Experimental investigation of contextual robustness and coherence in state discrimination

Abstract

Given that contextuality and coherence are significant resources in quantum physics, exploring the intricate interplay between these two factors presents a compelling avenue for research. Here, an experiment is presented to investigate the nuanced relationship between contextual robustness and coherence in a scenario of state discrimination. Specifically, two types of noises—depolarizing and dephasing—are introduced in the measurement procedure to assess the robustness of contextuality. By varying the overlap between the states to be discriminated, we explore the variations in contextual robustness across different levels of coherence. Importantly, our findings demonstrate that contextuality can persist under any degree of coherence. Notably, when coherence approaches zero (while remaining nonzero), contextuality can still endure under arbitrary amounts of partial dephasing. These results underscore the pivotal role of coherence in contextual phenomena, offering significant insights into the intricate interplay between coherence and contextuality.

Full text

PHYSICAL REVIEW A 109, 052208 (2024)
Experimental investigation of contextual robustness and coherence in state discrimination
Jingyan Lin,1,2Kunkun Wang,3Lei Xiao,2Dengke Qu,4Gaoyan Zhu,5Yong Zhang ,1,*and Peng Xue 2,†
1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
2School of Physics, Southeast University, Nanjing 211189, China
3School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
4Beijing Computational Science Research Center, Beijing 100084, China
5College of Electronic and Information Engineering, Shangdong University of Science and Technology, Qingdao 266590, China
(Received 19 September 2023; revised 15 April 2024; accepted 19 April 2024; published 8 May 2024)
Given that contextuality and coherence are significant resources in quantum physics, exploring the intricate
interplay between these two factors presents a compelling avenue for research. Here, an experiment is presented
to investigate the nuanced relationship between contextual robustness and coherence in a scenario of state dis-
crimination. Specifically, two types of noises—depolarizing and dephasing—are introduced in the measurement
procedure to assess the robustness of contextuality. By varying the overlap between the states to be discriminated,
we explore the variations in contextual robustness across different levels of coherence. Importantly, our findings
demonstrate that contextuality can persist under any degree of coherence. Notably, when coherence approaches
zero (while remaining nonzero), contextuality can still endure under arbitrary amounts of partial dephasing.
These results underscore the pivotal role of coherence in contextual phenomena, offering significant insights
into the intricate interplay between coherence and contextuality.
DOI: 10.1103/PhysRevA.109.052208
I. INTRODUCTION
Distinguishing nonclassical from classical phenomena is
instrumental in effectively harnessing quantum resources to
achieve superior performance in practical tasks. The concept
of noncontextuality, initially proposed by Bell [1], Kochen
and Specker [2], and subsequently refined by Spekkens [3],
represents a manifestation of classical phenomena. On the
contrary, contextuality, manifested through the violation of
noncontextuality inequalities, has emerged as a resource that
provides quantum advantages in various domains such as
cryptography [4,5], communication [6–9], and computation
[10–13]. Quantum coherence [14,15] as a valuable resource
in quantum information [16–18] is another significant non-
classical phenomenon that piques our interest.
Kochen-Specker noncontextuality [2] originates from the
noncontextual hidden variable model. The historical defini-
tion of context has confined the exploration of contextuality
to Hilbert spaces with dimensions three and above, and to
measurements limited to sharp measurements [19–30]. How-
ever, Spekkens [3] introduced modifications to the concept
of context and noncontextuality, building upon noncontex-
tual ontological models. This innovative approach allows for
the extension of contextuality to two-dimensional Hilbert
spaces and unsharp measurements, significantly broadening
the scope for exploring contextuality in a wider range of
quantum systems [31–34].
*zhyong98@bupt.edu.cn
†gnep.eux@gmail.com
Given that both contextuality and coherence are quantum
resources with the potential to offer quantum advantages in
various tasks, exploring the relationship between these two
phenomena is inherently intriguing. Furthermore, when con-
sidering the preparation contextuality defined by Spekkens,
it becomes necessary to account for at least two states,
implying the existence of state-independent coherence. Con-
sequently, basis-independent results can be established for
both contextuality and coherence [35,36]. Thus, investigating
basis-independent relationships between contextuality and co-
herence represents a compelling avenue of research.
Building upon Spekkens’ extension of noncontextuality
[3], we focus on the realm of minimum-error state discrimina-
tion within two-dimensional Hilbert spaces. In this scenario, it
has been theoretically recognized that contextuality provides
advantages in the successful discrimination of nonorthogonal
states [37]. Subsequently, measurement errors are introduced
by considering angular deviations from the ideal basis, leading
to experimental proof of the contextual advantage for state
discrimination in the presence of errors [38]. Note that the
discrimination of nonorthogonal states inherently introduces
nonzero coherence under the basis transformation. We further
investigate both contextuality and coherence in the presence
of experimentally applied depolarizing [37,39,40] and de-
phasing [41] noise.
Considering that quantum state discrimination serves as a
cornerstone in quantum information processing tasks, it has
garnered significant attention in both theoretical [42–45] and
experimental [46–48] studies. Moreover, this critical aspect
finds applications in various quantum cryptography proto-
cols [49] and communication protocols [50]. In this case,
the experimental exploration of contextual robustness and
2469-9926/2024/109(5)/052208(10) 052208-1 ©2024 American Physical Society

JINGYAN LIN et al. PHYSICAL REVIEW A 109, 052208 (2024)
Depolarizing Measurements
Dephasing Measurements
r
r
Dephasing Measurements
r
Z
X
Preparations
(a) (b)
(c) (d)
FIG. 1. Preparations and measurements in the zx plane of the
Bloch sphere. (a) Preparations without noise. (b) Depolarizing mea-
surements with the Helstrom measurement performed along the z
axis. (c) Dephasing measurements with the Helstrom measurement
performed along the zaxis. (d) Dephasing measurements with the
Helstrom measurement performed along the xaxis.
coherence in quantum state discrimination assumes substan-
tial significance.
In this work, by considering adjustable noise in mea-
surements and coherence of prepared states within the
noncontextuality framework, our experimental findings unveil
a nuanced relationship. We derive two conclusions: Firstly,
contextuality implies the presence of coherence in the system,
while the reverse is not necessarily true under sufficient noise
levels. Additionally, even when coherence approaches zero,
a system can still exhibit contextuality in the presence of
dephasing, even with arbitrarily large noise intensities. These
findings contribute valuable insights into the complex inter-
relationship between contextual robustness and coherence,
contributing to an enhanced understanding of quantum state
discrimination.
II. CONTEXTUALITY ROBUSTNESS AND COHERENCE
IN MINIMUM-ERROR STATE DISCRIMINATION
We focus on the scenario of minimum-error state dis-
crimination (MESD) in two-dimensional Hilbert spaces [37],
specifically under depolarizing and dephasing [41]. We con-
sider a system prepared in one of two nonorthogonal states,
|ψand |φ, each having an equal prior probability. The mea-
surement Mg≡{Egψ,Egφ}is used to determine the state of
the system. The probability of successfully discriminating the
system’s state scan be quantified by s≡1
2Tr[Egψ|ψψ|]+
1
2Tr[Egφ|φφ|]. When choosing Mgto be the Helstrom mea-
surement [51], it implies that the direction of the measurement
basis {Egψ,Egφ}aligns with the direction of the vector (|φ−
|ψ) on the Bloch sphere, as depicted in Fig. 1. This particular
measurement ensures that the error rate of the two-state dis-
crimination is minimized, fulfilling the requirement of MESD.
TABLE I. The probability of specific measurement outcomes oc-
curring for particular states. Since D(I−E)=I−D(E), the table
is sufficient to obtain all measurement outcomes.
|φφ||ψψ||
¯
φ ¯
φ||
¯
ψ ¯
ψ|
D(Eφ)1−c1−c
D(Eψ)c1−1−c
D(Egφ)s1−s1−ss
Without loss of generality, we can confine the preparations
and measurements of the system states to the zx plane of
the Bloch sphere, as illustrated in Fig. 1. The Helstrom mea-
surement satisfies the symmetric relation Tr[Egψ|ψψ|]=
Tr[Egφ|φφ|], allowing us to calculate sas
s=Tr[Egψ|ψψ|]=Tr[Egφ|φφ|].(1)
To study contextuality, the orthogonal states of |ψand |φ
are taken into account in Fig. 1to establish the preparation
equivalence
1
2|ψψ|+1
2|¯
ψ ¯
ψ|=1
2|φφ|+1
2|¯
φ ¯
φ|=I
2.(2)
The measurement {Eψ,E¯
ψ,Eφ,E¯
φ}represents the projective
measurements on the corresponding subscript state. The con-
cept of context is defined as the set of features that are
not specified by specifying the equivalence class. Therefore,
preparation noncontextuality implies that two states belonging
to the same preparation equivalence class cannot be distin-
guished by any quantity. This enables us to formulate the
noncontextuality inequalities, which serve as tools for assess-
ing whether the system exhibits contextual or noncontextual
behavior. The noncontextuality inequality in MESD is given
by [37]
s1−c
2,(3)
where c=Tr[|ψψ||φφ|]=|ψ|φ|2represents the over-
lap between |ψand |φand indicates the level of con-
fusability between these two states. When the process of
state discrimination is influenced by noise introduced through
imperfect preparations or measurements, the inequality is
relaxed to [37]
s1−c−
2,(4)
where denotes the noise intensity. The violation of the
inequality is denoted as s=s−1+c−
2, which can be
utilized to assess the contextuality of the system.
Here, we only consider noise introduced by imperfect mea-
surements with errors, represented by D(E). The probability
of a specific measurement outcome occurring for a particular
state is documented in Table I. We first focus on the mea-
surement with depolarizing. As illustrated in Fig. 1(b),the
imperfect measurements can be described as
Ddepol
r(E)≡(1 −r)E+r
2I,(5)
where rsignifies the intensity of depolarizing, and it is linked
to the noise intensity through the equation =r
2. The mini-
mum level of noise rpreventing the system from violating the
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EXPERIMENTAL INVESTIGATION OF CONTEXTUAL … PHYSICAL REVIEW A 109, 052208 (2024)
noncontextuality inequality can be readily quantified as [41]
rdepol
min =1−1
sin2θ+cos θ,(6)
serving as a metric for contextual robustness. It serves as
the threshold for rthat demarcates the boundary between
contextuality and noncontextuality under depolarizing noise.
When r<rdepol
min , the noncontextuality inequality (4)isvio-
lated, revealing the advantage of quantum measurements over
noncontextual measurements.
For dephasing, the imperfect measurements can be de-
scribed as
Ddeph
r(E)≡(1 −r)E+r
i∈{0,1}i|E|i|ii|,(7)
where {|i}i∈{0,1}represents the basis states along the zaxis.
Similarly, rsignifies the intensity of dephasing, and it is linked
to the noise intensity through the equation =r
2sin2θ.We
consider two measurement schemes shown in Figs. 1(c) and
1(d), respectively. The first scheme is similar to the depolar-
ization case, with the Helstrom measurement performed along
the zaxis. The contextual robustness can be expressed as [41]
rdeph
min =1−1−cos θ
sin2θ,(8)
where rdepol
min serves as the threshold for rthat demarcates the
boundary between contextuality and noncontextuality under
dephasing measurements. In the second scheme, the direction
of the Helstrom measurement is rotated from the zaxis to the
xaxis. The contextual robustness can be expressed as [41]
rdeph
min =1−sin θ. (9)
When r<rdeph
min , the noncontextuality inequality (4)isvio-
lated, revealing the advantage of quantum measurements over
noncontextual measurements.
As we exclude noise in the preparation procedure illus-
trated in Fig. 1(a),thel1norm of coherence can be quantified
as [15]
C(ρ)=min
σ∈Iρ−σ1=
i=j|ρij|.(10)
Here, due to the symmetry among the states, coherence of |ψ
and |φsatisfies
C(|ψψ|)=C(|φφ|)=sin θ, (11)
which is governed by θ∈[0,π
2]. Hence, manipulating the
system state via changes in θcan influence the coherence
of states. Noting that it is also valuable to consider the ef-
fect of noise on coherence, we present the relevant results
in Appendix Cby examining effective coherence. Given
that coherence serves as a defining characteristic distinguish-
ing quantum behavior from the classical realm, it becomes
intriguing to explore contextual robustness under different
amounts of coherence.
III. EXPERIMENTAL EXPLORATION OF
CONTEXTUALITY ROBUSTNESS AND COHERENCE
We employ a single-photon source [52–54] to prepare the
qubit states and measure them under arbitrary intensity of
405nm
Laser
HWP0
HWP1
HWP2
HWP3
HWP4
PPKTP BD
PBS
PBS
BD
Lens Lens QWP1
QWP2
APD
APD
FIG. 2. Experimental setup. Polarized photon pairs, character-
ized by orthogonal polarization directions, are generated through
a periodically poled potassium titanyl phosphate (PPKTP) crystal
pumped by a 405-nm laser. Following the second polarization beam
splitter’s (PBS) division of these photon pairs, the vertically polar-
ized photons are utilized as the trigger and detected by a single
photon avalanche diode (APD). Simultaneously, the horizontally
polarized photons serve as the signal, encoding the qubit in our
experimental setup. Subsequently, a half-wave plate (HWP) denoted
as HWP1 is employed to prepare the state. The measurement pro-
cedure, enclosed within a delineated black dashed box, facilitates
measurements amidst the presence of noise.
depolarizing and dephasing within an optical system shown
in Fig. 2. The qubit is encoded by the polarization state of
the photons as {|0≡|H,|1≡|V}, where |H(|V) de-
notes the horizontal (vertical) polarization. The initial state
|ψ=cos θ
2|H+sin θ
2|Vis obtained by passing the photon
through HWP1. The key point of our experiment is to achieve
a measurement process with adjustable noise intensity, as de-
picted within the black dashed box in Fig. 2. By manipulating
the angle of HWP2, we can achieve the correct or incorrect
discrimination results in the middle path of the output. Fur-
thermore, by manipulating the angles of HWP3 and HWP4
between the two beam displacers (BD), measurements can
be conducted under varying noise intensities. Details of the
experiment are presented in Appendix A.
For the measurements under depolarizing, we select the
Helstrom measurement along the zaxis, as shown in Fig. 1(b).
Varying ten angles of θin states and eight depolarizing
noisy intensities rdepol in measurements, the experimental
results are illustrated in Fig. 3(a). To fulfill the rigor-
ous requirements mentioned in Ref. [37], we postprocessed
the experimental data through data analysis, as detailed
in Appendix B. Note that despite considering four states
{|ψψ|,|¯
ψ ¯
ψ|,|φφ|,|¯
φ ¯
φ|} during the construction of
preparation equivalence in Eq. (2), the resulting noncon-
textuality inequality (4) can be related to only two states
{|ψψ|,|φφ|}. This simplification is attributed to the high
symmetry present in Table I. However, since achieving
this symmetry experimentally is challenging, we relax it to
Table II using the method outlined in Appendix B. In this case,
TABLE II. The probability of specific measurement outcomes
occurring for particular states with no symmetries assumed.
P|φP|ψP|¯
φP|¯
ψPyP¯y
Mφ1−φcψ¯
φ1−φ+¯
φ−cψ0.5 0.5
Mψcφ1−ψ1−c¯
φcφ−c¯
φ+ψ0.5 0.5
Mgφsφ1−sψ1−s¯
φsφ−s¯
φ−sψ0.5 0.5
My0.5 0.5 0.5 0.5 1 −yy
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JINGYAN LIN et al. PHYSICAL REVIEW A 109, 052208 (2024)
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
s
/2
/3
/6
0
r
deph
0.0
0.4
0.6
0.8
0.2
1.0
(a) (b)
(c) (d)
0.2
-0.5 0
0
0.0 0.5 1.0
-0.4
-0.2
0.0
0.2
s
r
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
s
0.2
/6 /3 /2
0
r
depol
0.4
0.6
0.8
1.0
0.0
2
s
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
s
r
deph
/2
/3
/6
0
0.4
0.0
0.6
0.8
1.0
0.2
0.0 0.2 0.4 0.6 0.8 1.0
-0.01
0.00
0.01
0.02
0.03
s
r
FIG. 3. Experimental tests of noncontextuality and contextuality.
Each cylinder in (a)–(c) represents the chosen states θand noisy
intensity rdepol (rdeph) in our experiment. The color filling inside the
cylinder represents the violation of the noncontextuality inequality
s. The experimental (theoretical) boundaries, represented by the
red (blue) curve, mark the transition from noncontextuality to con-
textuality. (a) Helstrom measurement performed along the zaxis
under depolarizing. The robustness of contextual reaches a maximum
of rdeol
min =0.1976 ±0.0008 at θ=π
3. (b) Helstrom measurement
performed along the zaxis under dephasing noise. The robust-
ness of contextual reaches a maximum of rdeph
min =0.504 ±0.001 at
θ=π/25, higher than under depolarizing measurements. (c) Hel-
strom measurement performed along the xaxis under dephasing
noise. The robustness of contextual reaches a maximum of rdeph
min =
0.938 ±0.001 at θ=π/50. (d) The maximum contextual robustness
in (a)–(c). The circles represent the experimental results, and the lines
depict the corresponding theoretical values. The black circles and
line in the inset depict the situation of (a). Only one black circle falls
within the scope of the main figure. Blue and red circles and line
correspond to the situation of (b) and (c), respectively. Error bars
indicate the statistical uncertainty
the noncontextual inequalities are no longer reducible to the
single inequality (4). Consequently, we actually calculate the
violation of all inequalities related to the probability of suc-
cessful discrimination as shown in inequalities (B11). If the
violation of these eight noncontextual inequalities is denoted
as {si}8
i=1, we just need to study the one with the biggest
violation, which is s=1
2max{si}8
i=1. This is because the
violation of any one of these inequalities signifies the contex-
tual nature of the system.
When s<0, the preparation and measurement proce-
dures can be described by noncontextual models. Conse-
quently, the probability of successfully discriminating states
in this scenario does not exceed that of the classical case.
Conversely, when s>0, contextuality is present within
the system. This contextual nature enhances the success
probability of state discrimination beyond the classical case.
In particular, when s=0, a curve aligned with Eq. (6)
emerges, illustrating the boundary between noncontextuality
and contextuality.
The boundary curve is closely connected to the coherence
of prepared states in Eq. (11), determined by θ, and the con-
textuality robustness, denoted by rdepol
min . Thus, it also serves
to depict the robustness of contextuality with varying degrees
of coherence. As θ=0 and θ=π
2correspond to orthogonal
(fully distinguishable) and parallel (completely indistinguish-
able) states, respectively, the classical and quantum boundary
coincide, resulting in no violation of the noncontextuality
inequality. Consequently, the boundary curve demonstrates
that contextuality can exist under any degree of coherence,
excluding 0 and 1.
However, the robustness under depolarizing is not particu-
larly high. Even under the scenario where rdepol
min is maximized
(θ=π
3), the robustness only reaches 0.1976 ±0.0008. Fur-
thermore, as coherence tends to zero, contextual robustness
also approaches zero. It is important to note that the pres-
ence of contextuality guarantees the existence of coherence,
but the reverse is not necessarily true. This is because if
rdepol >rdepol
min , the system remains noncontextual regardless of
the degree of coherence.
To compare the relationship between contextual robustness
and coherence under different types of noise, we employ a
controlled variable approach. Without altering the direction
of Helstrom measurement Mg, we change depolarizing to de-
phasing, as illustrated in Fig. 1(c).InFig.3(b), the theoretical
contextual robustness (blue curve) would exhibit a discontinu-
ity near θ=0, where the robustness drops directly from 0.5
to 0. However, based on the experimental results (red curve),
the violation of inequality snear this point is too small
to be reliably detected within the allowed error range (the
standard deviation is 0.001). Consequently, the discontinuity
point occurs near θ=π/25.
It can be observed that for θ>π/25, the contextual
robustness monotonically decreases as coherence increases.
This can be understood as smaller coherence of prepared
states approaching identity measurement in dephasing mea-
surements, resulting in better robustness against noise. The
highest robustness is achieved at θ=π/25, where rdeph
min =
0.504 ±0.001. Comparing Figs. 3(a) and 3(b), contextuality
exhibits different levels of robustness to depolarizing and
dephasing. Specifically, the optimal robustness under dephas-
ing (rdeph
min =0.504 ±0.001) surpasses that under depolarizing
(rdepol
min =0.1976 ±0.0008).
To maximize the robustness of contextuality, we rotate
the Helstrom measurement Mgfrom the zaxis to xunder
dephasing conditions, as illustrated in Fig. 1(d).InFig.3(c),
the theoretical contextual robustness (blue curve) indicates
that as θapproaches 0, rdeph
min converges toward 1. However,
constrained by the limited precision of our experiment, we are
only able to set θ=π/50 to achieve the maximum contextual
robustness of rdeph
min =0.938 ±0.001. This implies that when
the system’s coherence approaches zero and dephasing can
be of any magnitude (up to 0.938 ±0.001), the violation of
the noncontextuality inequality can still occur, indicating the
persistence of contextuality.
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EXPERIMENTAL INVESTIGATION OF CONTEXTUAL … PHYSICAL REVIEW A 109, 052208 (2024)
Finally, to depict the transition of the system from contex-
tuality to noncontextuality under the influence of noise more
clearly, we have chosen representative points from Figs. 3(a)–
3(c), where the contextual robustness achieves its maximum
value, as illustrated in Fig. 3(d). For depolarizing measure-
ments, represented by the black circles and line, we have
θ=π
3. The system intersects the boundary (dashed black
line) at r=0.1976 ±0.0008 with a violation quantity s=
0±0.001. For dephasing measurements with the Helstrom
measurement performed along the zaxis, depicted by the
blue circles and line, we set θ=π/25. The system intersects
the boundary at r=0.504 ±0.001 with s=0±0.001.
For dephasing measurements with the Helstrom measurement
conducted along the xaxis, shown by the red circles and
line, we have θ=π/50. The system intersects the boundary
(dashed black line) at r=0.938 ±0.001 with s=0.002 ±
0.001.
IV. CONCLUSION
Our investigation centers on the scenario of minimum-error
state discrimination within two-dimensional Hilbert spaces.
We conduct an experiment involving depolarizing and dephas-
ing measurements with adjustable noise intensity to explore
the relationship between contextual robustness and coherence.
The experimental results demonstrate that, for both depolariz-
ing and dephasing, contextuality can persist under any degree
of coherence, excluding the extreme cases of coherence 0 and
1. Particularly, in the scenario of dephasing, the robustness of
contextuality is superior to that in the case of depolarizing.
Remarkably, as the coherence approaches zero (yet remains
nonzero), contextuality can exist under arbitrary dephasing.
However, it is important to note that while the presence of
contextuality implies the existence of coherence, the oppo-
site relationship does not always apply. Furthermore, these
conclusions also hold under the basis-independent coher-
ence presented in Appendix C. Consequently, contextuality
can serve as a reliable basis-independent coherence witness.
Finally, the conclusion that the failure of noncontextuality
cannot be achieved without set coherence of all the states
(or all the measurements) bears similarity to the relationship
between entanglement and nonlocality. It appears that coher-
ence plays a pivotal role in inducing contextuality, akin to
the role of entanglement in nonlocality. Hence, we would like
to provide evidence or quantitative discussion to support this
claim in future investigation.
ACKNOWLEDGMENTS
This work is supported by the National Key R&D Program
of China (Grant No. 2023YFA1406701) and the National Nat-
ural Science Foundation of China (Grants No. 12025401, No.
92265209, No. 12104009, No. 12104036, and No. 12305008).
D.K.Q. acknowledges the support from the China Postdoc-
toral Science Foundation (Grant No. 2023M730198) and the
fellowship of China National Postdoctoral Program for Inno-
vative Talents (Grant No. BX20230036).
APPENDIX A: EXPERIMENTAL DETAILS
Here, we describe the procedure for adjusting the angle
of the wave plate to prepare the desired states and conduct
measurements with adjustable noise intensity. By choosing
θ=4ϕ1, where ϕ1represents the angle of HWP1, we can
prepare the state |ψ=cos θ
2|H+sin θ
2|V. This state is em-
ployed in the three scenarios outlined in the main text.
Utilizing noise-free measurements, we conduct quantum
state tomography processes [55] on the prepared states to
obtain their density matrix. This involves the projection of
the prepared states onto four basis {|H,|V,|H−|V
√2,|H−i|V
√2}
and recording the number of photons counted by detectors.
Subsequently, we compare the experimentally derived density
matrix with theoretical values to assess the quality of the
prepared states based on fidelity. Across all measured states,
our experimental results consistently demonstrate a fidelity
exceeding 99.1% ±0.3%. Additionally, coherence Cand set
coherence R1can be deduced by plugging the experimentally
derived density matrix into Eqs. (10) and (C1).
However, unlike the preparation phase, measurements with
adjustable noise intensity exhibit variation across these three
cases. Fortunately, we have identified a universal approach
for determining the wave plate angle that is applicable to
any trace preserved type of noise. Consider the noise-free
projection basis as |ξξ|; in the presence of noise, it can be
denoted as D(|ξξ|). The trace preserved property ensures
that D(|ξξ|) can always be diagonalized and expressed as
D(|ξξ|)=(1 −λ)|ζζ|+λ|¯
ζ¯
ζ|, where λrepresents the
noise intensity, and |ζζ|and |¯
ζ¯
ζ|form an orthogonal
basis.
Subsequently, we elaborate on the implementation of this
form of noise measurement by utilizing the dashed box seg-
ment of the experimental setup as depicted in the main
text. Through the adjustment of the angles of QWP2 and
HWP2, a simultaneous transformation of {|ζζ|,|¯
ζ¯
ζ|}to
{|HH|,|VV|} is achievable. As the photon traverses the
first BD, we attain the projection onto |ζζ|in the upper
path and onto |¯
ζ¯
ζ|in the middle path. Subsequently, by
employing HWP3 (HWP4), the photon is directed through
the second BD into the middle path with a probability of
1−λ(λ), facilitating the acquisition of the desired imperfect
measurement in this path.
This approach allows us to obtain imperfect measurements
for all elements in Table II, enabling the assessment of the
contextuality of the system. Consequently, we experimentally
derive the corresponding noise intensity , overlap c, and the
success probability of state discrimination s. Additionally, the
noise intensity rmentioned in the main text can be derived
from .
Furthermore, to calculate effective coherence, we con-
duct quantum state tomography processes [55]onthe
prepared states with imperfect measurements. This in-
volves the projection of the prepared states onto four basis
{D(|H),D(|V),D(|H−|V
√2),D(|H−i|V
√2)}. The effective co-
herence Ceff and set coherence Reff can then be deduced by
plugging the effective density matrix into Eqs. (10) and (C1).
The resulting outcome is theoretically equivalent to the sce-
nario of a perfect measurement preceded by a noisy channel.
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JINGYAN LIN et al. PHYSICAL REVIEW A 109, 052208 (2024)
This approach enables us to investigate decoherence in the
Appendix C.
Finally, we provide a concrete example to illustrate the
imperfect measurement process. Let us consider measure-
ments under dephasing noise with the Helstrom measurement
along the xaxis. The Helstrom measurement is expressed
as Ddeph
r(|++|)=(1 −rdeph
2)|++| + rdeph
2|−−|, where
|+ = 1
√2(|H+|V) and |− = 1
√2(|H−|V). By adjust-
ing HWP2 to an angle of ϕ2=22.5◦, the transformation
from {|++|,|−−|} to {|HH|,|VV|} is achieved. Sub-
sequently, the angles of HWP3 and HWP4 are chosen such
that ϕ4=π
4−ϕ3, enabling measurements under the noise
intensity rdeph =2sin
2(2ϕ3). Two counts are considered: the
count of correctly identified photons n0and the count of incor-
rectly identified photons n1. These counts are measured with
two different angles of HWP2: ϕ2=22.5◦and ϕ2=−22.5◦.
The success probability of state discrimination is determined
as s=n0/(n0+n1). Applying the same method, we can de-
termine the noise intensity and the overlap cfor imperfect
measurements.
APPENDIX B: DATA ANALYSIS
In adherence to the rigorous standards of Ref. [37],
we adopt the data-processing method initially proposed in
Ref. [31], comprising two sequential steps. The first step,
known as the primary procedure, assumes the experimental
data to be noncontextual and applies fitting through the gener-
alized probability theory (GPT) [56]. This approach facilitates
the verification of contextuality without reliance on the valid-
ity of quantum theory. The second step, termed the secondary
procedure, ensures that the data conform to the requirement of
preparation equivalence.
By relaxing the assumed symmetries among the state
preparations, we redefine Table Iin the main text within the
framework of GPT, as illustrated in Table II. It is noteworthy
that two extra preparations along the yaxis are introduced,
creating an additional dimension to bring the data in the
primary procedure closer to our raw experimental data. More-
over, the observable σyis included to ensure the completeness
of tomographic measurement.
All entries in the table are experimentally obtained and
utilized to construct a 6 ×4matrixDr, defined as Dr=
⎛
⎜
⎜
⎜
⎜
⎝
Pr
|φ,MφPr
|ψ,MφPr
|¯
φ,MφPr
|¯
ψ,MφPr
y,MφPr
¯y,Mφ
Pr
|φ,MψPr
|ψ,MψPr
|¯
φ,MψPr
|¯
ψ,MψPr
y,MψPr
¯y,Mψ
Pr
|φ,MgφPr
|ψ,MgφPr
|¯
φ,MgφPr
|¯
ψ,MgφPr
y,MgφPr
¯y,Mgφ
Pr
|φ,MyPr
|ψ,MyPr
|¯
φ,MyPr
|¯
ψ,MyPr
y,MyPr
¯y,My
⎞
⎟
⎟
⎟
⎟
⎠
.
(B1)
In the primary procedure, according to the proposition in
Supplementary Note 4 of Ref. [31], a matrix Dpcan emerge
from a GPT with three two-outcome measurements that are
tomographically complete if and only if every column of
Dpsatisfies the subsequent equation for j∈{φ, ψ, ¯
φ, ¯
ψ,y,¯y}
and i∈{φ, ψ, gφ, y},
aPp
pj,M1+bPp
pj,M2+cPp
pj,M3+dPp
pj,M4−1=0,(B2)
where Pp
pj,Mirepresents an element of matrix Dp, and
{a,b,c,d}are real numbers. Geometrically, the proposition
stipulates that the eight columns of Dpare situated on a three-
dimensional hyperplane defined by the constants {a,b,c,d}.
To determine the GPT of best fit, we fit a three-dimensional
hyperplane to the eight four-dimensional points constitut-
ing the columns of Dr. Subsequently, each column of Dris
mapped to its nearest point on the hyperplane, with these eight
points forming the columns of Dr.
The GPT of best fit is established by minimizing the
weighted distance χi, which gauges the deviation between raw
data and primary data, defined as
χi=



4

j=1
Pp
pj,Mi−Pr
pj,Mi
Pr
pj,Mi2.(B3)
Here, Pr
pj,Midenotes the statistical uncertainty in the ex-
periment. Thus, the primary procedure is formulated as the
following minimization problem:
min
{Pp
pj,Mi,a,b,c,d}χ2=min
{a,b,c,d}
6

i=1
χ2
i
s.t.aPp
pj,M1+bPp
pj,M2+cPp
pj,M3+dPp
pj,M4−1=0,
∀j=1,...,6.(B4)
Using the method of Lagrange multipliers [57], this problem
can be reformulated as a four-variable optimization problem:
min
{a,b,c,d}χ2=min
{a,b,c,d}
6

i=1
χ2
i
s.t.aPp
pj,M1+bPp
pj,M2+cPp
pj,M3+dPp
pj,M4−1=0,
∀j=1,...,6,(B5)
where χ2
i=
aPp
pj,M1+bPp
pj,M2+cPp
pj,M3+dPp
pj,M4−12
aPp
pj,M12+bPp
pj,M22+cPp
pj,M32+dPp
pj,M4−12.
(B6)
The χ2parameter resulting from the fitting procedure serves
as a metric for the goodness of fit of the hyperplane to the
data. Given that we are fitting eight data points to a hyperplane
defined by four fitting parameters a,b,c,d, we anticipate the
χ2parameter to follow a χ2distribution with four degrees of
freedom [58], with a mean of 4. If the fitting procedure had
yielded significantly higher χ2values, it would suggest that
the theoretical depiction of the preparation and measurement
procedures necessitated more than three degrees of freedom.
Conversely, if the fitting had produced an average χ2substan-
tially lower than 4, it would indicate an overestimation of the
uncertainty in data.
In the secondary procedure, since the columns of the Dp
matrix define the GPT states, we denote the vector defined by
the ith column as Pp
i. The secondary preparation is character-
ized by a probabilistic mixture of the primary preparations.
Consequently, the GPT state of the secondary preparation is
represented by a vector Ps
i, which is a probabilistic mixture of
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EXPERIMENTAL INVESTIGATION OF CONTEXTUAL … PHYSICAL REVIEW A 109, 052208 (2024)
the Pp
i, defined as
Ps
i=
6

i=1
ui
iPp
i,(B7)
where the ui
iare the weights in the mixture.
Therefore, the 4 ×4matrixDsis formed by the linear
transformation of Dp, denoted as
Ds=UD
p,(B8)
where Uis a 4 ×6 matrix representing the linear transforma-
tion. The preparation equivalence is enforced by the condition
Ps
|φ,Mi+Ps
|¯
φ,Mi=Ps
|ψ,Mi+Ps
|¯
ψ,Mi,∀j=1,2,3,(B9)
which is essential for establishing noncontextual inequalities.
Here, Ps
|φ,Mi,Ps
|¯
φ,Mi,Ps
|ψ,Mi,Ps
|¯
ψ,Miare elements of Ds.Ad-
ditionally, we aim to ensure that the new states formed by
linear combination are as close to the initial states as possible.
Therefore, the secondary procedure can be summarized as the
following maximization problem:
max Dps =max 1
4
4

v=1
Uvv
s.t.Ps
|φ,Mi+Ps
|¯
φ,Mi=Ps
|ψ,Mi+Ps
|¯
ψ,Mi,
∀j=1,2,3.(B10)
Solving this problem yields all elements of Ds. The first three
rows of the matrix correspond to rows 2–4 and columns 2–5 in
Table II. Consequently, it becomes feasible to compute all the
elements {φ,¯
φ,
ψ,cφ,c¯
φ,cψ,sφ,s¯
φ,sψ}required for non-
contextual inequalities given afterward based on this relation.
In Fig. 4(a), for clarity, we depict the construction of sec-
ondary preparations within the zx plane of the Bloch sphere.
Indeed, to refine the primary data toward the raw data, we
incorporate processes within the bulk of the Bloch sphere
using a similar methodology. Equation (B9) dictates that all
preparations need not converge to the center of the Bloch
sphere, but rather to the same state. Consequently, the two
pairs need not be coplanar in the Bloch sphere. Based on
this principle, supplementing the original set with just the two
eigenstates of σyoffers a suitable compromise between main-
taining a low number of preparations and ensuring proximity
of the secondary preparations to the ideal. As the σyeigen-
states are maximally distant from the zx plane, they facilitate
moving any point close to that plane in the ±ydirection, while
inducing only modest motion within the zx plane.
Given the absence of assumed symmetry, the noncontex-
tual inequalities in this scenario are not reducible to a single
inequality, as detailed in the full set of 15 noncontextuality
inequalities provided in Appendix D of Ref. [37]. Conse-
quently, we calculate the violation of all inequalities related to
the probability of successful discrimination listed below and
FIG. 4. The results of data analysis for preparation equivalence.
Raw preparations (blue triangles) are derived from experimental
data. Primary preparations (orange dots) represent the outcomes of
fitting with the GPT theory. Secondary preparations (green rectan-
gles) are specifically chosen to satisfy the assumption of preparation
equivalence. (a) The formation of secondary preparations occurs
within the zx plane of the Bloch sphere. If the four primary prepa-
rations are achievable, then any preparation within their convex hull
can be simulated. The scenario where preparation equivalence is met
corresponds to the alignment of the center point of the black dotted
line between two pairs of states. In (b)–(d), the secondary data sat-
isfies PMi=0, where PMi=Ps
|φ,Mi+Ps
|¯
φ,Mi−Ps
|ψ,Mi−Ps
|¯
ψ,Mi.
Each point in (b)–(d) corresponds to one preparation and measure-
ment setup within the contextual region of Fig. 3in the main text.
identify the largest violation.
0cψ+s¯
φ−sψ+φ
0cψ−s¯
φ+sψ+φ
0−cψ+sφ+sψ+¯
φ
0cφ+s¯
φ−sψ+ψ
0−c¯
φ+sφ+sψ+φ
0cφ−s¯
φ+sψ+φ
02−cψ−sφ−sψ+¯
φ
02−c¯
φ−sφ−sψ+ψ.
(B11)
For consistency with the noncontextual inequality violations
mentioned in the main text, we multiply the violations of
these equations by one-half. The violation of any one of these
inequalities signifies the contextual nature of the system.
In situations where the noise intensity approaches 1, the
secondary process shows substantial deviation from the raw
data at certain points, primarily because the preparations and
measurements are very close together. Consequently, while
all data undergo primary processing, secondary processing
is selectively applied to points that reveal contextuality, pre-
serving the rigor of our conclusions. Additionally, for the
scenario where both |φφ|and |ψψ|are close to the zaxis
in the zx plane of the Bloch sphere in the main text, two
052208-7

JINGYAN LIN et al. PHYSICAL REVIEW A 109, 052208 (2024)
extra preparations along the xaxis are introduced to bring
the data in the primary procedure closer to the raw data. Data
analysis for the 88 points revealing contextuality is presented
in Figs. 4(b)–4(d), where all secondary points satisfy the con-
dition of preparation equivalence.
APPENDIX C: COHERENCE, SET COHERENCE,
AND EFFECTIVE COHERENCE
The definition of contextuality used in the main text is
independent of the basis, whereas coherence is basis depen-
dent. This distinction might raise concerns about the potential
influence of coordinate choices on our conclusions. To address
this, we illustrate below that our two main conclusions in the
main text can still be drawn when examining the relationship
between contextuality and set coherence, both of which are
independent of bases.
The concept of set coherence, as introduced in Ref. [59],
offers a basis-independent quantification of coherence by min-
imizing the coherence of states in the set across all possible
basis choices. For a pair of qubit states ρ1and ρ2, the mean
robustness of set coherence R1characterizes the intrinsic co-
herence of this set and is given by the expression
R1(
ρ)=min

p∈S2
1
2
q1|sin( 
p,
q1)|+1
2
q2|sin( 
p,
q2)|,(C1)
where 
q1(
q2) is the Bloch vector of ρ1(ρ2), and ( 
p,
q1)
represents the angle between the Bloch vector 
pand 
q1.For
pairs of pure qubit states, the minimum in Eq. (C1) is achieved
when 
pcoincides with either 
q1or 
q2. In the case of mixed
states, the optimal 
paligns with the Bloch vector of the purest
state, corresponding to the longest Bloch vector.
Firstly, Fig. 5(a) explores the relationship between contex-
tuality and set coherence without the influence of noise. The
horizontal axis signifies the system state, while the vertical
axis represents the values of each physical quantity. The ex-
pressions for these four physical quantities in Fig. 5(a) can be
formulated as
R1=1
2sin(2θ),
C=sin θ,
sz=1
2(cos θ+sin2θ−1),
sx=1
2(sin θ+cos2θ−1).
(C2)
When comparing the connection between contextuality and
set coherence in both scenarios, despite the difference in
coordinate selection, the relationship remains consistent. To
emphasize this point, Fig. 5(b) offers a clearer depiction of the
association between contextuality and set coherence, where
the experimental data in two cases align closely with the same
theoretical values. It is noteworthy that a R1value corresponds
to two s, while the values of Cand shave a one-to-one
correspondence. Therefore, investigating the relationship be-
tween coherence and contextuality is more advantageous for
drawing conclusions, which is a central focus of our research
in the main text.
The conclusion, similar to the main text, can still be drawn
here: contextuality implies the presence of set coherence in
FIG. 5. Contextuality and different measures of coherence-
coherence C, set coherence R1, and efficient coherence Ceff (set
coherence Reff). All data points in the figure correspond to exper-
imental results. For all states measured, the fidelity consistently
remains above 99.1% ±0.3%. (a) The system with two coordinate
choices in the absence of noise. Blue points represent the case of
Helstrom measurement performed along the zaxis, while red points
represent the case of the Helstrom measurement performed along
the xaxis. Different shapes represent different physical quantities:
circles represent the violation of noncontextual inequalities s,
triangles represent R1, and squares represent C. The curve repre-
sents the corresponding theoretical values of each physical quantity.
(b) The relationship between contextuality and set coherence under
the change of parameter θ. (c) The system with one coordinate
choice in the present of two types of noise. The system state θis
set to 1.0472, near the maximum of sin the absence of noise.
Black represents the case of depolarizing noise, and blue represents
the case of dephasing noise. The solid line represents Reff, and the
dashed line represents Ceff. (d) The relationship between contextual-
ity and efficient coherence (set coherence) under the change of noise
intensity r.
the system, as depicted in Figs. 5(a) and 5(b). This rela-
tionship is also theoretically established in Appendix B of
Ref. [41]. However, the reverse is not necessarily true under
sufficient levels of noise. Additionally, it is noteworthy that the
conclusion holds when set coherence approaches zero (while
remaining nonzero); contextuality can endure under arbitrary
amounts of partially dephasing, as observed by comparing the
set coherence curve in Fig. 5(a) here and the contextuality
boundary curve in Fig. 3(c) in the main text.
In the subsequent analysis shown in Fig. 5(c), we investi-
gate decoherence through effective coherence (set coherence)
under two types of noise, specifically focusing on the case of
the Helstrom measurement performed along the zaxis. The
effective coherence Ceff (set coherence Reff) is deduced from
the density matrix obtained through state tomography with
imperfect measurements in our scenario. This is equivalent
to quantifying the coherence (set coherence) of the system
after it has undergone the same type of noisy channel. The
equations for Reff,Ceff, and sunder depolarizing noise are
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EXPERIMENTAL INVESTIGATION OF CONTEXTUAL … PHYSICAL REVIEW A 109, 052208 (2024)
as follows:
Rdepol
eff =1
2(1 −r)sin(2θ),
Cdepol
eff =(1 −r)sinθ, (C3)
sdepol
z=1
2[(1 −r)(cos θ+sin2θ)−1].
The expressions for Reff,Ceff, and sunder dephasing noise
are formulated as
Rdeph
eff =(1 −r)sin(2θ)
2cos2θ+(1 −r)2sin2θ
,
Cdeph
eff =(1 −r)sinθ,
sdeph
z=1
2[cos θ+(1 −r)sin
2θ−1].
(C4)
As the intensity of noise increases, the efficient coherence
Ceff (set coherence Reff) decreases, as illustrated in Fig. 5(c).
Building upon this observation of decoherence, we further
study the performance of contextuality in Fig. 5(d). The be-
haviors of Reff,Ceff, and sexhibit both linear and nonlinear
relationships, depending on the type of noise. In the case of
decoherence, this implies that the smaller the efficient co-
herence (set coherence), the less pronounced the violation
of the noncontextuality inequality, making the existence of
contextuality less likely.
APPENDIX D: ADDITIONAL DISCUSSION REGARDING
CONTEXTUALITY AND COHERENCE
Here, we explore the feasible method to establish a
basis-independent relationship between contextuality and co-
herence in the context of Spekkens’ categorization [3].
Spekkens distinguishes between preparation noncontextual-
ity and measurement noncontextuality, with Kochen-Specker
noncontextuality falling into the latter category.
For preparation noncontextuality, the construction of
preparation equivalence, akin to the first equality in Eq. (2)
in the main text, is essential. This implies that the derived
noncontextual inequalities involve at least two states and their
orthogonal counterparts. In such cases, the basis-independent
definition of coherence, known as set coherence, becomes rel-
evant and can be employed to explore the basis-independent
relationship between contextuality and set coherence of all the
states, as demonstrated in our work.
For measurement noncontextuality, the construction of
measurement equivalence, similar to preparation equivalence,
is essential. This implies that the resulting noncontextual in-
equalities involve at least two measurements. In such cases,
the set coherence of the measurement basis becomes rele-
vant and can be employed to explore the basis-independent
relationship between contextuality and set coherence for all
measurements. For example, when verifying Kochen-Specker
contextuality, at least one noncommutation pair of operators is
required. In this context, it is intriguing to investigate the set
coherence of the measurements. Moreover, the set coherence
of states can also be investigated by constructing a set of states
to test Kochen-Specker contextuality.
Finally, the conclusion that the failure of noncontextuality
cannot be achieved without set coherence of all the states
(or all the measurements), strictly proven in Ref. [41], bears
similarity to the relationship between entanglement and non-
locality. We speculate that coherence plays a pivotal role in
inducing contextuality, akin to the role of entanglement in
nonlocality. Therefore, we aim to establish a link between
coherence and contextuality.
[1] J.S.Bell,Rev. Mod. Phys. 38, 447 (1966).
[2] S. Kochen and E. P. Specker, J. Math. Mech. 17, 59 (1967).
[3] R. W. Spekkens, Phys. Rev. A 71, 052108 (2005).
[4] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and
G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009).
[5] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, New J.
Phys. 18, 045003 (2016).
[6] D. Saha, P. Horodecki, and M. Pawłowski, New J. Phys. 21,
093057 (2019).
[7] S. Gupta, D. Saha, Z.-P. Xu, A. Cabello, and A. S. Majumdar,
Phys.Rev.Lett.130, 080802 (2023).
[8] A. Hameedi, A. Tavakoli, B. Marques, and M. Bourennane,
Phys.Rev.Lett.119, 220402 (2017).
[9] S.A.YadavalliandR.Kunjwal,Quantum 6, 839 (2022).
[10] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Nature
(London) 510, 351 (2014).
[11] D. Schmid, H. Du, J. H. Selby, and M. F. Pusey, Phys. Rev. Lett.
129, 120403 (2022).
[12] R. Raussendorf, J. Bermejo-Vega, E. Tyhurst, C. Okay, and M.
Zurel, Phys.Rev.A101, 012350 (2020).
[13] A. K. Daniel and A. Miyake, Phys. Rev. Lett. 126, 090505
(2021).
[14] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys.Rev.Lett.
113, 140401 (2014).
[15] A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89,
041003 (2017).
[16] A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
[17] J. H. Selby and C. M. Lee, Quantum 4, 319 (2020).
[18] F. Ahnefeld, T. Theurer, D. Egloff, J. M. Matera, and M. B.
Plenio, Phys.Rev.Lett.129, 120501 (2022).
[19] D. Qu, K. Wang, L. Xiao, X. Zhan, and P. Xue, npj Quantum
Inf. 7, 154 (2021).
[20] P. Wang, J. Zhang, C.-Y. Luan, M. Um, Y. Wang, M. Qiao,
T. Xie, J.-N. Zhang, A. Cabello, and K. Kim, Sci. Adv. 8,
eabk1660 (2022).
[21] S. B. van Dam, J. Cramer, T. H. Taminiau, and R. Hanson, Phys.
Rev. Lett. 123, 050401 (2019).
[22] Z.-H. Liu, H.-X. Meng, Z.-P. Xu, J. Zhou, J.-L. Chen, J.-S.
Xu, C.-F. Li, G.-C. Guo, and A. Cabello, Phys. Rev. Lett. 130,
240202 (2023).
[23] P. Xue, L. Xiao, G. Ruffolo, A. Mazzari, T. Temistocles, M. T.
Cunha, and R. Rabelo, Phys.Rev.Lett.130, 040201 (2023).
[24] R. W. Spekkens, Phys.Rev.A75, 032110 (2007).
[25] A. Cabello, Phys. Rev. Lett. 101, 210401 (2008).
052208-9

JINGYAN LIN et al. PHYSICAL REVIEW A 109, 052208 (2024)
[26] E. Amselem, M. Rådmark, M. Bourennane, and A. Cabello,
Phys.Rev.Lett.103, 160405 (2009).
[27] X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue,
Phys.Rev.Lett.116, 090401 (2016).
[28] X. Zhan, E. G. Cavalcanti, J. Li, Z. Bian, Y. Zhang, H. M.
Wiseman, and P. Xue, Optica 4, 966 (2017).
[29] X. Zhan, P. Kurzy´
nski, D. Kaszlikowski, K. Wang, Z.
Bian, Y. Zhang, and P. Xue, Phys. Rev. Lett. 119, 220403
(2017).
[30] D. Qu, P. Kurzy´
nski, D. Kaszlikowski, S. Raeisi, L. Xiao, K.
Wang, X. Zhan, and P. Xue, Phys. Rev. A 101, 060101(R)
(2020).
[31] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W.
Spekkens, Nat. Commun. 7, ncomms11780 (2016).
[32] J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R.
Kunjwal, and R. W. Spekkens, Phys. Rev. Lett. 130, 230201
(2023).
[33] G. Sharma, S. Sazim, and S. Mal, Phys. Rev. A 104, 032424
(2021).
[34] A. Chaturvedi, M. Farkas, and V. J. Wright, Quantum 5, 484
(2021).
[35] R. Wagner, R. D. Baldijão, A. Tezzin, and B. Amaral, J. Phys.
A: Math. Theor. 56, 505303 (2023).
[36] R. Wagner, A. Camillini, and E. F. Galvão, Quantum 8, 1240
(2024).
[37] D. Schmid and R. W. Spekkens, Phys.Rev.X8, 011015
(2018).
[38] Q. Zhang, C. Zhu, Y. Wang, L. Ding, T. Shi, X. Zhang,
S. Zhang, and W. Zhang, Chin. Phys. Lett. 39, 080301
(2022).
[39] R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403
(2015).
[40] I. Marvian, arXiv:2003.05984.
[41] V. P. Rossi, D. Schmid, J. H. Selby, and A. B. Sainz, Phys. Rev.
A108, 032213 (2023).
[42] M. Lugli, P. Perinotti, and A. Tosini, Phys.Rev.Lett.125,
110403 (2020).
[43] H. Lee, K. Flatt, C. R. i Carceller, J. B. Brask, and J. Bae, Phys.
Rev. A 106, 032422 (2022).
[44] D. Ha and J. S. Kim, Phys.Rev.A105, 032421 (2022).
[45] S. M. Cohen, Phys.Rev.A107, 012401 (2023).
[46] I. A. Burenkov, M. V. Jabir, A. Battou, and S. V. Polyakov, PRX
Quantum 1, 010308 (2020).
[47] G. Zhu, O. Kálmán, K. Wang, L. Xiao, D. Qu, X. Zhan, Z. Bian,
T. Kiss, and P. Xue, Phys. Rev. A 100, 052307 (2019).
[48] X. W. Wang, G. Y. Zhu, L. Xiao, X. Zhan, and P. Xue, Quantum
Inf. Process 23, 8 (2024).
[49] C. H. Bennett, Phys.Rev.Lett.68, 3121 (1992).
[50] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, Phys. Rev.
Lett. 104, 230503 (2010).
[51] C. W. Helstrom, J. Stat. Phys. 1, 231 (1969).
[52] P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C.
Sanders, Phys.Rev.Lett.114, 140502 (2015).
[53] Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P.
Xue, Phys. Rev. Lett. 114, 203602 (2015).
[54] K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, B. C. Sanders, W.
Yi, and P. Xue, Nat. Commun. 10, 2293 (2019).
[55] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White,
Phys. Rev. A 64, 052312 (2001).
[56] J. Barrett, Phys.Rev.A75, 032304 (2007).
[57] M. Krystek and M. Anton, Meas. Sci. Technol. 18, 3438
(2007).
[58] W. H. Press, Numerical Recipes 3rd Edition: The Art of Scientific
Computing (Cambridge University, Cambridge, 2007).
[59] S. Designolle, R. Uola, K. Luoma, and N. Brunner, Phys. Rev.
Lett. 126, 220404 (2021).
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