# Simultaneous All-versus-Nothing Refutation of Local Realism and
Noncontextuality by a Single System

Min-Gang Zhou  National Laboratory of Solid State Microstructures and School
of Physics, Nanjing University, Nanjing 210093, China  Hua-Lei Yin
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and School
of Physics, Nanjing University, Nanjing 210093, China  Zeng-Bing Chen
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and
School of Physics, Nanjing University, Nanjing 210093, China

###### Abstract

Bell’s theorem and the Kochen-Specker theorem, respectively, capture two
wonderful phenomena that cannot be described by any classical theory, namely,
quantum nonlocality and quantum contextuality. Each of these two phenomena has
withstood extensive experimental verification and is considered to be an
important quantum resource capable of revolutionizing the field of information
and computing. Here we present an all-versus-nothing proof that can be used to
simultaneously demonstrate Bell’s theorem and the Kochen-Specker theorem. The
proof makes use of three singlet pairs and the Greenberger-Horne-Zeilinger-
state analyzer, and thus can be experimentally tested with the current linear
optics technology. Not only does the proposed scheme represent the strongest
conflict between quantum mechanics and hidden variable theories to date, it
also provides insightful guidance for deepening our understanding of the
relation between quantum nonlocality and quantum contextuality.

Although quantum mechanics (QM) is by far the most successful physical theory,
its counterintuitive nature still causes a conceptual obstacle to
understanding its foundations. Among those early efforts on quantum
foundations, Einstein, Podolsky, and Rosen (EPR) discovered one of the most
famous paradoxes einstein1935can  in physics: Under the presumption of local
realism, a maximally entangled two-party system seemingly allows one to infer
definite values of complementary variables (e.g., position and momentum) by
using perfect correlations, thus contradicting QM. The EPR paradox motivated
various hidden variable theories. Strikingly, two major theorems, Bell’s
theorem bell1964physics  and the Kochen-Specker (KS) theorem kochen1967 ;
simon2000 ; simon2001 ; cabello2008 ; yu ; context-nature ; chenjl , have been
developed to deny hidden variables in QM. The former denies local hidden
variables (LHV) based on EPR’s notion of local realism, while the latter poses
a more severe challenge to hidden variables. The KS theorem states that it is
impossible to assign definite values to each of not all commutative
observables in an individual system, known as quantum contextuality. This
means that even without the locality condition, the value for an observable is
dependent on the experimental context, thus negating noncontextual hidden
variables (NCHV). Klyachko et al. also derived inequalities Klyachko ; Kunjwal
that are valid for NCHV, but violated for individual quantum systems.

The original argument of Bell’s theorem has a strong statistical character,
and perfect two-qubit correlations would not violate Bell’s inequalities. To
underline the contradiction between QM and LHV for definite predictions,
“Bell’s theorem without inequalities”, also known as the Greenberg-Horne-
Zeilinger (GHZ) theorem, has been developed, which has been demonstrated for
the case of three greenberger1989bell ; greenberger1990bell ; mermin1990pt ;
pan2000experimental  or even two cabello2001all ; PhysRevLett.95.210401 ;
chen2003all ; yang2005all  spacelike separated observers. Because this proof
without inequalities involves perfect correlations, it directly refutes EPR’s
notion of local realism based on such correlations, implying a stronger
refutation of local realism.

From the perspective of quantum information science, both quantum nonlocality
(i.e., violations of Bell’s inequalities) and quantum contextuality empower
certain quantum information tasks context-nature ; Cabello2021 . Then it is
natural to seek the possible relations between the two nonclassical phenomena.
Since the works of Stairs Stairs , Heywood and Redhead Heywood and Redhead ,
and Mermin mermin1990simple , recent years have witnessed significant advances
on the interesting connections Cabello2021  between quantum contextuality and
nonlocality in terms of inequalities, such as converting local contextuality
into nonlocality Cabello2010 ; Cabello2021prl ; guo2016 , simultaneously
observing contextuality and nonlocality guo2018 ; xue2022 , and a trade-off
relation between nonlocality and contextuality Cabello2014 ; guo-pra ; xue2016
.

In this Letter, we prove for the first time that Bell’s theorem and the KS
theorem, both without inequalities, can be demonstrated by one and the same
system. Our proposal is inspired by, in addition to the above progress, the
proof of the KS theorem based on Pauli operators for three qubits due to
Mermin mermin1990simple , and by the “all-versus-nothing” proof of Bell’s
theorem chen2003all ; chen2005avn ; ghz2005 . In a sense, the present proposal
can be seen as a further development of Bell’s theorem and the KS theorem,
which represents the strongest conflict between QM and hidden-variable
theories to date as it rules out both local and noncontextual hidden variables
in an all-versus-nothing manner. In the context of the recent advances
mentioned above, our all-versus-nothing proof shows from a different
perspective the connection between quantum contextuality and nonlocality.

Consider the arrangement shown in Fig. 1. Initially, Debbie possesses three
pairs of antisymmetric singlet Bell states
$|\Psi\rangle_{123456}=\left|\psi^{-}\right\rangle_{12}\otimes\left|\psi^{-}\right\rangle_{34}\otimes\left|\psi^{-}\right\rangle_{56}$.
Here the singlet state

$\left|\psi^{-}\right\rangle_{ij}=\frac{1}{\sqrt{2}}(|01\rangle_{ij}-|10\rangle_{ij}),$
(1)

where $|01\rangle_{ij}=|0\rangle_{i}\otimes|1\rangle_{j}$. Let us suppose that
Debbie keeps qubits $2$, $4$, and $6$ in her laboratory, and sends qubits $1$,
$3$, and $5$ to Alice, Bob, and Charlie, respectively. Each of these four
observers are in a spacelike separated region from the other three observers.
Then, they perform local measurements on the qubits in their respective
laboratories. If Debbie performs the GHZ-state measurement, our arrangement
considered here is the GHZ-entanglement swapping as experimentally realized in
Ref. lu .

Figure 1: Setup for demonstrating simultaneous refutation of local realism and
noncontextuality. Debbie initially possesses three singlet pairs, and then
sends qubits $1$, $3$ and $5$ to Alice, Bob and Charlie, respectively. Each of
Alice, Bob, Charlie and Debbie is in a spacelike separated region from the
other three.

We first discuss the negation of local realism by our scheme. To make the
following discussion clearly, we define $|\Psi\rangle=|\Psi\rangle_{123456}$,
$x_{i}=\sigma_{xi}$, $y_{i}=\sigma_{yi}$ ($i=1,2,...,6$) and
$x_{2}y_{4}y_{6}=\sigma_{x2}\sigma_{y4}\sigma_{y6}$, etc., and apply $(\cdot)$
to separate local operators. Thus, one can easily check that the state
$|\Psi\rangle$ satisfies the following eigenequations:

$\displaystyle\ \left.x_{2}y_{4}y_{6}\cdot x_{1}\cdot y_{3}\cdot
y_{5}\left|\Psi\right\rangle=-\left|\Psi\right\rangle,\right.$ (2)
$\displaystyle\ \left.y_{2}x_{4}y_{6}\cdot y_{1}\cdot x_{3}\cdot
y_{5}\left|\Psi\right\rangle=-\left|\Psi\right\rangle,\right.$ (3)
$\displaystyle\ \left.y_{2}y_{4}x_{6}\cdot y_{1}\cdot y_{3}\cdot
x_{5}\left|\Psi\right\rangle=-\left|\Psi\right\rangle,\right.$ (4)
$\displaystyle\ \left.x_{2}x_{4}x_{6}\cdot x_{1}\cdot x_{3}\cdot
x_{5}\left|\Psi\right\rangle=-\left|\Psi\right\rangle,\right.$ (5)
$\displaystyle\ \left.x_{2}x_{4}x_{6}\cdot x_{2}y_{4}y_{6}\cdot
y_{2}x_{4}y_{6}\cdot
y_{2}y_{4}x_{6}\left|\Psi\right\rangle=-\left|\Psi\right\rangle.\right.$ (6)

The last equation [Eq. (6)] stems from the operator identity
$x_{2}x_{4}x_{6}\cdot x_{2}y_{4}y_{6}\cdot y_{2}x_{4}y_{6}\cdot
y_{2}y_{4}x_{6}=-I_{246}$, where $I_{246}$ is the identity operator for three
qubits $2$, $4$ and $6$.

Equations (2)-(6) include only local operators (Alice’s, Bob’s, Charlie’s or
Debbie’s), implying that any three parties among Alice, Bob, Charlie and
Debbie can assign values with certainty to local operators of the remaining
party by measuring their local observables without disturbing the system of
that party in any way. According to EPR’s notion of local realism
einstein1935can , “If, without in any way disturbing a system, we can predict
with certainty (i.e., with probability equal to unity) the value of a physical
quantity, then there exists an element of physical reality corresponding to
this physical quantity”. Therefore, from the perspective of those who believe
in local realistic theories, each local operator in Eqs. (2)-(6) separated by
$(\cdot)$ can be defined as EPR’s local “elements of reality”. Moreover, all
these elements of reality have predetermined values, which are denoted by
$v\left(x_{i}\right)$, $v\left(y_{i}\right)$, $v\left(x_{2}y_{4}y_{6}\right)$,
$v\left(y_{2}x_{4}y_{6}\right)$, $v\left(y_{2}y_{4}x_{6}\right)$, and
$v\left(x_{2}x_{4}x_{6}\right)$ with $v=\pm 1$. Therefore, Eqs. (2)-(6) can be
rewritten in a local realistic theory as

$\displaystyle\ \left.v(x_{2}y_{4}y_{6})v(x_{1})v(y_{3})v(x_{5})=-1,\right.$
(7) $\displaystyle\
\left.v(y_{2}x_{4}y_{6})v(y_{1})v(x_{3})v(y_{5})=-1,\right.$ (8)
$\displaystyle\ \left.v(y_{2}y_{4}x_{6})v(y_{1})v(y_{3})v(x_{5})=-1,\right.$
(9) $\displaystyle\
\left.v(x_{2}x_{4}x_{6})v(x_{1})v(x_{3})v(x_{5})=-1,\right.$ (10)
$\displaystyle\
\left.v(x_{2}x_{4}x_{6})v(x_{2}y_{4}y_{6})v(y_{2}x_{4}y_{6})v(y_{2}y_{4}x_{6})=-1.\right.$
(11)

However, one can easily check that Eqs. (7)-(11) are mutually incompatible.
Specifically, because each $v$-value appears twice on the left-hand sides of
Eqs. (7)-(11), the product of all the left-hand sides must equal $+1$, which
conflicts with the result $-1$ obtained by multiplying all the right-hand
sides. This contradiction reveals that any local realistic theory based on
EPR’s notion of local realism cannot reproduce the results of perfect
correlations predicted by QM as in Eqs. (2)-(6). This completes the proof of
all-versus-nothing nonlocality. Such a logical impossibility proof is often
called Bell’s theorem without inequalities. The experimental issues (in
particular, the measurement strategy) of our scheme will be discussed below.

Bell’s theorem without inequalities has been demonstrated for the case of
three greenberger1989bell ; greenberger1990bell ; mermin1990pt ;
pan2000experimental  or even two cabello2001all ; PhysRevLett.95.210401 ;
chen2003all ; yang2005all  spacelike separated observers. Therefore, it is not
surprising that our argument involving four spacelike separated observers can
demonstrate Bell’s theorem without inequalities. However, note that our
argument differs from the previous ones in that local realism and
noncontextuality can be denied simultaneously by one and the same experimental
system. Such an argument provides a new and instructive insight into the
relationship between Bell’s theorem and the KS theorem.

To deny noncontextuality in our scheme, we only need to focus on Debbie’s
system of three qubits $2$, $4$ and $6$. Consider the ten operators $x_{2}$,
$y_{2}$, $x_{4}$, $y_{4}$, $x_{6}$, $y_{6}$, $x_{2}y_{4}y_{6}$,
$y_{2}x_{4}y_{6}$, $y_{2}y_{4}x_{6}$, and $x_{2}x_{4}x_{6}$ of this system.
Now we have five operator identities mermin1990simple

$\displaystyle\ \left.x_{2}y_{4}y_{6}\cdot x_{2}\cdot y_{4}\cdot
y_{6}=I_{246},\right.$ (12) $\displaystyle\ \left.y_{2}x_{4}y_{6}\cdot
y_{2}\cdot x_{4}\cdot y_{6}=I_{246},\right.$ (13) $\displaystyle\
\left.y_{2}y_{4}x_{6}\cdot y_{2}\cdot y_{4}\cdot x_{6}=I_{246},\right.$ (14)
$\displaystyle\ \left.x_{2}x_{4}x_{6}\cdot x_{2}\cdot x_{4}\cdot
x_{6}=I_{246},\right.$ (15) $\displaystyle\ \left.x_{2}x_{4}x_{6}\cdot
x_{2}y_{4}y_{6}\cdot y_{2}x_{4}y_{6}\cdot y_{2}y_{4}x_{6}=-I_{246}.\right.$
(16)

These identities are quantum mechanical predictions that are independent of
the state of the system. Note that, in each of Eqs. (12)-(16), these operators
separated by $(\cdot)$ are mutually commuting. In an NCHV theory, these
operators separated by $(\cdot)$ have predetermined values, but here without
appealing to EPR’s local realism. Let us denote these predetermined values by
$v\left(x_{i}\right)$, $v\left(y_{i}\right)$, $v\left(x_{2}y_{4}y_{6}\right)$,
$v\left(y_{2}x_{4}y_{6}\right)$, $v\left(y_{2}y_{4}x_{6}\right)$, and
$v\left(x_{2}x_{4}x_{6}\right)$ with $v=\pm 1$. Therefore, Eqs. (12)-(16) can
be rewritten in an NCHV theory as

$\displaystyle\ \left.v(x_{2}y_{4}y_{6})v(x_{2})v(y_{4})v(y_{6})=1,\right.$
(17) $\displaystyle\
\left.v(y_{2}x_{4}y_{6})v(y_{2})v(x_{4})v(y_{6})=1,\right.$ (18)
$\displaystyle\ \left.v(y_{2}y_{4}x_{6})v(y_{2})v(y_{4})v(x_{6})=1,\right.$
(19) $\displaystyle\
\left.v(x_{2}x_{4}x_{6})v(x_{2})v(x_{4})v(x_{6})=1,\right.$ (20)
$\displaystyle\
\left.v(x_{2}x_{4}x_{6})v(x_{2}y_{4}y_{6})v(y_{2}x_{4}y_{6})v(y_{2}y_{4}x_{6})=-1.\right.$
(21)

Equations (17)-(20) can be explained by NCHV theories. NCHV theories hold that
the predetermined values of these operators could, in principle, be measured
without disturbance, independent of which context the operators are contained
in. In other words, no matter how an operator $A$ is measured for an
individual system, the result of measuring $A$ will always be
$v\left(A\right)$. Therefore, $v(x_{2}y_{4}y_{6})=v(x_{2})v(y_{4})v(x_{6})$ in
Eq. (17) because $x_{2}y_{4}y_{6}$ can be measured by separately measuring
$x_{2}$, $y_{4}$ and $y_{6}$ and multiplying their results. Or,
$x_{2}y_{4}y_{6}$, $x_{2}$, $y_{4}$ and $y_{6}$ can be co-measured by a single
apparatus as they are mutually commuting operators, yielding the product of
the four operators to be $+1$. Likewise, Eqs. (18)-(20) are also consequences
of noncontextuality. Finally, the left-hand side of Eq. (21) is the product of
mutually commuting operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$,
$y_{2}y_{4}x_{6}$ and $x_{2}x_{4}x_{6}$, which can be simultaneously measured
by a single measurement apparatus, to be considered below. Now, the
contradiction between NCHV theories and QM can be seen by combining Eqs.
(17)-(21).

Because each $v$-value appears twice on the left-hand sides of Eqs. (17)-(21)
and can only be assigned the value $+1$ or $-1$, the product of all the left-
hand sides must be $+1$, while the product of all the right-hand sides equals
to $-1$. Therefore, the attempt to assign values, as in Eqs. (17)-(21), to
these ten operators must fail. This failure means that any NCHV theory
contradicts quantum mechanical predictions in Eqs. (12)-(16). With emphasis,
the contradiction occurs in predictions for these operators on a
nonstatistical level. Therefore, our scheme also completes the demonstration
of the all-versus-nothing contextuality, without sepecifying any state of the
system. Namely, our proof of contextuality is state-independent.

Note that Mermin mermin1990simple , generalizing a state-dependent argument
due to Peres peres , made use of the same set of operator identities in Eqs.
(12)-(16) for his argument on the KS theorem. However, the three qubits
involved must be spacelike separated then to validate his GHZ argument. In
contrast, in our proof, the three qubits are always kept in Debbie’s side for
both the KS and GHZ arguments. Therefore, the present work represents
genuinely simultaneous all-versus-nothing refutation of local realism and
noncontextuality by a single system.

When demonstrating the negation of local realism in an ideal experimental
situation, each of Eqs. (2)-(6) would be proven true by experimental data.
This shows that QM can withstand the test of the experiment. Combined with the
above discussion, any local realistic theory, which is fundamentally
contradictory to QM, would be rejected by experiments. Unfortunately, it is
difficult to prepare the perfect correlations and ideal measurement devices.
Therefore, to address these less-than-ideal experimental situations, one can
introduce the operator $\mathcal{O}=x_{2}y_{4}y_{6}\cdot x_{1}\cdot y_{3}\cdot
y_{5}+y_{2}x_{4}y_{6}\cdot y_{1}\cdot x_{3}\cdot y_{5}+y_{2}y_{4}x_{6}\cdot
y_{1}\cdot y_{3}\cdot x_{5}+x_{2}x_{4}x_{6}\cdot x_{1}\cdot x_{3}\cdot x_{5}$,
in which $-x_{2}x_{4}x_{6}\cdot x_{2}y_{4}y_{6}\cdot y_{2}x_{4}y_{6}\cdot
y_{2}y_{4}x_{6}$ being an identity operator is not included. Combined with
Eqs. (2)-(6), one can easily check that $|\Psi\rangle$ is an eigenstate of the
operator $\mathcal{O}$, thus having

$\mathcal{O}|\Psi\rangle=-4|\Psi\rangle.$ (22)

Because of the operator identity in Eq. (16), one of the observed values of
these four operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$, $y_{2}y_{4}x_{6}$
and $x_{2}x_{4}x_{6}$ is always different from the other three. As a result,
any local realistic theory predicts the observed values of the operator
$\mathcal{O}$

$\left|\langle\mathcal{O}\rangle_{LRT}\right|\leq 2,$ (23)

which is a Bell-Mermin inequality mermin1990extreme  for $|\Psi\rangle$. The
inequality (23) also shows a contradiction with the prediction (22) of QM, and
can thus be verified in a less-than-ideal experimental situation. Suppose that
we have a noisy singlet state
$F\left|\psi^{-}\right\rangle$$\left\langle\psi^{-}\right|+\frac{1-F}{4}I$,
where $I$ is the unit operator in the two-qubit state space. Then the
violation of the inequality (23) can be observed as long as the fidelity $F$
of the noisy singlet state is greater than $\sqrt[3]{1/2}\approx 0.7937$.

Similarly, in an experimental demonstration of contextuality, one can define
the operator $\mathcal{O^{\prime}}=x_{2}y_{4}y_{6}\cdot x_{2}\cdot y_{4}\cdot
y_{6}+y_{2}x_{4}y_{6}\cdot y_{2}\cdot x_{4}\cdot y_{6}+y_{2}y_{4}x_{6}\cdot
y_{2}\cdot y_{4}\cdot x_{6}+x_{2}x_{4}x_{6}\cdot x_{2}\cdot x_{4}\cdot x_{6}$,
for which any NCHV theory must predict in a less-than-ideal experimental
situation

$\left|\langle\mathcal{O^{\prime}}\rangle_{NCHV}\right|\leq 2.$ (24)

On the other hand, the maximal value of quantum mechanical prediction of
$\left|\langle\mathcal{O^{\prime}}\rangle_{QM}\right|$ is 4. This difference
allows a statistical test of NCHV theories in a less-than-ideal experimental
situation.

We have discussed how our scheme demonstrates simultaneous refutation of local
realism and noncontextuality by one and the same experimental system in a
less-than-ideal experimental situation. Here we further emphasize that these
four operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$, $y_{2}y_{4}x_{6}$ and
$x_{2}x_{4}x_{6}$ are measured with one and the same apparatus, which is
crucial for demonstrating the negation of local realism chen2003all . In our
argument of the negation of local realism, each of the four operators
$x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$, $y_{2}y_{4}x_{6}$ and $x_{2}x_{4}x_{6}$
appears twice in Eqs. (2)-(6). For example, the operator $x_{2}y_{4}y_{6}$
appears in Eqs. (2) and (6), while Eqs. (2) and (6) involve two different
experimental contexts for measuring $x_{2}y_{4}y_{6}$. With emphasis, LHV does
not contain the assumption of noncontextuality. In other words, LHV only
emphasizes that all operators have predetermined values, but it does not deny
that measuring an operator may introduce some disturbance in the system,
thereby modifying the values of other operators. As a result, one who believes
in local realistic theory can argue that the contradiction in Eqs. (7)-(11)
arises from contextuality, not EPR’s local elements of reality. To avoid this
problem, one either adds the additional assumption of noncontextuality, or
measures the four operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$,
$y_{2}y_{4}x_{6}$ and $x_{2}x_{4}x_{6}$ with one and the same apparatus such
that they are measured within a single context. Note that our argument on the
negation of noncontextuality does not require such apparatuses, since
noncontextuality is already assumed in this argument.

Figure 2: Devices for measuring local operators in Eqs. (2)-(6). (a) A GHZ-
state analyzer realized with linear optics. Qubits $2$, $4$ and $6$ in Debbie
are incident in modes $A$, $B$ and $C$, respectively. According to different
threefold coincidences recorded by detectors, one can judge whether the three
photonic qubits collapse to the state $\left|\Phi_{0}^{+}\right\rangle$ or the
state $\left|\Phi_{0}^{-}\right\rangle$, and thus simultaneously measure the
values of the four operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$,
$y_{2}y_{4}x_{6}$ and $x_{2}x_{4}x_{6}$; (b) Device for measuring $x_{1}$,
$x_{3}$ and $x_{5}$; (c) Device for measuring $y_{1}$, $y_{3}$ and $y_{5}$.
PBS is polarizing beam splitter, HWP is half-wave plate, QWP is quarter-wave
plate, and $D_{H}$ and $D_{V}$ are trigger detectors for horizontal ($H$) and
vertical ($V$) polarizations, respectively.

As a nonlocality argument, one of course has to avoid the introduction of
additional assumption (i.e., noncontextuality). Then one needs to measure the
four operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$, $y_{2}y_{4}x_{6}$ and
$x_{2}x_{4}x_{6}$ with the same apparatus. A possible measurement strategy of
our nonlocality argument is to perform the GHZ state measurement
pan1998greenberger  on photonic qubits $2$, $4$ and $6$ of Debbie [Fig. 2(a)].
With emphasis, photonic GHZ-state discrimination realized with linear optics
can only identify two out of the eight maximally entangled GHZ states, namely,
$\Phi_{0}^{\pm}=\frac{1}{\sqrt{2}}(|0\rangle|0\rangle|0\rangle\pm|1\rangle|1\rangle|1\rangle)$,
where $|0\rangle=|H\rangle$ for horizontal polarization and
$|1\rangle=|V\rangle$ for vertical polarization. Fortunately, this does not
affect our argument, because one only needs to postselect the corresponding
experimental data when
$\Phi_{0}^{\pm}=\frac{1}{\sqrt{2}}(|0\rangle|0\rangle|0\rangle\pm|1\rangle|1\rangle|1\rangle)$
is measured. To verify that this post-selection is desirable and it does not
affect our conclusions, one can expand $\left|\Psi\right\rangle$ into the form
of the GHZ-entanglement swapping lu

$\displaystyle\left|\Psi\right\rangle$ $\displaystyle=$ (25)
$\displaystyle\frac{1}{\sqrt{8}}\left(\Phi_{0(2,4,6)}^{-}\otimes\Phi_{0(1,3,5)}^{+}-\Phi_{0(2,4,6)}^{+}\otimes\Phi_{0(1,3,5)}^{-}\right.$
$\displaystyle+\Phi_{1(2,4,6)}^{+}\otimes\Phi_{1(1,3,5)}^{-}-\Phi_{1(2,4,6)}^{-}\otimes\Phi_{1(1,3,5)}^{+}$
$\displaystyle+\Phi_{2(2,4,6)}^{+}\otimes\Phi_{2(1,3,5)}^{-}-\Phi_{2(2,4,6)}^{-}\otimes\Phi_{2(1,3,5)}^{+}$
$\displaystyle\left.+\Phi_{3(2,4,6)}^{+}\otimes\Phi_{3(1,3,5)}^{-}-\Phi_{3(2,4,6)}^{-}\otimes\Phi_{3(1,3,5)}^{+}\right),$

where
$\Phi_{1}^{\pm}=\frac{1}{\sqrt{2}}(|1\rangle|0\rangle|0\rangle\pm|0\rangle|1\rangle|1\rangle)$,
$\Phi_{2}^{\pm}=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle|0\rangle\pm|1\rangle|0\rangle|1\rangle)$,
$\Phi_{3}^{\pm}=\frac{1}{\sqrt{2}}(|0\rangle|0\rangle|1\rangle\pm|1\rangle|1\rangle|0\rangle)$,
and subscripts $(2,4,6)$ and $(1,3,5)$ represent qubits $2,4,6$ and qubits
$1,3,5$, respectively. It can be checked that each component of the expansion
in Eq. (25) satisfies Eqs. (2)-(6). Thus, whenever a GHZ state, e.g.,
$\Phi_{0(2,4,6)}^{-}$, is successfully detected by Debbie, she simultaneously
gets the measurement values of the four operators, whose eigenstate is the
detected GHZ state. This property enables one to demonstrate our scheme using
the GHZ post-selection method and linear optics fu , which is well within the
reach of current quantum optical technology. The fact that the post-selection
works here for our nonlocality argument is not surprising as the successful
GHZ detection by Debbie leaves qubits $1,3,5$ in an “event-ready” GHZ
entanglement lu , which can of course be used in the GHZ argument.

Figure 3: Devices for measuring local operators in Eqs. (12)-(15). Photonic qubits $2$, $4$ and $6$ in Debbie are projected into definite eigenstates and then incident in modes $A$, $B$ and $C$, respectively. Here the GHZ-state analyzer is the same as the experimental device shown in Fig. 2(a). Table 1: The results of jointly measuring $x_{2}x_{4}x_{6}$, $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$ and $y_{2}y_{4}x_{6}$ upon different GHZ states. | $x_{2}x_{4}x_{6}$ | $x_{2}y_{4}y_{6}$ | $y_{2}x_{4}y_{6}$ | $y_{2}y_{4}x_{6}$
---|---|---|---|---
$|\Phi^{+}_{0}\rangle$ | +1 | -1 | -1 | -1
$|\Phi^{-}_{0}\rangle$ | -1 | +1 | +1 | +1
$|\Phi^{+}_{1}\rangle$ | +1 | -1 | +1 | +1
$|\Phi^{-}_{1}\rangle$ | -1 | +1 | -1 | -1
$|\Phi^{+}_{2}\rangle$ | +1 | +1 | -1 | +1
$|\Phi^{-}_{2}\rangle$ | -1 | -1 | +1 | -1
$|\Phi^{+}_{3}\rangle$ | +1 | +1 | +1 | -1
$|\Phi^{-}_{3}\rangle$ | -1 | -1 | -1 | +1

Now let us consider the measurement strategy of our argument on the negation
of noncontextuality. Note that there is no way to measure all $10$ operators
in Eqs. (12)-(16) within a single context, as these operators are not all
commutative. However, this does not pose a problem because noncontextuality is
already assumed in the argument as stated above. To verify Eq. (16), Debbie
can measure therein the four operators $x_{2}y_{4}y_{6}$, $y_{2}x_{4}y_{6}$,
$y_{2}y_{4}x_{6}$ and $x_{2}x_{4}x_{6}$ via the GHZ-state measurement as
above. However, for Eqs. (12)-(15) she has to adopt a different measurement
strategy, namely, to test them in different measurement contexts. One possible
way to test Eq. (12) is to first project $x_{2}$, $y_{4}$ and $y_{6}$
separately into their definite eigenstates, and then feed qubits $2$, $4$ and
$6$ into the GHZ-state analyzer to measure $x_{2}y_{4}y_{6}$. Figure 3(a)
shows as an example the case where, before entering the GHZ-state analyzer,
qubits $2$, $4$ and $6$ collapse to the state $|+RR\rangle$, with $|+\rangle$
($|R\rangle$) being the eigenstate in the $X$ ($Y$) basis and subscripts
omitted for simplicity. This means that the measurement results of $x_{2}$,
$y_{4}$ and $y_{6}$ are all $+1$. Meanwhile, $|+RR\rangle$ can be expanded in
terms of the GHZ states

$|+RR\rangle=\frac{1}{2}(|\Phi^{-}_{0}\rangle+|\Phi^{-}_{1}\rangle+i|\Phi^{+}_{2}\rangle+i|\Phi^{+}_{3}\rangle).$
(26)

Combining this with Table 1 shows that the joint measurement of
$x_{2}y_{4}y_{6}$ for $|+RR\rangle$ must yield $+1$. Therefore, Eq. (12) is
verified as long as Debbie observes coincidence in the GHZ-state analyzer.
With emphasis, this verification method is not $100\%$ successful because the
probability of qubits $2$, $4$ and $6$ all entering the GHZ-state analyzer is
only $1/8$, and the GHZ-state analyzer can only identify two out of the eight
maximally entangled GHZ states. Again, this does not affect our argument.
Similarly, Fig. 3(b-d) are used to verify Eqs. (13)-(15), respectively. Before
entering the GHZ-state analyzer in Fig. 3(b-d), qubits $2$, $4$ and $6$
collapse into $|R+R\rangle$, $|RR+\rangle$, and $|+++\rangle$, respectively.
Here

$\begin{array}[]{l}|R+R\rangle=\frac{1}{2}(|\Phi^{-}_{0}\rangle+i|\Phi^{+}_{1}\rangle+|\Phi^{-}_{2}\rangle+i|\Phi^{+}_{3}\rangle)\\\
|RR+\rangle=\frac{1}{2}(|\Phi^{-}_{0}\rangle+i|\Phi^{+}_{1}\rangle+i|\Phi^{+}_{2}\rangle+|\Phi^{-}_{3}\rangle)\\\
|+++\rangle=\frac{1}{2}(|\Phi^{+}_{0}\rangle+|\Phi^{+}_{1}\rangle+|\Phi^{+}_{2}\rangle+|\Phi^{+}_{3}\rangle),\end{array}$
(27)

which, together with Table 1, confirm Eqs. (13)-(15). Thus, following the
above measurement strategy Eqs. (12)-(16) can be tested quantum mechanically,
but are not consistent with NCHV.

In summary, we have shown that local realism and noncontextuality can be
simultaneously negated by one and the same systems, namely, three singlet
pairs. In particular, by applying the GHZ-state measurement and the
aforementioned post-selection method, our scheme can be fully implemented with
linear optics. Not only does the proposed scheme represent the strongest
conflict between quantum mechanics and hidden variable theories to date, it
also sheds new light on the interesting relation between nonlocality and
contextuality, and on more effective use of them in certain quantum
information tasks.

We gratefully acknowledge the supports from the Natural Science Foundation of
Jiangsu Province (No. BK20211145), the Fundamental Research Funds for the
Central Universities (No. 020414380182), the Key Research and Development
Program of Nanjing Jiangbei New Aera (No. ZDYD20210101), and the Program for
Innovative Talents and Entrepreneurs in Jiangsu (No. JSSCRC2021484).

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