Max Planck Institute of Quantum Optics (MPQ)
Garching / Munich, Germany
Closing loopholes in Bell tests of local realism
Johannes Kofler
Workshop “Quantum Physics and the Nature of Reality”
International Academy Traunkirchen, Austria
22 November 2013
Overview
•
•
•
Assumptions in Bell’s theorem

Realism

Locality

Freedom of choice
Closing loopholes

Locality

Freedom of choice

Fair sampling

Coincidence time
Conclusion and outlook
Acknowledgements
Sae Woo Nam
Marissa Giustina Bernhard Wittmann
Rupert Ursin
Sven Ramelow
Anton Zeilinger
Jan-Åke Larsson
History
Quantum mechanics and hidden variables
1927
Kopenhagen interpretation
(Bohr, Heisenberg, etc.)
1932
Von Neumann’s (wrong) proof of nonpossibility of hidden variables
1935
Einstein-Podolsky-Rosen paradox
1952
De Broglie-Bohm (nonlocal) hidden
variable theory
1964
Bell’s theorem on local hidden variables
1972
First successful Bell test
(Freedman & Clauser)
Bohr and Einstein, 1925
Local realism
Classical world view:
• Realism:
Physical properties are (probabilistically) defined prior to and
independent of measurement
• Locality:
No physical influence can propagate faster than the speed of light
External world
Passive observers
Bell’s assumptions
Assumptions
Bell’s
1
Realism: Hidden variables determine global prob. distrib.: p(Aa1b1, Aa1b2, Aa2b1,…|λ)
2
Locality: (OI)Outcome independence:
(SI) Setting independence:
 factorizability:
p(A|a,b,B,λ) = p(A|a,b,λ)
p(A|a,b,λ) = p(A|a,λ)
& vice versa for B
& vice versa for B
p(A,B|a,b,λ) = p(A|a,λ) p(B|b,λ)
3
Freedom of choice:
1
(a,b|λ) = (a,b)  (λ|a,b) = (λ)
J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)
2 J. S. Bell, Physics 1, 195 (1964)
3
J. S. Bell, Speakable and Unspeakable in
Quantum Mechanics, p. 243 (2004)
Bell’s
Bell’s Assumptions
theorem
Realism + Locality + Freedom of choice + X  Bell’s inequality
Bell’s original derivation1 only implicitly assumed freedom of choice:
explicitly: A(a,b,B,λ) B(a,b,A,λ)
locality
freedom of choice
implicitly: (λ|a,b) A(a,λ) B(b,λ) – (λ|a,c) A(a,λ) B(c,λ)
Remarks:
1
2
original Bell paper1:
X = “Perfect anti-correlation”
CHSH2:
X = “Fair sampling”
J. S. Bell, Physics 1, 195 (1964)
J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)
Loopholes
Loopholes:
Why important?
maintain local realism
despite exp. Bell violation
– quantum foundations
– security of entanglement-based quantum cryptography
Three main loopholes:
• Locality loophole
hidden communication between the parties
closed for photons (19821,19982)
• Freedom-of-choice loophole
settings are correlated with hidden variables
closed for photons (20103)
• Fair-sampling (detection) loophole
measured subensemble is not representative
closed for atoms (20014), superconducting qubits
(20095) and for photons (20136)
1
A. Aspect et al., PRL 49, 1804 (1982)
2 G. Weihs et al., PRL 81, 5039 (1998)
3 T. Scheidl et al., PNAS 107, 10908 (2010)
4
E
M. A. Rowe et al., Nature 409, 791 (2001)
M. Ansmann et al., Nature 461, 504 (2009)
6 M. Giustina et al., Nature 497, 227 (2013)
5
Locality & freedom of choice
Tenerife
b,B
La Palma
E,A
a
E
La Palma
Locality:
Tenerife
A is space-like sep. from b and B
B is space-like sep. from a and A
Freedom of choice:
p(A,B|a,b,) = p(A|a,) p(B|b,)
a and b are random
a and b are space-like sep. from E
p(a,b|) = p(a,b)
T. Scheidl, R. Ursin, J. K., T. Herbst, L. Ratschbacher, X. Ma, S. Ramelow, T. Jennewein, A. Zeilinger,
PNAS 107, 10908 (2010)
Fair-sampling loophole
Fair sampling:
Local detection efficiency depends only on hidden variable:
A = A(), B = B()  observed outcomes faithfully
reproduce the statistics of all emitted particles
Unfair sampling:
Local detection efficiency is setting-dependent
A = A(a,), B = B(b,)  fair-sampling (detection) loophole1
• Local realistic models2,3
4
9
:
4
9
:
1
9
:
 
 
A ( a ,  )  sign( a   )
 
 
 A (a ,  )  | a   |
 
 A (a ,  )  1
 
 A (a ,  )  0
 
B (b ,  )
 
 B (b ,  )
 
 B (b ,  )
 
 B (b ,  )
 
 sign(  b   )
1
 
 | b  |
 
E (a , b ) 


S
d
2
2
A B
 
A B  a b
 0
Reproduces the quantum predictions of the singlet state with detection efficiency 2/3
• Detection efficiency is not optional in security-related tasks (device-independent
quantum cryptography): faked Bell violations4
1
P. M. Pearle, PRD 2, 1418 (1970)
2 F. Selleri and A. Zeilinger, Found. Phys. 18, 1141 (1988)
3 N. Gisin and B. Gisin, Phys. Lett. A 260, 323 (1999)
4 I.
Gerhardt, Q. Liu, A. Lamas-Linares, J. Skaar, V. Scarani,
V. Makarov, C. Kurtsiefer, PRL 107, 170404 (2011)
CHSH vs. CH/Eberhard inequality
CHSH inequality1
 two detectors per side
 correlation functions
E ( a1 , b1 )  E ( a1 , b 2 )  E ( a 2 , b1 )  E ( a1 , b1 )  2
 fair-sampling assumption used in derivation
 requires indep. verific. of tot > 82.8 %2
 maximally entangled states optimal
CH3 (Eberhard3) inequality
 only one detector per side
 probabilities (counts)
P ( a 1 , b1 )  P ( a 1 , b 2 )  P ( a 2 , b1 )
 no fair-sampling assumption in the derivation
 P ( a 2 , b 2 )  P ( a 1 )  P ( b1 )  0
 no requirement to measure tot
 impossible to violate unless tot > 66.7 %
 non-max. entangled states optimal
1
J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)
2 A. Garg and N. D. Mermin, PRD 35, 3831 (1987)
3
4
J. F. Clauser and M. A. Horne, PRD 10, 526 (1974)
P. H. Eberhard, PRA 47, 747 (1993)
Transition-edge sensors
Working principle
 Superconductor (200 nm thick tungsten film
at 100 mK) at transition edge
 Steep dependence of resistivity on
temperature
 Measurable temperature change by single
absorbed photon
Characteristics
 High efficiency > 95 %2
 Low noise < 10 Hz2
 Photon-number resolving
1
2
Picture from: Topics in Applied Physics 99, 63-150 (2005)
A. E. Lita, A. J. Miller, S. W. Nam, Opt. Express 16, 3032 (2008)
Superconducting
transition-edge sensors1
Setup
• Sagnac-type entangled
pair source
• Non-max. entangled
states

r

 HV
1
1 r
 r VH

2
• Fiber-coupling efficiency
> 90%
• Filters: backgroundphoton elimination > 99%
M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. K., J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. W. Nam, R. Ursin,
A. Zeilinger, Nature 497, 227 (2013)
Experimental results
 P ( a1 , b1 )  P ( a1 , b 2 )  P ( a 2 , b1 )  P ( a 2 , b 2 )  P ( a1 )  P ( b1 )
CH :
Eberhard
:
 0
J   C ( a 1 , b1 )  C ( a 1 , b 2 )  C ( a 2 , b 1 )  C ( a 2 , b 2 )  S A ( a 1 )  S B ( b 1 )  0
• Violation of Eberhard’s inequality1
• 300 seconds per setting combination
• Collection efficiency tot  75%
• No background correction etc.
Photon: only system for which all
main loopholes are now closed
(not yet simultaneously)
Exp. data1
Model2
Deviation
1
C(a1,b1)
C(a1,b2)
C(a2,b1)
C(a2,b2)
SA(a1)
SB(b1)
J
1 069 306
1 068 886
–0,04 %
1 152 595
1 152 743
0,01 %
1 191 146
1 192 489
0,11 %
69 749
68 694
–1,51 %
1 522 865
1 538 766
1,04 %
1 693 718
1 686 467
–0,43 %
–126 715
M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. K., J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. W. Nam, R. Ursin,
A. Zeilinger, Nature 497, 227 (2013)
2 J. K., S. Ramelow, M. Giustina, A. Zeilinger, arXiv:1307.6475 [quant-ph] (2013)
The coincidence-time loophole
Fair coincidences:
Local detection time depends only on hidden variable:
TA = TA(), TB = TB()  identified pairs faithfully
reproduce the statistics of all detected pairs
Unfair coincidences:
Detection time is setting-dependent
TA = TA(a,), TB = TB(b,)  coincidence-time loophole1
Local realistic model:
Standard “moving windows” technique:
coincidence if |TA(a,) –TB(b,)|  ½
a2b2 coincidences are missed,
CH/Eberhard violated
 C ( a 1 , b1 )  C ( a 1 , b 2 )  C ( a 2 , b1 )
 C ( a 2 , b 2 )  S A ( a 1 )  S B ( b1 )  0
1
J.-Å. Larsson and R. Gill, EPL 67, 707 (2004)
Closing the coincidence-time loophole
a) Moving windows
coincidence-time loophole open
b) Predefined fixed local time slots
coincidence-time loophole closed
c) Triple window for a2b2 coinc.
coincidence-time loophole closed
J.-Å. Larsson, M. Giustina, J. K., B. Wittmann, R. Ursin, S. Ramelow, arXiv:1309.0712 (2013)
Application to experimental data
Eberhard
inequ.
J   C ( a 1 , b1 )  C ( a 1 , b 2 )  C ( a 2 , b1 )  C ( a 2 , b 2 )  S A ( a 1 )  S B ( b1 )  0
Triple-window method
coinc.-time loophole closed
Fixed time slots
coinc.-time loophole closed
Moving windows
coinc.-time loophole open
 simultaneous closure of fair-sampling (detection) and coincidence-time loophole
J.-Å. Larsson, M. Giustina, J. K., B. Wittmann, R. Ursin, and S. Ramelow, arXiv:1309.0712 (2013)
Conclusion and outlook
Loophole:
How to close:
Locality
space-like separate A & b,B and B & a,A
a,b random
Freedom of
choice
space-like separate E & a,b
a,b random
Fair sampling
(detection)
use CHSH and also show  > 82.8%
or use CH/Eberhard
 C ( a 1 , b1 )  C ( a 1 , b 2 )  C ( a 2 , b1 )
 C ( a 2 , b 2 )  S A ( a 1 )  S B ( b1 )  0
Coincidencetime
use fixed time slots
or window-sum method
• Photons: each of the loopholes has been closed,
albeit in separate experiments
• Loophole-free experiment still missing but in reach
Loopholes hard/impossible to close
Futher loopholes:
Superdeterminism:
Common cause for E and a,b
Wait-at-the-source:
E is further in the past; pairs
wait before they start travelling
Wait-at-the setting:
a,b futher in the past; photons
used for the setting choice wait
before they start traveling
Wait-at-the-detector: A,B are farther in the future,
photons wait before detection,
“collapse locality loophole”
Actions into the past
…
E