1. Loophole-free test of Bell’s theorem with entangled photons
Johannes Kofler
Max Planck Institute of Quantum Optics (MPQ)
Garching/Munich, Germany
Indian Institute of Science Education and Research
Mohali, India
14 Oct. 2016

2. Introduction
Local realism: “objects have pre-existing definite properties & no action at a distance”  Bell’s inequality
Relevant for foundations of quantum mechanics and (security of) modern quantum information protocols
Quantum cryptography
Randomness amplification / expansion
Bell experiments have “loopholes”
Locality
Freedom of choice
Fair sampling
Coincidence time
Memory
Four “loophole-free” experiments in 2015
(Delft1, Boulder2, Vienna3, Munich)
John S. Bell (1928–1990)
1 B. Hensen et al., Nature 526, 682 (2015)
2 L. K. Shalm et al., PRL 115, (2015)
3 M. Giustina et al., PRL 115, (2015)

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 then Closing loopholes
Bohr and Einstein (1925)

4. Local realism Classical world view:
Realism: Physical properties are defined prior to and independent of measurement (via hidden variables)
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. Picture: Rev. Mod. Phys. 86, 419 (2014)
Derivation of the CHSH inequality
Dichotomic outcomes: Ai = A(ai,) = 1
Bj = B(bj,) = 1
A1 (B1 + B2) + A2 (B1 – B2) = 2
A1B1 + A1B A2B1 – A2B2 = 2
S := A1B1 + A1B2 + A2B1 – A2B2  2 =
locality
SQM = 22
Picture: Rev. Mod. Phys. 86, 419 (2014)
J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)

7. 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)

8. Loopholes
Loopholes:
maintain local realism despite exp. Bell violation
Relevance
– quantum foundations
– quantum cryptography, randomness amplification/expansion

9. Locality Loophole addressed by space-time arrangement:1,2
Locality
Loophole addressed by space-time arrangement:1,2
Space-like separation between the outcomes
(outcome independence)
Space-like separation between each outcome and the distant setting
(setting independence)
Remark:
Collapse locality loophole3 cannot be fully closed in principle
1 A. Aspect, P. Grangier, G. Roger, PRL 49, 91 (1982)
2 G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger, PRL 81, 5039 (1998)
3 A. Kent, PRA, (2005)

10. Freedom of choice Loophole addressed by space-time arrangement:1,2
Freedom of choice
Loophole addressed by space-time arrangement:1,2
Space-like separation of setting choice events a,b and the pair emission event E (assuming that hidden variables are created at E)
Remarks:
Superdeterminism can never be ruled out
Cosmic sources:3
1 T. Scheidl, R. Ursin, J.K., 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)
3 J. Gallicchio, A. S. Friedman, D. I. Kaiser, PRL 112, (2014)

11. Locality & freedom of choice
Locality & freedom of choice
Tenerife
b,B
La Palma
E,A a E
Optical Ground Station, Tenerife
Photo: ESA
T. Scheidl et al. PNAS 107, (2010)

12. Fair sampling
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 models with unfair sampling2,3
SQM = 22
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)

13. Fair sampling Two options to close the loophole:
Violate inequality that assumes fair sampling (e.g. CHSH) and show large total detection efficiency (> 82.8% for CHSH)
Atoms1, superconducting qubits2
Violate inequality that does not assume fair sampling
(e.g. CH, Eberhard, eff. 2/3)
Photons3,4
1 M. A. Rowe et al., Nature 409, 791 (2001)
2 M. Ansmann et al., Nature 461, 504 (2009)
3 M. Giustina et al., Nature 497, 227 (2013)
4 B. G. Christensen et al., PRL 111, (2013)

14. Coincidence time Unfair coincidences:
Detection time is setting-dependent
TA = TA(a,), TB = TB(b,)
 coincidence-time loophole1
Moving windows
coinc.-time loophole open
Predefined fixed local time slots2
coinc.-time loophole closed3,4,5
1 J.-Å. Larsson and R. Gill, EPL 67, 707 (2004)
3 M. B. Agüero et al., PRA 86, (2012)
4 B. G. Christensen et al., PRL 111, (2013)
5 M. Giustina et al., Nature 497, 227 (2013)
2 J.-Å. Larsson, M. Giustina, J.K., B. Wittmann, R. Ursin, S. Ramelow, PRA 90, (2014)

15. Memory Memory: k-th outcome A(k) can depend on history:
A(k) = A(k)(A(1),…,A(k–1); a(1),…,a(k); B(1),…,B(k–1); b(1),…,b(k–1); (1),…,(k))
similar for B(k)
 memory loophole1,2,3
Two solutions:
Space-like separated setups, used only once for each pair
(unfeasible / impossible)
.....
Drop assumption that trials are i.i.d. (independent and identically distributed)
cannot use “standard” standard-deviation approach
 “hypothesis testing”, e.g. supermartingales & Hoeffding‘s inequality
1 L. Accardi & M. Regoli, quant-ph/ ; quantph/ ; quant-ph/
2 R. Gill, quant-ph/ , quant-ph/
3 A. Kent, PRA 72, (2005)

16. The Vienna experiment
Source: polarization entangled photons, pulsed (1 MHz) type-II SPDC in Sagnac config.
Detectors: superconducting transition edge sensors

17. Closing the locality & freedom-of-choice loopholes

18. Closing fair sampling, coincidence-time, memory
CH-E inequality derived without the fair-sampling assumption:
Can be violated with non-maximally entangled states
Locally predefined fixed time-slots close the coincidence-time loophole
Excess predictability of settings:
  2  510–4
Requires adaptation of CH-E inequality:1
Closing memory loophole: Hoeffding’s inequality for J process
1 J. Kofler, M. Giustina, J.-Å. Larsson, M. W. Mitchell, PRA 93, (2016)

19. Results
trials at 1 MHz, 3510 s measurement time, i.e. 3.5 billion trials
one down-conversion pair in every 300 trials
total detection efficiency: 78.6% (Alice), 76.2% (Bob)
state: r  –2.9, visibility > 99% (for product and singlet state)
J-value: 7.2710–6
p-value: 3.7410–31
(probability that local realism could have produced the data by a random variation)
M. Giustina et al., PRL 115, (2015)

20. p-value versus excess predictability
p-value of 3.7410–31 for characterized excess predictability   2.410–4
p-value remains below “gold standard” of 10–6 (dashed line) for  up to 0.65%

21. Conclusion Bell experiment using entangled photons
Closing simultaneously the following loopholes:
Locality
Freedom of choice
Fair sampling
Coincidence time
Memory
Strong statistical violation
Still requires assumptions (no super- determinism, classical rules of logic, etc)

22. The team Marissa Giustina Marijn A. M. Versteegh Sören Wengerowsky
Johannes Handsteiner
Armin Hochrainer
Kevin Phelan
Fabian Steinlechner
Thomas Scheidl
Rupert Ursin
Bernhard Wittmann
Anton Zeilinger
Thomas Gerrits
Adriana E. Lita
L. Krister Shalm
Sae Woo Nam
Carlos Abellán
Waldimar Amaya
Valerio Pruneri
Morgan W. Mitchell
Jörn Beyer
Jan-Åke Larsson
Johannes Kofler
Reference: Phys. Rev. Lett. 115, (2015)