Towards "evidence based physics"
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The document discusses experiments testing Bell's theorem and quantum nonlocality. It notes that while experiments have shown violations of Bell inequalities for over 50 years, all previous experiments suffered from loopholes such as memory effects, detection inefficiency, and limited statistics. The text advocates for "evidence based physics" with rigorous experiments that close all loopholes through randomization, blinding procedures, intention-to-treat analysis, and treating individual time slots rather than particle pairs as the experimental unit. Upcoming experiments in Delft and jointly in Delft and Leiden aim to finally achieve a loophole-free test of Bell inequality violations.
![Christensen et al. 2013
nonical Clauser-Horne-
0] of 2:827 Æ 0:017—
ion allowed by quantum
alues are on par with the
Bell’s inequality ever
viously reported results
r postprocessing of any
vides not only the best
bits required a month of data collection. Here we show that
our setup can be used to implement DIRE much more
efficiently. The intrinsic randomness of the quantum sta-
tistics can be quantified as follows. The probability for any
observer (hence, for any potential adversary) to guess the
measurement outcome (of a given measurement setting) is
bounded by the amount of violation of the CH inequality:
pguess ½1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 À ð1 þ 2BÞ2
p
Š=2 [33], neglecting finite
size effects. In turn, this leads to a bound on the min-
entropy per bit of the output data, Hmin ¼ Àlog2ðpguessÞ.
Finally, secure private random bits can be extracted from
the data (which may, in general, not be uniformly random)
using a randomness extractor [37]. At the end of the pro-
tocol, a final random bit string of length L % NHmin À S is
produced, where N is the length of the raw output data and
S includes the inefficiency and security overhead of the
extractor.
Over the 4450 measurement blocks (each block
features 25 000 events), we acquire 111 259 682 data
points for 3 h of data acquisition. The average CH viola-
tion of B ¼ 5:4 Â 10À5
gives a min-entropy of Hmin ¼
7:2 Â 10À5
. Thus, we expect $8700 bits of private ran-
domness, of which one could securely extract at least 4350
bits [19]. The resultant rate (0:4 bits=s) improves by more
than 4 orders of magnitude over the bit rate achieved in
[33] (1:5 Â 10À5 bits=s). This shows that efficient and
practical DIRE can be implemented with photonic
systems.
We have presented a new entangled photon pair creation,
Bell parameter B0 [Eq. (3)]
d state. B0 > 1 (red dashed
stic theory. Data points in
ied in the state ðrjHHi þ
to a maximally entangled
ata point was measured for
he particular settings were
el of our source for each
Bell parameter we expect
improve the statistics, we
els cell. We see violations
measurements on entangled particles, and observing non-
local correlations between the outcomes of these measure-
ments, it is possible to certify the presence of genuine
randomness in the data in a device-independent way.
DIRE was recently demonstrated in a proof-of-principle
experiment using entangled atoms located in two traps
separated by 1 m [33]; however, the resulting 42 random
bits required a month of data collection. Here we show that
our setup can be used to implement DIRE much more
efficiently. The intrinsic randomness of the quantum sta-
tistics can be quantified as follows. The probability for any
observer (hence, for any potential adversary) to guess the
measurement outcome (of a given measurement setting) is
bounded by the amount of violation of the CH inequality:
pguess ½1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 À ð1 þ 2BÞ2
p
Š=2 [33], neglecting finite
size effects. In turn, this leads to a bound on the min-
entropy per bit of the output data, Hmin ¼ Àlog2ðpguessÞ.
Finally, secure private random bits can be extracted from
the data (which may, in general, not be uniformly random)
using a randomness extractor [37]. At the end of the pro-
tocol, a final random bit string of length L % NHmin À S is
Bell parameter B0 [Eq. (3)]
d state. B0 > 1 (red dashed
stic theory. Data points in
ied in the state ðrjHHi þ
s to a maximally entangled
ata point was measured for
P H Y S I C A L R E V I E W L E T T E R S
week ending
27 SEPTEMBER 2013
Application: DIRE
Device Independent Randomness Extraction](https://image.slidesharecdn.com/einstein-150306035102-conversion-gate01/85/Towards-evidence-based-physics-26-320.jpg)
![Christensen vs Giustina
11 12 21 22
dd 1054015 1140690 1178589 124696
dn 467388 377147 3509092 4557443
nd 639434 3336595 508696 4344146
nn 297803163 295109568 294767623 290937715
tot 299964000 299964000 299964000 299964000
11 12 21 22
dd 0.003513804989932130.003802756330759690.003929101492179060.00041570321771946
dn 0.001558146977637320.00125730754357190.011698377138590.0151932998626502
nd 0.002131702470963180.01112331813150910.001695856836153670.0144822245336107
nn 0.9927963455614670.9838166179941590.9826766645330770.96990877238602
11 12 21 22
dd 29173 34146 34473 1869
dn 16897 13931 116364 148844
nd 17029 112097 12975 142502
nn 27089919 28192171 27663499 27633773
tot 27153018 28352345 27827311 27926988
11 12 21 22
dd 0.00107439254082180.001204344825798360.001238818943016090.0000669245104412979
dn 0.0006222881007186750.0004913526553094640.004181647303255420.00532975485934967
nd 0.00062714943878430.003953711765287840.0004662685517835340.00510266270032415
nn 0.9976761699196750.9943505907536040.9941132652019450.989500657929885
> (sum(chris[2:3, 4]) - sum(chris[2:3, 1:3])) / sqrt(sum(chris[2:3, ]))
[1] 2.694237
> (sum(giust[2:3, 4]) - sum(giust[2:3, 1:3])) / sqrt(sum(giust[2:3, ]))
[1] 15.01396
Giustina et al: 5 times as many standard deviations (10 times as much data)
Christensen et al: 20 runs with total of 6000 random setting changes
Giustina et al: four runs with fixed settings](https://image.slidesharecdn.com/einstein-150306035102-conversion-gate01/85/Towards-evidence-based-physics-31-320.jpg)
Towards "evidence based physics"
- 1. Towards “evidence based physics” Richard Gill General Colloquium, 19 March 2015 Mathematical Institute Leiden Rutherford: If you need statistics you did the wrong experiment John Stewart Bell (1928–1990) Bell’s theorem 1964–2014 http://www.math.leidenuniv.nl/~gill/einstein.m4a
- 2. > 50 years after Bell (1964) – still no loophole-free experimental proof of quantum non-locality – but the experimenters are getting damn close! (thanks to statistics)
- 3. Those damned loopholes… • Memory loophole • Detection efficiency loophole • Locality loophole • Conspiracy loophole • Coincidence loophole • Finite statistics loophole
- 4. Weihs et al. (1998) Coming soon: Delft; Delft + Leiden 5.2. Verletzungder Bell-Ungleichungin einzelnenMessungen Bob 1 , + 1 , – 2 , + 2 , – Zählungen 104122 100144 93348 90841 1 , + 77988 313 1728 1636 179 1 , – 74935 1978 351 294 1143 2 , + 75892 418 1683 269 1100 Alice 2 , – 73456 1578 361 1386 156 Koinzidenzen Zeitverschiebung 3,8 ns QuantentheoretischeVorhersage Koinzidenzfenster 4,0 ns Koinzidenzengesamt 14573 ideal bei 94% -97% Kontrast ρ(1,1) -0,70 ± 0,01 -0,71 0,68 ± 0,02 ρ(2,1) -0,61 ± 0,01 -0,71 0,68 ± 0,02 ρ(1,2) 0,71 ± 0,01 0,71 0,68 ± 0,02 ρ(2,2) -0,71 ± 0,01 -0,71 0,68 ± 0,02 S 2,73 ± 0,02 2,82 2,72 ± 0,04 Verletzung 29,8 stddev lle 5.3.: Auswertungvonlongdist35, aufgenommen am22.4.1998. Zus®atzlichsindnochdiequantentheo- henVorhersagenf®ur dieWertederK orrelationsfunktionenangegeben,einmalf®ur einidealesE xperimenteinmal ziertum denerzieltenK ontrast (s. Tabelle5.2). Weil dieserK ontrast nichtf®ur alle Winkelstellungengleichist, Prob(≠): 0.85, 0.15, 0.81, 0.85 Prob(=): 0.15, 0.85, 0.19, 0.15
- 5. Weihs et al. (1998)5.2. Verletzungder Bell-Ungleichungin einzelnenMessungen Bob 1 , + 1 , – 2 , + 2 , – Zählungen 104122 100144 93348 90841 1 , + 77988 313 1728 1636 179 1 , – 74935 1978 351 294 1143 2 , + 75892 418 1683 269 1100 Alice 2 , – 73456 1578 361 1386 156 Koinzidenzen Zeitverschiebung 3,8 ns QuantentheoretischeVorhersage Koinzidenzfenster 4,0 ns Koinzidenzengesamt 14573 ideal bei 94% -97% Kontrast ρ(1,1) -0,70 ± 0,01 -0,71 0,68 ± 0,02 ρ(2,1) -0,61 ± 0,01 -0,71 0,68 ± 0,02 ρ(1,2) 0,71 ± 0,01 0,71 0,68 ± 0,02 ρ(2,2) -0,71 ± 0,01 -0,71 0,68 ± 0,02 S 2,73 ± 0,02 2,82 2,72 ± 0,04 Verletzung 29,8 stddev Tabelle 5.3.: Auswertungvonlongdist35, aufgenommen am22.4.1998. Zus®atzlichsindnochdiequantentheo- retischenVorhersagenf®ur dieWertederK orrelationsfunktionenangegeben,einmalf®ur einidealesE xperimenteinmal reduziertum denerzieltenK ontrast (s. Tabelle5.2). Weil dieserK ontrast nichtf®ur alle Winkelstellungengleichist, ergebensich gewisseFehler. AuflerdemmuflbeimVergleichderDatenmit denVorhersagenber®ucksichtigtwerden, Estimated Prob(≠): 0.85, 0.15, 0.81, 0.85 Estimated Prob(=): 0.15, 0.85, 0.19, 0.15
- 6. Ursin et al. (2006) Distance no worries for spooky particles Stephen Pincock ABC Science Online Friday,!8!December!2006 A message sent using entangled, or spooky, particles of light has been beamed across the ocean (Image: iStockphoto) Scientists have used quantum physics to zap an encrypted message more than 140 kilometres between two Spanish islands. Professor Anton Zeilinger from the University of Vienna and an international team of scientists used 'spooky' pulses of light to send the message. They say this an important step towards making international communications more secure. Zeilinger described the study this week at the Australian Institute Physics meeting in Brisbane. The photons they sent were linked together through a process known as quantum entanglement. This means that their properties remained tightly entwined or entangled, even when separated by large distances, a property Einstein called spooky. The group's achievement is important for the emerging field of quantum cryptography, which aims to use properties such as entanglement to send encrypted messages. Research groups around the world are working in this field. But until now they have only been able to send messages relatively shor distances, limiting their usefulness. Zeilinger's team wants to be able to beam the messages to satellites in space, so they could theoretically be relayed anywhere on the planet. To test their system, the team went to Tenerife in the Canary Islands, where the European Space Agency operates a telescope specifically desig to communicate with satellites. Instead of pointing the telescope at the stars, Zeilinger says, the scientists turned it to the horizontal and aimed towards a photon sending station 144 kilometres away on the neighbouring island of La Palma. "Very broadly speaking, we were able to establ a quantum communication connection," he says. "We worried a lot about whether atmospheric turbulence would destroy the quantum states. B turned out to work much better than we feared." The results suggest it should be possible to send encrypted photons to a satellite orbiting 300 o 400 kilometres above the Earth, he says. "This is our hope. We believe that such a system is feasible." The next step is to try the system out wit an actual satellite, a project which is likely to involve the European Space Agency and others. "This is about developing quantum communicati
- 8. J.S. Bell, “Bertlmann’s socks and the nature of reality” Timed settings Outputs (outcomes) Journal de Physique, Colloque C2, suppl. au no 3, Tome 42 (1981), C2 41–61
- 9. Timed settings α, β; values 1, 2 Outputs: outcomes A, B, values ± 1 Alice’s (binary) measurement outcome registered before Bob’s setting could become available, and vice-versa Binary settings chosen independently, completely at random α β A B
- 10. Local realism • Realism: we can define (in the model) what Alice and Bob’s outcomes would have been, for all possible values of their settings • Locality: Alice’s outcomes can’t depend on which setting is chosen by Bob, and vice-versa • Freedom: Alice and Bob have free choice of settings existence of counterfactual outcomes A1, A2, B1, B2; observed outcomes A, B determined by independent random setting choices (A = A1 or A2; B = B1 or B2)
- 12. Let A1, B1, A2, B2 be members of {"red", "green"}. Notice that (A1 = B2) & (B2 = A2) & (A2 = B1) (B1 = A1) Therefore (A1 ≠ B2) or (B2 ≠ A2) or (A2 ≠ B1) ⇐ (B1 ≠ A1 ) Therefore Prob(A1 ≠ B2) + Prob(B2 ≠ A2) + Prob(A2 ≠ B1) ≥ Prob(B1 ≠ A1) Just feasible: three LHS probabilities each equal to 25%, RHS equal to 75% QM allows three LHS probabilities equal to 15%, RHS equal to 85%. 15% = 0.1464…= ½ – ¼ √2; 85% = 0.8536…= ½ + ¼ √2 A1 A2 B1 B2
- 13. Bell’s inequality: Prob(A1 ≠ B2) + Prob(B2 ≠ A2) + Prob(A2 ≠ B1) ≥ Prob(B1 ≠ A1) More Bell inequalities by flipping outcomes, e.g. flipping B1 and B2 Prob(A1 = B2) + Prob(B2 = A2) + Prob(A2 = B1) ≥ Prob(B1 = A1) More inequalities by picking different side of square … Theorem (Fine, 1982). Set of 8 inequalities is complete and minimal A1 A2 B1 B2
- 14. S Tsirelson’s bound deterministic local-realistic model S’ 4 2 -2 -4 -4 -2 2 4 completely random 22 22- 22- 22 a generalized Bell inequality another another 2 x 2 x 2 p x q x r General case: p parties, each with q measurement settings, each measurement with r outcomes Generalised Bell inequalities (a cartoon)
- 15. How QM can do it • Random variables σx (1), σy (1), σx (2), σy (2) • Values ±1 • (σx (1)σy (2)) (σy (1)Sx (2)) (σy (1)σy (2)) = + (σx (1)σx (2))
- 16. How QM can do it • ∃ self-adjoint operators σx (1), σy (1), σx (2), σy (2) • eigenvalues ± 1 • “(1) operators” commute with “(2) operators” • “x operator” and “y operator” anti-commute • (σx (1)σy (2)) (σy (1)σx (2)) (σy (1)σy (2)) = – (σx (1)σx (2))
- 17. How QM can do it • (σx (1)σy (2)) (σy (1)σx (2)) (σy (1)σy (2)) = – (σx (1)σx (2)) • (σx (1)σy (2)), (σy (1)σx (2)), (σy (1)σy (2)), (σx (1)σx (2)) do not commute; have eigenvalues ±1 • Find vector as close as possible to an eigenvector with eigenvalue –1 of (σx (1)σy (2)), and of (σy (1)σx (2)), and of (σy (1)σy (2)), and as close as possible to an eigenvector of (σx (1)σx (2)) with eigenvalue +1 • The closest you can get leads to 0.85… Tsirelson, 1980
- 18. Why not better? Information causality! • Principle of information causality: if Bob sends m bits of information to Alice, then Alice gains at most m bits of information of Bob’s that she didn’t know before • No action at a distance = no-signalling = information causality with m = 0 • Pawlowski et al. (2009; Nature). (1) QM satisfies the principle of information causality. (2) No physical theory which satisfies the principle of information causality can do better than 0.85…
- 19. Why more experiments after Weihs 1998? Or even: after Aspect 1982? • Memory loophole • Detection efficiency loophole • Locality loophole • Conspiracy loophole • Coincidence loophole • Finite statistics loophole 5.2. Verletzungder Bell-Ungleichungin einzelnenMes Bob 1 , + 1 , – 2 , + 2 , – Zählungen 104122 100144 93348 90841 1 , + 77988 313 1728 1636 179 1 , – 74935 1978 351 294 1143 2 , + 75892 418 1683 269 1100 Alice 2 , – 73456 1578 361 1386 156 Koinzidenzen Zeitverschiebung 3,8 ns QuantentheoretischeVorhersage Koinzidenzfenster 4,0 ns Koinzidenzengesamt 14573 ideal bei 94% -97% Kontrast ρ(1,1) -0,70 ± 0,01 -0,71 0,68 ± 0,02 ρ(2,1) -0,61 ± 0,01 -0,71 0,68 ± 0,02 ρ(1,2) 0,71 ± 0,01 0,71 0,68 ± 0,02 ρ(2,2) -0,71 ± 0,01 -0,71 0,68 ± 0,02 S 2,73 ± 0,02 2,82 2,72 ± 0,04 Verletzung 29,8 stddev Tabelle 5.3.: Auswertungvonlongdist35, aufgenommen am22.4.1998. Zus®atzlichsindnochdiequan retischenVorhersagenf®ur dieWertederK orrelationsfunktionenangegeben,einmalf®ur einidealesE xperime reduziertum denerzieltenK ontrast (s. Tabelle5.2). Weil dieserK ontrast nichtf®ur alleWinkelstellungeg ergebensich gewisseFehler. AuflerdemmuflbeimVergleichderDatenmit denVorhersagenber®ucksichtig dafl dieimE xperimentrealisiertenAnalysatorwinkelnichtexakt denidealenWinkelnentsprechen. Beobachter-Stationen die A nalysatorstellungen durcheine zus®atzliche G leichspannungumein stantenWinkelvon rotiert. Wenn dieseR otationbei Bob angewandtwird,dannentspre verschiedenenModulatorstellungendenfolgendenWinkeln: Prob(≠): 0.85, 0.15, 0.81, 0.85 Prob(=): 0.15, 0.85, 0.19, 0.15
- 20. Towards “Evidence based physics” • Randomisation, “blinding” deals with memory, conspiracy, finite statistics, … • “Intention to treat principle” deals with detection inefficiency, coincidence loophole, … • “Experimental unit” is time-slot, not: pair of entangled particles
- 21. 1(B1 ≠ A1) ≤ 1(A1 ≠ B2) + 1(B2 ≠ A2) + 1(A2 ≠ B1) Define Alice and Bob’s settings, α, β ∈ {1, 2 }, completely random Counterfactual outcomes A1, A2, B1, B2 fixed ∈ {–1, +1} Define Iij = 1(α = i & β = j) E( I12 1(A1 ≠ B2) + I22 1(A2 ≠ B2) + I21 1(A2 ≠ B1) – I11 1(A1 ≠ B1) ) ≥ 0 Deterministic physics for measurement outcomes; random measurement settings! Towards “evidence based physics” (1) Randomisation A1 A2 B1 B2
- 22. Theorem (Gill 2003) • N consecutive trials, new random settings in each trial, arbitrary “local realistic” physics for outcomes (arbitrary memory effects, …) • Define Z = N12(≠) + N22(≠) + N21(≠) – N11(≠) • We expect Z to be positive • Using martingale Hoeffding inequality: for c > 0, Prob( Z / N ≤ – c /√N) ≤ exp(– ½ c2) QM best (in expectation): Z ≈ (0.15 + 0.15 + 0.15 – 0.85) N / 4
- 23. A new generation of experiments – and a priority dispute • Giustina et al (2013, Nature) • Christensen et al. (2013, PRL) Towards “evidence based physics” (2) “Intention to treat principle” Experimental unit = time-slot, not particle pair
- 24. Giustina et al. 2013
- 25. 3/6/2015 Physics - Closing Quantum Loopholes American Physical Society Log in | Create Account (what's this?) Home About Browse APS Journals SEARCH B. G. Christensen et al., Phys. Rev. Lett. (2013) DetectionLoopholeFree Test of Quantum Nonlocality, and Applications B. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calkins, T. Gerrits, A. E. Lita, A. Miller, L. K. Shalm, Y. Zhang, S. W. Nam, N. Brunner, C. C. W. Lim, N. Gisin, and P. G. Kwiat Phys. Rev. Lett. 111, 130406 (2013) Published September 26, 2013 Synopsis: Closing Quantum Loopholes The oddness of quantum mechanics came to the fore in an epic faceoff between Einstein and Bohr in the 1930s. On one side, Einstein said that quantum mechanics discards accepted notions of localism (a measurement couldn’t be influenced by a remote experiment) and realism (outcomes of measurements only depend on the thing being measured, not on the measurer). On the other side, Bohr said that’s right, and you’d better get used to it. Writing in Physical Review Letters, Bradley Christensen at the University of Illinois at UrbanaChampaign and colleagues report an experiment that solves one of the major experimental limitations that have denied a definitive win for quantum mechanics. In the 1960s, John Bell put the debate on a quantitative footing by publishing mathematical inequalities that, if violated, would conclusively rule out any classical theory, local and realistic, as an alternative to quantum mechanics and thus explain observations like entanglement. Many experiments have shown that quantum systems violate Bell’s inequalities, but they are all hampered by at least one of two loopholes. One, the locality loophole, might allow some kind of information to travel from one particle to another. The second is the detection loophole: if the measurement uses a lowefficiency detector, then you might not be sampling all the entangled photons properly. A recent paper by another group claiming to have closed the detection loophole is disputed by Christensen et al., who have carried out new measurements with a highefficiency source of entangled photons and highefficiency detectors. This eliminates assumptions about properly sampling the entangled photons and, the authors say, firmly closes the door on the detection loophole. And it’s not just of academic interest: Bell tests are currently used to verify the security of proposed quantum communications systems, so ensuring their reliability is important. – David Voss Article Options Printable Version Subject Areas Quantum Information Quantum Physics Related Articles More Quantum Information Quantum Mechanics Could Improve Radar Focus | Feb 27, 2015 Frequency Comb Produces an Entangled Web Viewpoint | Feb 2, 2015 All Quantum Information » More Quantum Physics Predicting the Quantum Past Synopsis | Mar 5, 2015 Quantum Mechanics Could Share/Email This Christensen et al. 2013
- 26. Christensen et al. 2013 nonical Clauser-Horne- 0] of 2:827 Æ 0:017— ion allowed by quantum alues are on par with the Bell’s inequality ever viously reported results r postprocessing of any vides not only the best bits required a month of data collection. Here we show that our setup can be used to implement DIRE much more efficiently. The intrinsic randomness of the quantum sta- tistics can be quantified as follows. The probability for any observer (hence, for any potential adversary) to guess the measurement outcome (of a given measurement setting) is bounded by the amount of violation of the CH inequality: pguess ½1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À ð1 þ 2BÞ2 p Š=2 [33], neglecting finite size effects. In turn, this leads to a bound on the min- entropy per bit of the output data, Hmin ¼ Àlog2ðpguessÞ. Finally, secure private random bits can be extracted from the data (which may, in general, not be uniformly random) using a randomness extractor [37]. At the end of the pro- tocol, a final random bit string of length L % NHmin À S is produced, where N is the length of the raw output data and S includes the inefficiency and security overhead of the extractor. Over the 4450 measurement blocks (each block features 25 000 events), we acquire 111 259 682 data points for 3 h of data acquisition. The average CH viola- tion of B ¼ 5:4 Â 10À5 gives a min-entropy of Hmin ¼ 7:2 Â 10À5 . Thus, we expect $8700 bits of private ran- domness, of which one could securely extract at least 4350 bits [19]. The resultant rate (0:4 bits=s) improves by more than 4 orders of magnitude over the bit rate achieved in [33] (1:5 Â 10À5 bits=s). This shows that efficient and practical DIRE can be implemented with photonic systems. We have presented a new entangled photon pair creation, Bell parameter B0 [Eq. (3)] d state. B0 > 1 (red dashed stic theory. Data points in ied in the state ðrjHHi þ to a maximally entangled ata point was measured for he particular settings were el of our source for each Bell parameter we expect improve the statistics, we els cell. We see violations measurements on entangled particles, and observing non- local correlations between the outcomes of these measure- ments, it is possible to certify the presence of genuine randomness in the data in a device-independent way. DIRE was recently demonstrated in a proof-of-principle experiment using entangled atoms located in two traps separated by 1 m [33]; however, the resulting 42 random bits required a month of data collection. Here we show that our setup can be used to implement DIRE much more efficiently. The intrinsic randomness of the quantum sta- tistics can be quantified as follows. The probability for any observer (hence, for any potential adversary) to guess the measurement outcome (of a given measurement setting) is bounded by the amount of violation of the CH inequality: pguess ½1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À ð1 þ 2BÞ2 p Š=2 [33], neglecting finite size effects. In turn, this leads to a bound on the min- entropy per bit of the output data, Hmin ¼ Àlog2ðpguessÞ. Finally, secure private random bits can be extracted from the data (which may, in general, not be uniformly random) using a randomness extractor [37]. At the end of the pro- tocol, a final random bit string of length L % NHmin À S is Bell parameter B0 [Eq. (3)] d state. B0 > 1 (red dashed stic theory. Data points in ied in the state ðrjHHi þ s to a maximally entangled ata point was measured for P H Y S I C A L R E V I E W L E T T E R S week ending 27 SEPTEMBER 2013 Application: DIRE Device Independent Randomness Extraction
- 27. Loopholes Giustina et al forgot about coincidence loophole. On re-analysing data appropriately, experiment fortunately still successful! (Larsson et al, 2014, PRA) • Memory loophole • Detection efficiency loophole • Locality loophole • Conspiracy loophole • Coincidence loophole • Finite statistics loophole
- 28. What is the experimental unit? • Physicist’s answer: the entangled pair of particles • Statistician’s answer: the time-slot! • Local coarse-graining forces binary outcome per time-slot • e.g.: if multiple events, take first; merge “–1” and “no event”
- 29. The coincidence loophole Events paired because detection times close enough Events not paired because detection times in different time-slots The problem with Giustina et al: initially analysed as above. Fortunately, after reanalysis as below, still a significant violation of Bell inequality Alice Bob Alice Bob
- 30. The detection loophole By imposing grid of fixed time-slots (preferably: new random settings per time slot) we have an experiment with (at least) ternary outcome: {“red”, “green”, “no detection”} Classical solution: discard time-slots without “coincident events” Correct solution: coarse-grain locally – also for “resolving” multiple events in one time-slot Alice Bob
- 31. Christensen vs Giustina 11 12 21 22 dd 1054015 1140690 1178589 124696 dn 467388 377147 3509092 4557443 nd 639434 3336595 508696 4344146 nn 297803163 295109568 294767623 290937715 tot 299964000 299964000 299964000 299964000 11 12 21 22 dd 0.003513804989932130.003802756330759690.003929101492179060.00041570321771946 dn 0.001558146977637320.00125730754357190.011698377138590.0151932998626502 nd 0.002131702470963180.01112331813150910.001695856836153670.0144822245336107 nn 0.9927963455614670.9838166179941590.9826766645330770.96990877238602 11 12 21 22 dd 29173 34146 34473 1869 dn 16897 13931 116364 148844 nd 17029 112097 12975 142502 nn 27089919 28192171 27663499 27633773 tot 27153018 28352345 27827311 27926988 11 12 21 22 dd 0.00107439254082180.001204344825798360.001238818943016090.0000669245104412979 dn 0.0006222881007186750.0004913526553094640.004181647303255420.00532975485934967 nd 0.00062714943878430.003953711765287840.0004662685517835340.00510266270032415 nn 0.9976761699196750.9943505907536040.9941132652019450.989500657929885 > (sum(chris[2:3, 4]) - sum(chris[2:3, 1:3])) / sqrt(sum(chris[2:3, ])) [1] 2.694237 > (sum(giust[2:3, 4]) - sum(giust[2:3, 1:3])) / sqrt(sum(giust[2:3, ])) [1] 15.01396 Giustina et al: 5 times as many standard deviations (10 times as much data) Christensen et al: 20 runs with total of 6000 random setting changes Giustina et al: four runs with fixed settings
- 32. Giustina et al: Prob(A1 ≠ B1), Prob(A1 ≠ B2), Prob(A2 ≠ B1), Prob(A2 ≠ B2) 37, 124, 134, 297 per thousand 37 + 124 + 134 = 295 < 297 Christensen et al: Prob(A1 ≠ B1), Prob(A1 ≠ B2), Prob(A2 ≠ B1), Prob(A2 ≠ B2) 12, 44, 46, 104 per 10 thousand 12 + 44 + 46 = 102 < 104 Why not 15%, 15%, 15%, 85% ?
- 33. Why not 15%, 15%, 15%, 85%? • “Detector efficiency” & “Less is more” • At QM’s best, we would need somewhat better detectors than when we’re rather far from QM’s best! We don’t have them yet. • A much less entangled state turns out to be much less sensitive to imperfections in measurement! • Giustina, Christensen aimed at best quantum state given their available detector efficiency
- 34. An infinite family of Bell inequalitiesRemember Bell inequality Prob(B1 ≠ A1) ≤ Prob(A1 ≠ B2) + Prob(B2 ≠ A2) + Prob(A2 ≠ B1) Rewrite as: Prob(A ≠ B | 11) ≤ Prob(A ≠ B | 12) + Prob(A ≠ B | 21) + Prob(A ≠ B | 22) We also have the “no-signalling” equalities Prob(A = “red” | 11) = Prob(A = “red” | 12) Prob(A = “red” | 21) = Prob(A = “red” | 22) Prob(B = “red” | 11) = Prob(B = “red” | 21) Prob(B = “red” | 12) = Prob(B = “red” | 22) Take linear combinations to get new inequalities Some of the resulting family have names: Clauser-Horne, Eberhard A statistician easily computes the optimal test (= Bell inequality minus linear combination of no-signalling equalities, with smallest variance) It seems that Giustina et al. and Christensen et al. each used a close to optimal test!
- 35. Conclusion • Maybe in a year or two, we will have a definitive (positive outcome) experiment • Maybe it will be done in Delft • The physicists have learnt that they do need sharp statistical thinking for this experiment
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