requirements for a loophole-free bell test using imperfect setting generators johannes kofler max...
TRANSCRIPT
Requirements for a loophole-free Bell test using
imperfect setting generators
Johannes Kofler
Max Planck Institute of Quantum Optics (MPQ)Garching/Munich, Germany
QuPoN
University of Vienna, 21 May 2015
Introduction
• Local realism: “objects have pre-existing definite properties & no action at a distance” Bell’s inequality
• Relevant for (security of) modern quantum information protocols
- Quantum cryptography
- Randomness amplification / expansion
• Bell experiments have “loopholes”
- Locality
- Freedom of choice
- Fair sampling
- Coincidence time
- Memory (joint work with Marissa Giustina)
• Loophole-free experiment on the horizon
John S. Bell (1928–1990)
Bell:1 Deterministic models: “Determinism”:
“Locality”:
Bell:2 Stochastic models:
“Local causality”:
“Freedom of choice”:3
(“measurement independence”)
Bell’s AssumptionsBell’s theorem
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)
Remarks: original Bell paper:1 X = “Perfect anti-correlation”: A(b,λ) = –B(b,λ)
CHSH:4 X = “Fair sampling”
4 J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)
Bell’s Assumptions“Realism”
An important moment in the history of quantum foundations
Nicolas and Anton agreeing on the definition of “realism”
Oxford, Sept. 2010
almost
Loopholes
Relevance
– quantum foundations– quantum cryptography, randomness amplification/expansion
Loopholes:
maintain local realism despite exp. Bell violation
Locality
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, 012107 (2005)
Loophole closed 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
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
1 T. Scheidl, R. Ursin, J.K., T. Herbst, L. Ratschbacher, X. Ma, S. Ramelow, T. Jennewein, A. Zeilinger, PNAS 107, 10908 (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)
Remarks:
Superdeterminism can never be ruled out
Cosmic sources:3
3 J. Gallicchio, A. S. Friedman, D. I. Kaiser, PRL 112, 110405 (2014)
Cosmic sources
Tenerife, Sept. 2013
Anton already searching for some (very bright) quasars?
Fair sampling
1 P. M. Pearle, PRD 2, 1418 (1970)
Fair sampling: Local detection efficiency depends only on hidden variable: A = A(), B = B() observed outcomes faithfully reproduce the statistics of all emitted particles
Two options to close the loophole:
1. Violate inequality that assumes fair sampling (e.g. CHSH) and show large total detection efficiency (> 82.8% for CHSH2)
Atoms3, superconducting qubits4
2. Violate inequality that does not assume fair sampling(e.g. CH, Eberhard, eff. 2/3)
Photons5,6
2 A. Garg & N. D. Mermin, PRD 35, 3831 (1987)
Unfair sampling: Local detection efficiency is setting-dependentA = A(a,), B = B(b,) fair-sampling (detection) loophole1
3 M. A. Rowe et al., Nature 409, 791 (2001)4 M. Ansmann et al., Nature 461, 504 (2009)
5 M. Giustina et al., Nature 497, 227 (2013)6 B. G. Christensen et al., PRL 111, 130406 (2013)
Coincidence-time
2 J.-Å. Larsson, M. Giustina, J.K., B. Wittmann, R. Ursin, S. Ramelow, PRA 90, 032107 (2014)
Moving windowscoinc.-time loophole open
Predefined fixed local time slots2
coinc.-time loophole closed3,4,5
Unfair coincidences:
Detection time is setting-dependentTA = TA(a,), TB = TB(b,)
coincidence-time loophole1
1 J.-Å. Larsson and R. Gill, EPL 67, 707 (2004) 3 M. B. Agüero et al., PRA 86, 052121 (2012)4 B. G. Christensen et al., PRL 111, 130406 (2013)5 M. Giustina et al., Nature 497, 227 (2013)
Memory
Memory: k-th outcome A(k) can depend on history:
A(k) = A(k)(A(1),A(2),…,A(k–1);a(1),a(2),…,a(k–1);B(1),B(2),…,B(k–1);b(1),b(2),…,b(k–1))
similar for B(k)
memory loophole1,2,3
1 L. Accardi & M. Regoli, quant-ph/0007005; quantph/0007019; quant-ph/0110086
2 R. Gill, quant-ph/0110137, quant-ph/0301059
3 A. Kent, PRA 72, 012107 (2005)
Two solutions:
1. Space-like separated setups, used only once for each pair
(unfeasible / impossible)
2. 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
.....
Towards a loophole-free Bell test
(At least) 3 groups:
Delft1 NV centers
Munich2 atoms
Vienna photons
1 W. Pfaff, B. Hensen, H. Bernien, S. B. van Dam, M. S. Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen, R. Hanson, Science 345, 532 (2014)
2 J. Hofmann, M. Krug, N. Ortegel, L. Gérard, M. Weber, W. Rosenfeld, H. Weinfurter, Science 337, 72 (2012)
Hofburg Vienna, June 2014
heralded entanglement
Imperfect setting generators
Setting generators always have non-zero correlation into the past predictability
Needs to be adapted:
Normalized Eberhard (CH) inequality
Det. efficiency:
Pairs per pulse:
Experimental runtime
Hoeffding inequality:
Eberhard value after trials:
–J is a supermartingale:
Case: Local realism (LR),
Case: Local realism + pred. ( LR)
–J is no longer a supermartingale:
But –K is a supermartingale:
Hoeffding inequality:
Runtime of the experiment:
for a statistically significant test closing the memory loophole
Rescue: Doob’s optional stopping theorem
Diluted process: “stopping times” must be chosen without looking into the future
Simple in LR:1 stop only at non-empty trials:
More tricky in LR: empty trials ( ) contribute to –K:
Solution:2
1 R. Gill, quant-ph/0301059
2 J.K. & M. Giustina, arXiv:1411.4787
Choose stopping times
Stop at: 1. non-empty trials:
2. after a street of length of empty trials
Range of increments from to in diluted sequence:
Conclusion
• Loopholes relevant from foundational & technological perspective
- Locality
- Freedom of choice
- Fair sampling
- Coincidence time
- Memory
• All loopholes closed in individual experiments
• Loophole-free Bell test in reach
- within reasonable assumptions (no superdeterminism, validity of rules of logic, etc.)
• For photons essential (with today’s technology):
- avoid CHSH
- Doob’s stopping theorem
Looking three steps ahead…