Necessary adaptation of the CH/Eberhard inequality bound for a loophole-free Bell test Quantum Theory: from Problems to Advances – QTPA Linnaeus University, Växjö, Sweden 10 June 2014 Johannes Kofler Max Planck Institute of Quantum Optics (MPQ) Garching/Munich, Germany

Contents Brief review of Bell’s assumptions and loopholes Eberhard/CH inequality Theorist’s view: Requirements for a definitive (photonic) Bell test (Q)RNGs Loopholes impossible to close Conclusion and outlook

Bell: 1 Determinism  Localitydeterministic LHVA(a,λ), B(b,λ) Bell: 2 Local causalitystochastic LHVp(A,B|a,b,λ) = p(A|a,λ) p(B|b,λ) Freedom of choice: 3  (a,b|λ) =  (a,b)   (λ|a,b) =  (λ) Bell’s Assumptions Bell’s local realism 1 J. S. Bell, Physics 1, 195 (1964) 3 J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, p. 243 (2004) 2 J. S. Bell, Epistemological Lett. 9 (1976) Local causality  Freedom of choice  Bell’s inequality Remarks:original Bell paper: 1 perfect anti-correlation CHSH: 4 fair sampling 4 J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, PRL 23, 880 (1969)

Loopholes Why important? – quantum foundations – quantum cryptography, randomness amplification/expansion Main loopholes: Locality loophole closed for photons (1982 1, ) Freedom-of-choice loophole closed for photons ( ) Fair-sampling (detection) loophole closed for atoms ( ), superconducting qubits ( ) and for photons ( ) Coincidence-time loophole closed for photons 7,8,6 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, (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 7 M. B. Agüero et al., PRA 86, (2012). 8 B. G. Christensen et al., PRL 111, (2013).

Definitive (photonic) Bell test Photons: each of the loopholes has been closed, albeit in separate experiments Alternatives 2 Loophole-free test still missing Loophole:How to close: Localityspace-like separate A & b,B and B & a,A a,b random Freedom ofspace-like separate E & a,b choicea,b random Fair samplinguse CH/Eberhard inequality (detection)violation requires  tot > 2/3 Coincidence-use fixed time slots timeor window-sum method 1 1 J.-Å. Larsson, M. Giustina, JK, B. Wittmann, R. Ursin, S. Ramelow, arXiv: (2013) 2 J. Hofmann, M. Krug, N. Ortegel, L. Gérard, M. Weber, W. Rosenfeld, and H. Weinfurter, Science 337, 72 (2012)

Eberhard inequality N photon pairs in each of the 4 setting combinations (  i,  j ) … “fate”: o, e, u o e o e 11 22  1 1  2 2 LR bound: 0 Quantum bound:– N Logical bound: – N Singles: Only one detector (o) per side: 1 P. H. Eberhard, PRA 47, 747 (1993) Eberhard inequality: 1 No-signaling conditions:

Eberhard and CH Relax assumption of N photon pairs per setting combination (  i,  j ) N ij trials for combination (  i,  j ) Normalized counts (probabilities): Normalized Eberhard inequality: Suppress index o, multiply by –1, write p instead of  CH inequality 1 LR bound: 0 Quantum bound:– Logical bound: – 1 1 J. F. Clauser and M. A. Horne, PRD 10, 526 (1974)

Trials To close locality and freedom-of-choice loopholes: Alice and Bob need new and random settings for every photon pair Definition of trial: At certain appropriate space-time intervals: (i)the source is in principle active (ii)Alice’s and Bob’s settings are generated (iii)their settings are applied (iv)their outcomes (including “undetected”) are recorded. Without trials: No normalized counts, i.e. probabilities 2 A. Khrennikov, S. Ramelow, R. Ursin, B. Wittmann, JK, and I. Basieva, arxiv: [quant-ph] (2014) 1 JK, S. Ramelow, M. Giustina, and A. Zeilinger, arxiv: [quant-ph] (2013) Normalization using production rate 1,2 not possible in loophole-free test Pulsed (synchronized) exp. more feasible Large distance reduces det. efficiency  measurement times T A, T B short (cut late photons) (Q)RNGs: best candidates for stochastic local realistic RNGs

Estimates Total collection efficiencies: 1 JK et al., in preparation Non-max. ent. state optimal: With optimal angles  1,  2,  1,  2,, and estimated dark/background counts: Pulsed experiment, pair population: Estimated experimental normalized Eberhard value 1 if every pulse has down-converted pair (recall: qu. bound –0.207) Achievable: slight mixture: Diagonal basis visibility:

Communication Bell inequ. with auxiliary communication 1 A fraction of settings occurs too early and is communicable to the other side Adapted bound: 1 Also “fates” uu can be relabeled!   detrimental 2 JK et al., in preparation 1 J. D. Bacon and B. F. Toner, PRL 90, (2003) Ex.: strategy for case: “  sent to Bob” Relabel “fates”:  1  2 Aliceoo Bob  1 oo  2 ou Contributions: n oo (  1,  1 ), n oo (  1,  2 ) n oo (  2,  1 ), n uu (  2,  2 ) Achieves logical bound: 

No-signaling general feature of pure strategies with communication (PR boxes are different) Example strategy violates no-signaling: 1 J. D. Bacon and B. F. Toner, PRL 90, (2003) Every quantum state can be simulated: Bacon/Toner 1  2. Alice picks setting, and outputs 1.Alice and Bob share random variables picked from i.e. with prob. 3. Alice sends to Bob 4.Bob outputs picked from i.e. with prob. Within the communication subensemble ( ): Pure comm. strategy signaling Comm.  max. qu. violation no-signaling PR boxes (no comm.) no-signaling Opt. no-sig. comm. strategy opt. ?

(Q)RNGs Humans 1 1 J. S. Bell, La nouvelle cuisine (1990) 2 J. Gallicchio et al., PRL 112, (2014) 3 T. Jennewein et al., Rev. Sci. Inst. 71, 1675 (2000) 4 M. Wayne et al., J. Mod. Opt. 56, 516 (2009) M. Fürst et al., Opt. Expr. 18, (2010) “[…] we can imagine these settings being freely chosen at the last second by two different experimental physicists or some other random devices.” Photon emission times 4 Distant quasars 2 Photons on a beam splitter 3

Autocorrelation 1 JK et al., in preparation Conservatively: 10 – 100 m 10 – 100 ns Requires ~14 autocorrelation times (likely too long) Alternative: discard runs without new photon (Q)RNG: Photons on a beam splitter: Randomness: beam splitter (Q)RNG: Photons emission times Randomness: emission times In theory: Poissonian In practice: detector dead time causes non-zero autocorrelation time Simultaneously: random numbers should be of high quality (“die harder” tests) Assessment: hard, but not impossible depends on local realistic model for the (Q)RNG

Loopholes hard/impossible to close 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

Conclusion and outlook All main loopholes closed individually for photons (within reasonable assumptions) Loophole-free test: trials are important uu matters (due to relabeling), population  matters Strict bounds for fraction of communicable settings Some loopholes can never be closed Definitive Bell test in reach (within reasonable assumptions)

Acknowledgments Jan-Åke Larsson Marissa Giustina Thomas Scheidl Rupert Ursin Bernhard Wittmann Anton Zeilinger Sven Ramelow Thomas Gerrits Sae Woo Nam Andrei Khrennikov