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 Locality deterministic LHV A(a,λ), B(b,λ)

Bell:2 Local causality stochastic LHV p(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 AssumptionsBell’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 loopholeclosed for photons (19821,19982)

• Freedom-of-choice loopholeclosed for photons (20103)

• Fair-sampling (detection) loopholeclosed for atoms (20014), superconducting qubits (20095) and for photons (20136)

• Coincidence-time loopholeclosed for photons7,8,6

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, 10908 (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, 052121 (2012).8 B. G. Christensen et al., PRL 111, 130406

(2013).

Definitive (photonic) Bell test

• Photons: each of the loopholes has been closed, albeit in separate experiments

• Alternatives2

• Loophole-free test still missing

Loophole: How to close:

Locality space-like separate A & b,B and B & a,Aa,b random

Freedom of space-like separate E & a,bchoice a,b random

Fair sampling use CH/Eberhard inequality(detection) violation requires tot > 2/3

Coincidence- use fixed time slotstime or window-sum method1

1 J.-Å. Larsson, M. Giustina, JK, B. Wittmann, R. Ursin, S. Ramelow, arXiv:1309.0712 (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, uoe

oe

1 2 1 2

LR bound: 0

Quantum bound: – 0.207 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)

Nij trials for combination (i,j)

Normalized counts (probabilities):

Normalized Eberhard inequality:

Suppress index o, multiply by –1, write p instead of

CH inequality1

LR bound: 0

Quantum bound: – 0.207

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:1403.2811 [quant-ph] (2014)

1 JK, S. Ramelow, M. Giustina, and A. Zeilinger, arxiv:1307.6475 [quant-ph] (2013)

Normalization using production rate1,2 not possible in loophole-free test

Pulsed (synchronized) exp. more feasible

Large distance reduces det. efficiency measurement times TA, TB 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 value1

if every pulse has down-converted pair(recall: qu. bound –0.207)

Achievable: slight mixture:Diagonal basis visibility:

Communication

Bell inequ. with auxiliary communication1

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, 157904 (2003)

Ex.: strategy for case: “ sent to Bob”Relabel “fates”: 1 2

Alice o oBob 1 o o

2 o u

Contributions: noo(1,1), noo(1,2)

noo(2,1), nuu(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, 157904 (2003)

Every quantum state can be simulated: Bacon/Toner1

2. Alice picks setting , and outputs

1. Alice and Bob share random variables picked fromi.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-signalingno-signaling

Opt. no-sig. comm. strategy

opt. ?

(Q)RNGs

Humans1

1 J. S. Bell, La nouvelle cuisine (1990)2 J. Gallicchio et al., PRL 112, 110405 (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, 13029 (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 times4

Distant quasars2

Photons on a beam splitter3

Autocorrelation

1 JK et al., in preparation

Conservatively:

10 – 100 m

10 –

100

nsRequires ~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 timesRandomness: 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 impossibledepends 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