1. Max Planck Institute of Quantum Optics (MPQ)
Garching/Munich, Germany
50 years after Bell’s theorem: Getting close to a loophole-free experiment
Johannes Kofler
50 years of Bell’s inequality
University of Valencia, Spain
3 November 2014

2. Contents Very brief history of hidden variables
Assumptions in Bell’s theorem
Loopholes
Theorist’s view: Requirements for a definitive (photonic) Bell test
(Q)RNGs
Loopholes impossible to close
Conclusion and outlook

3. History Quantum mechanics and hidden variables
Kopenhagen interpretation
(Bohr, Heisenberg, etc.)
1932 Von Neumann’s (wrong) proof of non-possibility of hidden variables
1935 Einstein-Podolsky-Rosen paradox
1952 De Broglie-Bohm (nonlocal) hidden variable theory
Bell’s theorem on local hidden variables
First successful Bell test
(Freedman & Clauser)
since 80s Closing loopholes
Bohr and Einstein (1925)

4. Local realism Classical world view:
Realism: Physical properties are defined prior to and independent of measurement
Locality: No physical influence can propagate faster than the speed of light
External world
Passive observers

5. Bell’s Assumptions Bell’s theorem
Bell:1 Deterministic LHV: “Determinism”:
“Locality”:
Bell:2 Stochastic LHV:
“Local causality”: 
“Freedom of choice”:3
(“measurement independence”) 
Local causality  Freedom of choice  Bell inequality
1 J. S. Bell, Physics 1, 195 (1964)
3 J. F. Clauser & M. A. Horne, Phys. Rev. D 10, 526 (1974)
2 J. S. Bell, Epistemological Lett. 9 (1976)

6. Bell’s Assumptions Freedom of choice
Local causality  Freedom of choice  X  specific Bell inequality
Bell’s original derivation1 only implicitly assumed freedom of choice:
explicitly:
A(a,b,λ)
B(a,b,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: original Bell paper:1 X = “Perfect anti-correlation”: A(b,λ) = –B(b,λ)
CHSH:2 X = “Fair sampling”
1 J. S. Bell, Physics 1, 195 (1964)
2 J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)

7. Loopholes Loopholes: Why important? Main loopholes:
maintain local realism despite exp. Bell violation
Why important?
– quantum foundations
– quantum cryptography, randomness amplification/expansion
Main loopholes:
Locality loophole
closed for photons (19821,19982)
Freedom-of-choice loophole
closed for photons (20103)
Fair-sampling (detection) loophole
closed for atoms (20014), superconducting qubits (20095) and for photons (20136,7)
Coincidence-time loophole
closed for photons8,7,6 E
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, (2010)
4 M. A. Rowe et al., Nature 409, 791 (2001)
5 M. Ansmann et al., Nature 461, 504 (2009)
6 M. Giustina et al., Nature 497, 227 (2013)
7 B. G. Christensen et al., PRL 111, (2013)
8 M. B. Agüero et al., PRA 86, (2012)

8. Locality & freedom of choice
Locality & freedom of choice
Violation of
CHSH inequality1
Tenerife
b,B
La Palma
E,A a E
La Palma
Tenerife
Locality: A is space-like sep. from b and B
B is space-like sep. from a and A
FoC: a and b are space-like sep. from E
Three-photon GHZ experiment2
1 T. Scheidl, R. Ursin, JK, T. Herbst, L. Ratschbacher, X. Ma, S. Ramelow, T. Jennewein, A. Zeilinger, PNAS 107, (2010)
2 C. Erven, E. Meyer-Scott, K. Fisher, J. Lavoie, B. L. Higgins, Z. Yan, C. J. Pugh, J.-P. Bourgoin, R. Prevedel, L. K. Shalm, L. Richards, N. Gigov, R. Laflamme, G. Weihs, T. Jennewein, K. J. Resch, Nature Photon. 8, 292 (2014)

9. 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
Reproduces the quantum predictions of the singlet state (with detection efficiency 2/3)
Detection efficiency is not optional in security-related tasks: faked Bell violations4
1 P. M. Pearle, PRD 2, 1418 (1970)
2 F. Selleri & A. Zeilinger, Found. Phys. 18, 1141 (1988)
3 N. Gisin & 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, (2011)

10. CHSH vs. CH/Eberhard inequality
CHSH inequality1
two detectors per side
correlation functions
fair-sampling assumption used in derivation
requires indep. verific. of tot > 82.8%2
maximally entangled states optimal
CH3 (Eberhard4) inequality
only one detector per side
probabilities (counts)
no fair-sampling assumption in the derivation
no requirement to measure tot
impossible to violate unless tot > 2/3 = 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 & N. D. Mermin, PRD 35, 3831 (1987)
3 J. F. Clauser & M. A. Horne, PRD 10, 526 (1974)
4 P. H. Eberhard, PRA 47, 747 (1993)

11. Photonic closure of fair-sampling loophole
tot  75%
C(a1,b1)
C(a1,b2)
C(a2,b1)
C(a2,b2)
SA(a1)
SB(b1) J
Exp. data1
69 749 –
Model2
68 694
Deviation
–0,04 %
0,01 %
0,11 %
–1,51 %
1,04 %
–0,43 %
1 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: [quant-ph] (2013)

12. 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
1 J.-Å. Larsson & R. Gill, EPL 67, 707 (2004)

13. 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, JK, B. Wittmann, R. Ursin, S. Ramelow, PRA 90, (2014)

14. Application to experimental data
Triple-window method
coinc.-time loophole closed
Fixed time slots
coinc.-time loophole closed
Moving windows
coinc.-time loophole open
 data1 simultaneously closed fair-sampling (detection) and coincidence-time loophole2
1 M. Giustina, A. Mech, S. Ramelow, B. Wittmann, JK, J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. W. Nam, R. Ursin, A. Zeilinger, Nature 497, 227 (2013)
2 J.-Å. Larsson, M. Giustina, JK, B. Wittmann, R. Ursin, S. Ramelow, PRA 90, (2014)

15. Schematic of a (photonic) loophole-free Bell test
To close locality and freedom-of-choice loopholes:
Alice and Bob need new and random settings for every photon pair
Definition of trial:
At certain appropriate space-time intervals:
(i) the source is in principle active
(ii) Alice’s and Bob’s settings are generated
(iii) their settings are applied
(iv) their outcomes (including “undetected”) are recorded.
Without trials: No normalized counts (probabilities)
Pulsed (synchronized) exp. more feasible
Distance: trade-off (det. efficiency vs. req. switching times), d = 10 to 100 m
(Q)RNGs: best candidates for stochastic local realistic RNGs

16. Estimates Total collection efficiencies: Non-max. ent. state optimal:
Achievable: slight mixture:
With optimal angles 1,2,1,2, and , and estimated dark/background counts, the normalized Eberhard inequality becomes:
if every pulse has down-converted pair
(quantum bound –0.207)
Pulsed experiment, pair population:
Estimated experimental normalized Eberhard value1
1 JK & M. Giustina, in preparation

17. (Q)RNGs
Humans1
“[…] we can imagine these settings being freely chosen at the last second by two different experimental physicists or some other random devices.”
Distant quasars2
Photons on a beam splitter3
Photon emission times4
3 T. Jennewein et al., Rev. Sci. Inst. 71, 1675 (2000)
1 J. S. Bell, La nouvelle cuisine (1990)
4 M. Wayne et al., J. Mod. Opt. 56, 516 (2009)
M. Fürst et al., Opt. Expr. 18, (2010)
2 J. Gallicchio et al., PRL 112, (2014)

18. One more loophole
“No memory” assumption: Trials are i.i.d. (independent and identically distributed)
Memory: k-th outcome Ak may depend on history: Ak-1,…,A1,ak-1,…,a1,Bk-1,…,B1,bk-1,…,b1
 Memory loophole1
Virtually impossible to close physically (would require new setup for every trial)
Does not change the local realistic bound, but the statistical analysis
Standard deviation cannot be used
Closure via p-value (Hoeffding), p = prob. that result can be explained by local realism
N … number of trials
“Scientific discovery”: p < 10–6, k > 5
Closed in every good Bell experiment to date, but non-trivial in a loophole-free test
Run-time of an experiment:2
R … rate of trials
1 J. Barrett, D. Collins, L. Hardy, A. Kent, S. Popescu, PRA 66, (2002)
2 JK & M. Giustina, in preparation

19. Summary of assumptions and loopholes
Minimal assumptions
Auxiliary assumptions
Loophole
Closure
Local causality
Locality
space-like separation of outcome
and (random) setting events
Freedom of
choice
Freedom of
choice
space-like separation of emission
and (random) setting events
Fair sampling
Fair sampling
(detection)
use CH/Eberhard inequality ( > 2/3)
or use CHSH ( > 82.8%)
Fair coincidences
Coincidence-
time
use fixed time slots
or window-sum method
No memory
Memory
sufficiently many trials

20. Loopholes hard/impossible to close
Superdeterminism:1 Common cause for E and a,b
Source: E is further in the past than believed; e.g.  is not created at down-conversion but passed on from previous events
Settings: a,b further in the past than believed; setting choice determined by earlier events
Detectors: A,B are further in the future than believed, e.g. photons wait before final detection, “collapse locality loophole”2
Actions into the past
Different rules of logic … E
1 J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, p (2004)
2 A. Kent, PRA 72, (2005)

21. Conclusion and outlook
Bell’s assumptions: local causality and freedom of choice
5 main loopholes: locality, freedom of choice, fair sampling, coincidence-time, memory
All main loopholes were closed individually for photons (within reasonable assumptions)
Some loopholes can never be closed
Alternative approaches are not purely photonic: atoms, NV centers
Definitive Bell test in reach (within reasonable assumptions)

22. Acknowledgments Marissa Giustina Thomas Scheidl Rupert Ursin
Bernhard Wittmann
Anton Zeilinger
Thomas Gerrits
Sae Woo Nam
Jan-Åke Larsson
Sven Ramelow
Andrei Khrennikov