This year’s Nobel prize in physics: homage to John Bell. Richard Gill Mathematical Institute, Leiden University. Focussing on statistical issues, I will first sketch the history initiated by John Bell’s landmark 1964 paper “On the Einstein Podolsky Rosen paradox”, which led to the 2022 Nobel prize awarded to John Clauser, Alain Aspect and Anton Zeilinger, https://www.nobelprize.org/prizes/physics/2022/press-release/ A breakthrough in the history was the four successful “loophole-free” Bell experiments of 2015 and 2016 in Delft, Munich, NIST and Vienna. These experiments pushed quantum technology to the limit and paved the way for DIQKD (“Device Independent Quantum Key Distribution”) and a quantum internet. They were the first successful implementations of the ideal experimental protocol described by John Bell in his 1981 masterpiece "Bertlmann's socks and the nature of reality", and depended on brilliant later innovations: Eberhard’s discovery that less entanglement could allow stronger manifestation of quantum non-locality, and Zeilinger’s discovery of quantum teleportation, allowing entanglement between photons to be transferred to entanglement between ions or atoms and ultimately to components of manufactured semi-conductors. I will also discuss reanalyses of the 2015+ experiments, which could have allowed the experimenters to claim even smaller p-values than the ones they published, https://arxiv.org/abs/2209.00702 "Optimal statistical analyses of Bell experiments"

1. Richard Gill, 18 October 2022
2022 Nobel prize for
physics
Alain Aspect, John F. Clauser, Anton Zeilinger
in homage to John S. Bell

2. Alain Aspect, John F. Clauser, Anton Zeilinger
2022 Nobel prize for physics

3. In homage to John S. Bell (1929–1990)

4. • EPR paradox
• Bell inequality, Bell’s theorem, Bell state
• Aspect experiment
• CHSH inequality
• GHZ, GHSZ state
• DIQKD
Einstein, Podolsky, Rosen
Clauser, Horne, Simony, Holt
Greenberger, Horne, (Shimony,) Zeilinger
Device independent quantum key distribution
Some acronyms
Personalities, philosophy, the times,
the connections, the coincidences

5. DIY QKD
(do it yourself quantum key distribution)
Take two apples and two oranges

6. https://images.app.goo.gl/PyLL8vZ4S4f2WhpRA
Take two bedside chests of drawers, each with two drawers

7. Alice Bob
Top Apple Orange
Bottom Orange Apple
Con
fi
guration 1 Con
fi
guration 2
Alice Bob
Top Apple Orange
Bottom Orange Apple
Secretly choose a con
fi
guration
completely at random
Alice and Bob, far apart, connected by a public phone line,
each get one chest of drawers.
They don’t know which one they got

8. • If one drawer is opened, the other instantaneously vanishes
• The initial con
fi
guration is secret
• Alice and Bob each toss a fair coin and open one drawer
• They announce whether they opened their top or bottom drawers
• They now jointly posses one secret random bit (they both know
whether they saw the same fruit, or di
ff
erent fruit, but nobody
else can know
BB84
Quantum Key Distribution

9. And can we trust the manufacturer?
Not very practical, for generating
long secret shared bit strings
This is where we enlist the help of quantum mechanics
1
2
( , ⟩ + , ⟩)
Alice’s qubit, Bob’s qubit + Alice’s qubit, Bob’s qubit
When Alice measures her qubit the “wave” will
collapse; she’ll see the
fi
rst state or the second state
and know what Bob would see if he measured his.

10. • They should create this state many times anew, measuring each
time, in several di
ff
erent ways
• They will then communicate using public, classical means, about
some of their results; keeping others secret
• They will either determine that their qubits were indeed entangled
and hence that nobody was watching, or that someone was
watching so much that the ‘secret’ results are unreliable
• With high probability, either they end up with many shared secret
bits, or they abort the mission

11. • Einstein, Podolsky, Rosen (1935) Assuming locality and realism, either
QM is wrong or it is incomplete
• John S. Bell (1964) Assuming locality and realism, QM is wrong
• EPR: the apples and oranges scenario. Alice and Bob can each
measure their qubit and will see either an apple or an orange
• Bell: introduce two more ways to look into those mysterious chests of
drawers
• QM says that certain correlations can *only* be achieved with certain
states and certain measurements. And that any measurements of those
states would destroy them. If they observe those correlations then they
know that nobody else saw the underlying measurement outcomes
How to be sure their qubits are in that state?
How to get their qubits into that state?

12. Define H = K = C2
|0i, |1i an o.n.b. of the complex vector space C2
|01i = |0i ⌦ |1i 2 H ⌦ K, etc.
|00i, |01i, |10i, |11i are an o.n.b. of H ⌦ K
So are the Bell basis ± = 1
p
2
(|01i ± |10i) and ± = 1
p
2
(|00i ± |11i)
Define = = 1
p
2
(|01i |10i)
The nitty-gritty

13. Alice and Bob can each choose an o.n.b. of H and K respectively
in which to measure.
Their measurements have as outcomes a binary variable indicating
which element of their basis was chosen by their qubit.
The probability that Alice’s outcome corresponds to |ai and Bob’s
to |bi is
†
|ai ⌦ |bi
2
where a and b parametrize in some convenient way the rays of C2
(the oriented one-complex-dimensional subspaces)
To every o.n.b. |0i, |1i of C2 there corresponds a Pauli spin matrix
|0ih0| |1ih1|: it is self-adjoint, idempotent, has eigenvalues +1,
1 and eigenvectors |0i, |1i.

14. Define the density matrix ⇢ = † 2 C4⇥4. It has trace 1, and
is nonnegative.
Define x =
✓
0 1
1 0
◆
, z =
✓
1 0
0 1
◆
, both 2 C2⇥2.
Suppose Alice and Bob do measurements corresponding to Pauli
spin matrices A and B. Let 1 denote the 2 ⇥ 2 identity matrix.
Then the mean values of Alice and Bob’s ±1-valued measurement
outcomes, and of their product, are trace ⇢ A ⌦ 1, trace ⇢ 1 ⌦ B,
and trace ⇢ A ⌦ B.
In a classical Bell experiment, Alice either measures x or z , Bob
either measures 1
p
2
( x + z ) or 1
p
2
( x z ), which are also two
Pauli spin matrices ( x and z anti-commute).

15. Indicate Alice’s choices by labels a = 1, 2, Bob’s by labels b = 1, 2
It turns out that the mean values of the measurement outcomes
under all these measurement choices are zero, and the correlations
(expectations of the products) are Eab = +1/
p
2 (first three cases:
ab = 11, 12, 21) and Eab = 1/
p
2 (fourth case, ab = 22).
Hence the sum of three correlations minus the fourth equals 2
p
2
Enough QM, now back to classical physics: Could there be a
classical physical explanation of these correlations?

16. Spatial-temporal disposition of one trial of
an ideal Bell experiment
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 that 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:
fi
gure 7 from J.S. Bell (1981), “Bertlmann’s socks and the nature of reality”

17. A causal model of one trial in a Bell experiment
NB a *classical* causal model, not a quantum causal model!
X Y
A B
(Hidden)
Experimenter
Settings A, B ∈ {1, 2}; outcomes X, Y ∈ {−1, + 1}

18. The graphical model says that X = f (A, hidden, X ) and
Y = f (B, hidden, Y ), where (A, B) is independent of all the
“lambda” variables, which are also all independent of one another.
More precisely, it says that that the joint distribution of
(A, B, X, Y ) can be represented in this way
There is no claim that the various hidden variables are physically
located in specific space-time locations
In Bell’s argument, = ( hidden, X , Y ) represents the initial
physical state of the measurement devices, source and transmission
lines between them, at some time point prior to the experimenter
fixing the settings of both measurement devices – as far as the
state is relevant to the physical mechanism creating the
measurement outcomes.

19. Now define random variables
X1 = f (1, ·), X2 = f (1, ·), Y1 = g(1, ·), Y2 = g(2, ·).
Notice that
(X1, X2, Y1, Y2) is independent of (A, B) and X = XA, Y = YB.
Conditional on A = a, B = b, we have X = Xa, Y = Yb.
Therefore Eab(XY ) := E (XY | A = a, B = b) = E(XaYb).

20. Recall
Eab(XY ) := E (XY | A = a, B = b) = E(XaYb)
Notice that
X1Y1+X1Y2+X2Y1 X2Y2 = X1(Y1+Y2)+X2(Y1 Y2) 2 { 2, +2}
Therefore
E11(XY ) + E12(XY ) + E21(XY ) E22(XY ) 2 [ 2, +2]
This is the inequality of Clauser, Horne, Shimony and Holt (1969),
a generalisation of an inequality of Bell (1964)

21. VOLUME 28, NUMBER 14 PHYSICAL REVIEW LETTERS 3 APRIL 1972
400-
200-
0
UJ
UJ
K
z —
200-
—
40Q-
'"'ll I ~
''s
II III lit
il,
I I I I
(o)
6 PRONGS
- Ip
OJ
O
0
- -IO
Fraser and Rudolph Hwa. He is indebted to the
following members of Group A at the Lawrence
Berkeley Laboratory for generously allowing him
to participate in the analysis of the K' exposure:
M. Alston-Ganjost, A. Barbaro-Galtieri, P. J.
Davis, S. M. Flatte, J. H. Friedman, G. R. Lynch,
M. J. Matison, J. J. Murray, M. S. Rabin, F. T.
Solmitz, N. J. Uyeda, V. Waluch, and R. %ind-
molders.
40Q-
- Ip
200-
0
p.~+
UJ
CO
—
200-
O
p
—
400-
(b)
8 PRONGS
-- ~ . -. - ~ - ~ . -- ~ -I
-4 -2 0 2 4
Y —
Y
2 I
- -10
F1G. 4. (a) G& and (b) G3 as defined in the text.
The statistics on the eight-prong data are not
good but show characteristics similar to those
for six-prong.
%e present this dramatic behavior of the two
*Work supported by the U. S. Atomic Energy Commis-
sion under Contract No. AT(04-8)-84 PA 191.
K. G. Wilson„Cornell University Report No. CLNS-
131, 1970 (to be published).
W. B.Fraser et al., to be published.
H. D. I. Abarbanel, Phys. Rev. D 8, 2227 (1971).
R. C. Hwa, to be published.
D. Z. Freedman, C. E. Jones, F. E. Low, and J.E.
Young, Phys. Rev. Lett. 26, 1197 (1971).
C. K. DeTar, Phys. Rev. D 8, 128 (1971).
VA. Bassetto, M. Toner, and L. Sertorie, Nucl. Phys.
B34, 1 (1971).
8A. Mueller, Phys. Rev. D 4, 150 (1971).
W. Ko and B. L. Lander, Phys. Bev. Lett. 26, 1064
(1971}.
J. Erwin, W. Ko, R. L. Lander, D. K. Pellett, and
P. M. Yager, Phys. Rev. Lett. 27, 1534 (1971).
The correlation length of about 2 is even shorter
than the short-range Mueller-Begge-theoretical value
I.R. C. Arnold, ANL Report No. ANL-HKP 7189, 1971
(unpublished), and Ref. 5J. In that theory a correlation
appreciates the many useful discussions with him,
David Pellett, and Philip Yager. He also benefit-
ted from stimulating conversations with William
dicted for fragment-center or fragment-fragment cor-
relations.
H. T. Nieh and J. M. Wang, to be published.
Experimental Test of Local Hidden-Variable Theories*
Stuart J. Freedman and John F. Clauser
Department of Physics and Lagerence Berkeley Laboratory, Unioersity of California, Berkeley, California 94720
(Received 4 February 1972)
We have measured the linear polarization correlation of the photons emitted in an atom-
ic cascade of calcium. It has been shown by a generalization of Bell's inequality that the
existence of local hidden variables imposes restrictions on this correlation in conflict
with the predictions of quantum mechanics. Our data, in agreement with quantum me-
chanics, violate these restrictions to high statistical accuracy, thus providing strong evi-
dence against local hidden-variable theories.
'Since quantum mechanics was first developed,
there have been repeated suggestions that its sta-
tistical features possibly might be described by
an underlying deterministic substructure. Such
features, then, arise because a quantum state
represents a statistical ensemble of "hidden-
variable states. " Proofs by von Neumann and
others, demonstrating the impossibility of a hid-
938
R,/RD=~(e„'+e '), (lb)
R,/RD =
2 (eu + e„'). (1c)
Here e„' (e ') is the transmittance of the ith po-
larizer for light polarized parallel (perpendicu-
lar) to the polarizer axis, and E,(8) is a function
of the half-angle 8 subtended by the primary 1ens-
es. It represents a depolarization due to noncol-
linearity of the two photons, and approaches unity
for infinitesimal detector solid angles. [For this
experiment, 8=30, and E,(30 ) =0.99.]
We make the following assumptions for any lo-
cal hidden-variable theory: (1) The two photons
propagate as separated localized particles. (2) A
binary selection process occurs for each photon
at each polarizer (transmission or no-transmis-
sion). This selection does not depend upon the
orientation of the distant polarizer.
In addition, we make the following assumption
to allow a comparison of the generalization of
Bell's inequality with out experiment: (3) All
photons incident on a detector have a probability
of detection that is independent of whether or not
the photon has passed through a polarizer. '
The above assumptions constrain the coinci-
R«)/RD =.(Eu'+ &.')(&u'+ E.')+ 4 (Eu'- '.')
while
x( — )E (0) o 2p, (la)
R,/RD=~(e„'+e '), (lb)
R,/RD =
2 (eu + e„'). (1c)
Here e„' (e ') is the transmittance of the ith po-
larizer for light polarized parallel (perpendicu-
lar) to the polarizer axis, and E,(8) is a function
of the half-angle 8 subtended by the primary 1ens-
es. It represents a depolarization due to noncol-
linearity of the two photons, and approaches unity
for infinitesimal detector solid angles. [For this
experiment, 8=30, and E,(30 ) =0.99.]
We make the following assumptions for any lo-
cal hidden-variable theory: (1) The two photons
propagate as separated localized particles. (2) A
binary selection process occurs for each photon
at each polarizer (transmission or no-transmis-
sion). This selection does not depend upon the
orientation of the distant polarizer.
In addition, we make the following assumption
to allow a comparison of the generalization of
Bell's inequality with out experiment: (3) All
…

22. • P. Pearle (1970). “Hidden-variable example based upon data
rejection”. Phys. Rev. D 2, 1418–1425. https://doi.org/10.1103/
PhysRevD.2.1418
• R.D. Gill (2020). “Pearle's Hidden-Variable Model Revisited”.
Entropy 22(1), 1–15. https://doi.org/10.3390/e22010001
“The fair sampling assumption”
The first loophole

23. “The locality loophole”
The second loophole
chaotic. " The purpose of this communica-
as to use an independent and exact result
he statistical-mechanical theory of d =1
m walks to test the randomness of the para-
map for parameter values where the exis-
of "true chaos" is still an open question.
4K. Tomita, in Pattern Formation by Dynamic Sys-
tems and Pattern Recognition, edited by H. Haken
(Springer-Verlag, Heidelberg, 1979), pp. 90-97.
5S. Thomae and S. Grossmann, J.Stat. Phys. 26,
485 (1981).
S. Qrossmann and S. Thomae, Z. Naturforsch. 32a,
1353 (1977).
E*perimen&al Tes& of Bell's Inequalities Using Time-Varying Analyzers
Alain Aspect, Jean Dalibard, ' and Gerard Roger
Institut d'Optique Theomque et APPliquee, F-9j406 Qxsay Cedex, France
(Received 27 September 1982)
Correlations of linear polarizations of pairs of photons have been measured with
time-varying analyzers. The analyzer in each leg of the apparatus is an acousto-opti-
cal switch followed by two linear polarizers. The switches operate at incommensurate
frequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps between
two orientations in a time short compared with the photon transit time. The results
are in good agreement with quantum mechanical predictions but violate Bell's inequal-
ities by 5 standard deviations.
PACS numbers: 03.65.8z, 35.80.+s
s inequalities apply to any correlated meas-
nt on two correlated systems. For in-
, in the optical version of the Einstein-
ky-Rosen-Bohm Gedankenexperiment, ' a
PM1 ~
I(a)
~ PM2
I I (b)
The settings should be chosen while the photons are in
fl
ight
However, Aspect’s experiment is still vulnerable to the “detection loophole”

24. • The physics may change in time. Also, subsequent trial outcomes
can be statistically dependent on the past history of the
experiment, including the past at the other measurement location
• Solution: randomisation, and base the statistical evaluation on the
complete randomisation of setting choices, instead of assumptions
of i.i.d. outcome pairs per given setting pair
• R.D. Gill (2001), “Accardi contra Bell (cum mundi): The Impossible
Coupling”, https://arxiv.org/abs/quant-ph/0110137 and IMS
Lecture Notes vol. 42
• RDG martingale approach improved by Delft group
(2015 experiment)
“The memory loophole”
The third loophole

25. • Delft, Munich, NIST, Vienna
• All three experimental loopholes closed (more or less)
• Remaining loopholes are metaphysical
• Superdeterminism
• Retrocausality
• Certainly, improvement is possible (better randomisers for setting
choice, larger samples, more e
ffi
cient statistical analysis…)
2015-16
Loophole-free experiments

26. • Delft and Munich used entanglement swapping in three party
Bell experiments
• Alice, Bob, Carol
• Study
• NIST and Vienna used Eberhard (1993): a less entangled state
can exhibit stronger quantum non-locality in the presence of
noisy measurement!
p (x, y ∣ z; a, b, c)
Delft, Munich vs. NIST, Vienna
What made this possible?

27. PHYSICAL REVIEW
LETTERS
VOLUME 80 4 MAY 1998 NUMBER 18
Experimental Entanglement Swapping: Entangling Photons That Never Interacted
Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger
Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria
(Received 6 February 1998)
We experimentally entangle freely propagating particles that never physically interacted with one
another or which have never been dynamically coupled by any other means. This demonstrates that
quantum entanglement requires the entangled particles neither to come from a common source nor to
have interacted in the past. In our experiment we take two pairs of polarization entangled photons and
subject one photon from each pair to a Bell-state measurement. This results in projecting the other two
outgoing photons into an entangled state. [S0031-9007(98)05913-4]
PACS numbers: 03.65.Bz, 03.67.–a, 42.50.Ar
Entanglement is one of the most fundamental features
of quantum mechanics. It is at the heart of the Einstein-
Podolsky-Rosen paradox, of Bell’s inequalities, and of
the discussions of the nonlocality of quantum mechanics.
Thus far, entanglement has been realized either by having
the two entangled particles emerge from a common source
[1], or by having two particles interact with each other [2].
Here jHl or jVl indicates the state of a horizontally or a
vertically polarized photon, respectively. The total state
describes the fact that photons 1 and 2 (3 and 4) are
entangled in an antisymmetric polarization state. Yet, the
state of pair 1-2 is factorizable from the state of pair 3-4;
that is, there is no entanglement of any of the photons 1 or
2 with any of the photons 3 or 4.
Entangling Photons That Never Interacted
arald Weinfurter, and Anton Zeilinger
uck, Technikerstrasse 25, A-6020 Innsbruck, Austria
February 1998)
particles that never physically interacted with one
upled by any other means. This demonstrates that
cles neither to come from a common source nor to
take two pairs of polarization entangled photons and
easurement. This results in projecting the other two
9007(98)05913-4]
Here jHl or jVl indicates the state of a horizontally or a
vertically polarized photon, respectively. The total state
describes the fact that photons 1 and 2 (3 and 4) are
entangled in an antisymmetric polarization state. Yet, the
state of pair 1-2 is factorizable from the state of pair 3-4;
that is, there is no entanglement of any of the photons 1 or
2 with any of the photons 3 or 4.
We now perform a joint Bell-state measurement on
photons 2 and 3; that is, photons 2 and 3 are projected onto
one of the four Bell states which form a complete basis for
FIG. 1. Principle of entanglement swapping. Two EPR
t is one of the most fundamental features
chanics. It is at the heart of the Einstein-
n paradox, of Bell’s inequalities, and of
of the nonlocality of quantum mechanics.
glement has been realized either by having
ed particles emerge from a common source
g two particles interact with each other [2].
ve possibility to obtain entanglement is to
rojection of the state of two particles onto
te. This projection measurement does not
uire a direct interaction between the two
n each of the particles is entangled with
r particle, an appropriate measurement, for
-state measurement, of the partner particles
lly collapse the state of the remaining two
entangled state. This striking application
n postulate is referred to as entanglement
], and in this Letter we report its first
alization.
o EPR sources, simultaneously emitting
ntangled particles (Fig. 1). In anticipation
ents we assume that these are polarization
ns in the state
34 ≠
1
2 sjHl1jVl2 2 jVl1jHl2d
3 sjHl3jVl4 2 jVl3jHl4d . (1)
Here jHl or jVl indicates the state of a horizontally or a
vertically polarized photon, respectively. The total state
describes the fact that photons 1 and 2 (3 and 4) are
entangled in an antisymmetric polarization state. Yet, the
state of pair 1-2 is factorizable from the state of pair 3-4;
that is, there is no entanglement of any of the photons 1 or
2 with any of the photons 3 or 4.
We now perform a joint Bell-state measurement on
photons 2 and 3; that is, photons 2 and 3 are projected onto
one of the four Bell states which form a complete basis for
FIG. 1. Principle of entanglement swapping. Two EPR
sources produce two pairs of entangled photons, pair 1-2
and pair 3-4. One photon from each pair (photons 2 and
3) is subjected to a Bell-state measurement. This results in
projecting the other two outgoing photons 1 and 4 onto an
entangled state. Change of the shading of the lines indicates
the change in the set of possible predictions that can be made.

28. ur experiments we assume that these are polarization
ngled photons in the state
jCl1234 ≠
1
2 sjHl1jVl2 2 jVl1jHl2d
3 sjHl3jVl4 2 jVl3jHl4d . (1)
FIG. 1. Principle o
sources produce two
and pair 3-4. One
3) is subjected to a
projecting the other
entangled state. Cha
the change in the set
0031-9007y98y80(18)y3891(4)$15.00 © 1998 The Amer
VOLUME 80, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S
the combined state of photons 2 and 3
jC6
l23 ≠
1
p
2
sjHl2jVl3 6 jVl2jHl3d ,
jF6
l23 ≠
1
p
2
sjHl2jHl3 6 jVl2jVl3d .
(2)
This measurement projects photons 1 and 4 also onto a Bell
state, a different one depending on the result of the Bell-
state measurement for photons 2 and 3. Close inspection
shows that for the initial state given in Eq. (1) the emerging
state of photons 1 and 4 will be identical to the one photons
2 and 3 collapsed into. This is a consequence of the fact
that the state of Eq. (1) can be rewritten as
jCl1234 ≠
1
2 sjC1
l14jC1
l23 1 jC2
l14jC2
l23
1 jF1
l14jF1
l23 1 jF2
l14jF2
l23d . (3)
In all cases photons 1 and 4 emerge entangled despite the
fact that they never interacted with one another in the past.
After projection of particles 2 and 3 one knows about the
which of the detected pho
ton 1, or with photon 4,
cannot be used in practice
of existing single-photon d
avalanche photodiodes as
times of about 500 fs).
The second possibility i
times of the interfering p
than the time interval wit
Then again, one cannot i
tected photons was created
photon 4, respectively. In
a duration of 200 fs are u
We then choose narrow ba
front of the detectors regis
sulting coherence time of a
than the pump pulse dura
fiber couplers acting as sp
tee good mode overlap of
Figure 2 is a schemati

29. Where we are now
Bob (Lab 2)
Alice (Lab 1)
Device 1 Device 2
Input Xi
SM fibre
SM fibre
Fibre
BS
Input Yi
High-NA
Objective
High-NA
Objective
Ready signal
795 nm
Read-out
795 nm
Read-out
780 nm
Excitation
780 nm
Excitation
BSM
BS
SPDs SPD
Output Ai Output Bi
TTL to
optical
Optical
to TTL
700 m
Fibre channel
X X
Spectral
filter
Shutter
QRNG
Storage
Storage
QRNG
Ready signal
Experimental device-independent quantum key distribution between distant users
Wei Zhang et al. Nature 609, 687 (2022)
Device-independent quantum key distribution (DIQKD) is the art of using untrusted devices to establish secret keys over an untrusted
channel. So far, the real-world implementation of DIQKD remains a major challenge, as it requires the demonstration of a loophole-free Bell
test across two remote locations with very high quality entanglement to ensure secure key exchange. Here, we demonstrate for the
f
irst time
the distribution of a secure key -- based on asymptotic security estimates -- in a fully device-independent way between two users separated
by 400 metres. The experiment is based on heralded entanglement between two independently trapped single Rubidium 87 atoms. The
implementation of a robust DIQKD protocol indicates an expected secret key rate of r=0.07 per entanglement generation event and r>0
with a probability error of 3%. Furthermore, we analyse the experiment's capability to distribute a secret key with
f
inite-size security against
collective attacks.
BS = Beam Splitter; BSM = Bell-state measurement; SM = single mode; NA = numerical aperture; TTL = transistor-transistor logic
Shanghai -> Amsterdam -> Munich

30. • Answer to question 1: “entanglement swapping”
How to get their qubits into that state?
How to be sure their qubits are in that state?

31. Up to constants of proportionality
(00 + 11)(00 + 11) = 0000 + 0011 + 1100 + 1111
= 0 ((00 + 11) + (00 - 11)) 0 +
0 ((01 + 10) + (01 - 10)) 1 +
1 ((01 + 10) - (01 - 10)) 0 +
1 ((00 + 11) - (00 - 11)) 1
= 0 (00 + 11) 0 + 1 (00 + 11) 1 +
0 (00 - 11) 0 - 1 (00 - 11) 1 +
0 (01 + 10) 1 + 1 (01 + 10) 0 +
0 (01 - 10) 1 - 1 (01 - 10) 0
So on measuring the inside two qubits in the Bell basis, conditional
on finding “00 + 11”, the outside two qubits are in the state “00
+ 11”, etc.

32. Now with two stationary electrons, atom, or ions
Delft: stationary qubit = electron spin associated with a single Nitrogen-vacancy defect in diamond
Munich: stationary qubit = excitement level of a single Rubidium atom in an atom trap

33. Zhang et al, Bell test, raw counts
+ – + –
+ 178 44 222 199 36 235
– 29 183 212 28 160 188
207 227 434 227 196 423
+ 160 47 207 38 160 198
– 31 151 182 166 39 205
191 198 389 204 199 403
> 178+183 + 199+160 + 160+151 + 160+166
[1] 1357
> 434 + 423 + 389 + 403
[1] 1649
> pbinom(1356, 1649, 3/4, lower.tail = FALSE)
[1] 8.042943e-13
> 1357 / 1649
[1] 0.822923
> (sqrt(2)+2)/4
[1] 0.8535534

34. 1357 1649
+ – + –
+ 0,41 0,10 0,51 0,47 0,09 0,56
– 0,07 0,42 0,49 0,07 0,38 0,44
0,48 0,52 1,00 0,54 0,46 1,00
+ 0,41 0,12 0,53 0,09 0,40 0,49
– 0,08 0,39 0,47 0,41 0,10 0,51
0,49 0,51 1,00 0,51 0,49 1,00
P(X= + | 11) = 0.51, P(X = + | 12) = 0.56
P(X= + | 21) = 0.53, P(X = + | 22) = 0.49
P(Y= + | 11) = 0.48, P(Y = + | 21) = 0.49
P(Y= + | 21) = 0.54, P(Y = + | 22) = 0.51
r11 = 0.66, r12 = 0.70, r21 = 0.60, r22 = –0.62 S = 2.58
Zhang et al, relative frequencies

35. • Conventional multinomial + normal approximation analysis
S = 2.577832 (0.07540739)
z = 7.662799
p = 9.10 x 10–15
• Optimized (RDG) multinomial + normal approximation analysis
S = 2.577653 (0.07534677)
z = 6.986411
p = 1.41 x 10–12
• Bell game (martingale based, Delft group (2015) improvement on RDG (2001)
p = 5.151435 x 10–13
Zhang et al, p-values

36. • https://arxiv.org/abs/2209.00702 Optimal statistical analyses of
Bell experiments
• R scripts and data: https://rpubs.com/gill1109 (includes Zhang et
al’s DIQKD experiment)
References

37. SHOW YOUR
CLASSICAL
APPARATUS
Delineating the border between the quantum realm ruled by the Schrodinger equation and the classical realm
ruled by Newton's laws is one of the unresolved problems of physics. Figure 1
minima of the effective potential.4
If macroscopic systems cannot always be safely placed
by these two viewpoints nevertheless becomes apparent
when we ask the obvious question "Why do I, the observer,
Zurek, 1991. https://physicstoday.scitation.org/doi/10.1063/1.881293
What does
it all mean?

38. • GB: “I argue that no experiment whose purpose is to con
fi
rm the predictions of
quantum theory can possibly be used as an argument in favour of nonlocality
because any theory of physics that does not allow instantaneous signalling to
occur and has reversible dynamics (such as unitary quantum theory) can be
explained in a purely local and realistic universe.”
• RDG: How to have your cake and eat it: there is only Schrödinger’s equation and
unitary evolution of the wave function of the universe, but we must add a
Heisenberg cut to separate the past from the future (separate particles from
waves): Belavkin’s eventum mechanics
• The past is a commuting sub-algebra A of the algebra of all observables B, and in
the Heisenberg picture, the past history of any observable in A is also in A
• Particles have de
fi
nite trajectories back into the past; Eventum Mechanics de
fi
nes
the probability distributions of future given past
• https://arxiv.org/abs/0905.2723 Schrödinger's cat meets Occam's razor (version 3:
10 Aug 2022); to appear in Entropy; https://www.slideshare.net/gill1109/
schrdingers-cat-meets-occams-razor
Debate with Gilles Brassard (2nd “B” in BB84)
Appendix

39. • GB: “We provide a framework that describe all local-realistic theories and all no-
signalling theories. We show that every local-realistic theory is a no-signalling
theory. We also show that every no-signalling theory with invertible dynamics
has a local-realistic model. This applies in particular to unitary quantum theory.”
• RDG: you have re
fi
ned “local” and “realistic”; I understand your reasoning but I
don’t like your de
fi
nitions
• RDG: The usual purpose of Bell experiments is to disprove Local Realism
(essentially: determinism). A good Bell experiment moreover provides good
evidence that the principle of “invertible dynamics” is wrong. IMHO: “collapse”
is real, fundamental, intrinsic, stochastic; one of the foundations of the
physical universe is irreducible non-local quantum randomness.
• The past is particles, the future is a wave
• Yesterday is history, tomorrow is a mystery, today is a gift
Appendix (continued)
Debate with Gilles Brassard (2nd “B” in BB84)

40. What I think it all means
Now
The Past
The Future

41. https://arikegill.nl/wordpress/