vaxjo2023rdg.pdf
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- The authors performed a loophole-free Bell test experiment using superconducting circuits to violate Bell's inequality. They entangled pairs of qubits over a 30 meter distance and measured them in randomly chosen bases, accumulating over 1 million trials. - The average S value obtained was 2.0747 ± 0.0033, violating Bell's inequality with a p-value smaller than 10-108, demonstrating non-local quantum correlations between the spatially separated qubits. This establishes superconducting circuits as a viable platform for foundational tests of quantum physics and applications in quantum information.
![• Evaluating more than 1 million experimental trials, we
fi
nd an
average S value of 2.0747 ± 0.0033, violating Bell’s inequality
with a p–value smaller than 10–108
• For the
fi
nal Bell test with an optimal angle θ (see main text), we
performed n = 220 Bell trials and obtained c = 796228 wins in the
Bell game. With these values we
fi
nd p ≤ 10–108
Notice how close they are …
Two log10 p–values
> pnorm(747/33, lower.tail = FALSE, log.p = TRUE) / log(10)
[1] –113.0221
> pbinom(796228 – 1, 2^20, lower.tail = FALSE, prob = 3/4,
+ log.p = TRUE) / log(10)
[1] –108.6195
2^20 = 1,048,576](https://image.slidesharecdn.com/vaxjo2023rdg-230602104303-4abb6e5e/85/vaxjo2023rdg-pdf-5-320.jpg)
![10 3
3 10
10 3
3 10
10 3
3 10
3 10
10 3
Three correlations equal to 14 / 26 = 1 / 2 + 1 / 26 = 0.54
One equal to –14 / 26 = –0.54
S = 4 * 14 / 26 = 2 + 4 / 26 = 2 + 2 / 13 = 2.15
J = (S – 2) / 4 = 1 / 26 = 0.038
Probability of winning Bell game = 20 / 26 = 10 / 13 = 0.77 > 0.75
The random setting generators seems perfect
> ABdsn <-
+ c(sum(table11), sum(table12), sum(table21), sum(table22))
> ABdsn
[1] 262304 262369 262200 261703
> N <- sum(ABdsn)
> N
[1] 1048576
> expected <- N * c(1, 1, 1, 1) / 4
> expected
[1] 262144 262144 262144 262144
> chisquare <- sum( (ABdsn - expected)^2 / expected)
> chisquare
[1] 1.044624
> pchisq(1, 3, lower.tail = TRUE)
[1] 0.198748](https://image.slidesharecdn.com/vaxjo2023rdg-230602104303-4abb6e5e/85/vaxjo2023rdg-pdf-7-320.jpg)
![> pnorm((CHSHopt – 2)/ sqrt(varCHSHopt), lower.tail = FALSE)
[1] 7.727839e–112
> pchisq(2 * (– optim(ThetaOpt, negloglik)$value + optim(ThetaOptL[1:7],
+ negloglikL)$value), 1, lower.tail = FALSE) / 2
[1] 5.668974e–110
> pbinom(796228 – 1, 2^20, lower.tail = FALSE, prob = 3/4,
+ log.p = TRUE) / log(10)
[1] –108.6195
> CHSH
[1] 2.074724
> CHSHopt
[1] 2.074882](https://image.slidesharecdn.com/vaxjo2023rdg-230602104303-4abb6e5e/85/vaxjo2023rdg-pdf-9-320.jpg)
vaxjo2023rdg.pdf
- 1. Richard D. Gill (Leiden University), 13 June 2023, QIP Växjö Statistical analysis of the recent Bell experiments “If your experiment needs statistics, you ought to have done a better experiment”
- 2. • RD Gill, Optimal Statistical Analyses of Bell Experiments, AppliedMath 2023, 3(2), 446-460; https://doi.org/10.3390/ appliedmath3020023 • Storz, S., Schär, J., Kulikov, A. et al. Loophole-free Bell inequality violation with superconducting circuits. Nature 617, 265–270 (2023). https://doi.org/10.1038/s41586-023-05885-0 • Giustina M. Superconducting qubits cover new distances. Nature 617 (7960), 254-256. https://doi.org/10.1038/d41586-023-01488-x • I have promised Marian Kupczynski not to talk about: RD Gill & JP Lambare, Kupczynski’s Contextual Locally Causal Probabilistic Models Are Constrained by Bell’s Theorem, Quantum Reports 2023; https://arxiv.org/abs/2208.09930 Storz et. al. – ETH Zürich – Nature 617, 265–270 (2023) Loophole-free Bell inequality violation with superconducting circuits
- 3. Richard D. Gill, 13 June 2023 On: “Loophole–free Bell inequality violation with superconducting circuits”
- 4. Loophole-freeBellinequalityviolationwith superconductingcircuits Simon Storz1✉,Josua Schär1 ,Anatoly Kulikov1 ,Paul Magnard1,10 ,Philipp Kurpiers1,11 , Janis Lütolf1 ,Theo Walter1 ,Adrian Copetudo1,12 ,Kevin Reuer1 ,Abdulkadir Akin1 , Jean-Claude Besse1 ,Mihai Gabureac1 ,Graham J. Norris1 ,Andrés Rosario1 ,Ferran Martin2 , José Martinez2 ,Waldimar Amaya2 ,Morgan W. Mitchell3,4 ,Carlos Abellan2 ,Jean-Daniel Bancal5 , Nicolas Sangouard5 ,Baptiste Royer6,7 ,Alexandre Blais7,8 &Andreas Wallraff1,9✉ Superposition,entanglementandnon-localityconstitutefundamentalfeaturesof quantumphysics.Thefactthatquantumphysicsdoesnotfollowtheprincipleoflocal causality1–3 canbeexperimentallydemonstratedinBelltests4 performedonpairsof spatiallyseparated,entangledquantumsystems.AlthoughBelltests,whicharewidely regardedasalitmustestofquantumphysics,havebeenexploredusingabroadrange ofquantumsystemsoverthepast50years,onlyrelativelyrecentlyhaveexperiments freeofso-calledloopholes5 succeeded.Suchexperimentshavebeenperformedwith spinsinnitrogen–vacancycentres6 ,opticalphotons7–9 andneutralatoms10 .Herewe demonstratealoophole-freeviolationofBell’sinequalitywithsuperconducting circuits,whichareaprimecontenderforrealizingquantumcomputingtechnology11 . ToevaluateaClauser–Horne–Shimony–Holt-typeBellinequality4 ,wedeterministically entangleapairofqubits12 andperformfastandhigh-fidelitymeasurements13 along randomlychosenbasesonthequbitsconnectedthroughacryogeniclink14 spanning adistanceof30 metres.Evaluatingmorethan1 millionexperimentaltrials,wefindan averageSvalueof2.0747 ± 0.0033,violatingBell’sinequalitywithaPvaluesmallerthan 10−108 .Ourworkdemonstratesthatnon-localityisaviablenewresourceinquantum informationtechnologyrealizedwithsuperconductingcircuitswithpotential applicationsinquantumcommunication,quantumcomputingandfundamental physics15 . Oneoftheastoundingfeaturesofquantumphysicsisthatitcontradicts ourcommonintuitiveunderstandingofnaturefollowingtheprinciple oflocalcausality1 .Thisconceptderivesfromtheexpectationthatthe causes of an event are to be found in its neighbourhood (see Supple- mentaryInformationsectionIforadiscussion).In1964,JohnStewart Bellproposedanexperiment,nowknownasaBelltest,toempirically demonstratethattheoriessatisfyingtheprincipleoflocalcausalitydo notdescribethepropertiesofapairofentangledquantumsystems2,3 . cannotdependoninformationavailableatthelocationofpartyBand vice versa, and measurement independence, the idea that the choice between the two possible measurements is statistically independent from any hidden variables. AdecadeafterBell’sproposal,thefirstpioneeringexperimentalBell tests were successful16,17 . However, these early experiments relied on additionalassumptions18 ,creatingloopholesintheconclusionsdrawn fromtheexperiments.Inthefollowingdecades,experimentsrelyingon https://doi.org/10.1038/s41586-023-05885-0 Received: 22 August 2022 Accepted: 24 February 2023 Published online: 10 May 2023 Open access Check for updates Nature | Vol 617 | 11 May 2023 | 265 ToevaluateaClauser–Horne–Shimony–Holt-typeBellinequality4 ,wedeterministically entangleapairofqubits12 andperformfastandhigh-fidelitymeasurements13 along randomlychosenbasesonthequbitsconnectedthroughacryogeniclink14 spanning adistanceof30 metres.Evaluatingmorethan1 millionexperimentaltrials,wefindan averageSvalueof2.0747 ± 0.0033,violatingBell’sinequalitywithaPvaluesmallerthan 10−108 .Ourworkdemonstratesthatnon-localityisaviablenewresourceinquantum informationtechnologyrealizedwithsuperconductingcircuitswithpotential applicationsinquantumcommunication,quantumcomputingandfundamental physics15 . Oneoftheastoundingfeaturesofquantumphysicsisthatitcontradicts ourcommonintuitiveunderstandingofnaturefollowingtheprinciple oflocalcausality1 .Thisconceptderivesfromtheexpectationthatthe causes of an event are to be found in its neighbourhood (see Supple- mentaryInformationsectionIforadiscussion).In1964,JohnStewart Bellproposedanexperiment,nowknownasaBelltest,toempirically demonstratethattheoriessatisfyingtheprincipleoflocalcausalitydo notdescribethepropertiesofapairofentangledquantumsystems2,3 . In a Bell test4 , two distinct parties A and B each hold one part of an entangledquantumsystem,forexample,oneoftwoqubits.Eachparty then chooses one of two possible measurements to perform on their qubit, and records the binary measurement outcome. The parties repeat the process many times to accumulate statistics, and evaluate aBellinequality2,4 usingthemeasurementchoicesandrecordedresults. Systems governed by local hidden variable models are expected to obey the inequality whereas quantum systems can violate it. The two underlyingassumptionsinthederivationofBell’sinequalityarelocality, theconceptthatthemeasurementoutcomeatthelocationofpartyA cannotdependoninformationavailableatthelocationofpartyBand vice versa, and measurement independence, the idea that the choice between the two possible measurements is statistically independent from any hidden variables. AdecadeafterBell’sproposal,thefirstpioneeringexperimentalBell tests were successful16,17 . However, these early experiments relied on additionalassumptions18 ,creatingloopholesintheconclusionsdrawn fromtheexperiments.Inthefollowingdecades,experimentsrelyingon fewerandfewerassumptionswereperformed19–21 ,untilloophole-free Bell inequality violations, which close all major loopholes simultane- ously,weredemonstratedin2015andthefollowingyears6–10 ;seeref.22 for a discussion. Inthedevelopmentofquantuminformationscience,itbecameclear that Bell tests relying on a minimum number of assumptions are not only of interest for testing fundamental physics but also serve as a key resource in quantum information processing protocols. Observ- ing a violation of Bell’s inequality indicates that the system possesses non-classical correlations, and asserts that the potentially unknown 1 Department of Physics, ETH Zurich, Zurich, Switzerland. 2 Quside Technologies S.L., Castelldefels, Spain. 3 ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), Spain. 4 ICREA - Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain. 5 Institute of Theoretical Physics, University of Paris-Saclay, CEA, CNRS, Gif-sur-Yvette, France. 6 Department of Physics, Yale University, New Haven, CT, USA. 7 Institut quantique and Départment de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada. 8 Canadian Institute for Advanced Research, Toronto, Ontario, Canada. 9 Quantum Center, ETH Zurich, Zurich, Switzerland. 10 Present address: Alice and Bob, Paris, France. 11 Present address: Rohde and Schwarz, Munich, Germany. 12 Present address: Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore. ✉e-mail: simon.storz@phys.ethz.ch; andreas.wallraff@phys.ethz.ch
- 5. • Evaluating more than 1 million experimental trials, we fi nd an average S value of 2.0747 ± 0.0033, violating Bell’s inequality with a p–value smaller than 10–108 • For the fi nal Bell test with an optimal angle θ (see main text), we performed n = 220 Bell trials and obtained c = 796228 wins in the Bell game. With these values we fi nd p ≤ 10–108 Notice how close they are … Two log10 p–values > pnorm(747/33, lower.tail = FALSE, log.p = TRUE) / log(10) [1] –113.0221 > pbinom(796228 – 1, 2^20, lower.tail = FALSE, prob = 3/4, + log.p = TRUE) / log(10) [1] –108.6195 2^20 = 1,048,576
- 6. Counts Na,b,x,y x = +1 y = +1 x = +1 y = -1 x = -1 y = +1 x = -1 y = -1 a = 0, b = 0 100,529 31,780 29,926 99,965 a = 0, b = 1 30,638 101,342 96,592 33,131 a = 1, b = 0 94,661 30,018 35,565 102,060 a = 1, b = 1 96,291 29,186 32,104 104,788 ABLE SV. Raw counts of the individual occurrences for al Bell test for fixed o↵set angle ✓ = ⇡/4 with the m tistics (220 trials), presented in the main text. e correlators 10 3 3 10 3 10 10 3 10 3 3 10 10 3 3 10 In ten thousands, rounded
- 7. 10 3 3 10 10 3 3 10 10 3 3 10 3 10 10 3 Three correlations equal to 14 / 26 = 1 / 2 + 1 / 26 = 0.54 One equal to –14 / 26 = –0.54 S = 4 * 14 / 26 = 2 + 4 / 26 = 2 + 2 / 13 = 2.15 J = (S – 2) / 4 = 1 / 26 = 0.038 Probability of winning Bell game = 20 / 26 = 10 / 13 = 0.77 > 0.75 The random setting generators seems perfect > ABdsn <- + c(sum(table11), sum(table12), sum(table21), sum(table22)) > ABdsn [1] 262304 262369 262200 261703 > N <- sum(ABdsn) > N [1] 1048576 > expected <- N * c(1, 1, 1, 1) / 4 > expected [1] 262144 262144 262144 262144 > chisquare <- sum( (ABdsn - expected)^2 / expected) > chisquare [1] 1.044624 > pchisq(1, 3, lower.tail = TRUE) [1] 0.198748
- 8. a = 0, b = 0 a = 0, b = 1 a = 1, b = 0 a = 1, b = 1 x = +1, y = +1 100529 30638 94661 96291 x = +1, y = –1 31780 101342 30018 29186 x = –1, y = +1 29926 96592 35565 32104 x = –1, y = –1 99965 33131 102060 104788 Zürich, transposed a’ = 0, b = 0 a’ = 0, b = 1 a’ = 1, b = 0 a’ = 1, b = 1 x = +1, y = +1 94661 96291 100529 30638 x = +1, y = –1 30018 29186 31780 101342 x = –1, y = +1 35565 32104 29926 96592 x = –1, y = –1 102060 104788 99965 33131 Zürich, transposed & Alice’s setting fl ipped
- 9. > pnorm((CHSHopt – 2)/ sqrt(varCHSHopt), lower.tail = FALSE) [1] 7.727839e–112 > pchisq(2 * (– optim(ThetaOpt, negloglik)$value + optim(ThetaOptL[1:7], + negloglikL)$value), 1, lower.tail = FALSE) / 2 [1] 5.668974e–110 > pbinom(796228 – 1, 2^20, lower.tail = FALSE, prob = 3/4, + log.p = TRUE) / log(10) [1] –108.6195 > CHSH [1] 2.074724 > CHSHopt [1] 2.074882
- 10. Inputs (binary) Outputs (binary) Time Distance (left to right) is so large that a signal travelling from one side to the other at the speed of light takes longer than the time interval between input and output on each side One “go = yes” trial has binary inputs and outputs; model as random variables A, B, X, Y Image: figure 7 from J.S. Bell (1981), “Bertlmann’s socks and the nature of reality”
- 12. The long box in the middle
- 17. • Suppose multinomial distribution, condition on 4 counts N(a, b) • We expect p(x | a, b) does not depend on b, and p(y | a, b) does not depend on a • If so, 4 empirical “deviations from no-signalling” are noise (mean = 0). That noise is generally correlated with the noise in the CHSH statistic S • Reduce noise in S by subtracting prediction of statistical error, given statistical errors in no-signalling equalities: 2SLS with plug- in estimates of variances and covariances Reduce data to 16 counts N(a, b, x, y) ∧ Theory (Method 1) ∧ ∧
- 18. • Estimate the 4 sets of 4 tetranomial probabilities p(x, y | a, b) by maximum likelihood assuming no-signalling (four linear constraints) (a) Assuming local realism (8 linear inequalities) and (b) Without assuming local realism • Test null hypothesis of local realism using Wilks’ log likelihood ratio test Theory (Method 2) Reduce data to 16 counts N(a, b, x, y) ∧
- 19. • Use martingale test (“Bell game”) • Suppose all p(a, b) = 0.25 • Compute N( = | 1, 1) + N( = | 1, 2) + N( = | 2, 1) + N( ≠ | 2, 2); compare to Bin(N, 3/4) Theory (Method 3) Reduce data to 16 counts N(a, b, x, y)
- 20. • The 3 tests are asymptotically equivalent if their model assumptions are satis fi ed and the probabilities p(x, y | a, b) have the expected symmetries Comparison of the 3 p-values Theory f g g f f g g f f g g f g f f g