Wang
et al
.,
Sci. Adv.
8
, eabk1660 (2022) 9 February 2022
SCIENCE ADVANCES
|
RESEARCH ARTICLE
1 of 7
PHYSICS
Significant loophole-free test of
Kochen-Specker
contextuality using two species of
atomic ions
Pengfei Wang
1,2
†, Junhua
Zhang
3
†, Chun-Yang
Luan
1
, Mark
Um
1
, Ye
Wang
4
, Mu
Qiao
1
, Tian
Xie
5
,
Jing-Ning Zhang
2
, Adán
Cabello
6,7
*, Kihwan
Kim
1,2,8
*
Quantum measurements cannot be thought of as revealing preexisting results, even when they do not disturb
any other measurement in the same trial. This feature is called contextuality and is crucial for the quantum
advantage in computing. Here, we report the observation of quantum contextuality simultaneously free of
the detection, sharpness, and compatibility loopholes. The detection and sharpness loopholes are closed by adopting
a hybrid two-ion system and highly efficient fluorescence measurements offering a detection efficiency of 100%
and a measurement repeatability of >98%. The compatibility loophole is closed by targeting correlations between
observables for two different ions in a Paul trap, a
171
Yb
+
ion and a
138
Ba
+
ion, chosen so measurements on each
ion use different operation laser wavelengths, fluorescence wavelengths, and detectors. The experimental
results show a violation of the bound for the most adversarial noncontextual models and open a way to certify
quantum systems.
INTRODUCTION
In everyday life, whenever the measurements of two observables
A
and
B
yield the same values (
a
for
A
and
b
for
B
) when the measure-
ments are repeated in any order, we attribute it to the measured
system having preexisting values revealed by every measurement
and which persist after the measurements. However, this assump-
tion fails in quantum mechanics. Quantum systems can produce
correlations (
1
,
2
) between measurements that do not disturb each
other and yield the same result when repeated and that, however,
cannot be explained by models on the basis of the assumption of
outcome noncontextuality that states that the result is the same no
matter which other compatible observables are measured in the
same trial. This phenomenon, called Kochen-Specker contextuality
or contextuality for sharp measurements, is rooted in the Bell-
Kochen-Specker theorem (
3
,
4
) of impossibility of hidden variables
in quantum mechanics and is behind the power of quantum
computers to outperform classical computers (
5
–
9
).
Contextual correlations between sequential measurements have
been observed in experiments with photons (
10
–
17
), neutrons (
18
),
ions (
19
–
21
), molecular nuclear spins (
22
), superconducting sys-
tems (
23
), and nuclear spins (
24
). However, these experiments have
“loopholes,” as noncontextual models assisted by mechanisms that
exploit the experimental imperfections can simulate the observed
correlations.
Three main loopholes have been considered. The sharpness
loophole follows from the observation (
25
,
26
) that the assumption
of outcome noncontextuality, on which the bound of the non-
contextuality inequalities is derived (
1
,
2
), can only be justified for
the case of sharp measurements, defined (
27
,
28
) as those that yield
the same result when repeated and do not disturb compatible
[i.e., jointly measurable (
29
)] observables. The detection loophole
(
30
,
31
) exploits the lack of perfect detection efficiency and is
common to Bell inequality experiments (
32
–
35
). The compatibility
loophole (
36
–
39
) exploits that, in experiments with sequential mea-
surements on the same system, the assumption that the measured
observables are compatible cannot be verified.
Loophole-free Bell inequality tests (
32
–
35
) can be thought as
contextuality tests that simultaneously close the detection and
compatibility loopholes. However, as tests of noncontextual models,
they leave open the sharpness loophole. On the other hand, there
are contextuality tests free of the detection loophole and whose cor-
relations cannot be produced by specific mechanisms exploiting the
lack of perfect repeatability (
19
,
40
). However, they suffer from the
compatibility loophole, as they require sequential measurements
performed on the same system. Therefore, a pending challenge is
closing all three loopholes simultaneously in the same experiment.
For this aim, we choose a composite system of two different ions
(
41
–
46
), one
171
Yb
+
ion and one
138
Ba
+
ion. This dual-species
system allows us to perform sequential repeatable highly efficient
single-shot fluorescence measurements on each of the ions. In the
system, the detection loophole is naturally addressed because of the
detection efficiency of 100%, that is, no missing results in all trials of
the experiments. We note that the detection fidelity, the probability
of obtaining a correct result from a measurement, is not necessary
to be perfect to close the detection loophole.
To close the compatibility loophole, we target a “Bell-like”
(
10
,
11
) noncontextuality inequality, in which only two observ-
ables are measured per context, and each of the observables is
defined on a different ion. Therefore, these two observables are
“trivially compatible (i.e., they are simultaneously measurable in
an uncontentious sense)” (
47
). In addition, the compatibility is
enforced by choosing ions of different species requiring different
1
State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics,
Tsinghua University, Beijing 100084, People’s Republic of China.
2
Beijing Academy
of Quantum Information Sciences, Beijing 100193, People’s Republic of China.
3
Shenzhen Institute for Quantum Science and Engineering, Southern University of
Science and Technology, Shenzhen 518055, People’s Republic of China.
4
Depart-
ment of Electrical and Computer Engineering, Duke University, Durham, NC 27708,
USA.
5
Kavli Nanoscience Institute and Thomas J.
Watson Sr., Laboratory of Applied
Physics, California Institute of Technology, Pasadena, CA 91125, USA.
6
Departa-
mento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain.
7
Instituto
Carlos I de Física Teórica y Computacional, Universidad de Sevilla, E-41012 Sevilla,
Spain.
8
Frontier Science Center for Quantum Information, Beijing 100084, People’s
Republic of China.
*Corresponding author. Email: adan@us.es (A.C.); kimkihwan@mail.tsinghua.
edu.cn (K.K.)
†These authors contributed equally to this work.
Copyright © 2022
The Authors, some
rights reserved;
exclusive licensee
American Association
for the Advancement
of Science. No claim to
original U.S. Government
Works. Distributed
under a Creative
Commons Attribution
NonCommercial
License 4.0 (CC BY-NC).
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Wang
et al
.,
Sci. Adv.
8
, eabk1660 (2022) 9 February 2022
SCIENCE ADVANCES
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RESEARCH ARTICLE
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operation laser wavelengths, fluorescence wavelengths, and
detectors.
RESULTS
The noncontextuality inequality we focus on is the only tight
(i.e., strictly separating noncontextual from contextual correlations)
noncontextuality inequality in the four-cycle contextuality scenario
(
48
) shown in Fig. 1. This is the scenario involving the smallest
number of measurements that allows for contextuality for sharp
measurements, as follows from a theorem by Vorob’ev (
49
,
50
).
This noncontextuality inequality is algebraically identical to the
Clauser-Horne-Shimony-Holt (CHSH) Bell inequality (
51
). It can
be written as
C
= 〈
ˆ
O
0
ˆ
O
1
〉 + 〈
ˆ
O
1
ˆ
O
2
〉 + 〈
ˆ
O
2
ˆ
O
3
〉 − 〈
ˆ
O
3
ˆ
O
0
〉 ≤ 2
(1)
where each of the four observables
ˆ
O
i
has possible results either −1
or +1 and
〈
ˆ
O
i
ˆ
O
j
〉 denotes the mean value of the product of the
results of
ˆ
O
i
and
ˆ
O
j
. Unlike the CHSH Bell inequality, testing
inequality (
1
) neither requires spacelike separation (
52
) nor assigning
the observables to two parties. Instead, it requires the measure-
ments to be sharp.
The test of inequality (
1
) is performed on a two-qubit system in
which each qubit is encoded in a different atomic ion. One of
171
Yb
+
and the other of
138
Ba
+
both trapped in a four-rod Paul trap (
53
), as
shown in Fig. 2. The first qubit is encoded in two hyperfine levels of
the
2
S
1/2
manifold of the
171
Yb
+
ion. The corresponding states are
denoted ∣0〉
Yb
≡
∣
F
= 0,
m
F
= 0〉 and ∣1〉
Yb
≡
∣
F
= 1,
m
F
= 0〉. The
energy gap between the two states is
f
Yb
= 12.64281 GHz. The
second qubit is encoded in the two Zeeman levels of the
2
S
1/2
manifold of the
138
Ba
+
ion. The corresponding states are denoted
∣0〉
Ba
≡
∣
m
= 1/2〉 and ∣1〉
Ba
≡
∣
m
= −1/2〉. The energy gap is
f
Ba
=
16.8 MHz in an external magnetic field of 6.0 G.
The two-ion system is initially prepared in state
∣
〉 =
1
_
√
_
2
(∣ 00〉 +
i
∣
11〉)
. The state of each qubit can be measured with a fluo-
rescence detection technique. For the
171
Yb
+
ion, the cyclic transi-
tion between ∣
F
= 1〉 states in
2
S
1/2
and ∣
F
= 0,
m
F
= 0〉 in
2
P
1/2
is
excited with a 370-nm laser beam so that only ∣1〉
Yb
scatters pho-
tons. The error of detecting ∣1〉
Yb
for ∣0〉
Yb
is 0.96%, and the other
error is 2.25%. For the
138
Ba
+
ion, we first transfer the population
of ∣0〉
Ba
to
2
D
5/2
with a 1762-nm laser beam before exciting the 493-nm
transition between
2
S
1/2
and
2
P
1/2
levels. The error of detecting ∣1〉
Ba
for ∣0〉
Ba
is 2.10%, and the other error is below 0.01%. A 1064-nm
picosecond-pulsed laser is used for the coherent quantum operations
of the two qubits. Two beams from its 532-nm frequency-
doubled
output are used to generate a stimulated Raman process to control
the
138
Ba
+
ion, and another two beams from its 355-nm frequency-
tripled output are used for the ion (
54
). The schematic diagram of
the arrangement of both Raman laser beams is shown in Fig. 2C.
State ∣
〉 is generated through the Mølmer-Sørensen (M-S)
interaction mediated by the axial out-of-phase (OOP) mode of the
two ions with a frequency of
f
z
= 1.67 MHz (
44
).
Fig. 1. The four observables and their compatibility relations.
ˆ
O
0
and
ˆ
O
2
are
measured on the
171
Yb
+
ion, and
ˆ
O
1
and
ˆ
O
3
are measured on the
138
Ba
+
ion.
Connected observables are compatible (jointly measurable). Here,
ˆ
x
,
ˆ
y
are Pauli
operators, and
ˆ
I
2
is the identity operator.
B
=6
.0
G
138
Ba
+
171
Yb
+
x
z
2
P
1/2
2
S
1/2
2
P
3/2
66 THz
34 THz
12.64281
GHz
355nm
370n
m
2
P
1/2
2
S
1/2
44 THz
94 THz
2
P
3/2
493 nm
532nm
614n
m
650 nm
1762
nm
5
D
3/2
5
D
5/2
f
(MHz)
Radial
OO
P
Radial
IP
Ax
ial IP
Ax
ial OO
P
16.80MHz
PMT
2
PMT
1
AB
C
D
F
= 0
F
= 1
F
= 0
F
= 1
F
= 1
Fig. 2. Experimental setup.
(
A
and
B
) are the energy level diagrams of
171
Yb
+
and
138
Ba
+
ion, respectively. Only relevant Raman transitions are shown here. (
C
) Ion
trap in the octagon chamber and schematic diagram for Raman beams. Two different
photomultiplier tubes (PMTs) with different spectral responses and filters are used
to detect two ions fluorescence independently, which are located at the top and
the bottom of the chamber in the actual experimental system. Solid and dashed
arrows indicate the directions and the polarizations of 532-
and 355-nm laser
beams, respectively. In the figure,
f
Yb
and
f
Ba
are the qubit frequencies of
171
Yb
+
and
138
Ba
+
, respectively;
f
z
= 1.67
MHz is the frequency of the axial out-of-phase (OOP)
mode; and
is the detuning of the laser from the OOP mode sideband, when
is
zero, then the Raman transition is directly red and blue sideband transitions. For
the M-S gate,
should match to the sideband Rabi frequency and determines the
duration of the M-S interaction as 1/
.
= 22.0
kHz here. (
D
) Frequencies of vibra-
tional modes of a single
171
Yb
+
and a single
138
Ba
+
ions. Axial OOP mode is used for
the Mølmer-Sørensen (M-S) interaction. IP, in-phase mode.
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Wang
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The average phonon number of axial OOP mode is cooled down
to 0.04, and the in-phase (IP) mode is cooled down to 0.11 with
Doppler cooling, electromagnetically induced transparency (EIT)
cooling (
55
), and Raman sideband cooling (
56
). We note that EIT
cooling and Raman sideband cooling are performed only by
138
Ba
+
ion, which sympathetically cools the
171
Yb
+
ion. The time evolution
of the M-S interaction is shown in Fig. 3A. After the M-S gate, we
apply
/2 rotations to both ions with varying phases and obtain the
parity oscillation signal as shown in Fig. 3B. According to the state
population after the M-S gate and the contrast of the parity oscilla-
tion, we obtain a fidelity of the generated entangled state ∣
〉 of
0.939 ± 0.014. Gate errors mainly come from parameter drifts due
to the long-term calibration process and imperfect cooling of axial
IP mode.
Results and analysis of the loopholes
After the generation of the entangled state, one of the four contexts
is chosen and measured.
For each ion, a
/2 rotation is first performed to map the corre-
sponding observable to the
ˆ
z
basis, and then the fluorescence
detection is performed. The experiment is repeated 40,000 times.
The acquired data with SEs are shown in Table 1.
The validity of the assumption of outcome noncontextuality that
leads to the bound of inequality (Eq. 1) relies on the assumption
that measurements are sharp (
25
). That is, they yield the same
outcome when repeated and do not disturb measurements in the
same context (
27
,
28
).
In our experiment, measurement repeatability is checked by
measuring the same observable two times in a single experimental
run. For each observable, this is repeated 1000 times.
We define the repeatability
R
i
as the fraction of measurement
runs in which the observable
ˆ
O
i
is measured twice and both of the
outcomes are the same. For perfectly sharp measurements,
R
i
should be 1 for all
i
= 0...3. In our experiment, the average value for
the four observable is 98.4 ± 0.4%. Within our experimental error
bars, the imperfection in the repeatability can be explained mainly
by the detection infidelity of dark states for both ions, which is 1.5 ±
0.4% in average. The repeatability for each of the four observables is
shown in Fig. 4. The sequence used for testing the repeatability is
discussed in Materials and Methods.
Nondisturbance between measurements in the same context is
enforced by choosing trivially compatible observables. The devia-
tion form perfect nondisturbance is attributable to finite statistics.
Repeatability could be further improved, up to 99.9%, by adopt-
ing closer-to-ideal equipment. To show that a repeatability of
~98.4% is enough to close the sharpness loophole, we consider three
types of noncontextual models that exploit this imperfection to in-
crease the value of
C
beyond the limit for the ideal case. For these
models, the bound of inequality (Eq. 1) has to be corrected as follows
C
= 〈
ˆ
O
0
ˆ
O
1
〉 + 〈
ˆ
O
1
ˆ
O
2
〉 + 〈
ˆ
O
2
ˆ
O
3
〉 − 〈
ˆ
O
3
ˆ
O
0
〉 ≤ 2 +
(2)
where
depends on the way that the noncontextual models may
take advantage of the imperfections. We focus on three types
of models.
The models considered in (
57
) are based on the assumption that
outcome noncontextuality holds only for a fraction
f
of trials where
the assumption of repeatability is satisfied for both measurements,
while for the other fraction, 1 –
f
, the worst case scenario is assumed.
That is, with probability 1 –
f
, the hidden variables can conspire
to achieve the maximum algebraic value of
C
. In our experiment,
f
= 0.984
2
= 0.97, and the maximum algebraic value of
C
is 4. There-
fore,
= 0.06.
The “maximally noncontextual models” (
40
,
58
) are defined as
those in which outcome noncontextuality holds with the maximum
probability allowed by the observed marginals, that is, models that are
only as conspiratorial as needed to account for the disturbance observed
A
P
11
P
00
P
10
P
01
0
30
60
90
120
150
180
45.4
0.
0.2
0.4
0.6
0.8
1.
t
(μs)
Population
P
11
+P
00
=0
.960
B
0.
0.5
1.
1.5
2.
−1.
−0.5
0.
0.5
1.
Parity
2 × 0.919
Fig. 3. Evolution of the M-S interaction and oscillation of parity signal.
Each data
point is the average of 100 repetitions, and all the error bars are SDs. (
A
) The time
evolution of the M-S interaction.
P
ij
is the population of state ∣
ij
〉, where ∣ij〉 = ∣i
Yb
〉∣j〉
Ba
.
The duration of a single M-S gate is 45.4
s, and
P
11
+
P
00
= 0.960 ± 0.018 at the end
of the gate. (
B
) The parity scan of the entangled state. Parity of a state is defined as
P
11
+
P
00
−
P
10
−
P
01
, which is the population difference between the two qubits
being in same or opposite states. Parity contrast is 0.919 ± 0.021.
Table 1. Experiment settings and results of mean values and
SEM.
Each setting repeats 10,000 times.
〈
ˆ
O
i
j
〉 is the expectation value of
observable
ˆ
O
i
measured jointly with observable
ˆ
O
j
.
{
i
,
j
}
〈
ˆ
O
i
ˆ
O
j
〉
〈
ˆ
O
i
j
〉
〈
ˆ
O
j
i
〉
{0,1}
0.6164 ± 0.0079
−0.0008 ± 0.0100
0.1096 ± 0.0099
{1,2}
0.625 ± 0.0078
0.1066 ± 0.0099
0.1236 ± 0.0099
{2,3}
0.6678 ± 0.0074
0.1356 ± 0.0099
0.1078 ± 0.0099
{3,0}
−0.6166 ± 0.0079
0.1114 ± 0.0099
−0.0056 ± 0.0100
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Wang
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in the marginals. For these models,
=
∑
i
=0
3
∣ 〈
ˆ
O
i
i
⊕
1
〉 − 〈
ˆ
O
i
i
⊖1
〉 ∣ ,
⊕
is right shift (0 ↦ 1 ↦ 2 ↦ 3 ↦ 0), and ⊖ is left shift (0
↤
1
↤
2
↤
3
↤
0).
〈
ˆ
O
i
ˆ
O
j
〉 is the correlation between observable
ˆ
O
i
and
ˆ
O
j
, and
〈
ˆ
O
i
j
〉 is
the expectation value of observable
ˆ
O
i
measured jointly with ob
-
servable
ˆ
O
j
. Using the results in Table 1, for these models,
= 0.023
± 0.027.
In addition, we consider the models (
19
,
36
) that apply to experi
-
ments with sequential incompatible measurements. In this case, the
experimentally observed repeatability is used to estimate the distur-
bance that a measurement can cause to the result of the measure-
ment performed afterward and correct the bound for the ideal case.
With our repeatability, these models lead to
= 0.128 [see (
19
,
36
)
for details].
Using the data in Table 1 to evaluate
C
in inequality (Eq. 2), we
obtain
C
= 2.526 ± 0.016, which corresponds to a violation of in-
equality (Eq. 2) for any of the models considered. Therefore, our
experiment rules out noncontextual models maximally exploiting
the lack of perfect repeatability, maximally noncontextual models,
and we even consider a model that takes advantage of a lack of
compatibility, which does not apply to our system.
To close the compatibility loophole, we map trivially compatible
observables on separated ions of different species. Measurements
on each ion use different operation laser wavelengths, fluorescence
wavelengths, and detectors, as shown in Fig. 2. The 355-nm laser
beams perform coherent operations on the
171
Yb
+
ion, while the
532-nm laser beams perform coherent operations on the
138
Ba
+
ion.
Although, in principle, the laser beams can also influence the “wrong”
ion, this disturbance is too small to be detected, as it affects
C
at the
level of 10
−6
(see Materials and Methods).
To close the detection loophole, we adopt a scheme of 100%
detection efficiency, which produces two measurement outcomes
in every trial of the experiment. Therefore, the assumption of fair
sampling (
30
) is not needed, and the mere violation of the non-
contextuality inequality (Eq. 2) is enough to single out noncontextual
models. However, because of the detection infidelity, this strategy
leads to a reduction of the violation of inequality (Eq. 2) with
respect to the one predicted by quantum mechanics for ideal
equipment,
2
√
_
2
∼
2.828 (
59
).
DISCUSSION
Our experiment demonstrates that, as predicted by quantum me-
chanics, neither the persistency of a result when a measurement is
repeated nor the observation that measurements in the same trial
are not disturbing each other (as all of them yield the same outcome)
implies that measurements reveal “properties” had by the systems.
Our experiment shows, beyond any reasonable doubt, that nature
allows for correlations between the outcomes of sharp measure-
ments that cannot be explained by models on the basis of the as-
sumption of outcome noncontextuality. This result is contrary to
the deeply rooted conception in science that persistency and repeat-
ability of results imply the existence of properties revealed by the
measurements. Our test is “loophole-free” in the sense that it simul-
taneously closes the main loopholes affecting previous contextuality
tests. To the best of our knowledge, no other loopholes have been
pointed out for Kochen-Specker contextuality experiments. How-
ever, in principle, there could be more loopholes. Inspiration for
identifying them can be obtained in the following review paper on
loopholes for Bell nonlocality experiments (
60
).
One could have argued that the only way to guarantee perfect
compatibility is to spacelike-separate the measurements. However,
the same technical reasons (e.g., the finiteness of the experimental
statistics and the impossibility of implementing the same measure-
ment twice) that would make perfect nondisturbance and thus
perfect compatibility impossible in a spatially (but not spacelike)
separated experiment would also prevent any experiment with
spacelike separation to achieve perfect compatibility. In this sense,
an experiment with spatial (but not spacelike) separation in which
the deviation from the nondisturbance condition is statistically
negligible is as free of the compatibility loophole as any experiment
can be. On the other hand, both in classical mechanics and in non-
relativistic quantum mechanics, two observables,
A
on system 1 and
B
on a spatially separated system 2, are trivially compatible as there
are a third observable
C
(which, in this case, is trivial as it can be
measured by performing a measurement of
A
on system 1 and a
measurement of
B
on system 2) and functions
f
and
g
such that
A
=
f
(
C
) and
B
=
g
(
C
); thus, an outcome can be ascribed to both
A
and
B
by a single measurement of
C
. This argument has been used
in previous proposals and experiments closing the compatibility
loophole (
34
,
38
,
39
,
61
).
Our results have direct implications to quantum algorithms and
protocols running on devices where the assumption of locality
cannot be made, as it is the case of quantum computers (
52
). These
devices are not large enough to allow for spacelike-related events that
justify the assumption of locality. There, the possibility of pro
-
ducing loophole-free contextual correlations for sharp measurements
allows, without relying on locality, for testing whether a claimed
quantum computer is truly quantum (
62
), characterizing quantum
systems (
63
,
64
), self-testing quantum random number generation (
65
),
and blind quantum computation (
62
), among other applications.
MATERIALS AND METHODS
Repeatability test
We perform the repeatability test by measuring the same observable
two times in a single experimental run. The sequence used for
testing the repeatability is discussed in Fig. 5. There, a single qubit
rotation by
about the
cos(
)
ˆ
x
+ sin (
)
ˆ
y
axis is defined as
R
(
,
) =
⎛
⎜
⎝
cos
(
─
2
)
−
i
e
−
i
sin
(
─
2
)
−
i
e
i
sin
(
─
2
)
cos
(
─
2
)
⎞
⎟
⎠
(3)
Our projective measurements require different sequences de-
pending on the detected state because the fluorescence detection for
0
1
2
3
0.96
0.98
1.00
Repeatability
R
Fig. 4. Repeatability of the measurements.
The repeatability of each observable
is tested 1000 times. Error bars are the SEM.
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the ∣0〉 state (dark state) is ideal, but it is not ideal for the ∣1〉 state
(bright state). The bright-state detection is not ideal because of the
leakage to other states outside qubit space. We address this problem
by adopting postselection technique, where only results of the dark
state are collected. We first collect the data of only the ∣0〉 state
(dark state) and abandon the data of the ∣1〉 (bright state). We test
the case of ∣1〉 similar to that of ∣0〉 by inverting the ∣1〉 to the ∣0〉.
We note that there is no fundamental problem of postselection for
the repeatability test since we identify the not ideal data, whose first
outcome is ∣1〉 (bright state), just by looking at the first outcome
and without using any information about the second measurement
setting or outcome and exclude them from the experiment.
Cross-talk between qubits
The 355- and 532-nm Raman lasers are designed to drive the tran-
sition of the
171
Yb
+
and
138
Ba
+
ions, respectively. However, in
principle, they can also drive the
138
Ba
+
and
171
Yb
+
ions, respectively,
but this cross-talk is quite small. As shown in Fig. 6, when one of
the Raman lasers is applied to the system, the wrong ion does not
have any excitation other than fluctuations caused by detection
errors. This cross-talk is too small to be detected but easy to be
estimated in theory. For that, we first assume that the pulse laser
comb differences are resonant with the qubit transitions and only
consider the energy structure of the ions and laser wavelength. The
Raman transition strengths of
171
Yb
+
and
138
Ba
+
on the laser wave-
length are (
44
)
Yb
=
I
─
12
(
−
k
1
─
1
+
k
2
─
2
)
(4)
Ba
=
√
_
2
I
─
12
(
−
k
1
─
1
+
k
2
─
2
)
(5)
where subscripts 1 and 2 refer to the P
1/2
and P
3/2
levels, respectively.
I
is the laser intensity and
k
i
=
i
2
/
I
sat,
i
.
i
,
I
sat,
i
, and
i
are the natural
linewidth, saturation intensity, and detuning for corresponding level.
All related parameters are shown in Table 2 (
44
).
In our experiment, the transition strength
Yb,355
=
Ba,532
=
(2
) 0.18 MHz, which leads to
I
532
= 6.86 × 10
6
mW/cm
2
and
I
355
=
6.37 × 10
6
mW/cm
2
. Then, the unwanted cross-talk transition
strengths are
∣
Yb,532
∣ =
I
532
─
12
(
k
1
─
248 THz
−
k
2
─
347 THz
)
= ( 2
) 0.006 MHz
(6)
∣
Ba,355
∣ = −
√
_
2
I
355
─
12
(
−
k
1
─
187 THz
+
k
2
─
238 THz
)
= ( 2
) 0.009 MHz
(7)
The two ions are assumed to be uniformly illuminated by the
lasers. However, in the real experiment, both beams are aligned to
their respective target ions, which further reduces the cross-talk.
Second, we consider the comb difference between two lasers.
The repetition rate of our pulse laser is 80.097 MHz. The frequency
shift between two 355 (532)–nm beam combs is 12.5 (16.3) MHz to
M-
S
gate
PM
PM
PM
PM
PM
OPT
OPT
A
B
Fig. 5. Sequence used in the test of the repeatability of the measurements.
(
A
) The whole sequence for the test. The sequence includes six steps: (i) Pump two
qubits to ∣0〉 and then prepare the entangled state with an M-S gate. (ii) Rotate
the measurement basis
ˆ
z
to the observable basis
ˆ
O
i
=
R
+
(
_
2
,
)
ˆ
z
R
(
_
2
,
)
,
where
Yb
=
5
_
4
and
3
_
4
for observables
ˆ
O
0
and
ˆ
O
2
,
Ba
=
3
_
2
, and
for observables
ˆ
O
1
and
ˆ
O
3
. (iii) Projective measurement (PM). (iv) Rotate the measurement basis back.
(v) Rotate the measurement basis to the observable basis again. (vi) PM again.
R
(
_
2
,
Yb
)
in the purple box and
R
(
_
2
,
Ba
)
in the green box are
/2 rotations of the
171
Yb
+
and
138
Ba
+
qubits, respectively. Only rotations in the purple box will be
applied when observable
ˆ
O
0
or
ˆ
O
2
are measured since they only performed on the
171
Yb
+
ion. Similarly, only rotations in the green box will be applied for observables
ˆ
O
1
and
ˆ
O
3
. (
B
) Scheme of PM with postselection for the test of repeatability. The PM
measurement of the ∣0〉 state is realized by a single-shot fluorescence measure-
ment and an optical pumping (OPT) pulse. The OPT pulse is used to recover the
measured ∣0〉 state. The PM measurement of the ∣1〉 state is realized by
rotation
before and after the single-shot fluorescence measurement and optical pumping
pulse. Here, for the measurement of both ∣0〉 and ∣1〉, the results of no fluores-
cence are selected.
A
Yb
Ba
0
5
10
0.
0.2
0.4
0.6
0.8
1.
T
Up-state population
B
Yb
Ba
0
5
10
0.
0.2
0.4
0.6
0.8
1.
T
Up-state population
Fig. 6. Experimental test of the cross-talk between qubits.
Each point is repeated
100 times. Error bars are the SEM. (
A
) The up-state population of both ions when
355-nm Raman laser is applied. (
B
) The up-state population of both ions when
532-nm Raman laser is applied.
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Wang
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.,
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, eabk1660 (2022) 9 February 2022
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meet the
171
Yb
+
(
138
Ba
+
) qubit splitting of 12642.8 (16.3) MHz.
Then, the undesirable Raman transitions have detuning of at least
∣
∣/ 2
= 16.8 MHz − 12.5 MHz = 4.3 MHz for both
138
Ba
+
and
171
Yb
+
qubit transitions. These far-detuned couplings cause limited
maximum population transfers in a single pulse of
P
max,Yb,532
=
Yb,532
2
─
2
+
Yb,532
2
=
0.006
2
─
4 . 3
2
+ 0.006
2
= 1.9 × 1
0
−6
(8)
P
max,Ba,355
=
Ba,355
2
─
2
+
Ba,355
2
=
0.009
2
─
4 . 3
2
+ 0.009
2
=
4.3 × 1
0
−6
(9)
This amount of cross-talk is negligible for our experiment.
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Acknowledgments:
We thank C.
Budroni for comments.
Funding:
This work was supported
by the National Key Research and Development Program of China under grant nos.
2016YFA0301900 and 2016YFA0301901, the National Natural Science Foundation of China
grant nos. 92065205 and 11974200, Project Qdisc (project no. US-15097) with FEDER funds,
QuantERA grant SECRET by MINECO (project no. PCI2019-111885-2), Guangdong Basic and
Applied Basic Research Foundation grant no. 2019A1515111135, and the Key-Area Research
and Development Program of Guangdong Province grant no. 2019B030330001.
Author
contributions:
P.W., J.Z., C.-Y.L., M.U., and Y.W. developed the experimental system with the
support of M.Q and T.X.
A.C. and K.K. conceived the work. J.-N.Z. and A.C. provided theoretical
support. P.W., J.Z., and C.-Y.L. led the date taking. P.W. analyzed the data. P.W., J.Z., C.-Y.L.
A.C.,
and K.K. wrote the manuscript with the participation of the other authors.
Competing
interests:
The authors declare that they have no competing interests.
Data and materials
availability:
All data needed to evaluate the conclusions in the paper are present in the paper
and/or the Supplementary Materials.
Submitted 25 June 2021
Accepted 16 December 2021
Published 9 February 2022
10.1126/sciadv.abk1660
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