1. 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

2. 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)

3. 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)

4. 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

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

6. 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 loophole 3 cannot be fully closed in principle

7. 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)

8. Cosmic sources Tenerife, Sept. 2013 Anton already searching for some (very bright) quasars?

9. 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, 130406 (2013)

10. Coincidence-time 2 J.-Å. Larsson, M. Giustina, J.K., B. Wittmann, R. Ursin, S. Ramelow, PRA 90, 032107 (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, 052121 (2012) 4 B. G. Christensen et al., PRL 111, 130406 (2013) 5 M. Giustina et al., Nature 497, 227 (2013)

11. 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/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.....

12. 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

13. 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:

14. 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

15. 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: 

16. 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

17. Looking three steps ahead…