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Closing loopholes in Bell tests of local realism
Workshop “Quantum Physics and the Nature of Reality”
International Academy Traunkirchen, Austria
22 November 2013
Johannes Kofler
Max Planck Institute of Quantum Optics (MPQ)Garching / Munich, Germany
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
Anton Zeilinger
Marissa Giustina Bernhard Wittmann
Sae Woo Nam
Rupert Ursin
Sven Ramelow
Jan-Åke Larsson
Quantum mechanics and hidden variables
Bohr and Einstein, 1925
1927 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
1964 Bell’s theorem on local hidden variables
1972 First successful Bell test(Freedman & Clauser)
History
Local realism
• 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
Classical world view:
Realism: Hidden variables determine global prob. distrib.: p(Aa1b1, Aa1b2, Aa2b1,…|λ)
Locality: (OI)Outcome independence: p(A|a,b,B,λ) = p(A|a,b,λ) & vice versa for B(SI) Setting independence: p(A|a,b,λ) = p(A|a,λ) & vice versa for B
factorizability: p(A,B|a,b,λ) = p(A|a,λ) p(B|b,λ)
Freedom of choice: (a,b|λ) = (a,b) (λ|a,b) = (λ)
Bell’s AssumptionsBell’s assumptions
1 J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)
1
2
3
3 J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, p. 243 (2004)2 J. S. Bell, Physics 1, 195 (1964)
Realism + Locality + Freedom of choice + X Bell’s inequality
Bell’s original derivation1 only implicitly assumed freedom of choice:
A(a,b,B,λ)
locality
(λ|a,b) A(a,λ) B(b,λ) – (λ|a,c) A(a,λ) B(c,λ)
freedom of choice
explicitly:
implicitly:
Bell’s AssumptionsBell’s theorem
1 J. S. Bell, Physics 1, 195 (1964)2 J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)
B(a,b,A,λ)
Remarks: original Bell paper1: X = “Perfect anti-correlation”
CHSH2: X = “Fair sampling”
Loopholes
Why important?
– quantum foundations– security of entanglement-based quantum cryptography
Three main loopholes:
• Locality loopholehidden communication between the parties
closed for photons (19821,19982)
• Freedom-of-choice loopholesettings are correlated with hidden variables
closed for photons (20103)
• Fair-sampling (detection) loopholemeasured 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 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)
Loopholes:
maintain local realism despite exp. Bell violation
E
Locality: A is space-like sep. from b and BB is space-like sep. from a and A
T. Scheidl, R. Ursin, J. K., T. Herbst, L. Ratschbacher, X. Ma, S. Ramelow, T. Jennewein, A. Zeilinger, PNAS 107, 10908 (2010)
Locality & freedom of choice
b,B
E,A
a
Tenerife
La Palma
Freedom of choice: a and b are randoma and b are space-like sep. from E
E
p(a,b|) = p(a,b)
p(A,B|a,b,) = p(A|a,) p(B|b,)
La Palma Tenerife
Fair-sampling loophole
Unfair sampling: Local detection efficiency is setting-dependentA = A(a,), B = B(b,) fair-sampling (detection) loophole1
• Local realistic models2,3
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)
• Detection efficiency is not optional in security-related tasks (device-independent quantum cryptography): faked Bell violations4
)sign(),(
aaA )sign(),(
bbB
||),(A
aabaBAbaE
S
BA2 2
d),(
1),(A
a
0),(A
a
1),(B
b
||),(B
bb
0),(B
b
:94
:94
:91
Reproduces the quantum predictions of the singlet state with detection efficiency 2/3
Fair sampling: Local detection efficiency depends only on hidden variable: A = A(), B = B() observed outcomes faithfully reproduce the statistics of all emitted particles
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
- fair-sampling assumption used in derivation
- requires indep. verific. of tot > 82.8 %2
- maximally 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)
0)()(),(
),(),(),(
1122
122111
bPaPbaP
baPbaPbaP
CH3 (Eberhard3) 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 > 66.7 %
- non-max. entangled states optimal
2),(),(),(),( 11122111 baEbaEbaEbaE
3 J. F. Clauser and M. A. Horne, PRD 10, 526 (1974)4 P. H. Eberhard, PRA 47, 747 (1993)
Transition-edge sensors
1 Picture from: Topics in Applied Physics 99, 63-150 (2005)2 A. E. Lita, A. J. Miller, S. W. Nam, Opt. Express 16, 3032 (2008)
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
Superconducting transition-edge sensors1
Characteristics
- High efficiency > 95 %2
- Low noise < 10 Hz2
- Photon-number resolving
Setup
• Sagnac-type entangled pair source
• Non-max. entangled states
• Fiber-coupling efficiency > 90%
• Filters: background-photon elimination > 99%
VHrHVr
r
21
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)
Experimental results
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:1307.6475 [quant-ph] (2013)
0)()(),(),(),(),(:Eberhard 1122122111 bSaSbaCbaCbaCbaCJ BA
Photon: only system for which all main loopholes are now closed(not yet simultaneously)
• Violation of Eberhard’s inequality1
• 300 seconds per setting combination
• Collection efficiency tot 75%
• No background correction etc.
C(a1,b1) C(a1,b2) C(a2,b1) C(a2,b2) SA(a1) SB(b1) J
Exp. data1 1 069 306 1 152 595 1 191 146 69 749 1 522 865 1 693 718 –126 715Model2 1 068 886 1 152 743 1 192 489 68 694 1 538 766 1 686 467Deviation –0,04 % 0,01 % 0,11 % –1,51 % 1,04 % –0,43 %
0)()(),(),(),(),(:CH 1122122111 bPaPbaPbaPbaPbaP
The coincidence-time loophole
1 J.-Å. Larsson and R. Gill, EPL 67, 707 (2004)
Unfair coincidences: Detection time is setting-dependentTA = TA(a,), TB = TB(b,) coincidence-time loophole1
Fair coincidences: Local detection time depends only on hidden variable: TA = TA(), TB = TB() identified pairs faithfully reproduce the statistics of all detected pairs
Standard “moving windows” technique: coincidence if |TA(a,) –TB(b,)| ½
0)()(),(
),(),(),(
1122
122111
bSaSbaC
baCbaCbaC
BA
a2b2 coincidences are missed, CH/Eberhard violated
Local realistic model:
Closing the coincidence-time loophole
J.-Å. Larsson, M. Giustina, J. K., B. Wittmann, R. Ursin, S. Ramelow, arXiv:1309.0712 (2013)
a) Moving windowscoincidence-time loophole open
b) Predefined fixed local time slotscoincidence-time loophole closed
c) Triple window for a2b2 coinc.coincidence-time loophole closed
Application to experimental data
J.-Å. Larsson, M. Giustina, J. K., B. Wittmann, R. Ursin, and S. Ramelow, arXiv:1309.0712 (2013)
Moving windowscoinc.-time loophole open
Fixed time slotscoinc.-time loophole closed
Triple-window methodcoinc.-time loophole closed
0)()(),(),(),(),(inequ. Eberhard 1122122111 bSaSbaCbaCbaCbaCJ BA
simultaneous closure of fair-sampling (detection) and coincidence-time loophole
Conclusion and outlook
• Photons: each of the loopholes has been closed, albeit in separate experiments
• Loophole-free experiment still missing but in reach
Loophole: How to close:
Locality space-like separate A & b,B and B & a,Aa,b random
Freedom of space-like separate E & a,bchoice a,b random
Fair sampling use CHSH and also show > 82.8%(detection) or use CH/Eberhard
Coincidence- use fixed time slotstime or window-sum method
0)()(),(
),(),(),(
1122
122111
bSaSbaC
baCbaCbaC
BA
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
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