Final version of slides for Växjö meeting QIP 2023.

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
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: 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 tens of 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)
• 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)
∧
∧ ∧ ∧
∧

18. • 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)
∧
∧

19. • 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)

20. • Suppose 16-nomial, i.e., only grand total = 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)

21. • 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

22. QIRIF (2019)
Déja vu?

23. • 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

24. 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
⌘

25. • 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

26. • David Oaknin: physicists are forbidden to look at
𝛼
– β
• Karl Hess, Hans de Raedt: physicists must condition 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

27. • 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
• 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