Self-testing with finite statistics enabling the certification of
a quantum network link
Jean-Daniel Bancal
1,2,3
, Kai Redeker
4
, Pavel Sekatski
3,2
, Wenjamin Rosenfeld
4,5
, and
Nicolas Sangouard
1,3
1
Université Paris-Saclay, CEA, CNRS, Institut de Physique Théorique, 91191, Gif-sur-Yvette, France
2
Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
3
Quantum Optics Theory Group, Universität Basel, CH-4056 Basel, Switzerland
4
Fakultät für Physik, Ludwig-Maximilians-Universität, 80799 München, Germany
5
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
16 February 2021
Self-testing is a method to certify devices
from the result of a Bell test. Although
examples of noise tolerant self-testing are
known, it is not clear how to deal effi-
ciently with a finite number of experimen-
tal trials to certify the average quality of
a device without assuming that it behaves
identically at each run. As a result, exist-
ing self-testing results with finite statistics
have been limited to guarantee the proper
working of a device in just one of all experi-
mental trials, thereby limiting their practi-
cal applicability. We here derive a method
to certify through self-testing that a de-
vice produces states on average close to a
Bell state without assumption on the ac-
tual state at each run. Thus the method
is free of the I.I.D. (independent and iden-
tically distributed) assumption. Applying
this new analysis on the data from a recent
loophole-free Bell experiment, we demon-
strate the successful distribution of Bell
states over 398 meters with an average fi-
delity of ≥55.50% at a confidence level of
99%. Being based on a Bell test free of
detection and locality loopholes, our cer-
tification is evidently device-independent,
that is, it does not rely on trust in the
devices or knowledge of how the devices
work. This guarantees that our link can be
integrated in a quantum network for per-
forming long-distance quantum communi-
cations with security guarantees that are
independent of the details of the actual im-
plementation.
1 Introduction
The distribution of entanglement over long
distances is a key challenge in extending the
range of quantum communication and building
quantum networks [1, 2]. The direct transmission
of entangled states through optical fibers is a
viable solution for short distances but is limited
by transmission loss. Quantum repeaters have
thus been proposed for entanglement distribution
over long distances
1
[4, 5]. The basic idea is to
divide the global distance into short elementary
network links. Entanglement is created in each
link and successive entanglement swapping
operations are used to combine links and extend
entanglement. The key to efficient quantum
repeater is to use elementary links where i) the
successful creation of entanglement is heralded
and ii) entanglement is stored so that it can be
created in each link independently. Impressive
progress along this line now allows one to
envision multipartite quantum networks where
entanglement is distributed between arbitrary
parties. [6–11]
At the heart of quantum networks lies the
ability to distribute but also certify an entangled
state between two distant locations. Although
entangled states have been produced between
remote locations forming an elementary network
link in multiple experiments [12–20], their suit-
ability for general purposes including – but not
1
For protocols that do not rely on entanglement cre-
ation in independent links, but are based exclusively on
error correction codes, see [3].
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 1
arXiv:1812.09117v4 [quant-ph] 1 Mar 2021
limited to – quantum key distribution, remains
unproven. Demonstrations based on the qubit
assumption for instance, stating that all elements
involved are of dimension two, are subject to
side-channels which completely corrupt the
security guarantees [21].
More generally, the identification of a quantum
state provides the most complete description of
a system. But the trace left by a state in the
measurement outcomes is as much influenced
by the state as by the measurement itself.
Consequently, it is challenging to obtain an
accurate description of a quantum state from
observed statistics without presuming a detailed
description of the measurement apparatus. Yet,
characterizing univocally a quantum resource by
identifying its quantum state constitutes a crucial
step to set quantum technologies on a solid stand.
The possibility of device-independent state
characterization which is not relying on assump-
tions on the dimension of the Hilbert space
and on the correct calibration or modelling
of the measurements [22] was first realized in
Ref. [23, 24]. There it was noted that the only
quantum states able to achieve a maximum vio-
lation of the Bell-CHSH inequality [25], are Bell
states – two-qubit maximally entangled states.
Interest in self-testing however only started
growing significantly after Mayers and Yao redis-
covered it and showed that it provides security
for quantum key distribution [26, 27]. Since then,
it has been understood that self-testing guaran-
tees the security of many quantum information
tasks, including randomness generation [28–30]
and delegated quantum computing [31]; see [32]
for more details. Therefore, self-testing a state
guarantees its direct applicability for a wide
range of applications. Motivated by this perspec-
tive, further theoretical self-testing results have
been obtained lately, addressing an increasing
range of states, and with improving tolerance to
noise [30, 33–41]. Moreover, self-testing has also
been extended to the characterization of quan-
tum measurements and channels [27, 31, 42–47].
In the case of Bell states, it is now known that
self-testing based on the Bell-CHSH inequality
is strongly resistant to noise [34, 36, 37]. Re-
cently, this led to the experimental estimation
of self-testing fidelities from the perspective
of hypothesis testing, in which the null hy-
pothesis to be rejected is that the source only
produces states with a fidelity below a fixed
threshold value [48]. Rejection of this hypothesis
then implies that at least one state produced
by the source had a fidelity higher than the
threshold value. However, the implications
of hypothesis testing to practical protocols is
not clear since no statement on the average
fidelity is provided. As an example, data from
the experiment involving two individual ions
separated by ≈ 1 m [29] were shown to lead
to a significant rejection of the null hypothesis
even though the average Bell violation that
could be certified with the methods presented
in [29] at 99% confidence level is lower than 2.05,
hence precluding a conclusion on the average
Bell state fidelity [49]. A higher Bell violation
was demonstrated between two ions separated
by about 340 µm within a trap [48], but this
short-distance setting is not directly applicable
for quantum networks. The recent advent [50–53]
of loophole-free Bell tests [54] involving large
separations between entangled particles opens a
new perspective for device-independent certifi-
cation of states distributed in quantum networks.
We here derive a method that provides a
confidence interval on the
average violation of
any binary Bell inequality without assuming
that the trials are independent and identically
distributed. Applying this method to the CHSH-
Bell test allows us to certify that a source has the
capability of producing on average states close
to a Bell state without making any assumption
on the actual state at each trial. This changes
the status of self-testing from a mere theoretical
tool to a practical certification technique. We
show this by considering the data used on the
loophole-free Bell inequality violation reported
in Ref. [53] where entanglement is distributed
in a heralded way and stored in two single
atoms trapped at two locations separated by
398m before a Bell test is performed. We first
optimize the heralding conditions using an
ab-initio model of the entanglement generation
process, hence improving on the entanglement
fidelity of the data set in [53]. We then apply
our new statistical tool accounting for finite
experimental statistics and imperfections of
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 2
the random number generators. From the
observed Bell-CHSH value, we certify the suc-
cessful distribution over 398 m of an entangled
state with a Bell state fidelity of 55.50% at
a confidence level of 99%. This constitutes
the first result where a statistically relevant
bound on the average fidelity of the distributed
state is obtained directly from the Bell-CHSH
value and the first device-independent certifica-
tion of an elementary link for a quantum network.
2 Device-independent assumptions
The scenario we consider involves three protago-
nists, colloquially referred to as Alice, Bob and
Charlie, see Fig. 1. Charlie holds a prepara-
tion device which indicates when the experiment
is ready: it heralds the start of every measure-
ment procedure. The two other parties each hold
one measurement device and one random number
generator device. Upon heralding, the random
number generators are used by Alice and Bob to
choose a measurement setting which is applied
to their measurement devices. Measurement set-
tings and outcomes are recorded locally for later
analysis. The claim of self-testing for the state
measured is based on a number of assumptions
that we review now.
1. The experiment admits a quantum descrip-
tion. Essentially, the state of the system can
be represented in terms of a density operator,
and the measurements as operators acting on
the same Hilbert space with the appropriate
tensor structure.
2. All devices mentioned above are well iden-
tified in space and operate sequentially in
time. In particular, the separation between
the parties Alice and Bob is clear, as well
as between the random number generators
and the measurement devices of each party.
Moreover, results are recorded before going
to the next round, hence we know exactly
when a round is going on (two rounds don’t
happen simultaneously), when it is finished,
and we can monitor how many rounds hap-
pened in a given time.
3. The random number generators are indepen-
dent from all other devices and sample from
a well characterized probability distribution.
Hence, the measurements used are chosen
freely: the measured particles cannot influ-
ence this choice, nor vice versa. The random
number devices can be correlated to each
other, but not to the rest of the setup.
4. Finally, the classical and quantum communi-
cation between Alice and Bob is limited: no
communication (whether direct or indirect)
is allowed between the measurement boxes
once the settings choices are received and un-
til the measurement outcomes are produced.
Moreover, the random number generators
only provide the choice of measurement set-
ting when required, and to their respective
measurement device. (Note that space-like
separation can be used to guarantee the con-
dition of no communication between Alice
and Bob.)
Apart from the first assumption, which has
not been challenged by any experiment so
far, note that all three remaining assumptions
concern the relation between the various devices
involved in the experiment rather than their
internal working. This approach is thus often
called “black-box" or “device-independent". For
a physical setup permitting for self-testing these
assumptions are requirements. The settings and
requirements for self testing are sufficient to
test a Bell inequality and they have been used
recently in Ref. [53] to perform a loophole-free
violation of the Bell-CHSH inequality. We briefly
present this experiment in the next section.
3 Event-ready CHSH-Bell test with
neutral atoms
In our experiment, Alice and Bob’s stations
are made each with a single
87
Rb atom stored
in an optical dipole trap, see Fig. 1. The two
setups are independently operated, that is, they
are equipped with their own laser and control
systems. Two Zeeman states |m
F
= ±1i of the
ground state manifold 5
2
S
1/2
are used as 1/2-
spin states. After an initial state preparation,
the atoms are optically excited to emit a photon
whose polarization is entangled with the atomic
spin states, see Fig. 3(a). The photons are
sent to Charlie’s station that is located close to
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 3
fiber
BS
PBS
APD
ready
1
2
1
2
Charlie
BSM
device 1
setting
choice
RNG
local
storage
CEM
1e
CEM
1i
Alice
outcome
x
i
a
i
398 m
CEM
2e
CEM
2i
y
i
setting
choice
RNG
local
storage
Bob
outcome
device 2
b
i
single-mode fibers
Figure 1: Sketch of self-testing based on violation of Bell’s inequality with entangled atoms separated by a large
distance. Each "device" of Alice and Bob is an independent apparatus for trapping and manipulating single atoms.
Entanglement between the atoms is generated by entangling the spin of each atom with polarization of a single photon.
The photons are coupled into single mode fibers and overlapped at a fiber beamsplitter. Coincident detection of
two photons in Charlie’s device heralds the entanglement. Alice and Bob then use their random number generators
(RNGs) to select a measurement setting for fast and efficient read-out of the atomic state based on state selective
ionization using particle detectors (CEMs) to detect the created ions (i) and electrons (e).
Alice’s location where a Bell state measurement
is implemented with a beamsplitter followed by
a polarizing beamsplitter at each output port
and four single photon detectors, see Fig. 1. The
atom excitation procedure is synchronized on a
timescale that is much shorter than the photon
duration. Careful adjustment of experimental
parameters ensures a spectral, temporal and
spatial mode overlap of photons close to unity
[19]. This allows us to achieve a high two-photon
interference quality limited mostly by two-
photon emission effects of a single atom. The
joint measurement performed on these photons
distinguishes two out of the four Bell states
and ideally projects the atoms into either of
the two states |ψ
±
i = (|↑i
x
|↓i
x
± |↓i
x
|↑i
x
)/
√
2
according to the outcome. Depending on the
loading rate of the traps, 1 to 2 successful Bell
state measurements are obtained per minute.
At each success, a signal is sent to Alice and
Bob and triggers setting choices. The analysis
basis is selected by the output of a fast quantum
random number generator, which is based on
counting photons emitted by an LED with a
photo-multiplier tube [53, 55]. The measurement
outcome is obtained by a spin-state dependent
ionization with a fidelity of 97% on a timescale
≤ 1.1µs. Given that Alice’s and Bob’s locations
are separated by 398 m, this warrants space-like
separation of the measurements. Although
(strict) space-like separation is not a necessary
condition for self-testing, it is a strong guarantee
that Alice and Bob’s measurement devices are
indeed separated from each other and that
information about the setting of one party is not
available to the other one upon measurement,
i.e. for assumptions 2 and 4 above.
4 CHSH Bell inequality
Let us label the measurement settings x = 0, 1
and y = 0, 1 for Alice and Bob respectively, with
outcomes a = 0, 1 and b = 0, 1 for each spin
measurement. For each pair of settings, we de-
fine the correlator E
xy
=
P
ab
(−1)
a+b
P (a, b|x, y)
where P (a, b|x, y) is the conditional probability
of observing outcome a and b when choosing the
settings x and y. This allows us to define the
Bell-CHSH value, given by
S = E
00
+ E
01
+ E
10
− E
11
. (1)
The latter is upper bounded by 2 for any local
causal theory [25]. A significant violation of
this bound can thus rule out this possibility,
as conclusively demonstrated earlier [53], see
also [50–52]. Note that here the values of 0 and
1 for the settings and outcomes were assigned
arbitrarily, therefore, any of the 8 relabellings
of Eq. (1) equally qualifies as a valid definition
of the quantity S [56]. With fixed measurement
settings, such equivalent rewritings of the CHSH
expression may be necessary to obtain a violation
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 4
of the local bound with different Bell states.
5 Self-testing a Bell state
Given assumption 1 above, we can associate to
each measurement of Alice and Bob quantum ob-
servables A
x
and B
y
acting on two Hilbert spaces
H
A
and H
B
of unknown dimension. Also, we
can define the quantum state shared by the two
parties as ρ
AB
∈ L(H
A
⊗ H
B
). We emphasize
that the internal functioning of the source and
measurement boxes do not need to be known.
We simply attribute a quantum state and mea-
surement operators to the actual implementation.
Our aim is to identify the actual state ρ
AB
from
the observed statistics only. More precisely, we
wish to estimate its fidelity with respect to a max-
imally entangled state of two qubits, that is
F (ρ
AB
) = max
Λ
A
,Λ
B
Tr ((Λ
A
⊗ Λ
B
)[ρ
AB
], |ψ
−
ihψ
−
|),
(2)
where the maximization is over all local trace-
preserving maps Λ
A/B
: H
A/B
→ C
2
. The role of
these maps Λ
A/B
is to identify the subsystems in-
side the unknown Hilbert spaces H
A/B
in which
ρ
AB
can be compared to the desired state. Given
an observed Bell-CHSH value S, the Bell state
self-testing fidelity is defined as the minimum fi-
delity of the unknown quantum state ρ
AB
which
is compatible with the violation, i.e.
F = min
ρ
AB
,A
x
,B
y
F (ρ
AB
) (3)
s.t. E
00
+ E
01
+ E
10
− E
11
= S,
where the correlators are now given by
E
xy
= Tr (ρ
AB
A
x
B
y
). This quantity cap-
tures the relation between ρ
AB
and the singlet
state |ψ
−
i, one representative Bell state, that
can be inferred from observed statistics: if the
quantum state is separable, then F ≤
1
2
; on the
other hand, if F = 1, then we have the guarantee
that local maps exist which identify perfectly a
Bell state within the state ρ
AB
, because this is
the case for all admissible quantum realizations.
It has been shown that the self-testing fidelity
F can be directly related to the sole knowledge of
the Bell-CHSH value S [34]. The tightest known
relation is given by [37]
F ≥ f(S) =
max
40, 12 + (4 + 5
√
2)(5S − 8)
80
.
(4)
6 Statistical analysis
The previous formula holds in the limit where the
CHSH value S is known exactly. In order to an-
alyze a real experiment with finite statistics, we
consider that each run i = 1, . . . , n is character-
ized by an (unknown) CHSH value S
i
and fidelity
F
i
. This fidelity could be different at each round,
and depend on past events. We are then inter-
ested in making a claim on the average fidelity
F =
1
n
n
X
i=1
F
i
. (5)
Other works have considered a different figure of
merit for the certification of states in a non-I.I.D.
setting [32]. In Appendix C, we show that the
two approaches are equivalent to each other and
that the numerical value provided by the average
fidelity (5) has the advantage of being a direct
quantifier of the source quality.
Assuming that the measurement settings are
chosen independently by both parties and with a
maximum bias τ with respect to a uniform distri-
bution, i.e. 1/2 − τ ≤ P (x), P (y) ≤ 1/2 + τ, we
show in the Appendix B that
ˆ
S = 8
I
−1
α
(nt − 1, n(1 − t) + 2) − τ − τ
2
− 4
(6)
is a lower bound on the average CHSH violation
S =
1
n
P
i
S
i
with confidence level 1 − α. This
allows us to conclude that [
ˆ
F, 1] with
ˆ
F = f (
ˆ
S) (7)
is a one-sided confidence interval for F with
confidence level 1 − α. Here, t = (4 + S
u
)/8
with S
u
is the average CHSH value observed
over the n rounds assuming a uniform sampling
of the settings, i.e. following Eq. (40) from the
Appendix B, and I
−1
is the inverse regularized
incomplete Beta function, i.e. I
y
(a, b) = x for
y = I
−1
x
(a, b). We emphasize that this bound on
the average Bell state fidelity does not rely on
the I.I.D. assumption.
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 5
Figure 2: Expected self-testing fidelity
ˆ
F resulting from
the pre-selection model (lines) for different confidence
levels (CL) as a function of the starting time of the ac-
ceptance time window for heralding events t
s
(at a fixed
t
e
). The optimal start times for for different confidence
levels according to the model are shown in Tab. 1. In
comparison the self-testing fidelity for the measured data
(|ψ
−
i, symbols) evaluated with the acceptance time
window starting with t
s
. Note that the model for the
pre-selection is ab-initio and not a fit for the data. For
details of the model and a comparison to the measured
data see Appendix.
7 Preselection
In contrast to results presented in Ref. [53],
where all registered events were taken into
account, here we use a pre-selected set of events
to compute the Bell-CHSH violation and the
subsequent self-testing fidelities of heralded
atomic states. This selection is based on a phys-
ical model which takes into account detrimental
two-photon emission effects of a single atom
and allows to define pre-selection criteria, here
a time-window for acceptance of photons in the
BSM, to improve the fidelity of the entangled
atom-atom state. Details can be found in the
Appendix A. Importantly, these considerations
are not based on the results observed during
the experiment. They are based on an ab-initio
model of the underlying excitation and emission
processes. Therefore, these considerations allow
for the determination of a significance level
and desired amount of data prior to the data
acquisition stage, in agreement with the require-
ments of confidence intervals construction. This
selection can then be seen as a pre-selection of
the data, or equivalently, as a state preparation.
In particular, it does not open the detection
loophole or introduce expectation bias [57].
8 Results
For the evaluation we use the data of events
heralding the |ψ
−
i state from the loophole-
free Bell test [53] taken between 05.02.2016 and
24.06.2016 (25211 events). Fig. 2 shows the re-
sulting lower bound of the average fidelity
ˆ
F for
the ab-initio model and the data set using the
same pre-selection as a function of the acceptance
window starting time t
s
for different confidence
levels. The model allows to determine the accep-
tance time window start time t
s
and end time
t
e
for an optimal expected lower bound for the fi-
delity
ˆ
F for each confidence level shown in Fig. 2.
The results for the pre-selected data are shown in
Tab. 1. For calculation of the lower bound of the
fidelity
ˆ
F we consider bias of the RNGs bounded
by τ = 6.3 × 10
−4
(arsing from the “paranoid”
model for the predictability [53]).
The lower bound of fidelity exceeding the value
of 0.5 (at a confidence level of up to 1−1.0×10
−7
for t
s
= 746 ns) represents the first device-
independent certification of a distributed entan-
gled state. Moreover, an evaluation of the full
data set without any pre-selection yields a Bell
state fidelity of 0.5061 at 99% confidence and a
Bell state fidelity larger than 0.5 can be certified
even at a significance level as high as 99.7%.
As a comparison, note that a confidence in-
terval could also be constructed from Hoeffding’s
inequality [58], yielding
ˆ
S
H
= 8
t −
s
log(1/α)
2n
− τ − τ
2
− 4. (8)
The conclusion obtained with this inequality
would however be significantly weaker
2
. For in-
stance, the claim that the fidelity F is nontriv-
ial (i.e. ≥ 1/2) for the whole data set would
not be statistically significant. Indeed, the corre-
sponding statistical level is α = 6.5%, i.e. about
20 times larger than guaranteed by our bound.
The average fidelity guaranteed with pre-selection
would also be significantly lower. Using t
s
=
748 ns, for instance, the lower bound on the fi-
delity at 99% confidence level is 0.5291, i.e. about
2
Perhaps counter-intuitively, the improved tail bound
given in Eq. (2.1) of [58] is not of much help here since it
relies on the knowledge of the average winning probabil-
ity, but confidence intervals must hold for all (unknown a
priori) winning probabilities.
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 6
CL
ˆ
F t
s
n S
u
99% 0.5550 748 ns 13141 2.2589
99.9% 0.5407 746 ns 14807 2.2554
99.99% 0.5272 745 ns 15671 2.2505
Table 1: Fidelity
ˆ
F at different confidence levels. For
each the data is pre-selected with an optimal start time
for the acceptance time window [t
s
, t
e
= 895 ns] result-
ing in the corresponding n events and an average CHSH
value S
u
.
twice closer to the trivial value of 1/2 compared
to 0.5550.
Additionally, we applied our method to the
data of the table-top Bell test performed between
two ions separated by 1 meter reported in [29].
Our statistical analysis yields an average Bell vio-
lation of 2.2715 or higher at 99% confidence level,
clearly above the threshold 2.11. This sets a lower
bound on the Bell state fidelity of 61.46% at this
confidence level, hence guaranteeing for the first
time that the states distributed in this experi-
ment had a significant average Bell state fidelity.
Finally, we applied our method to data
from [59] obtained at a distance of 1.3 km.
Due to the limited number of events (545), the
method can only guarantee a fidelity larger than
50% with a confidence of ∼ 94.2%. Still, this
demonstrates that the method can be used in
different systems without the need of knowing
their specific details.
9 Discussion
We have derived a bound on the average fidelity
of a measured state with respect to a Bell
state from the sole knowledge of the observed
Bell-CHSH value which is free of the I.I.D.
assumption. This bound was achieved by con-
structing a non-I.I.D. confidence interval for the
sum of n independent binary random variables.
We used it to quantify device-independently the
quality of a bipartite state distributed over 398 m
in a real-world elementary quantum network
link. These results guarantee that this link is
suitable for an integration in a quantum network,
either directly or as a part of a quantum repeater.
Acknowledgments
J.-D.B., P.S. and N.S. acknowledge funding by
the Swiss National Science Foundation (SNSF),
through the Grant PP00P2-179109 and 200021-
175527, by the Army Research Laboratory Cen-
ter for Distributed Quantum Information via the
project SciNet and from the European Union’s
Horizon 2020 research and innovation programme
under grant agreement No 820445 and project
name Quantum Internet Alliance. K.R. and
W.R. acknowledge funding by the German Fed-
eral Ministry of Education and Research via the
project Q.com-Q.
A Preselection of heralding events
To allow filtering as a preselection we have devel-
oped an ab-initio physical model independently
of our measurement results to find an optimum
filtering based on the model only. This model
describes the photon emission process of a single
atom excited by a short laser pulse and takes into
account all important processes within its multi-
level structure. Thereby we are able to calculate
the expected fidelity for the entangled state of two
atoms heralded by a two-photon coincidence at a
certain time. The full description of the model
goes far beyond the focus of the present work and
can be found in [60, 61], here we only present a
brief sketch of it.
For generation of a photon whose polarization
is entangled with the atomic spin state, the atom
is excited by a laser pulse resonant to the transi-
tion 5
2
S
1/2
, F = 1 → 5
2
P
3/2
, F
0
= 0. The tem-
poral shape of the pulse is approximately Gaus-
sian with a FWHM of 22 ns, see Fig. 4. After
the successful emission of a photon, ideally, the
atom should not interact with the excitation laser
due to selection rules, see Fig. 3(a). In practice,
however, the atomic state remains weakly sensi-
tive to the excitation laser due to two reasons.
First, there unavoidably are small polarization
misalignments of the excitation laser, i.e. its po-
larization is not perfectly aligned along the quan-
tization axis (imperfect π-polarization), allowing
for a reexcitation of the 5
2
P
3/2
, F
0
= 0, m
F
= 0
level, see Fig. 3(b). Second, off-resonant scat-
tering via the 5
2
P
3/2
, F
0
= 1 level is possible
(Fig. 3(c),(d)). Moreover, before the emission of a
photon into the desired mode takes place, there is
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 7
5
2
P
3/2
F=1
5
2
S
1/2
F=2
F'=0
F'=1
(a)
(b)
5
2
P
3/2
F=1
5
2
S
1/2
F=2
F'=0
F'=1
(c)
(d)
Figure 3: Structure of the relevant levels in
87
Rb, ex-
citation an decay processes. (a) Generation of atom-
photon entanglement in spontaneous decay of the ex-
cited 5
2
P
3/2
, F
0
= 0, m
F
= 0 level. The excitation laser
is shown in orange. Photons polarized linearly along the
quantization axis (π-decays, gray) are not detected in
our system. After the first decay, a second excitation
is possible due to polarization misalignment (b), or off-
resonant excitation (c). If the 5
2
P
3/2
, F
0
= 1 is excited,
also decay to 5
2
S
1/2
, F = 2 level is possible (d).
a finite probability that the atom emitted a first
photon in a π-transition, which is not collected
by the optics. These multiple photon emissions
are detrimental for the quality of the atomic state
announced by detection of the photons in the Bell
state measurement in two different ways. On the
one hand, the state of the atom can be changed
by scattering additional photons. On the other
hand, the interference quality of photons is re-
duced since the temporal shape and coherence of
the photonic wavepackets is affected.
B Finite statistics analysis
In this section, we detail the construction of the
confidence interval on the average singlet fidelity
reported in the main text.
B.1 Model
In the experimental situation described in the
main text, the settings used after the i
th
herald-
ing event can be described by two random vari-
ables X
i
and Y
i
. These variables follow a global
probability distribution
P (
~
X = ~x,
~
Y = ~y), (9)
Figure 4: Time histogram of single photon detection
(red) for excitation of a single atom by a short resonant
laser pulse (blue). t
s
and t
e
define the acceptance time
window for coincidences.
Importantly, the unwanted multiphoton pro-
cesses happen predominantly during the excita-
tion. Thus, to filter them, we only accept the
detection events for which the first detector click
is obtained after a time t
s
when the excitation
laser pulse is essentially off, see Fig. 4. Addition-
ally, to reduce the dark counts contribution to the
heralding event we define a maximal time t
e
for
the detection of the second photon. A later start
time t
s
increases the entanglement swapping fi-
delity and by this the expected S-value for the
CHSH inequality but at the expense of the ob-
tained events number, see Fig. 5. Since the mea-
sured S-values will depend not only on the entan-
glement swapping fidelity but also on other prop-
erties, e.g., the atomic state measurement fidelity
and the coherence time of the entangled states,
we use the experimental parameters as specified
in [53] to predict the experiment’s S-value.
The optimal selection of the time window of
[t
s
= 748 ns, t
e
= 895 ns], considering Eq. (7) for
a 99% confidence interval, reduces the number of
heralding events by approximately a factor of 2
but the atoms are expected to be in an entangled
state of a higher quality.
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 8
Figure 5: S-value for CHSH inequality (orange line) re-
sulting from the ab-initio model as a function of the
start time t
s
of the acceptance time window. The or-
ange circles show the values obtained from experimental
data by applying the same filtering criteria. The black
dashed line and squares respectively show the predicted
and measured fraction of remaining events after filter-
ing. The optimal calculated start time for the accep-
tance time window is t
s
= 748 ns according to Eq. (7)
for 99% confidence. The contribution of t
e
to the fidelity
was found to be negligible, it was fixed at t
e
= 890 ns.
where
~
X = (X
1
, X
2
, . . .),
~
Y = (Y
1
, Y
2
, . . .) and
~x, ~y ∈ {0, 1}
n
for a binary choice of settings.
Similarly, two random variables A
i
and B
i
can
be used to describe the outcomes observed upon
measuring the state in the i
th
round. By assump-
tions 2-4 of the main text, the settings and out-
comes follow a joint probability distribution of
the form
P (
~
A = ~a,
~
B =
~
b,
~
X = ~x,
~
Y = ~y) =
P (
~
X = ~x,
~
Y = ~y)
n
Y
i=1
P
i
(a
i
, b
i
|x
i
, y
i
, past
i
).
(10)
Here, ~a,
~
b ∈ {0, 1}
n
are the possible outcome
strings in the binary case,
P
i
(a, b|x, y, past
i
) =
P (A
i
= a, B
i
= b|X
i
= x, Y
i
= y, Past
i
= past
i
)
(11)
describes the behavior sampled in the i
th
round
and past
i
3 {a
j
, b
j
, x
j
, y
j
}
j<i
stands for any infor-
mation available from the past of round i such as
the previous settings and outcomes. The settings
distribution can be decomposed into the measure-
ment rounds in a similar fashion as
P (
~
X = ~x,
~
Y = ~y) =
n
Y
i=1
P
i
(x
i
, y
i
|past
i
) (12)
with
P
i
(x, y|past
i
) = P (X
i
= x, Y
i
= y|Past
i
= past
i
).
(13)
We associate to each measurement round i the
CHSH value
S
i|past
i
=
X
a,b,x,y
(−1)
a+b+xy
P
i
(a, b|x, y, past
i
).
(14)
This quantity S
i|past
i
is the expectation value for
the S parameter given in Eq. (1) for the given
round i. We also define the singlet fidelity F
i|past
i
,
which is bounded according to Eq. (4) as
F
i|past
i
≥ f (S
i|past
i
). (15)
Note that these statistical parameters may be dif-
ferent for all rounds i and may depend on past
events.
B.2 Estimation
Before focusing on the fidelity, our figure of merit,
let us estimate the Bell contribution correspond-
ing to a given round i. For this, we introduce the
statistic
T
i|past
i
=
1
4
·
χ(A
i
⊕ B
i
= X
i
Y
i
)
P
i
(X
i
, Y
i
|past
i
)
, (16)
where χ is the indicator function,
i.e. χ(condition) = 1 if the condition is true
and χ(condition) = 0 for a false condition, ⊕
is the addition modulo 2, and the term in the
denominator
P
i
(X
i
,Y
i
|past
i
)
=
P
i
(0, 0|past
i
), if X
i
= 0, Y
i
= 0
P
i
(0, 1|past
i
), if X
i
= 0, Y
i
= 1
P
i
(1, 0|past
i
), if X
i
= 1, Y
i
= 0
P
i
(1, 1|past
i
), if X
i
= 1, Y
i
= 1
(17)
refers to the probability with which the observed
settings have been sampled, a notation custom-
ary in statistics (c.f. the usual definition of Fisher
information for instance). The expectation value
of this estimator is directly related to the CHSH
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 9
violation on round i given the past:
E(T
i|past
i
)
=
X
~a,
~
b,~x,~y
T
i|past
i
P (
~
A = ~a,
~
B =
~
b,
~
X = ~x,
~
Y = ~y)
=
1
4
X
a,b,x,y
χ(a ⊕ b = xy)
P (X
i
= x, Y
i
= y|Past
i
= past
i
)
× P (A
i
= a, B
i
= b, X
i
= x, Y
i
= y|Past
i
= past
i
)
=
1
4
X
a,b,x,y
χ(a ⊕ b = xy) P
i
(a, b|x, y, past
i
)
=
4 + S
i|past
i
8
.
(18)
This expression thus provides a good estimation
of the Bell violation contribution of round i. Note
that the relation (18) is valid for all distribution
of the settings which is independent from A and
B’s behavior according to Eq. (10).
In the case where the settings are chosen uni-
formly, i.e. P
i
(x
i
, y
i
|past
i
) =
1
4
, the random vari-
able T
i|past
i
is a Bernoulli variable whose only
possible values are 0 and 1. It can then be in-
terpreted as a binary game which is either won
(if T
i|past
i
= 1) or lost (if T
i|past
i
= 0). The
CHSH contribution of round i can then be re-
interpreted in terms of the winning probability
q
i|past
i
= P (T
i|past
i
= 1) of this game, such that
8q
i|past
i
= 4 + S
i|past
i
. (19)
B.3 Settings choice bias
In practice, it may be difficult to guarantee that
the choice of settings is exactly uniform. One
can then resort to a partial characterization of
the settings’ distribution. For instance, consider
the case where the settings of Alice and Bob are
chosen independently as
P
i
(x
i
, y
i
|past
i
) = P (x
i
|τ
x
i
)P (y
i
|τ
y
i
) (20)
with
P (x
i
|τ
x
i
) =
1
2
+ (−1)
x
i
τ
x
i
(21)
P (y
i
|τ
y
i
) =
1
2
+ (−1)
y
i
τ
y
i
, (22)
and we only have the guarantee that the local bi-
ases are bounded |τ
x
i
|, |τ
y
i
| ≤ τ by some maximal
value τ ≤
1
2
. In this case the statistic (16) as well
as the CHSH value Eq. (14) cannot be evaluated
directly. We can nevertheless bound its behavior.
For this, let us then consider the statistic that
would correspond to a uniform choice of settings
T
u
i|past
i
= χ(A
i
⊕ B
i
= X
i
Y
i
).
(23)
As mentioned before, this statistic is a Bernoulli
random variable taking value either 0 or 1. Its
winning probability is q
u
i|past
i
= E(T
u
i|past
i
), and
can be evaluated without the knowledge of the
settings distribution. It’s expectation value is
given by
E(T
u
i|past
i
)
=
X
a,b,x,y
χ(a ⊕ b = xy)P
i
(a, b|x, y, past
i
) (24)
× P (x|τ
x
)P (y|τ
y
)
=
X
a,b,x,y
χ(a ⊕ b = xy)P
i
(a, b|x, y, past
i
)
×
1
4
+
(−1)
x
2
τ
x
+
(−1)
y
2
τ
y
+ (−1)
x+y
τ
x
τ
y
=
1
4
X
a,b,x,y
χ(a ⊕ b = xy)P
i
(a, b|x, y, past
i
)
| {z }
=E(T
i|past
i
)
(25)
+
X
a,b,x,y
χ(a ⊕ b = xy)P (a, b|x, y, past
i
)
×
(−1)
x
2
τ
x
+
(−1)
y
2
τ
y
+ (−1)
x+y
τ
x
τ
y
Defining f
xy
=
P
a,b
χ(a ⊕ b =
xy)P (a, b|x, y, past
i
) ∈ [0, 1] we write
E(T
u
i|past
i
) − E(T
i|past
i
) (26)
=
X
x,y
f
xy
(−1)
x
2
τ
x
+
(−1)
y
2
τ
y
+ (−1)
x+y
τ
x
τ
y
.
Let us now consider this sum. Without loss of
generality we set 0 ≤ τ
y
≤ τ
x
≤ τ , all the other
cases directly follow by a permutation of the out-
comes or the exchange of x and y. One has
2
X
x,y
f
xy
(−1)
x
2
τ
x
+
(−1)
y
2
τ
y
+ (−1)
x+y
τ
x
τ
y
= f
00
(τ
x
+ τ
y
+ 2τ
x
τ
y
) + f
10
(−τ
x
+ τ
y
− 2τ
x
τ
y
)
+ f
01
(τ
x
− τ
y
− 2τ
x
τ
y
) + f
11
(−τ
x
− τ
y
+ 2τ
x
τ
y
)
≤ τ
x
+ τ
y
+ 2τ
x
τ
y
+ f
01
(τ
x
− τ
y
− 2τ
x
τ
y
) ,
(27)
where we used f
xy
∈ [0, 1], τ
x
+ τ
y
+ 2τ
x
τ
y
≥ 0,
−τ
x
+ τ
y
−2τ
x
τ
y
≤ 0 and −τ
x
−τ
y
+ 2τ
x
τ
y
≤ 0.
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 10
For the last term one finds
f
01
(τ
x
− τ
y
− 2τ
x
τ
y
)
≤
(
τ
x
− τ
y
− 2τ
x
τ
y
τ
y
<
τ
x
1+2τ
x
0 τ
y
≥
τ
x
1+2τ
x
,
(28)
leading to
X
x,y
f
xy
(−1)
x
2
τ
x
+
(−1)
y
2
τ
y
+ (−1)
x+y
τ
x
τ
y
≤
(
τ
x
τ
y
<
τ
x
1+2τ
x
1
2
(τ
x
+ τ
y
) + τ
x
τ
y
τ
y
≥
τ
x
1+2τ
x
(29)
≤ τ + τ
2
, (30)
which holds for all allowed values of τ
x
and τ
y
.
Plugging this inequality in Eq. (26) then gives
E(T
u
i|past
i
) − E(T
i|past
i
) ≤ τ + τ
2
. (31)
Therefore, we obtain
q
i|past
i
≥ q
u
i|past
i
− τ − τ
2
, (32)
meaning that a lower bound on the winning prob-
ability q
u
i|past
i
of the uniform statistic T
u
i|past
i
gives
rise to a lower bound on q
i|past
i
. In order to esti-
mate q
i|past
i
with a distribution of settings which
is not fully known, we can thus safely estimate
the CHSH value with the statistic (23), effectively
assuming that the settings are chosen uniformly,
and then correct the winning probability q
u
i|past
i
according to the value of τ , as expressed in (32).
This provides a lower bound on the actual win-
ning probability q
i|past
i
of (16).
To simplify the notation, we now drop the ex-
plicit conditioning on the past, and thus simply
write e.g. S
i
, q
i
, T
i
and F
i
for the quantities in-
troduced above.
B.4 Bounding the fidelity
We construct a statistical parameter for the whole
experiment corresponding to the average Bell
state fidelity:
F =
1
n
X
i
F
i
. (33)
Thanks to the convex relation between the CHSH
violation and the singlet fidelity Eq. (4), this
quantity can be bounded from the average CHSH
violation S as
F =
1
n
X
i
F
i
(34)
≥
1
n
X
i
f(S
i
) (35)
≥ f
1
n
X
i
S
i
!
= f(S), (36)
or equivalently, using relation (19), from the av-
erage winning probability q =
1
n
P
i
q
i
as
F ≥ f(8q − 4). (37)
In particular, a left-sided confidence interval for
q gives rise to a left-sided confidence interval for
the singlet fidelity. By relation (32), a left-sided
confidence interval for q
u
=
P
i
q
u
i
gives rise to a
left-sided confidence interval for q, and thus also
for the singlet fidelity:
F ≥ f
8(q
u
− τ − τ
2
) − 4
. (38)
Let us thus focus now on the average winning
probability q
u
.
B.5 A confidence interval for the average win-
ning probability
The random variables T
u
i
in Eq. (23) being esti-
mators for the parameters q
u
i
, we use their aver-
age
T
u
=
1
n
n
X
i=1
T
u
i
(39)
to estimate q
u
. This gives rise to the following
effective CHSH value
S
u
= 8T
u
− 4 (40)
which can be evaluated in practice directly from
the observed data, without assumption on the
distribution of measurement settings. Note that
each random variable T
u
i
is a Bernoulli vari-
able with parameter q
u
i
. Therefore, T
u
is a so-
called (renormalized) Poisson binomial random
variable. The distribution probability of such a
random variable in terms of the average param-
eter q
u
has been characterized by Hoeffding in
1956 [62]. We recall this result here.
Theorem B.1 (Hoeffding, 1956). Let T =
1
n
P
n
i=1
T
i
be the average of n independent
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 11
Bernoulli variables T
i
with parameters q
i
. If c
and d are two integers such that
0 ≤ c ≤ nq ≤ d ≤ n (41)
for q =
1
n
P
n
i=1
q
i
, then
P (c ≤ nT ≤ d) ≥
d
X
k=c
n
k
!
q
k
(1 − q)
n−k
. (42)
This theorem says that within all sets of n
choices of Bernoulli variables {T
i
}
n
i=1
with a fixed
average parameter q =
1
n
P
i
q
i
, the one produc-
ing the largest tail distribution for the average
variable T is the set of n identically-distributed
Bernoulli variables with q
i
= q ∀i. The tail prob-
ability then follows a binomial distribution. Since
q
i
= q ∀i is an admissible parameter value, this
bound is tight.
We recall that the Binomial cumulative distri-
bution can be expressed in terms of the regular-
ized incomplete Beta function I
x
(a, b) as
d
X
k=0
n
k
!
q
k
(1−q)
n−k
= 1−I
q
(d+ 1, n −d), (43)
or equivalently
n
X
k=c
n
k
!
q
k
(1 − q)
n−k
= I
q
(c, n − c + 1). (44)
We then denote the inverse regularized incom-
plete Beta function by I
−1
, i.e. I
y
(a, b) = x for
y = I
−1
x
(a, b).
Note that even if Thm. B.1 applies to inde-
pendent (though not necessarily identically dis-
tributed) random variables, it is still useful in
our context, in which we do not wish to assume
rounds to be either independent or identically dis-
tributed. The main reason for that is that our
figure of merit is given by the average winning
probability q
u
defined as the average of individ-
ual winning probabilities q
u
i
= q
u
i|past
i
conditioned
on past events. Hence, even if the physical pro-
cess under study may depend on previous rounds,
the random variables on which we would like to
apply the theorem are statistically independent
from each other: P (T
u
i
= t
i
, T
u
i+1
= t
i+1
) =
P (T
u
i
= t
i
)P (T
u
i+1
= t
i+1
). Indeed, the random
variable T
u
i
of each round i is fully characterized
by a single and well-definied parameter q
u
i
. We
can thus use this result to construct a confidence
interval for the average CHSH winning probabil-
ity q
u
.
Theorem B.2. Given Bernoulli random vari-
ables T
i
with parameter q
i
for i = 1, . . . , n and
0 ≤ α ≤ 1/2, the interval [ˆq, 1] with the random
variable
ˆq =
(
I
−1
α
(nT − 1, n(1 − T ) + 2) if nT ≥ 1
0 if nT = 0
(45)
is a confidence interval for q =
1
n
P
i
q
i
with con-
fidence level 1 − α.
Proof. We need to show that
P (ˆq ≤
q) ≥ 1 − α (46)
for all possible sets of Bernoulli variables charac-
terized by parameters 0 ≤ q
i
≤ 1 with
1
n
P
i
q
i
=
q. Condition (46) states that whatever the un-
known distribution of the Bernoulli variables T
i
happens to be, the value of ˆq computed from
them (the random variable ˆq depends on the ob-
served T ) can be higher than the actual param-
eter q only with probability at most α. ˆq then
constitutes a reasonable lower bound for the pa-
rameter q.
The case for which nT = 0 is clear. We can
thus assume that nT ≥ 1. But before starting,
let us introduce the function
g :R → R
z 7→ I
−1
α
(z, n − z + 1).
(47)
This function describes the trade-off between the
parameters q and z in I
q
(z, n − z + 1). Let
us remember that the incomplete beta function
I
q
(α, β) is the cumulative distribution function
for the beta distribution with parameter α and β.
Therefore, I
q
(z, n−z +1) is strictly increasing on
q ∈ [0, 1] even for non-integer values of n and z.
At the same time, 1−I
q
(z, n−z+1) seen as a func-
tion of z ∈ R is a cumulative distribution for the
continuous binomial distribution with parameter
q, so it strictly increases with z ∈ [0, n + 1] [63].
In other words, I
q
(z, n −z + 1) strictly decreases
with z. Since g(z) is defined as the value of q
which leaves I
q
(z, n −z + 1) invariant (and equal
to α) when z changes, it is a strictly increasing
function of z ∈ [0, n + 1]: increasing z increases
I
q
(z, n − z + 1) unless q increases as well.
Accepted in Quantum 2021-01-28, click title to verify. Published under CC-BY 4.0. 12