of 8
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
Ad
́
an Cabello,
6
,
7
,
Kihwan Kim,
1
,
2
,
8
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
Department 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, California 91125, USA
6
Departamento de F
́
ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain
7
Instituto Carlos I de F
́
ısica Te
́
orica y Computacional, Universidad de Sevilla, E-41012 Sevilla, Spain
8
Frontier Science Center for Quantum Information, Beijing 100084, People’s Republic of China
These two authors contributed equally
adan@us.es
kimkihwan@mail.tsinghua.edu.cn
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 ad-
vantage in computing. Here, we report the first 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
>
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 new way to
certify quantum systems.
I. INTRODUCTION
In everyday life, whenever the measurements of two ob-
servables
A
and
B
yield the same values (
a
for
A
and
b
for
B
) when the measurements are repeated in any order, we at-
tribute it to the measured system possessing preexisting val-
ues revealed by every measurement and which persist after
the measurements. However, this assumption fails in quan-
tum mechanics. Quantum systems can produce correlations
[1, 2] between measurements which do not disturb each other
and yield the same result when repeated and which, however,
cannot be explained by models based on 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 con-
textuality 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 comput-
ers [5–9].
Contextual correlations between sequential measurements
have been observed in experiments with photons [10–17] neu-
trons [18], ions [19–21], molecular nuclear spins [22], super-
conducting systems [23], and nuclear spins [24]. However,
these experiments have “loopholes”, as noncontextual models
assisted by mechanisms that exploit the experimental imper-
fections can simulate the observed correlations.
Three main loopholes have been considered. The sharpness
loophole follows from the observation [25, 26] that the as-
sumption of outcome noncontextuality, on which the bound of
the noncontextuality 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]) observ-
ables. 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] ex-
ploits that, in experiments with sequential measurements on
the same system, the assumption that the measured observ-
ables 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 noncontex-
tual models, they leave open the sharpness loophole. On the
other hand, there are contextuality tests free of the detection
loophole and whose correlations cannot be produced by spe-
cific mechanisms exploiting the lack of perfect repeatability
[19, 40]. However, they suffer from the compatibility loop-
hole, 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 differ-
ent ions [41–46], one
171
Yb
+
ion and one
138
Ba
+
ion. This
dual-species system allows us performing sequential repeat-
able highly efficient single-shot fluorescence measurements
on each of the ions. In the system, the detection loophole
arXiv:2112.13612v1 [quant-ph] 27 Dec 2021
2
is naturally addressed due to 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 per-
fect to close the detection loophole.
To close the compatibility loophole, we target a “Bell-like”
[10, 11] noncontextuality inequality in which only two ob-
servables are measured per context, and each of the observ-
ables is defined on a different ion. Therefore, these two ob-
servables are “trivially compatible (
i.e.
, they are simultane-
ously measurable in an uncontentious sense)” [47]. In addi-
tion, the compatibility is enforced by choosing ions of dif-
ferent species requiring different operation laser wavelengths,
fluorescence wavelengths and detectors.
ˆ
O
0
=
1
2
σ
x
ˆ
σ
y
)
ˆ
I
2
ˆ
O
1
=
ˆ
I
2
ˆ
σ
x
ˆ
O
2
=
1
2
σ
x
+ ˆ
σ
y
)
ˆ
I
2
ˆ
O
3
=
ˆ
I
2
⊗−
ˆ
σ
y
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
σ
x
,
ˆ
σ
y
are Pauli operators and
ˆ
I
2
is the
identity operator.
II. RESULTS
The noncontextuality inequality we focus on is the only
tight (i.e., strictly separating noncontextual from contextual
correlations) noncontextuality inequality in the 4-cycle con-
textuality scenario [48] shown in Fig. 1. This is the sce-
nario involving the smallest number of measurements that
allows for contextuality for sharp measurements, as follows
from a theorem by Vorob’yev [49, 50]. This noncontextual-
ity 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 require spacelike sep-
aration [52] nor adscribing the observables to two parties. In-
stead, it requires the measurements to be sharp.
The test of inequality (1) is performed on a two-qubit sys-
tem 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
mani-
fold of the
171
Yb
+
ion. The corresponding states are denoted
B = 6.0 Gauss
138
Ba
+
171
Yb
+
x
z
F=0
F=1
F=0
F=1
2
P
1/2
2
S
1/2
2
P
3/2
F=1
66
THz
34
THz
12.64281 GHz
355 nm
370 nm
2
P
1/2
2
S
1/2
44
THz
94
THz
2
P
3/2
493 nm
532 nm
614 nm
650 nm
1762 nm
5
D
3/2
5
D
5/2
f
/ MHz
Radial OOP
Radial IP
Axial IP
Axial OOP
16.80 MHz
PMT2
PMT1
FIG. 2.
Experimental setup.
(a) and (b) are the energy level dia-
grams 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 response 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 po-
larizations of 532 nm 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
δ
is the detuning of the laser from the OOP
mode sideband, when
δ
is zero, then the Raman transition is directly
red and blue sideband transitions. For the M-S gate,
δ
should match
to the sideband Rabi frequency and determines the duration of the
M-S interaction as
1
~
δ
.
δ
=
22
.
0
kHz here. (d), Frequencies of vi-
brational 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
stands for in-phase mode.
S
0
Yb
S
F
=
0
,m
F
=
0
and
S
1
Yb
S
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
S
0
Ba
S
m
=
1
~
2
and
S
1
Ba
S
m
=
1
~
2
. The
energy gap is
f
Ba
=
16
.
8 MHz
in an external magnetic field of
6
.
0 Gauss
.
The two-ion system is initially prepared in state
S
ψ
=
1
2
(S
00
+
i
S
11
⟩)
. The state of each qubit can be measured
with a fluorescence detection technique.
For the
171
Yb
+
3
ion, the cyclic transition between
S
F
=
1
states in
2
S
1
~
2
and
S
F
=
0
,m
F
=
0
in
2
P
1
~
2
is excited with a
370 nm
laser beam
so that only
S
1
Yb
scatters photons. The error of detecting
S
1
Yb
for
S
0
Yb
is 0.96% and the other error is 2.25%. For the
138
Ba
+
ion, we first transfer the population of
S
0
Ba
to
2
D
5
~
2
with a
1762 nm
laser beam before exciting the
493 nm
tran-
sition between
2
S
1
~
2
and
2
P
1
~
2
levels. The error of detecting
S
1
Ba
for
S
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 stim-
ulated Raman process to control the
138
Ba
+
ion, and another
two beams from its
355 nm
frequency-tripled output are used
for the
171
Yb
+
ion [54]. The schematic diagram of the ar-
rangement of both Raman laser beams is shown in Fig. 2(c).
State
S
ψ
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]. The average phonon number of axial OOP mode is
cooled down to
0.04, and the 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 cool-
ing are performed only by
138
Ba
+
ion, which sympathetically
cools the
171
Yb
+
ion. The time evolution of the M-S interac-
tion is shown in Fig. 3(a). After the M-S gate, we apply
π
~
2
rotations to both ions with varying phases and obtain the par-
ity oscillation signal as shown in Fig. 3(b). According to the
state population after the M-S gate and the contrast of the par-
ity oscillation, we obtain a fidelity of the generated entangled
state
S
ψ
of
0
.
939
±
0
.
014
.
Gate errors mainly come from parameter drifts due to the
long time 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
π
~
2
rotation is first performed to
map the corresponding observable to the
ˆ
σ
z
basis, and then
the fluorescence detection is performed. The experiment is
repeated
40000
times. The acquired data with standard errors
are shown in Table I.
The validity of the assumption of outcome noncontextual-
ity that leads to the bound of inequality (1) relies on the as-
sumption that measurements are sharp [25]. That is, they yield
the same outcome when repeated and do not disturb measure-
ments in the same context [27, 28].
In our experiment, measurement repeatability is checked by
measuring the same observable two times in a single experi-
mental run. For each observable, this is repeated
1000
times.
We define the repeatability
R
i
as the fraction of measure-
ment runs in which the observable
ˆ
O
i
is measured twice and
both of the outcomes are the same. For perfectly sharp mea-
surements,
R
i
should be
1
for all
i
=
0
...
3
. In our exper-
iment, 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 by mainly 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
(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
(
μ
s
)
Population
P
11
+
P
00
=
0.960
(b)
0.
0.5
1.
1.5
2.
-
1.
-
0.5
0.
0.5
1.
φ
(
π
)
Parity
2
×
0.919
FIG. 3.
Evolution of the M-S interaction and oscillation of par-
ity signal.
Each data point is the average of 100 repetitions and
all the error bars are standard deviations. (a) The time evolution
of the M-S interaction.
P
ij
is the population of state
S
ij
, where
S
i,j
=
S
i
Yb
S
j
Ba
. The duration of a single M-S gate is 45.4 μs 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
.
shown in Fig. 4. The sequence used for testing the repeatabil-
ity is discussed in the Materials and Methods.
O
0
O
1
O
2
O
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 standard
error of the mean (SEM).
Nondisturbance between measurements in the same context
is enforced by choosing trivially compatible observables. The
deviation form perfect nondisturbance is attributable to finite
statistics.
Repeatability could be further improved, up to
99
.
9%
, by
adopting closer to ideal equipment. To show that a repeata-
bility
98
.
4%
is enough to close the sharpness loophole, we
consider three types of noncontextual models that exploit this
imperfection to increase the value of
C
beyond the limit for
the ideal case. For these models, the bound of inequality (1)
4
TABLE I. Experiment settings and results of mean values and standard error of the mean (SEM). Each setting repeats 10000 times.
ˆ
O
j
i
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
j
i
ˆ
O
i
j
{
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
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
+
ε,
(2)
where
ε
depends on the way the noncontextual models may
take advantage of the imperfections. We focus on three types
of models.
The models considered in [57], 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 al-
gebraic value of
C
. In our experiment,
f
=
0
.
984
2
=
0
.
97
and
the maximum algebraic value of
C
is
4
. Therefore,
ε
=
0
.
06
.
The “maximally noncontextual models” [40, 58] defined as
those in which outcome noncontextuality holds with the max-
imum probability allowed by the observed marginals. That is,
models that are only as conspiratorial as needed to account for
the disturbance observed in the marginals. For these models,
ε
=
3
i
=
0
T
ˆ
O
i
1
i
ˆ
O
i
1
i
T
,
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
j
i
is
the expectation value of observable
ˆ
O
i
measured jointly with
observable
ˆ
O
j
. Using the results in Table I, for these models
ε
=
0
.
023
±
0
.
027
.
In addition, we consider the models [19, 36] which apply
to experiments with sequential
incompatible
measurements.
In this case, the experimentally observed repeatability is used
to estimate the disturbance that a measurement can cause to
the result of the measurement performed afterwards and cor-
rect the bound for the ideal case. With our repeatability, these
models lead to
ε
=
0
.
128
(see [19, 36] for details).
Using the data in Table I to evaluate
C
in inequality (2), we
obtain
C
=
2
.
526
±
0
.
016
, which corresponds to a violation
of inequality (2) for any of the models considered. Therefore,
our experiment rules out noncontextual models maximally ex-
ploiting the lack of perfect repeatability, maximally noncon-
textual models, and we even consider a model which takes
advantage of a lack of compatibility, which does not apply to
our system.
To close the compatibility loophole, we map trivially com-
patible observables on separated ions of different species.
Measurements on each ion use different operation laser wave-
lengths, 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 co-
herent operations on the
138
Ba
+
ion. Although, in principle,
the laser beams can also influence the “wrong” ion, this dis-
turbance 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 as-
sumption of fair sampling [30] is not needed and the mere vi-
olation of the noncontextuality inequality (2) is enough to sin-
gle out noncontextual models. However, due to the detection-
infidelity, this strategy leads to a reduction of the violation of
inequality (2) with respect to the one predicted by quantum
mechanics for ideal equipment,
2
2
2
.
828
[59].
III. DISCUSSION
Our experiment demonstrates that, as predicted by quantum
mechanics, neither the persistency of a result when a measure-
ment is repeated nor the observation that measurements in the
same trial are not disturbing each other (as all of them yield
the same outcome) imply that measurements reveal “proper-
ties” possessed by the systems. Our experiment shows, be-
yond any reasonable doubt, that nature allows for correlations
between the outcomes of sharp measurements that cannot be
explained by models based on the assumption of outcome
noncontextuality. This result is contrary to the deeply-rooted
conception in science that persistency and repeatability of re-
sults imply the existence of properties revealed by the mea-
surements. Our test is “loophole-free” in the sense that it
closes simultaneously the main loopholes affecting previous
contextuality tests. To the best of our knowledge, no other
loopholes have been pointed out for KS contextuality experi-
ments. However, in principle, there could be more loopholes.
Inspiration for identifying them can be obtained in the fol-
lowing review paper on loopholes for Bell nonlocality experi-
ments [60].
One could have argued that the only way to guarantee per-
fect compatibility is to spacelike separate the measurements.
However, the same technical reasons (e.g., the finiteness of
the experimental statistics and the impossibility of implement-
ing the same measurement twice) that would make perfect
non-disturbance 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 spa-
tial (but not spacelike) separation in which the deviation from
the non-disturbance condition is statistically negligible is as
free of the compatibility loophole as any experiment can be.
5
On the other hand, both in classical mechanics and in nonrel-
ativistic quantum mechanics, two observables,
A
on system 1
and
B
on a spatially separated system 2, are trivially compati-
ble as there is 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 ex-
periments 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 lo-
cality 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 producing loophole-free contextual correla-
tions for sharp measurements allows, without relying on local-
ity, 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.
IV. MATERIALS AND METHODS
Repeatability test
We perform the repeatability test by measuring the same
observable two times in a single experimental run. The se-
quence used for testing the repeatability is discussed in Fig. 5.
A single qubit rotation by
θ
about the
cos
(
φ
)
ˆ
σ
x
+
sin
(
φ
)
ˆ
σ
y
axis is defined as
R
(
θ,φ
)
=
Œ
cos
(
θ
2
)
ie
sin
(
θ
2
)
ie
sin
(
θ
2
)
cos
(
θ
2
)
‘
.
(3)
Our projective measurements require different sequences de-
pending on the detected state because the fluorescence detec-
tion for the
S
0
state (dark state) is ideal but it is not ideal
for the
S
1
state (bright state). The bright-state detection is not
ideal due to the leakage to other states outside qubit space. We
address this problem by adopting post-selection technique,
where only results of the dark state are collected. We first col-
lect the data of only the
S
0
state (dark state) and abandon the
data of the
S
1
(bright state). We test the case of
S
1
similar to
that of
S
0
by inverting the
S
1
to the
S
0
. We note that there is
no fundamental problem of post-selection for the repeatability
test, since we identify the not ideal data, whose first outcome
is
S
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.
Crosstalk between qubits
The 355 nm and 532 nm Raman lasers are designed to drive
the transition of the
171
Yb
+
and
138
Ba
+
ions, respectively.
However, in principle, they can also drive the
138
Ba
+
and
M
-
S
gate
PM
PM
PM
PM
PM
OPT
OPT
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
S
0
and then prepare the
entangled state with a M-S gate. (ii) Rotate the measurement ba-
sis
ˆ
σ
z
to the observable basis
ˆ
O
i
=
R
+
(
π
2
)
ˆ
σ
z
R
(
π
2
)
, where
φ
Yb
=
5
π
4
and
3
π
4
for observables
ˆ
O
0
and
ˆ
O
2
,
φ
Ba
=
3
π
2
and
π
for observables
ˆ
O
1
and
ˆ
O
3
. (iii) Projective measurement (PM). (iv)
Rotate the measurement basis back. (v) Rotate the measurement ba-
sis to the observable basis again. (vi) PM again.
R
‰
π
2
Yb
Ž
in the
purple box and
R
‰
π
2
Ba
Ž
in the green box are
π
~
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 projective measurement (PM) with post-selection for the test of
repeatability. The PM measurement of the
S
0
state is realized by a
single-shot fluorescence measurement and an optical pumping (OPT)
pulse. The OPT pulse is used to recover the measured
S
0
state. The
PM measurement of the
S
1
state is realized by
π
-rotation before and
after the single-shot fluorescence measurement and optical pumping
pulse. Here for the measurement of both
S
0
and
S
1
, the results of no
fluorescence are selected.
171
Yb
+
ions, respectively. But this crosstalk is quite small. As
shown in Fig 6, When one of the Raman lasers is applied to
the system, the “wrong” ion doesn’t have any excitation other
than fluctuations caused by detection errors. This crosstalk is
too small to be detected but easy to be estimated in theory. For
that, we assume, firstly, 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 Ra-
man transition strengths of
171
Yb
+
and
138
Ba
+
on the laser
wavelength and 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
=
γ
2
i
~
I
sat,i
.
γ
i
,
I
sat,i
and
i
are the natural linewidth, saturation intensity,
and detuning for corresponding level. All related parameters
are shown in table II [44].
In our experiment, the transition strength
Yb
,
355
=
Ba
,
532
=
(
2
π
)
0
.
18
MHz, which leads to
I
532
=
6
.
86
×