1. INTRODUCTION

There are a number of proposals for tests that pit quantum mechanics against alternative views of reality, including the theorems of Bell [1] and of Kochen and Specker (KS) [2]. Corresponding experimental tests [3–8] have been performed and support the validity of quantum mechanics. Bell’s theorem refers to a situation with two or more spatially separate particles and states that local hidden variable theories are incompatible with the statistical predictions of quantum mechanics. The KS theorem has the advantage of applying to a single system, and states that noncontextual hidden variable theories are incompatible with quantum predictions, under the assumption that the measurements can be described by projectors. A qutrit (three-level system) and five projectors are required for a proof of the traditional KS contextuality in a state-dependent manner [9,10], while a qutrit and 13 projectors for such a proof in a state-independent manner [11–17].

To find simpler proofs of contextuality, applicable to a qubit (two-level system), generalizations of KS noncontextuality have been proposed [18–21]. These all utilize generalized measurements, described by positive operator-valued measures (POVMs). It has been argued, however [22], that these works make an unwarranted assumption of determinism for unsharp measurements.

More recently, Liang, Spekkens, and Wiseman (LSW) ([23], Section 7.3) followed a different approach to derive noncontextuality inequalities for a particular class of non-projective measurements. The relevant class is the unsharp projective measurements, in which each of the set of orthogonal projectors is mixed in some ratio with other projectors from the same set, in order to make the POVM. (Thus each element of the POVM commutes with each other element, just as for a projective measurement.) The LSW assumption is that the response function is likewise a mixture of the deterministic response functions assumed by KS for projective measurements, in the same ratios. Using this principle, LSW derived a generalized noncontextuality inequality involving three different unsharp projective measurements on a qubit. Subsequently, Kunjwal and Ghosh [24] found a triple of unsharp observables that, according to the predictions of quantum mechanics, would give a significant violation of the LSW inequality, in a state-dependent manner.

Here, we experimentally violate the LSW inequality for the first time, via three unsharp binary qubit measurements that are pairwise jointly measurable. We use a photon polarization qubit, and the scheme of Ref. [24]. Our work verifies experimentally that even a single qubit is enough to demonstrate quantum contextuality, under the weak assumptions of Ref. [23]. As we assume the validity of operational quantum theory for the error analysis, our work demonstrates that our results cannot be reproduced by a noncontextual fragment of quantum theory—an important experimental benchmark. We exceed the LSW bound by many standard deviations, in an experimentally verified regime of validity for the inequality.

We note that an independent experimental demonstration of contextuality with qubit systems, following techniques complementary to the present work, is reported in [25]. There, the state preparations and measurements are realized with time-sharing methods, and the problem of noises in measurements is solved with a technique derived within the framework of generalized probabilistic theories.

2. THEORETICAL IDEA

A. Scheme for Violating the LSW Inequality

A generalized noncontextual model, referred to as a LSW model, can be realized using noisy spin-$\frac{1}{2}$ observables [23]. Specifically, three such observables, ${\mathcal{M}}_k(k=1,2,3)$, are required, each described by a two-outcome POVM, ${\mathcal{M}}_k=\{E_{+}^k,E_{-}^k\}$, of the form [23]

Testing the LSW inequality for a quantum mechanical violation requires a special kind of joint measurability, denoted by joint measurability contexts $\{\{{\mathcal{M}}_1,{\mathcal{M}}_2\},\{{\mathcal{M}}_2,{\mathcal{M}}_3\},\{{\mathcal{M}}_1,{\mathcal{M}}_3\}\}$. That is, the three observables ${\mathcal{M}}_k$ ($k=1,2,3$) are pairwise jointly measurable, for all three pairs, but not triply jointly measurable. Pairwise joint measurability is possible only if $\eta \le (\sqrt{3}-1)\approx 0.732$ [23]. Triple-wise joint measurability—which would eliminate any possibility of contextuality since the entire experiment could be performed using a single context $\{{\mathcal{M}}_1,{\mathcal{M}}_2,{\mathcal{M}}_3\}$—is possibly only if $\eta <2/3$ [23]. Here we restrict our consideration of $\eta$ to the narrow range $2/3<\eta \le (\sqrt{3}-1)$.

The joint measurability context $\{{\mathcal{M}}_i,{\mathcal{M}}_j\}$ means that there exists a POVM ${\mathcal{J}}_{ij}\equiv \{G_{++}^{ij},G_{+-}^{ij},G_{-+}^{ij},G_{--}^{ij}\}$ satisfying the marginal condition that $\sum _{ϵ}{G}_{\epsilon ϵ}^{ij}=E_{\epsilon }^i$, and $\sum _{\epsilon }{G}_{\epsilon ϵ}^{ij}=E_{ϵ}^j$, where $\epsilon$, $ϵ\in \{+1,-1\}$. We follow Ref. [24] in using joint POVMs with the following general form:

The LSW inequality is the following [23]:

In quantum theory, where ${\mathcal{J}}_{ij}$ is described by a joint POVM as defined above, the average anticorrelation probability $R_3$ takes the form [24]

In our experiment, we aim for $\eta =0.670$, strictly within the range $[2/3,0.732]$ but close to the optimum at 2/3.

B. Implementation of Joint POVMs

For the pairwise joint measurements described above, each element of the POVM is rank one, and can be rewritten as $G_{\epsilon ϵ}^{ij}={\lambda }_{\epsilon ϵ}|{\xi }_{\epsilon ϵ}^{ij}⟩⟨{\xi }_{\epsilon ϵ}^{ij}|$ with $\epsilon$, $ϵ\in \{+1,-1\}$. Here, ${\lambda }_{++}={\lambda }_{--}=(2-{\eta }^2)/4$ and ${\lambda }_{+-}={\lambda }_{-+}=(2+{\eta }^2)/4$. We propose a scheme for implementing the joint POVMs in three stages, each of which is a single-qubit rotation followed by a two-outcome measurement. In each, the positive result (i.e., a detector click) has a POVM element proportional to the appropriate projector, while the null result qubit is fed into the next stage. The null result qubit from the third stage is then also detected.

To be more specific, the three single-qubit rotations are designed as

3. EXPERIMENTAL REALIZATION

A. Experimental Violation of the LSW Inequality

We perform the test of the LSW inequality with single photons. The basis states of the qubit, $|0⟩$ and $|1⟩$, are encoded by the polarizations of single photons, $|H⟩$ and $|V⟩$. We generate contextual quantum correlations by performing the four-outcome joint POVM on this qubit.

The experimental setup shown in Fig. 1 involves preparing the specific state (preparation stage) and then performing the joint POVM (measurement stage). In the preparation stage, polarization-degenerate trigger-herald photon pairs are produced and are registered by a coincidence count at two single-photon avalanche photodiodes (APDs) with a 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 of order ${10}^{-4}$, which is negligible. The second-order correlation $g^{(2)}$ is measured as $0.0089\pm 0.0018$, which shows that the single-photon source is extremely non-classical [26]. The heralded single photons are prepared in state $|{\varphi }_0⟩=(|H⟩+i|V⟩)/\sqrt{2}$ after passing through a polarizing beam splitter (PBS), a half-wave plate (HWP, H0), and a quarter-wave plate (QWP, Q0).

 

Fig. 1. Experimental setup. Single photons are created via Type-I spontaneous parametric down-conversion (SPDC) in a 0.5 mm thick nonlinear-$\beta$-barium-borate (BBO) crystal, which is pumped by a CW diode laser with 80 mW of power. One photon in the pair is detected to herald the other photon, which is injected into the optical network. To realize the joint POVM ${\mathcal{J}}_{ij}$, the single-qubit rotations $U_{\epsilon ϵ}$ 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 PBS. Detecting heralded single photons means in practice registering coincidences between single photon detectors: D0 and each of D1–D4.

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In the measurement stage, to implement the two-outcome measurements, partially projecting polarizing elements are added to the setup and 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 $T_V$ and the same transmission probability for horizontal polarization $T_H=1$. This allows us to project the state onto $|V⟩$ on the reflected port of the PPBSs.

After passing through each QWP–HWP–QWP set and the following PPBS, the photons are detected by APDs on the reflected port, in coincidence with the trigger photons. The transmitted photons go into the next QWP–HWP–QWP set and PPBS (or, at the final stage, a PBS that can be regarded as a special PPBS with equal transmission and reflection probabilities). The relative detection efficiencies of detectors D2–D4 with respect to D1 are measured as $0.9499\pm 0.0070$, $0.9199\pm 0.0069$, and $0.9801\pm 0.0063$, respectively and these figures are used to correct the coincidence counts (see Supplement 1). The probability of measuring the photons is obtained by normalizing the corrected coincidence counts on each mode with respect to the total corrected coincidence counts. The overall detection efficiency of the heralded photons in our experiment is approximately 11%. Thus, we make the fair-sampling assumption: that the event selected out by the photonic coincidence is an unbiased representation of the whole sample.

The probabilities of photons being measured on the reflected ports (clicks on detectors D1–D3) correspond to those of joint POVM elements $G_{+-}^{ij}$, $G_{-+}^{ij}$, and $G_{++}^{ij}$, whereas the probability of photons being measured on the transmitted port of the PBS (click on the detector D4) corresponds to that of the element $G_{--}^{ij}$. We can estimate the matrix forms of the joint POVM elements from the measured probabilities (see Subsection 3.B). The negligible difference from the theoretical prediction guarantees successful experimental realization of joint POVMs by taking into account all of the imperfections of the experimental setup.

In Table 1, we present the measured probabilities and the outcomes of the joint POVM with noise parameter $\eta =0.67$ on the specific state $|{\varphi }_0⟩$. The result of measured average probability of anticorrelations is ${\tilde{R}}_3^Q=0.8125\pm 0.0010$. Here, and below, the tilde relates to the experimentally implemented POVMs, as opposed to the theoretical ones aimed for; see Subsection 3.B. This ${\tilde{R}}_3^Q$ violates the bound set by the noncontextual hidden variable theory $1-\eta /3=0.7767$ by 35 standard deviations. Furthermore, in our experiment the noise parameter can be estimated by the experimental data (see Subsection 3.B). The average value of the estimated noise parameters in the experiment is $\overline{\tilde{\eta }}=0.6690\pm 0.0019$. Using this value, rather than the aimed-for 0.670, makes almost no difference in the LSW bound: the bound set by the noncontextual hidden variable theory can be calculated as $1-\overline{\tilde{\eta }}/3=0.7770\pm 0.0006$ compared to $1-0.670/3=0.7767$. Even including the uncertainty in the former bound, the experimentally measured average probability of anticorrelation, ${\tilde{R}}_3^Q=0.8125\pm 0.0010$, still implies a violation of this experimental bound of the LSW inequality $1-\overline{\tilde{\eta }}/3$ by 22 standard deviations. In Subsection 3.B we give an alternate way of comparing the correlations and the bound, which also gives a violation by many standard deviations. Here, we finish by noting that the experimental value ${\tilde{R}}_3^Q$ is in agreement (1.6 standard deviations) with its theoretical prediction $0.8087\pm 0.0022$, predicted via the estimated noise parameter $\overline{\tilde{\eta }}$.

Table 1. Experimental Results^a

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B. Evaluating the Quality of Experimental Realization of POVM

We consider the effect on the implementation of the joint POVM due to all the important imperfections, namely, in the PPBSs ($T_V^1=0.3904\pm 0.0045$, $T_V^2=0.2897\pm 0.0050$), WPs (typical retardance accuracy $<2.67\mathrm{nm}$), PBSs (typical extinction ratio $\sim {10}^5\text{:}1$), and detectors. We define a modified 2-norm distance $D(A,B)$ between the matrix form of the theoretical prediction of POVM element $A$ and that of experimental implementation of the corresponding POVM element $B$ as

For the particular forms of the POVM described in our paper, the distance ranges between 0 for a perfect match and $\sqrt{2}$ for a complete mismatch. For example, we use the distance $D(G_{\epsilon ϵ}^{ij},{\tilde{G}}_{\epsilon ϵ}^{ij})$ to measure the mismatch between the theoretical prediction of $G_{\epsilon ϵ}^{ij}$ with $i\ne j\in \{1,2,3\}$, $\epsilon$, $ϵ\in \{+1,-1\}$, and the corresponding experimental implementation ${\tilde{G}}_{\epsilon ϵ}^{ij}$.

To obtain the distance, we perform measurement tomography [27,28]. Single photons, prepared in the states $|H⟩$, $|V⟩$, $|R⟩=(|H⟩+i|V⟩)/\sqrt{2}$, and $|D⟩=(|H⟩+|V⟩)/\sqrt{2}$, are passed through the optical circuit and are detected by APDs in coincidence with the trigger photons. After correcting for the relative efficiencies of the different detectors, the photon counts give the measured probabilities. From these we can obtain the matrix forms of all 12 elements of the joint POVMs $\tilde{G}$ via maximum-likelihood estimation.

In our experiment, the accuracy of the experimental implementation of the measurements described by the POVM ${\tilde{E}}_{\epsilon }^{i(j)}$—the noisy version of the projective measurements—are more important. Here ${\tilde{E}}_{\epsilon }^{i(j)}\approx \sum _{ϵ}{G}_{\epsilon ϵ}^{ij}$, so the $(j)$ superscript indicates any dependence on the context in which it is performed (see below). The element ${\tilde{E}}_{\epsilon }^{i(j)}$ is estimated by minimizing the 2-norm distance $D({\tilde{E}}_{\epsilon }^{i(j)},\sum _{ϵ}{\tilde{G}}_{\epsilon ϵ}^{ij})$, which is defined in Eq. (8). This minimization is subject to the constraints that the sum ${\tilde{E}}_{+}^{i(j)}+{\tilde{E}}_{-}^{i(j)}$ equals the identity operator, while the difference is traceless, as per Eq. (1). The minimum distances found by this procedure are very small (all less than $6.2\times {10}^{-5}$), which justifies our approach. The theoretical prediction of the element satisfies the marginal condition $E_{\epsilon }^{i(j)}=\sum _{ϵ}{G}_{\epsilon ϵ}^{ij}=E_{\epsilon }^{i(k)}=\sum _{ϵ}{G}_{\epsilon ϵ}^{ik}$ ($i\ne j\ne k\in \{1,2,3\}$). However, because of the imperfections in the experiment, there might be a slight difference between $\sum _{ϵ}{\tilde{G}}_{\epsilon ϵ}^{ij}$ and $\sum _{ϵ}{\tilde{G}}_{\epsilon ϵ}^{ik}$. Hence, we use superscripts $i$ and $j$ to represent the estimated element of POVM ${\tilde{E}}_{\epsilon }^{i(j)}$, which is estimated by ${\tilde{G}}_{\epsilon ϵ}^{ij}$ and corresponds to the joint measurable context $\{{\mathcal{M}}_i,{\mathcal{M}}_j\}$. The difference between the elements ${\tilde{E}}_{\epsilon }^{i(j)}$ and ${\tilde{E}}_{\epsilon }^{i(k)}$ can also be measured by the 2-norm distance $D({\tilde{E}}_{\epsilon }^{i(j)},{\tilde{E}}_{\epsilon }^{i(k)})$ defined in Eq. (8). Thus, the distances satisfy $D({\tilde{E}}_{+}^{i(j)},{\tilde{E}}_{+}^{i(k)})=D({\tilde{E}}_{-}^{i(j)},{\tilde{E}}_{-}^{i(k)})$. As shown in Fig. 2, all the distances $D({\tilde{E}}^{i(j)},{\tilde{E}}^{i(k)})$ are smaller than 0.0006, which validates the experimental realizations of pairwise jointly measurable POVMs.

 

Fig. 2. Distance $D({\tilde{E}}^{i(j)},{\tilde{E}}^{i(k)})$ between the estimated POVM elements for different contexts $\{{\mathcal{M}}_i,{\mathcal{M}}_j\}$ and $\{{\mathcal{M}}_i,{\mathcal{M}}_k\}$. Error bars indicate the statistical uncertainty, obtained from Monte Carlo simulations assuming Poissonian photon-counting statistics.

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We also compare the estimated element ${\tilde{E}}_{\epsilon }^{i(j)}$ with the theoretical ideal $E_{\epsilon }^i$ by calculating the 2-norm distance $D(E_{\epsilon }^i,{\tilde{E}}_{\epsilon }^{i(j)})$. Since the distances satisfy $D(E_{+}^i,{\tilde{E}}_{+}^{i(j)})=D(E_{-}^i,{\tilde{E}}_{-}^{i(j)})$, we show only six values of the distances $D(E^i,{\tilde{E}}^{i(j)})$ in Fig. 3. All the distances are smaller than 0.0007, which shows the successful experimental realizations of the POVMs with the chosen noise parameter $\eta =0.67$.

 

Fig. 3. Distance $D(E^i,{\tilde{E}}^{i(j)})$ between the aimed-for POVM element $E^i$ for context $\{{\mathcal{M}}_i,{\mathcal{M}}_j\}$ and the estimated POVM elements from the experiment ${\tilde{E}}^{i(j)}$. Error bars indicate the statistical uncertainty, obtained from Monte Carlo simulations assuming Poissonian photon-counting statistics.

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For determining the LSW bound used in Subsection 3.A it is important to know the noise parameter of $\tilde{\eta }$ associated with the POVM. This can be estimated as

The condition ${\tilde{E}}_{+}^{i(j)}+{\tilde{E}}_{-}^{i(j)}=1$ guarantees that we have the same values of ${\tilde{\eta }}_{+}^{i(j)}$ and ${\tilde{\eta }}_{-}^{i(j)}$. Compared to the value of the noise parameter we aimed for in the experiment $\eta =0.67$, all the differences $|{\tilde{\eta }}^{i(j)}-\eta |$ are smaller than 0.0015. The average value of the estimated noise parameters in the experiment is $\overline{\tilde{\eta }}=\frac{1}{6}\sum _{i(j)}{\tilde{\eta }}^{i(j)}=0.6690\pm 0.0019$.

Finally, the value $R_3^Q$ corresponding to the ideal POVMs can also be bounded, as follows. An arbitrary qubit POVM element $G$ can be written as $G=a\mathbb{1}+b\widehat{n}·\overrightarrow{\sigma }$, where $\widehat{n}$ is a unit vector and $a$ and $b$ are nonnegative numbers satisfying $b\le a$ and $b\le 1-a$. An arbitrary qubit density operator can be written as $\rho =\frac{1}{2}(1+\overrightarrow{r}·\overrightarrow{\sigma })$, where $|\overrightarrow{r}|\le 1$. The probability of obtaining the outcome corresponding to $G$ for a POVM containing $G$ on state $\rho$ is given by $\mathrm{Pr}(G)=\mathrm{Tr}[G\rho ]=\frac{1}{2}(a+b\overrightarrow{r}·\widehat{n})$. Let $\overrightarrow{g}=(a,b{n}_x,b{n}_y,b{n}_z)$ and $\overrightarrow{s}=\frac{1}{2}(1,r_x,r_y,r_z)$. Then $\mathrm{Pr}(G)=\overrightarrow{g}·\overrightarrow{s}$. Let $\tilde{G}$ denote the experimentally realized POVM element corresponding to $G$, and likewise $\overrightarrow{\tilde{g}}$ to $\overrightarrow{g}$. Then $|\mathrm{Pr}(G)-\mathrm{Pr}(\tilde{G})|=|(\overrightarrow{g}-\overrightarrow{\tilde{g}})·\overrightarrow{s}|\le |\overrightarrow{g}-\overrightarrow{\tilde{g}}||\overrightarrow{s}|\le \frac{1}{\sqrt{2}}|\overrightarrow{g}-\overrightarrow{\tilde{g}}|$. Thus we obtain the bound

Let ${\tilde{g}}_{+-}^{ij}({\tilde{g}}_{-+}^{ij})$ be the vector representation of ${\tilde{G}}_{+-}^{ij}({\tilde{G}}_{-+}^{ij})$ in our experiment, obtained above by tomography. Then from Eq. (10) we obtain a lower bound for the ideal value:

The uncertainty here is larger than that in ${\tilde{R}}_3^Q$ because of uncertainties in the $\tilde{g}s$ that contribute to the correction term in Eq. (11). Now, the appropriate point of comparison is the ideal noncontextual bound of 0.7767, from the aimed-for $\eta =0.67$, because we are inferring the correlations from an ideal measurement with this $\eta$. The value of the bound in Eq. (12) implies a violation of this ideal bound by at least 16 standard deviations.

Note that as we assume the validity of quantum mechanics, there is no need to establish operational equivalences between the measured POVM elements in different contexts, as is done in Ref. [25].

4. DISCUSSION

Any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. This provides a compelling reason to test contextuality applicable to unsharp measurements. Here we test the generalized noncontextuality inequality for the unsharp measurements of LSW [23]. For unsharp measurements that can be jointly performed, correlated noise could allow correlations to be generated by a noncontextual hidden variable model. The LSW inequality takes such correlations into account by setting a higher bound. Thus a violation of the LSW inequality certifies nonclassicality that cannot be attributed to hidden variables associated with noise in the unsharp measurements.

Our experimental results show convincing violation of the LSW inequality with single-photon qubits. That is, it is a demonstration of contextuality for the simplest type of quantum system. It is also the first experiment to apply the LSW argument to rule out noncontextuality within quantum theory.

The experimental confirmation of quantum contextuality in its simple and fundamental form sheds new light on the contradiction between quantum mechanics and noncontextual realistic models. Furthermore, we realize joint POVMs of noisy spin-$\frac{1}{2}$ observables on a single-qubit system, which is the key point to implement the unsharp measurements, paving the way for further developments, such as real-time estimation [29], monitoring of the Rabi oscillations of a single qubit in a driving field [30], and understanding the relation between information gain and disturbance [31].