Why is quantum nonlocality limited Perspectives from nonlocality swapping

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Why is quantum non-locality limited?
Perspectives from non-locality swapping

• Nicolas Brunner
• Paul Skrzypczyk Sandu Popescu
• University of Bristol

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Entanglement versus non-locality
No-signalling
NON-LOCALITY
???
ENTANGLEMENT (QM)
???
Post-quantum theories
Why is quantum non-locality limited ?
3
Outline

• Post-quantum theories
• Non-locality swapping
• Overview

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Non-locality / CHSH game
Correlation (same)
Anti-correlation (opposite)
Score 1 if correct -1 if wrong
Classical systems
Quantum Mechanics
Non-signalling
lt
Best score 2
Best score 2v2
Best score 4
???
Bell inequality
Tsirelsons bound
PR box
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Popescu-Rohrlich (PR) box, NL box

• Non-signalling
• Maximally Non-local CHSH 4

Why does the PR-box not exist in Nature ?
Popescu Rohrlich, Found. Phys. 1994
Barrett et al. PRA 2005
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Black-box
Joint probability distribution
Here
Linear Constraints
Polytope of non-signaling correlations

• Positivity
• No-signalling

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Geometry
PR
PL
PL
Non-local
No-Signaling polytope

• 8-dimensional
• 24 vertices
• a. 16 deterministic local boxes
• b. 8 PR non-local boxes

Local
PR2
Barrett et al. PRA 2005
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Geometry
Q boundary (Tsirelson,Landau, Masanes,
Navascues, Pironio, Acin, Doherty, Liang,Toner,
Wehner)
PR
PR
PL
PL
Non-local
PL
PL
Local
PR2
Id
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Separation Communication Complexity
PR
Communication Complexity becomes trivial in
specific Post-quantum theories
???
PL
PL

• PR (van Dam, 2005)

• Noisy PR boxes
• (Brassard et al., 2006)

• Correlated NL boxes
• (Brunner, Skrzypczyk, 2009)

Id
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Dynamics?
QM allows for rich dynamics. What about
Post-quantum models ??
Entanglement swapping
Does a similar protocol exist for PR boxes ? Can
joint measurements be implemented ?
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Non-locality swapping
New device in the theory COUPLER (analogue of
Q joint meas.)

• Genuine boxes
• Joint meas.
  Rich dynamics
• Most general No-Signaling
• No Joint Meas.
  Poor dynamics

(here)
(Short, Barrett, Popescu, Gisin (2006)
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Emergence of quantum boundary
Is NL swapping still possible with noisy PR boxes?
Swapping is possible iff the box is
post-quantum !
Skrzypczyk, Brunner, Popescu, PRL 102, 110402
(2009)
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Similarities with QM

• No-cloning, no-broadcasting (Barnum et al.
  2007)
• Monogamy of correlations (Masanes et al. 2006)
• Secure Key distribution (Barrett et al. 2005,
  Acin et al. 2006)
• Information-Disturbance trade-off (Scarani et
  al. 2006)
• Dynamics (Non-locality Swapping) (Skrzypczyk
  et al. 2009)

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Separation from QM

• Communication power
• Trivial Communication Complexity
• (vanDam, Brassard et al. 2006, Brunner
  Skrzypcyzk 2009)
• Advantage for non-local computation
  (Linden et al. 2007)
• Violation of Information Causality
  (Pawlowski et al. 2009)
• Non-locality Swapping (Skrzypczyk et al. 2009)
• Macroscopic Locality (Navascues Wunderlich
  2009)

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Thank you!
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Non-locality swapping
PR
PR
A theory limited only by non-signalling has poor
dynamics
Short Popescu Gisin (2006) Barrett (2006)
Trade-off between states and dynamics
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Closure of non-signaling theories
Consistent theory
Set of correlations closed under wirings
Theory
Set of correlations
Ex. Local, NS, QM
Finding other closed sets is not an easy task!

• Origin of Q boundary
• Relation between information theoretic tasks
• Theories beyond QM

Allcock, NB, Linden, Popescu, Skrzypczyk, Vertesi
arxiv 2009
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Genuine boxes
Intuition Some boxes are more fundamental than
others
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Wirings
Bi
Bf
Bi
By wiring several boxes together one obtains a
new box
Bi Bi ? Bf
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Moving around in the polytope
PR
Distillation of Isotropic boxes
??
??

• No 2-copy distillation
• (Short, PRL 2009)

Pc
Pc

• Very limited for Q boxes
• (Djukaric, Wolf 2006)

What about other boxes?
Correlated NL boxes
Can be distilled up to CHSH3 (Forster et al. PRL
2009)
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Non-locality distillation
Correlated NL boxes
Linearity
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Non-locality distillation PR PR
PR
PR
Box 1 PR
Box 2 PR
Overall
PR PR
PR
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Non-locality distillation PR PR
PR PR
PR
PR P1
PR
PR
P1 PR
where
P1 P1
P1
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Non-locality distillation
All correlated NL boxes are distilled to the PR
box
Brunner Skrzypczyk, PRL 2009
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Discrete map
PR
Pc
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Choice of genuine boxes
1. All PR boxes genuine
2. Single genuine PR box

• Poor dynamics
• No joint measurement

• Rich dynamics
• Joint measurements

Short et al. Barrett (2006)
Here
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Thank you!
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Emergence of quantum boundary
Is NL swapping still possible with noisy PR boxes?
Noisy PR boxes
When can the noisy box used for swapping?
When
Swapping is possible iff the box is
post-quantum !
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Emergence of quantum boundary
2D slice of the polytope
When can noisy boxes be swapped?
iff the boxes are post-quantum
Boundary of swapping Tsirelson-Landau-Masanes
criteria for quantumness of correlations

Skrzypczyk, Brunner, Popescu, PRL 102, 110402
(2009)
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Consistency
1. Bob cannot signal by applying the coupler
PR
PR
PR
2. Coupler can be applied directly to any
allowed box
Single genuine PR box works !
All 8 PR boxes genuine
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Connection to Bell inequalities
Bell Inequality
Coupler
Coupler CHSH
Coupler reveals a joint property of the box
Skrzypczyk, Brunner, arxiv08120758
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Supp. 1 Bells inequality


a1,-1
b1,-1
-
-
Bells inequality (CHSH)
Clauser-Horne-Shimony-Holt (CHSH), PRL 1969
Bells inequality holds for ANY local theory
Quantum Correlations
QM violates Bells inequality
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Supp. 2 Post-quantum model
Non-local box of Popescu and Rohrlich (PR-box)
Popescu Rohrlich, Found. Phys. 1994
Barrett et al. PRA 2005

• Non-signalling
• Maximally Non-local (more NL than QM 2v2)
  CHSH ? 4
• CHSH C00 C01 C10 C11

Why does the PR-box not exist in Nature ?