11 August 2009; revised manuscript received 27 August 2009; published 11 December 2009 PHYSICAL REVIEW A 80, 062107 2009

SLIDE 9

Outline

Introduction to non-local boxes and wirings Two measures of correlation with the tensorization property

Maximal correlation Hypercontractivity ribbon

Main result: maximal correlation and hypercontractivity ribbon are monotone under wirings Example: simulation of isotropic boxes with each other

Resolves a conjecture of Lang, V´ ertesi, Navascu´ es ’14

Computability of the above invariants

SLIDE 10

Local measurements on bipartite physical systems

SLIDE 11

Local measurements on bipartite physical systems

SLIDE 12

Local measurements on bipartite physical systems

x y

SLIDE 13

Local measurements on bipartite physical systems

x y a b

SLIDE 14

Local measurements on bipartite physical systems

x y a b x a y b

SLIDE 15

Local measurements on bipartite physical systems

x y a b x a y b p(a, b|x, y) = the probability of outcomes a, b under measurement settings x, y

SLIDE 16

Local measurements on bipartite physical systems

x y a b x a y b p(a, b|x, y) = the probability of outcomes a, b under measurement settings x, y No-signaling: instantaneous signaling is impossible

p(a|xy) is independent of y p(b|xy) is independent of x

SLIDE 17

Isotropic boxes

x a y b

Example: x, y, a, b ∈ {0, 1}, and 0 ≤ η ≤ 1 PRη(a, b|x, y) :=

1+η

4

if a ⊕ b = xy,

1−η 4

therwise.

SLIDE 18

Wirings

1 1 2 2

SLIDE 19

Wirings

1 1 2 2

SLIDE 20

Wirings

1 1 2 2

x0 y0 b0 a0

SLIDE 21

Wirings

1 1 2 2

x0 y0 b0 a0

Wirings are the local operations in the box world [Allcock et al. ’09] The set of physical non-local boxes is closed under wirings

SLIDE 22

Wirings

1 1 2 2

x0 y0 b0 a0

Wirings are the local operations in the box world [Allcock et al. ’09] The set of physical non-local boxes is closed under wirings Problem: 1/2 ≤ η′ < η ≤ 1. Can we generate PRη from some copies of PRη′ under wirings?

No if there are two [Short ’09] or at most nine [Forster ’11] copies of PRη′ available

SLIDE 23

Tensorization of measures of correlation

A1 B1 An Bn

A0

B0

Problem: Given some samples of pAB can we generate one sample from qA′B′ under local operations?

SLIDE 24

Tensorization of measures of correlation

A1 B1 An Bn

A0

B0

Problem: Given some samples of pAB can we generate one sample from qA′B′ under local operations? Measures of correlation are monotone under local operations I(A, B)p < I(A′, B′)q ⇒ No

SLIDE 25

Tensorization of measures of correlation

A1 B1 An Bn

A0

B0

Problem: Given some samples of pAB can we generate one sample from qA′B′ under local operations? Measures of correlation are monotone under local operations I(A, B)p < I(A′, B′)q ⇒ No NOT quite right! I(An, Bn)pn = nI(A, B)p.

SLIDE 26

Tensorization of measures of correlation

A1 B1 An Bn

A0

B0

Problem: Given some samples of pAB can we generate one sample from qA′B′ under local operations? Measures of correlation are monotone under local operations I(A, B)p < I(A′, B′)q ⇒ No NOT quite right! I(An, Bn)pn = nI(A, B)p. [Tensorization]: Is there a measure of correlation ρ such that ρ(An, Bn)pn = ρ(A, B)p?

SLIDE 27

Tensorization of measures of correlation

A1 B1 An Bn

A0

B0

Problem: Given some samples of pAB can we generate one sample from qA′B′ under local operations? Measures of correlation are monotone under local operations I(A, B)p < I(A′, B′)q ⇒ No NOT quite right! I(An, Bn)pn = nI(A, B)p. [Tensorization]: Is there a measure of correlation ρ such that ρ(An, Bn)pn = ρ(A, B)p?

Maximal correlation Hypercontractivity ribbon

SLIDE 28

Maximal correlation

Bipartite distribution pAB ρ(A, B) := max Cov(f, g)

Var[fA]Var[gB]

fA : A → R, gB : B → R

SLIDE 29

Maximal correlation

Bipartite distribution pAB ρ(A, B) := max Cov(f, g)

Var[fA]Var[gB]

fA : A → R, gB : B → R 0 ≤ ρ(A, B) ≤ 1, ρ(A, B) = 0 iff pAB = pA · pB

SLIDE 30

Maximal correlation

Bipartite distribution pAB ρ(A, B) := max Cov(f, g)

Var[fA]Var[gB]

fA : A → R, gB : B → R 0 ≤ ρ(A, B) ≤ 1, ρ(A, B) = 0 iff pAB = pA · pB [Tensorization]: ρ(An, Bn) = ρ(A, B) [Data processing]: ρ(·, ·) is monotone under local operations

SLIDE 31

Maximal correlation

Bipartite distribution pAB ρ(A, B) := max Cov(f, g)

Var[fA]Var[gB]

fA : A → R, gB : B → R 0 ≤ ρ(A, B) ≤ 1, ρ(A, B) = 0 iff pAB = pA · pB [Tensorization]: ρ(An, Bn) = ρ(A, B) [Data processing]: ρ(·, ·) is monotone under local operations Maximal correlation for non-local boxes: ρ(A, B|X, Y) := max

x,y ρ(A, B|X = x, Y = y)

SLIDE 32

Maximal correlation under wirings

x a y b

Lemma: For any no-signaling box p(ab|xy) we have ρ(A, B) ≤ max{ρ(A, B|X, Y), ρ(X, Y)}.

SLIDE 33

Maximal correlation under wirings

x a y b

Lemma: For any no-signaling box p(ab|xy) we have ρ(A, B) ≤ max{ρ(A, B|X, Y), ρ(X, Y)}. Proof:

E[fg] = EXYEAB|XY[fg] ≤EXY

EA|XY[f] · EB|XY[g] + ρ
VarA|XY[f] · VarB|XY[g]
= EXY
EA|X[f] · EB|Y[g]
+ ρEXY
VarA|X[f] · VarB|Y[g]
≤EXEA|X[f] · EYEB|Y[g] + ρ
VarXEA|X[f] · VarYEB|Y[g] + ρEXY
VarA|X[f] · VarB|Y[g]
≤EXEA|X[f] · EYEB|Y[g] + ρ
VarXEA|X[f] · VarYEB|Y[g] + ρ
EXVarA|X[f] · EYVarB|Y[g]

≤EAX[f] · EBY[g] + ρ

VarXEA|X[f] + EXVarA|X[f]

VarYEB|Y[g] + EYVarB|Y[g]

=EAX[f] · EBY[g] + ρ
VarAX[f]VarBY[g].

SLIDE 34

Maximal correlation under wirings

1 1 2 2

x0 y0 b0 a0

c

d Theorem Maximal correlation of no-signaling boxes does not increase under wirings.

SLIDE 35

Maximal correlation under wirings

1 1 2 2

x0 y0 b0 a0

Theorem Maximal correlation of no-signaling boxes does not increase under wirings. The proof doesn’t work for these types of wirings! We need new tools.

SLIDE 36

Hypercontractivity ribbon

[Ahlswede, G´ acs ’76] (λ1, λ2) ∈ R(A, B) iff E[fAgB] ≤ fA 1

λ1 gB 1 λ2 ,

∀fA, gB Schatten norm: fA 1

λ1 = E

|fA|1/λ1λ1

SLIDE 37

Hypercontractivity ribbon

[Ahlswede, G´ acs ’76] (λ1, λ2) ∈ R(A, B) iff E[fAgB] ≤ fA 1

λ1 gB 1 λ2 ,

∀fA, gB Schatten norm: fA 1

λ1 = E

|fA|1/λ1λ1

[Nair ’14] (λ1, λ2) ∈ R(A, B) iff: I(U; AB) ≥ λ1I(U; A) + λ2I(U; B), ∀pU|AB

SLIDE 38

Hypercontractivity ribbon

(1, 1)

[Ahlswede, G´ acs ’76] (λ1, λ2) ∈ R(A, B) iff E[fAgB] ≤ fA 1

λ1 gB 1 λ2 ,

∀fA, gB Schatten norm: fA 1

λ1 = E

|fA|1/λ1λ1

[Nair ’14] (λ1, λ2) ∈ R(A, B) iff: I(U; AB) ≥ λ1I(U; A) + λ2I(U; B), ∀pU|AB R(A, B) = [0, 1]2 iff A, B are independent

SLIDE 39

Hypercontractivity ribbon

(1, 1)

[Ahlswede, G´ acs ’76] (λ1, λ2) ∈ R(A, B) iff E[fAgB] ≤ fA 1

λ1 gB 1 λ2 ,

∀fA, gB Schatten norm: fA 1

λ1 = E

|fA|1/λ1λ1

[Nair ’14] (λ1, λ2) ∈ R(A, B) iff: I(U; AB) ≥ λ1I(U; A) + λ2I(U; B), ∀pU|AB R(A, B) = [0, 1]2 iff A, B are independent [Tensorization]: R(An, Bn) = R(A, B) [Data processing]: R(·, ·) expands under local operations

SLIDE 40

Hypercontractivity ribbon under wirings

Hypercontractivity ribbon for non-local boxes: R(A, B|X, Y) :=

x,y

R(A, B|X = x, Y = y).

SLIDE 41

Hypercontractivity ribbon under wirings

Hypercontractivity ribbon for non-local boxes: R(A, B|X, Y) :=

x,y

R(A, B|X = x, Y = y). Theorem Suppose that a no-signaling box p(a′b′|x′y′) can be generated from some copies of a box p(ab|xy) under wirings. Then R(A, B|X, Y) ⊆ R(A′, B′|X′, Y′).

SLIDE 42

Hypercontractivity ribbon under wirings

Hypercontractivity ribbon for non-local boxes: R(A, B|X, Y) :=

x,y

R(A, B|X = x, Y = y). Theorem Suppose that a no-signaling box p(a′b′|x′y′) can be generated from some copies of a box p(ab|xy) under wirings. Then R(A, B|X, Y) ⊆ R(A′, B′|X′, Y′). Proof: Chain rule!

SLIDE 43

Example: Isotropic boxes

PRη(a, b|x, y) :=

1+η

4

if a ⊕ b = xy,

1−η 4

therwise.

ρ(PRη) = η

SLIDE 44

Example: Isotropic boxes

PRη(a, b|x, y) :=

1+η

4

if a ⊕ b = xy,

1−η 4

therwise.

ρ(PRη) = η Corollary For 0 ≤ η′ < η ≤ 1, using an arbitrary number of copies of PRη′, a single copy of PRη cannot be generated under wirings.

SLIDE 45

Example: Isotropic boxes

PRη(a, b|x, y) :=

1+η

4

if a ⊕ b = xy,

1−η 4

therwise.

ρ(PRη) = η Corollary For 0 ≤ η′ < η ≤ 1, using an arbitrary number of copies of PRη′, a single copy of PRη cannot be generated under wirings. For 1/ √ 2 ≤ η′ < η ≤ 1, using an arbitrary number of copies of PRη′, a single copy of PRη cannot be generated under wirings with shared randomness.

SLIDE 46

Ribbon in terms of a lower convex envelope

Computation of maximal correlation is easy. How about computation of the ribbon?

SLIDE 47

Ribbon in terms of a lower convex envelope

Computation of maximal correlation is easy. How about computation of the ribbon? Define Υ(·) on the probability simplex by qAB → Υ(qAB) = λ1H(qA) + λ2H(qB) − H(qAB)

SLIDE 48

Ribbon in terms of a lower convex envelope

Computation of maximal correlation is easy. How about computation of the ribbon? Define Υ(·) on the probability simplex by qAB → Υ(qAB) = λ1H(qA) + λ2H(qB) − H(qAB) Let Υ be the lower convex envelope of Υ

distributions

Υ

e Υ

SLIDE 49

Ribbon in terms of a lower convex envelope

Computation of maximal correlation is easy. How about computation of the ribbon? Define Υ(·) on the probability simplex by qAB → Υ(qAB) = λ1H(qA) + λ2H(qB) − H(qAB) Let Υ be the lower convex envelope of Υ

distributions

Υ

e Υ

Lemma For every distribution pAB, we have (λ1, λ2) ∈ R(A, B) if and only if Υ(pAB) = Υ(pAB).

SLIDE 50

Maximal correlation ribbon

distributions

Υ

e Υ

Definition: (λ1, λ2) ∈ S(A, B) if Var[f] ≥ λ1VarAEB|A[f] + λ2VarBEA|B[f], ∀fAB

SLIDE 51

Maximal correlation ribbon

distributions

Υ

e Υ

Definition: (λ1, λ2) ∈ S(A, B) if Var[f] ≥ λ1VarAEB|A[f] + λ2VarBEA|B[f], ∀fAB R(A, B) ⊆ S(A, B)

SLIDE 52

Maximal correlation ribbon

distributions

Υ

e Υ

Definition: (λ1, λ2) ∈ S(A, B) if Var[f] ≥ λ1VarAEB|A[f] + λ2VarBEA|B[f], ∀fAB R(A, B) ⊆ S(A, B) [Tensorization]: S(An, Bn) = S(A, B) [Data processing]: S(·, ·) expands under local operations

SLIDE 53

Maximal correlation ribbon for non-local boxes

Maximal correlation ribbon for non-local boxes: S(A, B|X, Y) :=

x,y

S(A, B|X = x, Y = y).

SLIDE 54

Maximal correlation ribbon for non-local boxes

Maximal correlation ribbon for non-local boxes: S(A, B|X, Y) :=

x,y

S(A, B|X = x, Y = y). Theorem Suppose that a no-signaling box p(a′b′|x′y′) can be generated from some copies of box p(ab|xy) under wirings. Then S(A, B|X, Y) ⊆ S(A′, B′|X′, Y′).

SLIDE 55

Maximal correlation ribbon for non-local boxes

Maximal correlation ribbon for non-local boxes: S(A, B|X, Y) :=

x,y

S(A, B|X = x, Y = y). Theorem Suppose that a no-signaling box p(a′b′|x′y′) can be generated from some copies of box p(ab|xy) under wirings. Then S(A, B|X, Y) ⊆ S(A′, B′|X′, Y′). Theorem ρ2(A, B) = inf

1−λ1

λ2

(λ1, λ2) ∈ S(A, B)

SLIDE 56

Maximal correlation ribbon for non-local boxes

Maximal correlation ribbon for non-local boxes: S(A, B|X, Y) :=

x,y

S(A, B|X = x, Y = y). Theorem Suppose that a no-signaling box p(a′b′|x′y′) can be generated from some copies of box p(ab|xy) under wirings. Then S(A, B|X, Y) ⊆ S(A′, B′|X′, Y′). Theorem ρ2(A, B) = inf

1−λ1

λ2

(λ1, λ2) ∈ S(A, B)
Maximal correlation is monotone under wirings

SLIDE 57

Summary

Introduced hypercontractivity ribbon for non-local boxes and Showed that it expands under wirings Defined Maximal correlation ribbon Showed that maximal correlation ribbon expands under wirings Characterized maximal correlation in terms of maximal correlation ribbon Maximal correlation is monotone under wirings There is a continuum of closed sets of boxes