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…