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 Assumptions Bell’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 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, (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 loophole 3 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, (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, (2014)

Cosmic sources Tenerife, Sept 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 CHSH 2 ) Atoms 3, superconducting qubits 4 2.Violate inequality that does not assume fair sampling (e.g. CH, Eberhard, eff. 2/3) Photons 5,6 2 A. Garg & N. D. Mermin, PRD 35, 3831 (1987) Unfair sampling:Local detection efficiency is setting-dependent  A =  A (a, ),  B =  B (b, )  fair-sampling (detection) loophole 1 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, (2013)

Coincidence-time 2 J.-Å. Larsson, M. Giustina, J.K., B. Wittmann, R. Ursin, S. Ramelow, PRA 90, (2014) Moving windows coinc.-time loophole open Predefined fixed local time slots 2 coinc.-time loophole closed 3,4,5 Unfair coincidences: Detection time is setting-dependent T A = T A (a, ), T B = T B (b, )  coincidence-time loophole 1 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)

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 loophole 1,2,3 1 L. Accardi & M. Regoli, quant-ph/ ; quantph/ ; quant-ph/ R. Gill, quant-ph/ , quant-ph/ A. Kent, PRA 72, (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: Delft 1 NV centers Munich 2 atoms Viennaphotons 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/ J.K. & M. Giustina, arXiv: 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…