Violation of a generalized non-contextuality inequality with single photons
Xiang Zhan
Jian Li
Hao Qin
Zhihao Bian
Yongsheng Zhang\footnote{yshzhang@ustc.edu.cn
Peng Xue\footnote{gnep.eux@gmail.com
Is a violation of Kochen-Specker (KS) non-contextuality inequality a tight bound to rule out the non-contextual hidden variable theory? A test designed by Liang, Spekkens and Wiseman (LSW) provides a more stringent bound and a violation of a generalized non-contextuality inequality such as LSW inequality rules out more generalized non-contextual models including the traditional KS non-contextual models. It is proven that three unsharp binary qubit measurements are enough to violate LSW inequality in a state-dependent manner. We show such measurements can be implemented through joint positive operator-valued measures (POVMs) of noisy spin-$\frac{1{2$ observables and report an experimental realization of the joint POVMs on photonic qubits. We demonstrate an experimental violation of LSW inequality with single photons and provide the experimental evidence of a more stringent bound to rule out the non-contextual models without unwarranted idealizations such as noiseless measurements.
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\begin{document}

\title{Violation of a generalized non-contextuality inequality with single photons}
\author{Xiang Zhan}
\affiliation{Department of Physics, Southeast University, Nanjing
211189, China}
\author{Jian Li}
\affiliation{Department of Physics, Southeast University, Nanjing
211189, China}
\author{Hao Qin}
\affiliation{Department of Physics, Southeast University, Nanjing
211189, China}
\author{Zhihao Bian}
\affiliation{Department of Physics, Southeast University, Nanjing
211189, China}
\author{Yongsheng Zhang\footnote{yshzhang@ustc.edu.cn}}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, China}
\affiliation{Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China}
\author{Peng Xue\footnote{gnep.eux@gmail.com}}
\affiliation{Department of Physics, Southeast University, Nanjing
211189, China}

\begin{abstract}
Is a violation of Kochen-Specker (KS) non-contextuality inequality a tight bound to rule out the non-contextual hidden variable theory? A test designed by Liang, Spekkens and Wiseman (LSW) provides a more stringent bound and a violation of a generalized non-contextuality inequality such as LSW inequality rules out more generalized non-contextual models including the traditional KS non-contextual models. It is proven that three unsharp binary qubit measurements are enough to violate LSW inequality in a state-dependent manner. We show such measurements can be implemented through joint positive operator-valued measures (POVMs) of noisy spin-$\frac{1}{2}$ observables and report an experimental realization of the joint POVMs on photonic qubits. We demonstrate an experimental violation of LSW inequality with single photons and provide the experimental evidence of a more stringent bound to rule out the non-contextual models without unwarranted idealizations such as noiseless measurements.
\end{abstract}

\pacs{42.50.Xa, 03.65.Ta, 03.65.Ud}

\maketitle

{\it Introduction:-}There are a number of proposals for tests which pit quantum mechanics against alternative views of reality, for example the theorems of Bell~\cite{Bell} and of Kochen and Specker (KS)~\cite{KS}. Corresponding experimental tests~\cite{exp1,exp2,K+09,ARBC09,B+09,MRCL10} have been performed and support the necessity of quantum mechanics. Bell's theorem refers to a situation with two particles and the statistical predictions of quantum mechanics. The KS theorem states non-contextual theories are incompatible with quantum mechanics. A qutrit and five projectors are required for a proof of the traditional KS-contextuality in state-dependent manner~\cite{L+11,KCBS08}, while a qutrit and thirteen projectors for that in state-independent manner~\cite{P91,B96,BBCP09,YO12,CAB+12,ZWD+12,ZUZ+13}. Generalizations of KS non-contextuality for a two-level system have been proposed in~\cite{C03,B03,CFMR04,LSW,YO13,KG14}. Recently, Liang, Spekkens and Wiseman (LSW)~\cite{LSW} derive a generalized non-contextuality inequality by assuming certain linearity for unsharp measurements and Yu and Oh prove LSW inequality can be established only based on the assumption of non-contextuality~\cite{YO13}. Unsharp measurements introduce some kind of measurement contextuality, which however is not enough to explain all the quantum predictions on a qubit if LSW inequality is violated. A theoretical proof of a state-dependent violation of LSW inequality with three unsharp measurements was found in~\cite{KG14}.

A violation of LSW inequality is interesting because the benchmark for non-classicality set by generalized
non-contextuality is more stringent than that set by the traditional KS non-contextuality. Moreover, a test of LSW contextuality does not require unwarranted idealizations such as noiseless preparation and measurement. Thus a violation of LSW inequality rules out the possibility and certifies genuine nonclassicality that can not be attributed to hidden variables associated with either the preparation or noise in measurement.

In this paper we experimentally demonstrate the unsharp binary qubit measurements can be realized via joint positive operator-valued measures (POVMs) of noisy spin-$\frac{1}{2}$ observables. Our work provides the first experimental evidence of a violation of LSW inequality~\cite{LSW}, in a state-dependent manner. Furthermore, our experimental test of noncontextuality is robust to noise and experimental imprecisions.

{\it Idea:-}A generalized non-contextual model referred to LSW model can be realized using noisy spin-$\frac{1}{2}$ observables~\cite{LSW}. If one does not assume outcome-determinism for three POVMs $M_k=\{E_+^k,E_-^k\}$, $k=1,2,3$ of noisy spin-$\frac{1}{2}$ observables in the forms
\begin{equation}
E_{\pm}^k\equiv \frac{1}{2}\one\pm \frac{\eta}{2} \vec{\sigma}\cdot \hat{n}_k=\frac{1-\eta}{2}\one+\eta \Pi_{\pm}^k
\end{equation} and models them stochastically but non-contextually, then this generalized non-contextual model obeys a bound of $1-\eta/3$, where $\eta\in \left[0,1\right]$ is the sharpness associated with each observable. The observables $E_{\pm}^k$ are noisy versions of the projectors and POVM $M_k$ is therefore an unsharp version of the projective measurement $P_k=\{\Pi_+^k,\Pi_-^k\}$ with the projectors $\Pi_\pm^k=(\one\pm \vec{\sigma}\cdot \hat{n}_k)/2$ (here $\one$ is $2 \times 2$ identity matrix, $\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ with $\sigma_i$ the Pauli matrix and $\hat{n}_k$ the axis for measurement).

Testing LSW inequality for a quantum mechanical violation requires a special kind of joint measurability, denoted by joint measurable contexts $\{\{M_1,M_2\},\{M_2,M_3\},\{M_1,M_3\}\}$, i.e. pairwise but no triplewise joint measurability. In our experiment, we choose a special case of trine spin axes
\begin{equation}
\hat{n}_1=(0,0,1),\hat{n}_2=(\frac{\sqrt{3}}{2},0,-\frac{1}{2}),\hat{n}_3=(-\frac{\sqrt{3}}{2},0,-\frac{1}{2})
\end{equation} in the $z$-$x$ plane and note that this condition for special joint measurability is necessary and sufficient~\cite{LSW}.

The joint POVM $G_{ij}\equiv\{G_{++}^{ij},G_{+-}^{ij},G_{-+}^{ij},G_{--}^{ij}\}$ for the context $\{M_i, M_j\}$ constructed in~\cite{KG14} satisfy the marginal condition $\sum_{X_{j(i)}} G_{X_i,X_j}^{ij}=E_{X_{i(j)}}^{i(j)}$, and consist of valid qubit effects $0\leq G_{X_i,X_j}^{ij}\leq \one$ and $\sum_{X_i,X_j} G_{X_i,X_j}^{ij}=\one$, where $X_i,X_j\in\{+1,-1\}$ are the outcomes of the POVMs $M_i$ and $M_j$ respectively. Thus $G^{ij}_{X_i,X_j}$ is the joint measurement effect corresponding to the effects $E^i_{X_i}$ and $E^j_{X_j}$. The joint POVM has taken the general form
\begin{align}
&G_{++}^{ij}=\frac{1}{2}\left\{\frac{\alpha_{ij}}{2}\one+\vec{\sigma}\cdot\frac{1}{2}\left[\eta\left(\hat{n}_i+\hat{n}_j\right)-\vec{a}_{ij}\right]\right\},\nonumber\\
&G_{+-}^{ij}=\frac{1}{2}\left\{\left(1-\frac{\alpha_{ij}}{2}\right)\one+\vec{\sigma}\cdot\frac{1}{2}\left[\eta\left(\hat{n}_i-\hat{n}_j\right)+\vec{a}_{ij}\right]\right\},\nonumber\\
&G_{-+}^{ij}=\frac{1}{2}\left\{\left(1-\frac{\alpha_{ij}}{2}\right)\one+\vec{\sigma}\cdot\frac{1}{2}\left[\eta\left(-\hat{n}_i+\hat{n}_j\right)+\vec{a}_{ij}\right]\right\},\nonumber\\
&G_{--}^{ij}=\frac{1}{2}\left\{\frac{\alpha_{ij}}{2}\one+\vec{\sigma}\cdot\frac{1}{2}\left[\eta\left(-\hat{n}_i-\hat{n}_j\right)-\vec{a}_{ij}\right]\right\},
\end{align}
where $\alpha_{ij}\in \mathbb{R}$ and $\vec{a}_{ij}\in \mathbb{R}^3$. LSW inequality concerns the average probability
\begin{equation}
R_3\equiv\frac{1}{3}\sum_{(ij)\in\left\{(12),(23),(13)\right\}}\text{Pr}(X_i\neq X_j|G_{ij})\leq1-\frac{\eta}{3},
\end{equation}
where $\text{Pr}(X_i\neq X_j|G_{ij})$ denotes the probability of obtaining anticorrelated outcomes in a joint POVM $G_{ij}$. In quantum theory, for pairwise joint POVM $G_{ij}$ the average probability $R_3$ takes the form
\begin{equation}
R^Q_3=\frac{1}{3}\sum_{(ij)\in\{(12),(23),(13)\}}\text{Tr}\left[\left(G_{+-}^{ij}+G_{-+}^{ij}\right)\ket{\phi_0}\bra{\phi_0}\right],
\end{equation}
where $\ket{\phi_0}$ is the qubit state being measured.
A necessary condition for state-dependent violation of LSW inequality is
Tr$\left[\sum_{ij}(\alpha_{ij}\one-\vec{\sigma}\cdot\vec{a}_{ij})\ket{\phi_0}\bra{\phi_0}\right]<2\eta$.

LSW inequality in Eq.~(4) is different from the traditional KS inequality with the bound $2/3$ and has a more stringent bound instead. A violation of LSW inequality rules out the generalized non-contextual models of the measurements which were previously conjectured to exist. Remarkably, this class of models includes the traditional KS non-contextual models. %Hence a violation of LSW inequality rules out the traditional KS non-contextuality.

The optimal violation of LSW inequality for measurements along trine spin axes in the $z$-$x$ plane occurs for the specific state $\ket{\phi_0}=(\ket{0}+i\ket{1})/\sqrt{2}$. The noise parameter $\eta$ satisfies $2/3<\eta\leq(\sqrt{3}-1)$~\cite{LSW}. The parameter of the joint POVM $G_{ij}$ in Eq.~(3) is $\alpha_{ij}=1+\eta^2\hat{n}_i\cdot\hat{n}_j$ and the vector $\vec{a}_{ij}$ satisfies $\vec{a}_{ij}=(0,\sqrt{1+\eta^4(\hat{n}_i\cdot\hat{n}_j)^2-2\eta^2},0)$. For measurements along trine spin axes, the optimal violation of LSW inequality in a state-dependent manner~\cite{KG14} is achieved with $\eta\rightarrow 2/3$, so that $\alpha_{ij}\rightarrow 7/9$ and $|\vec{a}_{ij}|\rightarrow \sqrt{13}/9$ for any $(ij)\in \{(12),(23),(13)\}$, which satisfies the necessary condition of the violation. The quantum average probability of anticorrelation is $R^Q_3\rightarrow 0.8114$ and exceeds LSW non-contextual bound $7/9\approx0.7778$.

In our experiment, we choose $\eta=0.67$ which satisfies $\eta>2/3$ and $\eta\rightarrow 2/3$.

\begin{figure*}
   \includegraphics[width=.7\textwidth]{newsetup}
   \caption{Experimental setup.
Single photons are created via type-I SPDC in
a BBO crystal. One photon in the pair is detected to herald the other photon, which is injected into the optical network. To realize the joint POVM $G_{ij}$, the single-qubit rotations $U_{\epsilon\varepsilon}$ are realized by sandwich-type QWP-HWP-QWP sets with different angles placed in different optical modes. The first two projectors are realized by PPBSs with different transmission probabilities for vertical polarization. The last projector is realized by a $50:50$ PBS. Detecting heralded single photons means in practice registering coincidences between single photon detectors: D0 and each of D1, D2, D3 and D4.}
 \label{setup}
\end{figure*}

\begin{table}[htbp]%[tbp]
%\tiny
%\scriptsize
\caption{The setting angles of WPs and the transmission probabilities for vertical polarization $T^{1(2)}_V$ of the two PPBSs for realization of joint POVMs.}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
$G_{ij}$ & Q1/Q2 & H1 & Q3/Q4 & H2 & Q5/Q6 & H3 & $T^1_V$ & $T^2_V$ \\
\hline\hline
$G_{12}$ & $-7.3^\circ$ & $32.9^\circ$ & $1.4^\circ$ & $-29.5^\circ$  & $22.4^\circ$ & $69.2^\circ$ & $0.39$ & $0.29$\\ \hline
$G_{23}$ & $21.7^\circ$ & $60.2^\circ$ & $-4.9^\circ$ & $-36.5^\circ$ & $21.7^\circ$ & $60.2^\circ$ & $0.39$  & $0.29$ \\ \hline
$G_{13}$ & $7.3^\circ$ & $47.5^\circ$ & $-1.4^\circ$ & $-32.2^\circ$ & $-112.4^\circ$ & $110.8^\circ$ & $0.39$  & $0.29$\\ \hline
\end{tabular}
\end{table}

{\it Implementation of joint POVMs:-}The joint POVM of noisy spin-$\frac{1}{2}$ observables is the key point for testing LSW inequality. For each joint POVM $G_{ij}$ has four elements and can be rewritten as $G^{ij}_{\epsilon\varepsilon}=\lambda_{\epsilon\varepsilon}\ket{\xi_{\epsilon\varepsilon}}\bra{\xi_{\epsilon\varepsilon}}$  with $\epsilon, \varepsilon\in\{+,-\}$. We propose a scheme for implementing the joint POVMs via three single-qubit rotations and three two-outcome projectors. Each element of the joint POVM is proportional to a projector onto the state of the qubit of interest at this stage. After each projector following by a single-qubit rotation, one of the clicks of the projector implies one of the elements has been implemented with the probability of the joint POVM on the system of interest $\text{Tr}(G^{ij}_{\epsilon\varepsilon}\ket{\phi_0}\bra{\phi_0})$ and the other outcome goes for the next rotation and projector.

The three single-qubit rotations are designed as
\begin{equation}
U_{\epsilon\varepsilon}=\ket{0}\bra{\phi'^\perp_{\epsilon\varepsilon}}+\ket{1}\bra{\phi'_{\epsilon\varepsilon}},
\epsilon\varepsilon\in\{+-,-+,++\},
\end{equation}
where $\ket{\phi'_{\epsilon\varepsilon}}$ is the normalized state of the qubit of interest at this stage if the qubit is initially prepared in $\ket{\xi^\perp_{\epsilon\varepsilon}}$. This guarantees after this rotation the join POVM element is proportional to $\ket{\xi_{\epsilon\varepsilon}}\bra{\xi_{\epsilon\varepsilon}}$. The projectors take the form $\{P^0_{\epsilon\varepsilon},P^1_{\epsilon\varepsilon}\}$, where
\begin{equation}
P^0_{\epsilon\varepsilon}=\ket{0}\bra{0}+(1-\chi_{\epsilon\varepsilon})\ket{1}\bra{1},
P^1_{\epsilon\varepsilon}=\chi_{\epsilon\varepsilon}\ket{1}\bra{1}.
\end{equation}
The parameter $\chi_{\epsilon\varepsilon}$ is chosen such that after the projector the probability of the click of $P^1_{\epsilon\varepsilon}$ is $\lambda_{\epsilon\varepsilon}\text{Tr}(\ket{\xi_{\epsilon\varepsilon}}\bra{\xi_{\epsilon\varepsilon}}\phi_0\rangle\bra{\phi_0})$ with $\ket{\phi_0}$ the state being measured. Each projector $\{P^0_{\epsilon\varepsilon},P^1_{\epsilon\varepsilon}\}$ following by a single-qubit rotation $U_{\epsilon\varepsilon}$ can be used to design an element of the joint POVM $G^{ij}_{+-}$, $G^{ij}_{-+}$, $G^{ij}_{++}$ respectively on the system of interest. Due to $\sum_{\epsilon\varepsilon}G^{ij}_{\epsilon\varepsilon}=\one$, the click of $P^0_{++}$ corresponds to the implementation of $G^{ij}_{--}$. Given the parameters of the joint POVM $\eta$, $\vec{n}_{i(j)}$, the single-qubit rotations and projectors can be obtained via the above procedure.

\begin{table*}[htbp]
\scriptsize
\caption{Experimental results. The ``element" columns contain the elements of joint POVM corresponding to heralded single-click events. The ``probability" columns contain the measured probabilities of joint POVM. The ``condition" column contains assigned
measurement values. The ``value" column contains the measured probabilities of the anticorrelations. Rows 1-3 correspond to the results of joint POVMs on the state $\ket{\phi_0}$. Row 4 presents the measured $R^Q_3$. Error bars indicate the statistical uncertainty.}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hline
\multicolumn{2}{c}{D4} & \multicolumn{2}{c}{D3} & \multicolumn{2}{c}{D2} & \multicolumn{2}{c}{D1} & \multicolumn{2}{c}{Calculated contribution}\\
\hline
element & probability & element & probability & element & probability & element & probability & condition & value \\
\hline\hline
$G^{12}_{+-}$ & 0.4042(28) & $G^{12}_{-+}$ & 0.4065(28) & $G^{12}_{++}$ & 0.0948(13) & $G^{12}_{--}$ & 0.0944(13) & $\text{Pr}(X_1\neq X_2|G_{12})$ & 0.8108(56)\\
\hline
$G^{23}_{+-}$ & 0.4048(28) & $G^{23}_{-+}$ & 0.4067(28) & $G^{23}_{++}$ & 0.0953(14) & $G^{23}_{--}$ & 0.0931(13) & $\text{Pr}(X_2\neq X_3|G_{23})$ & 0.8116(56)\\
\hline
$G^{13}_{+-}$ & 0.4073(28) & $G^{13}_{-+}$ & 0.4078(28) & $G^{13}_{++}$ & 0.0931(13) & $G^{13}_{--}$ & 0.0919(13) & $\text{Pr}(X_1\neq X_3|G_{13})$ & 0.8150(56)\\
\hline
- & - & - & - & - & - & - & - & $R_3^{Q}$ & 0.8125(56)\\ \hline
\end{tabular}
\end{table*}

{\it Experiment:-}We perform the test of LSW inequality with single photons. The states of the system of interest $\ket{0}$ and $\ket{1}$ are encoded by the polarizations of single photons $\ket{H}$ and $\ket{V}$. We generate contextual quantum correlations by performing joint POVM of the polarizing qubits of single photons.

The experimental setup shown in Fig.~\ref{setup} involves preparing the specific state (preparation stage) and then joint POVM on the system of interest (measurement stage). In the preparation stage, polarization-degenerate photon pairs of $801.6$nm wavelength are produced in a type-I spontaneous parametric down-conversion (SPDC) process where a $0.5$mm-thick nonlinear-$\beta$-barium-borate (BBO) crystal is pumped by a CW diode laser with $80$mW of power. The pump is filtered out with the help of an interference filter which restricts the photon bandwidth to $3$nm. With the detection of trigger photon the signal photon is heralded in the measurement setup. In practice, it means we record pairwise coincidence between the heralding detector and any one of the four detectors in the measurement apparatus (heralded single clicks). Experimentally this trigger-herald pair is registered by a coincidence count at two single-photon avalanche photodiodes (APDs) with $7$ns time window. Total coincidence counts are about $10^5$ over a collection time of $60$s, and the probability of randomly creating more than one simultaneous photon pair is thus less than a negligible $10^{-4}$. The heralded single photons are prepared in $\ket{\phi_0}=(\ket{H}+i\ket{V})/\sqrt{2}$ after passing through a polarizing beam splitter (PBS) and the following half-wave plate (HWP) and quarter-wave plate (QWP).

In the measurement stage, the single-qubit rotations $U_{\epsilon\varepsilon}$ can be realized by a combination of a QWP-HWP-QWP sequence, so-called a sandwich-type wave plate (WP) set, with certain setting angles depending on the parameters of the joint POVM. The setting angles of the WPs used to realize the corresponding elements of joint POVMs are shown in Table I. To implement the two-outcome measurements, partially projecting polarizing elements are added to the set-up to allow us to produce the required projectors with the appropriate weights. We employ partially polarizing beam splitters (PPBSs) with specific transmission probabilities for vertical polarization $T_{V}$ (shown in Table I) and same transmission probability for horizontal polarization $T_{H}=1$. This allows us to implement the measurement $\bra{V}$ on the reflected port of the PPBSs.

After passing through the first sandwich-type QWP-HWP-QWP set and the following PPBS, the photons are detected by APDs on the transmitted port, in coincidence with the trigger photons. The transmitted photons go into the next QWP-HWP-QWP set and PPBS (or PBS which can be regard as a special PPBS with equal transmission and reflection probabilities). The probability of the photons being measured is obtained by normalizing the coincidence counts on each mode with respect to the total counts. The probabilities of photons being measured on the reflected ports (clicks on the detectors D1, D2, and D3) correspond to those of the joint POVM elements $G^{ij}_{+-}$, $G^{ij}_{-+}$, and $G^{ij}_{++}$. Whereas the probability of photons being measured on the transmitted port of the last PBS (click on the detector D4) corresponds to that of the element $G^{ij}_{--}$ on the polarization state of single photons.

Photon loss opens up a detection efficiency loophole in our experiment. Thus fair-sampling assumption is taken here which assumes that the event selected out by the photonic coincidence is an unbiased representation of the whole sample.

In Table II, we present the measured probabilities and the outcomes of the joint POVM with noise parameter $\eta=0.67$ on the specific state $\ket{\phi_0}$. The result of measured average probability of anticorrelations $R^Q_3=0.8125\pm0.0056$ violates the boundary set by the non-contextual hidden variable theory $1-\eta/3=0.7767$ by $6.4$ standard deviations and is in a great agreement with quantum prediction $0.8075$.

To prove the validity of the constructed joint POVM $G_{ij}$, we test its joint measurability. For different qubit states, we analyze the experimental results of the joint POVM and test whether the marginal condition $\sum_{X_{j(i)}} G_{X_i,X_j}^{ij}=E_{X_{i(j)}}^{i(j)}$ is satisfied. Without loss of generality, we choose the four states $|H\rangle$, $|V\rangle$, $|R\rangle=(|H\rangle+i|V\rangle)/\sqrt{2}$, and $|D\rangle=(|H\rangle+|V\rangle)/\sqrt{2}$ as states being measured. The results are shown in Table III.

\begin{table}[htbp]
\scriptsize
\caption{Experimental results of $\sum_{X_j}G^{ij}_{X_i,X_j}$ on the four different single-qubit states compared to the theoretical predictions of the probabilities of $E_{X_i}^i$ on these states.}
\begin{tabular}{c|c|c|c|c}
\hline
\quad & $|H\rangle$ & $|V\rangle$ & $|R\rangle$ & $|D\rangle$\\
\hline\hline
$G_{++}^{12}+G_{+-}^{12}$ & 0.8357(53) & 0.1607(24) & 0.4990(41) & 0.4958(44)\\ \hline
$G_{++}^{13}+G_{+-}^{13}$ & 0.8353(53) & 0.1699(25) & 0.5003(41) & 0.5029(38)\\ \hline
$E_+^1\text{(Theory)}$ & 0.8350 & 0.1650 & 0.5000 & 0.5000\\
\hline\hline
$G_{-+}^{12}+G_{--}^{12}$ & 0.1642(24) & 0.8393(53) & 0.5009(41) & 0.5042(38)\\ \hline
$G_{-+}^{13}+G_{--}^{13}$ & 0.1647(24) & 0.8301(53) & 0.4997(41) & 0.4971(44)\\ \hline
$E_-^1\text{(Theory)}$ & 0.1650 & 0.8350 & 0.5000 & 0.5000\\
\hline\hline
$G_{++}^{12}+G_{-+}^{12}$ & 0.3155(32) & 0.6747(44) & 0.5013(41) & 0.7818(54)\\ \hline
$G_{++}^{23}+G_{+-}^{23}$ & 0.3179(31) & 0.6748(52) & 0.5001(41) & 0.7788(48)\\ \hline
$E_+^2\text{(Theory)}$ & 0.3325 & 0.6675 & 0.5000 & 0.7901\\
\hline\hline
$G_{+-}^{12}+G_{--}^{12}$ & 0.6845(45) & 0.3253(33) & 0.4987(41) & 0.2182(28)\\ \hline
$G_{-+}^{23}+G_{--}^{23}$ & 0.6821(52) & 0.3252(31) & 0.4999(41) & 0.2212(25)\\ \hline
$E_-^2\text{(Theory)}$ & 0.6675 & 0.3325 & 0.5000 & 0.2099\\
\hline\hline
$G_{++}^{13}+G_{-+}^{13}$ & 0.3454(34) & 0.6652(41) & 0.5008(41) & 0.2083(27)\\ \hline
$G_{++}^{23}+G_{-+}^{23}$ & 0.3448(32) & 0.6519(51) & 0.4020(41) & 0.2102(25)\\ \hline
$E_+^3\text{(Theory)}$ & 0.3325 & 0.6675 & 0.5000 & 0.2099\\
\hline\hline
$G_{+-}^{13}+G_{--}^{13}$ & 0.6546(44) & 0.3348(34) & 0.4992(41) & 0.7917(55)\\ \hline
$G_{+-}^{23}+G_{--}^{23}$ & 0.6551(51) & 0.3481(32) & 0.4980(41) & 0.7898(49)\\ \hline
$E_-^3\text{(Theory)}$ & 0.6675 & 0.3325 & 0.5000 & 0.7901\\
\hline
\end{tabular}
\end{table}

The measured probabilities Tr$(\sum_{X_j}G^{ij}_{X_i,X_j}\rho)$ are in agreement with the theoretical predictions of the probabilities Tr$(E_{X_i}^i\rho)$ of the POVM element $E_{X_i}^i$ on the state $\rho$, which proves the marginal condition is satisfied. Thus the constructed joint POVM $G_{ij}$ shows the joint measurability.

{\it Conclusion:-}Any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness, which provides a reason to test noncontextuality that applies to unsharp measurements, such as LSW noncontextuality. By using unsharp measurements instead of projective measurements, a violation of the LSW inequality rules out the possibility that the imperfect measurement apparatus could introduce anticorrelations that have nothing to do with hidden variables but would allow violation of the KS-noncontextual. Thus a violation of the LSW inequality certifies genuine nonclassicality that cannot be attributed to hidden variables associated with the noise from the imperfect measurement and preparation in experiments.

Our experimental results show a violation of LSW inequality with single photons for the first time, which represents a state-dependent experimental test of generalized quantum contextuality ruling out the possibility of a larger class of models than those ruled out by a violation of the traditional KS inequality.

The experimental confirmation of quantum contextuality in its simple and fundamental form sheds new light on the contradiction between quantum mechanics and non-contextual realistic models. Furthermore, we realize joint POVMs of noisy spin-$\frac{1}{2}$ observables on a single-qubit system for the first time which is the key point to implement the unsharp measurements and our results pave the way for further developments such as the real time estimation~\cite{CG09}, monitoring of the Rabi oscillations of a single qubit in a driving field~\cite{KU12} and understanding the relation between information gain and disturbance~\cite{W11}.

\acknowledgements We would like to thank R. Kunjwal for simulating discussion. This work has been supported by  National Natural Science Foundation of China, Grant Nos. 11174052, 11474049 and 61275122, 973 Program under 2011CB921203, and the CAST Innovation fund.
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\end{references}

\end{document}

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{KS S. Kochen, and E. P. Specker, J. Math. Mech. {17, 59-87 (1967).
1505.06825.extracted_bib
{exp1 G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. {81, 5039 (1998).
1505.06825.extracted_bib
{exp2 Y. F. Huang, C. F. Li, Y. S. Zhang, J. W. Pan, and G. C. Guo, Phys. Rev. Lett. {90, 250401 (2003).
1505.06825.extracted_bib
{K+09 G. Kirchmair et al., Nature (London) {460, 494 (2009).
1505.06825.extracted_bib
{ARBC09 E. Amselem, M. Radmark, M. Bourennane, A. Cabello, Phys. Rev. Lett. {103, 160405 (2009).
1505.06825.extracted_bib
{B+09 H. Bartosik, J. Klepp, C. Schmitzer, S. Sponar, A. Cabello, H. Rauch, and Y. Hasegawa, Phys. Rev. Lett. {103, 040403 (2009).
1505.06825.extracted_bib
{MRCL10 O. Moussa, C. A. Ryan, D. G. Cory, and R. Laflamme, Phys. Rev. Lett. {104, 160501 (2010).
1505.06825.extracted_bib
{L+11 R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wie\'{sniak and A. Zeilinger, Nature (London) {474, 490 (2011).
1505.06825.extracted_bib
{KCBS08 A. A. Klyachko, M. A. Can, S. Binicioglu, and A. S. Shumovsky, Phys. Rev. Lett. {101, 20403 (2008).
1505.06825.extracted_bib
{P91 A. Peres, J. Phys. A: Math. Gen. {24, L175 (1991).
1505.06825.extracted_bib
{B96 J. Bub, Found. Phys. {26, 787 (1996).
1505.06825.extracted_bib
{BBCP09 P. Badziag, I. Bengtsson, A. Cabello, and I. Pitowsky, Phys. Rev. Lett. {103, 050401 (2009).
1505.06825.extracted_bib
{YO12 S. Yu, and C. H. Oh, Phys. Rev. Lett. {108, 030402 (2012).
1505.06825.extracted_bib
{CAB+12 A. Cabello, E. Amselem, K. Blanchfield, M. Bourennane, and I. Bengtsson, Phys. Rev. A {85, 032108 (2012).
1505.06825.extracted_bib
{ZWD+12 C. Zu, Y. X. Wang, D. L. Deng, X. Y. Chang, K. Liu, P. Y. Hou, H. X. Yang, and L. M. Duan, Phys. Rev. Lett. {109, 150401 (2012).
1505.06825.extracted_bib
{ZUZ+13 X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D. L. Deng, C. Shen, L. M. Duan, and K. Kim, Phys. Rev. Lett. {110, 070401 (2013).
1505.06825.extracted_bib
{C03 A. Cabello, Phys. Rev. Lett. {90, 190401 (2003).
1505.06825.extracted_bib
{B03 P. Busch, Phys. Rev. Lett. {91, 120403 (2003).
1505.06825.extracted_bib
{CFMR04 C. M. Caves, C. A. Fuchs, K. K. Manne, and J. M. Renes, Found. Phys. {34, 193 (2004).
1505.06825.extracted_bib
{LSW L. S. Liang, R. W. Spekkens, and H. W. Wiseman, Phys. Rep. {506, 1 (2011).
1505.06825.extracted_bib
{YO13 S. Yu, and C. H. Oh, arXiv:1312.6470.
1505.06825.extracted_bib
{KG14 R. Kunjwal, and S. Ghosh, Phys. Rev. A {89, 042118 (2014).
1505.06825.extracted_bib
{CG09 B. Chase, and J. M. Geremia, Phys. Rev. A {79, 022314 (2009).
1505.06825.extracted_bib
{KU12 T. Konrad, and H. Uys, Phys. Rev. A {85, 012102 (2012).
1505.06825.extracted_bib
{W11 H. M. Wiseman, Nature (London) {470, 178-179 (2011). %
1505.06825.extracted_bib
{W13 %P. Kurzy\'{nski, and A. W\'{ojcik, Phys. Rev. Lett. {110, 200404 (2013). %
1505.06825.extracted_bib
{B15 %Z. H. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, Phys. Rev. Lett. {114, 203602 (2015).