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The document discusses a loophole-free violation of Bell's inequality achieved with superconducting circuits, presenting results from over one million experimental trials that demonstrate significant non-local correlations. The analysis evidences that non-locality can be harnessed as a resource for quantum information technology, with implications for quantum communication and computing. This work advances the understanding of quantum mechanics by addressing the principle of local causality and the role of entanglement.
![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”
https://www.slideshare.net/gill1109/presentations](https://image.slidesharecdn.com/vaxjo2023rdg-230613071538-1f07c22c/85/vaxjo2023rdg-pdf-1-320.jpg)
![• 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-230613071538-1f07c22c/85/vaxjo2023rdg-pdf-6-320.jpg)
![10 3
3 10
10 3
3 10
10 3
3 10
3 10
10 3
The random setting generators seem perfect
Three correlations ≈ +14 / 26 = 1 / 2 + 1 / 26 = 0.54
One correlation ≈ – 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
> 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-230613071538-1f07c22c/85/vaxjo2023rdg-pdf-8-320.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-230613071538-1f07c22c/85/vaxjo2023rdg-pdf-10-320.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” https://www.slideshare.net/gill1109/presentations
- 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 Rep. 2023, 5(2), 481-495; https://doi.org/10.3390/quantum5020032 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: [email protected]; [email protected]
- 5. 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 . 3-05885-0
- 6. • 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
- 7. 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 tens of thousands, rounded
- 8. 10 3 3 10 10 3 3 10 10 3 3 10 3 10 10 3 The random setting generators seem perfect Three correlations ≈ +14 / 26 = 1 / 2 + 1 / 26 = 0.54 One correlation ≈ – 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 > 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
- 9. 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
- 10. > 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
- 11. 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”
- 13. The long box in the middle
- 18. • Suppose multinomial distribution, condition on 4 counts N(a, b) • Compute the CHSH statistic S and estimate its standard deviation in the usual way (four independent binomial counts with variance estimated by plug-in) • Compare z-value = S / s.e.(S) with N(0, 1) • In some circumstances one would prefer Eberhard’s J • Are there more possibilities? Yes: a 4 dimensional continuum of alternatives. Why not just pick the best??? Reduce data to 16 counts N(a, b, x, y) Statistical analysis (Classical method) ∧ ∧ ∧ ∧ ∧
- 19. • 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) Statistical analysis (New method 1) ∧ ∧
- 20. • Suppose multinomial distribution, condition on 4 counts N(a, b) • Estimate the 4 sets of 4 tetranomial probabilities p( x, y | a, b ) by maximum likelihood assuming no-signalling (4 linear equalities) • (a) Also assuming local realism (8 linear inequalities) and • (b) Without also assuming local realism • Test null hypothesis H0 : local realism against H1 = ¬ H0 using Wilks’ log likelihood ratio test: Under H0, asymptotically, –2 ( max{ log lik(p) : p ∈ H0 } – max{ log lik(p) : p ∈ H0 ∪ H1 } ) ~ 1/2 𝜒 2(1) + 1/2 𝜒 2(0) Statistical analysis (New method 2) Reduce data to 16 counts N(a, b, x, y)
- 21. • Suppose 16-nomial, i.e., only grand total = N = 220 is fi xed • Suppose all p(a, b) = 0.25 • Use martingale test (“Bell game”) • Compute N( = | 1, 1) + N( = | 1, 2) + N( = | 2, 1) + N( ≠ | 2, 2); compare to Bin(N, 3/4) Statistical analysis (Bell game) Reduce data to 16 counts N(a, b, x, y)
- 22. • The 4 tests are asymptotically equivalent if their model assumptions are satis fi ed and the true probabilities p( x, y | a, b ), p(a, b), have the nice symmetries Comparison of the 4 p-values Theory e f f e e f f e e f f e f e e f g g g g
- 24. • Qutrits’ bases: | gA ⟩, | eA ⟩, | fA ⟩ ; | gB ⟩, | eB ⟩, | fB ⟩ • Ground state g , and fi rst two excited states e, f • We get qutrits AB into state ( | eA, gB ⟩ + | gA, fB ⟩ ) / √2 + noise • Channel has states | 0 ⟩, | 1 ⟩, | 2 ⟩ , … • First create Schrödinger cat in A, superposition of two excited states • One of the excited cats ( f ) emits a photon (microwave pulse) into the channel and returns to the ground state • The photon interacts with qutrit B and puts it into excited state f • Attenuation of microwave plus interaction with environment -> noise Actually, a tripartite system: qutritA ⨂ channel ⨂ qutritB The QM stuff
- 25. The QM stuff A tripartite system: qutritA ⨂ channel ⨂ qutritB Create Schrödinger cat in qutrit A (superposition of two excited states e, f) Actually done in two steps: g ➝ e ➝ (e + f) / √2 Cat “f” moves from qutrit A into channel Cat “f” moves from channel into qutrit B THE AUTHOR |gA, 0, gBi ! 1 p 2 ⇣ |eA, 0, gBi + |fA, 0, gBi ⌘ ! 1 p 2 ⇣ |eA, 0, gBi + |gA, 1, gBi ⌘ ! 1 p 2 ⇣ |eA, 0, gBi + |gA, 0, fBi ⌘
- 26. • Tim Maudlin, Sabine Hossenfelder, Jonte Hance; Gerard ’t Hooft • Restrict QM to a countable dense subset of states such that Bell experiment is impossible: at most three of the four experiments ( 𝛼 i, βj) “exists” • Paul Raymond-Robichaud • Careful de fi nitions of QM, distinguish two levels (maths, empirical observations) make it both local and realistic • The catch: probability distributions of measurement outcomes are in R-R’s model, individual outcomes are not Two over-complex solutions to the Bell theorem quandrary Lost in math
- 27. • David Oaknin: physicists are forbidden to look at 𝛼 – β • Karl Hess, Hans de Raedt: physicists must condition (post-select) on both photons being observed (the “photon identi fi cation loophole”) • Marian Kupczynski: physicists must use 4 disjoint probability spaces for the 4 sub-experiments • I am not a physicist!. As a mathematician, I do what I like. • In physics there are no moral constraints on thought experiments • Counterfactual reasoning is essential to (statistical) science, to law, to morality • As a statistician, I know that “all models are wrong, some are useful” Argumentum ab auctoritate Physics fatwas
- 28. • Convert a bug into a feature • The “Heisenberg cut” is for real; where it should be placed is up to the user of the QM framework • But it must be compatible with the underlying unitary evolution of the world: the past consists of particles with de fi nite trajectories, the future consists of waves of possibilities. Born’s rule gives us the probability law of the resulting stochastic process by compounding the conditional probabilities of the “collapse” in each next in fi nitesimal time step, given the history so far • A simple solution to what? • It’s a mathematical solution to the measurement problem; a mathematical resolution of the Schrödinger cat paradox Belavkin’s “eventum mechanics” A simple solution