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Abstract
The violation of the Bell inequalities implies that quantum mechanics cannot be interpreted using any local hidden variable theory. However, particular quantum states that can originally be described using a local hidden variable model surprisingly exhibit nonlocality by employing local filters before a standard Bell test is performed. This is referred to as hidden nonlocality. In this study, we provide the experimental demonstration of hidden nonlocality through linear optics towards the local states which are put forword by Hirsch et al., PRL 111, 160402 (2013). A class of local states is generated through a spontaneous parametric down-conversion process, and the violation of the Clauser–Horne–Shimony–Holt (CHSH)–Bell inequality is observed by applying local filters. Our experimental results confirm the superiority of local filters, and throw light on deep understanding the intriguing phenomena of hidden nonlocality and local states.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum entanglement plays an important role in both fundamental physics and the application of quantum information processing (QIP) [1]. In a scenario involving some entangled states with two spatially separated players, the observed statistics can confirm nonlocality by violating Bell’s inequality, which inhibits a reinterpretation of quantum theory using any local hidden variable (LHV) model [2,3]. Recently, such bipartite nonlocality has been verified in loophole-free experiments [4–6]. Furthermore, this can be exploited for device-independent QIP protocols [7–9].
Entanglement is not sufficient for the verification of nonlocality [10]. For the two-qubit case, all the entangled pure states are nonlocal [11,12]. However, Werner showed that there exists an LHV model for Werner mixed states with white noise of a certain strength [13]. Popescu proved that for some Werner-type states with a Hilbert space dimension larger than $5\times 5$, whose single-measurement correlations on each subsystem for two players can be simulated using an LHV model, the Bell inequality can be violated when two consecutive measurements are evolved [14]. This is called the demonstration of hidden nonlocality. Gisin showed that for some pure nonmaximally entangled states mixed with classical correlation, hidden nonlocality can be demonstrated with the aid of local filter operations before the measurement of the Clauser–Horne–Shimony–Holt (CHSH)–Bell inequality [15]. Kwiat $et\ al.$ and Wang, Z. W. et al. performed such a demonstration using photonic systems in sequence [1617]. Further, Masanes $et\ al.$ proved that for all bipartite entangled states, the CHSH inequality can be violated with the help of another bipartite state that also does not violate the inequality [18]. Hirsch $et\ al.$ demonstrated the existence of a genuine hidden nonlocality for a class of two-qubit states, for which a local model can be constructed for general local measurements, and the Bell inequality can be maximally violated by a sequence of measurements [19].
In this letter, we present the generation of a class of such states using a spontaneous parametric down-conversion (SPDC) process and linear optical elements. Notably, both the state preparation and local filters operation, we just need to adjust the rotation angle of a single half-wave plate to alter the parameter continuously. Significantly, with the help of local filters, the hidden nonlocality of some states is demonstrated via the violation of the CHSH inequality.
2. Bell scenario
Consider a bipartite state $\rho$ shared by two players, Alice and Bob (see Fig. 1(a)). Each player can select her/his own measurement setting $\{A_{x}\}$ /$\{B_{y}\}$, labeled as $x$ for Alice and $y$ for Bob. The output of each measurement is recorded, and labeled as $a$ and $b$ for Alice and Bob, respectively. All of the verification information is represented by the joint probability distribution $\{ p(a, b|x, y)\}$. The bipartite state is local, if for all kinds of measurement settings, the joint distribution can be written as a decomposition
where $\lambda$ is the LHV, which determines all the outputs of the local measurements on both subsystems; $\varPhi$ is the set of $\lambda$; and $\omega (\lambda ) \in [0, 1]$ is the distribution of $\lambda$ in $\varPhi$. Nonlocality can be illustrated via the violation of the Bell inequality. In the simplest example, both the players have two possible dichotomic observables $x,y\in \{0,1\}$ and $a,b\in \{0,1\}$, and the famous CHSH–Bell inequality is decomposed into four joint expectation values: where the joint expectation value $\langle A_{x}\otimes B_{y} \rangle \equiv \sum _{a,b}(-1)^{a\oplus b}p(a, b|x, y)$. If a distribution $\{p(a, b|x, y)\}$ can be decomposed in the form in Eq. (1) for all projective measurements, then we consider the entangled state $\rho$ shared by Alice and Bob to be local for projective measurements [19]. If it can be decomposed in the form in Eq. (1) for all POVMs, then we consider it to be local for POVMs [19].
Fig. 1. Bell scenario with two players, Alice and Bob. $a)$ Standard case: Each player performs a dichotomic measurement independently on their subsystem, based on the classical inputs $\{x,y\}$, and obtains outputs $\{a,b\}$. $b)$ Filtered case: Each player performs local filters on their subsystem before the Bell measurement. Here, the dashed lines indicate the distributions of the two quantum subsystems for the players: one for Alice and the other for Bob. The solid lines indicate the classical information, including the inputs and outputs for both parties.
3. Protocol
Here, we follow the protocol proposed by Hirsch $et\ al.$ [19]. The players are allowed to perform more general measurements, whereas only a single measurement is permitted in the standard Bell scenario described above. Similar to the schemes [15,16], hidden nonlocality can be demonstrated using a local filtering operation applied before local measurements at each side. Consider the two-qubit system in the state
Compared with the standard Bell scenario, our scenario involves local filter operations, which play a pivotal role in our experiment. As shown in Fig. 1(b), the players perform the Bell test after applying judicious local filters on their own subsystems. Here, we consider local filters for Alice and Bob of the form
Here, we consider projective measurements in a more general manner for the two-dimensional case:
where $\vec {n}$ is a unit vector on the Bloch sphere and $\vec {\sigma } = (\sigma _{x},\sigma _{y},\sigma _{z})$. Alice and Bob randomly and independently choose a measurement setting $x, y\in \{0,1\}$, and each performs a projective measurement for each spin along a corresponding direction $A_{x} ,B_{y}$. Here, we consider that the above directions follow the form where $\theta \in [-\pi ,,\pi ]$. Considering the case that $\epsilon \rightarrow 0$, the filtered state violates CHSH up to $I_{chsh}=2\sqrt {1+q}$. We simulate the following case for $\epsilon \rightarrow 0, \epsilon =0.2, \epsilon =0.4$. Figure 2 depicts the across-the-board results. In our simulation (Fig. 2), we stipulate that Alice and Bob employ the measurement directions in Eq. (7) with the optimal $\theta$ which satisfy the maximization for values of CHSH inequality.Fig. 2. Expected values of the Bell inequality with projective measurements. The $x$ axis represents the parameter $q$ of the state in Eq. (3), whereas the $y$ axis represents the value of the CHSH–Bell inequality in Eq. (2). The standard Bell scenario (black solid line) can be considered as an unfiltered case with $\epsilon =\delta =1$. Five cases with different filter parameters $\epsilon$ are depicted: $\epsilon \rightarrow 0$(red solid), $\epsilon =0.05$(green solid), $\epsilon =0.1$(black dashed), $\epsilon =0.2$(blue solid), and $\epsilon =0.4$(red dot-dashed), with $\delta =\epsilon /\sqrt {q}$. The magenta line represents the classical bound $2$.
4. Experimental implementation
The schematic of our experimental setup is shown in Fig. 3, which primarily consists of three modules. The first part is the generation of the state (Eq. (3), see Fig. 3(a)), which consists of a source of polarization-entangled photon-pairs in state$|\psi _{-}\rangle =\dfrac {1}{\sqrt {2}}(|HV\rangle -|VH\rangle )$, and a controllable depolarization channel [19] (Fig. 3(b)). The second part contains local tunable filters for each party, followed by the third one, which is the projective measurement for each player in the standard Bell test (Fig. 3(c)).
Fig. 3. Experimental setup. (a) State preparation: Two polarization-entangled photons are generated by the down-converter of two pieces of sandwich-type BBO crystal, cut for beam-like type-II phase matching and coupled into two single-mode fibers. One photon, $A$, is sent to the tunable decoherence channel (TDC), and the state $\rho _{q}$ is prepared. (b) TDC: This channel is based on a Sagnac-like interferometer followed by a completely dephasing operation on one output (path $2$) under the basis $\{|\pm \rangle \}$, and the recombination of the two outputs (using a beamsplitter). (c) Local filters and the Bell test: Alice and Bob perform local filters $F_{A}(\epsilon )$ and $F_{B}(\delta )$ on photons $A$ and $B$, respectively. Each filter consists of two beam displacers (BDs) and four HWPs. Subsequently, the CHSH–Bell test with projective measurements is performed. HWP: half-wavelength plate. QWP: quarter-wavelength. PBS: polarization beamsplitter. APD: avalanche photodiode (Excelitas: SPCM-AQRH-24-FC). IF: interference filter. CC: coincidence counter (UDQ-Logic-16).
The maximally entangled photon pair $|\psi _{-}\rangle$ is generated via the SPDC process. The mode-locked Ti:Sapphire pulsed laser (with the pulse width of approximately 100 $fs$, the repetition rate of 80 $MHz$, and the central wavelength of 780 $nm$) is converted into ultraviolet light as the pump of SPDC via a second-harmonic generation (SHG) process with a $\beta$-barium-borate ($\beta$-BBO) crystal cut for type-I phase-matching. The down-converter is a sandwich-type converter consisting of a pair of $\beta$-BBO crystals cut for beamlike type-II phase matching with a true-zero-order HWP between them (see details in [20–23]. The polarizations of the two photons generated, photons $A$ and $B$, are maximally entangled. Both photons are coupled into single-mode fibers.
The photon $A$ is collimated into a TDC (Fig. 3(b)). The main part of the tunable channel is a Sagnac-like interferometer. An HWP, $H_{1}$, is inserted in the interferometer to tune the ratio of the two outputs, (paths $1$ and $2$ in Fig. 3(b)), without reducing the entanglement between photons $A$ and $B$. The photon from the out-port $1$ is unchanged, whereas the one from the out-port $2$ is dephased in the basis $\{ |\pm \rangle = \frac {1}{\sqrt {2}}(|H\rangle \pm |V\rangle )\}$ by two pieces of birefringence crystal (one BBO and one LiNbO$_{3}$) and two pieces of HWP. A polarizer is used to block the $V$-polarized component. Hence, the photon pair $A$ in path $1$ and $B$ is maximally entangled, $|\psi _{-}\rangle$, whereas the pair $A$ in path $2$ and $B$ is separated, $|H\rangle \langle H|\otimes \frac {\textbf {I}_{2}}{2}$. Then, paths $1$ and $2$ are mixed via a beamsplitter. Hence, with the parameter $q$ tuned by the HWP $H_{1}$ in the Sagnac interferometer, the biphoton state in Eq. (3) is generated, where $A$ is sent to Alice and $B$ to Bob.
Before the standard Bell test, the local filters $F_{A}$ and $F_{B}$ are performed on photons $A$ and $B$, respectively. In [16], the filtering is realized by a series of coated glass slabs, tilted approximately at the Brewster angle. Here, we build a more flexible local filter with an interferometer consisting of two BDs, $BD_{a1,2}$, and four HWPs, $H_{2,3,4,a}$ for the photon $A$ at Alice’s side. The direction of light is unchanged after being transmitted through the BD, whereas only the path of vertical polarization is shifted away. As shown in Fig. 3(c), the first BD $BD_{a1}$ splits the photon $A$ into two paths. Through the two HWPs $H_{2}$ and $H_{4}$ at $pi/4$, flipping between the $|H\rangle$ and $|V\rangle$ polarizations, and a second BD $BD_{a2}$ recombining the two paths, the polarization of $A$ is unchanged from the fixed path when the difference of the upper and lower paths between the two BDs is tuned to be zero, i.e., the photon $A$ is under an identical operation. The local filter operation $F_{A}$ for the photon from the same path can be realized with two HWPs, $H_{a}$ at $\phi _{a}$ and $H_{3}$ at $0$, inserted in the upper and lower paths, respectively,
where the parameter of the filter is $\epsilon =\cos (2\phi _{a})$. Furthermore, the local filter for the photon $B$ at Bob’s side can be realized similarly.After the local filters, the standard Bell test can be performed. Here, we focus on the demonstration of $hidden\ nonlocality$, whereas the loopholes in the nonlocality verification are out of the scope of this study. Projective measurements are realized using polarization analyzers consisting of an HWP, a QWP, a polarization beamsplitter, and an interference filter, followed by an APD single-photon detector. The coincidence counts of clicking of both APDs are recorded by the UDQ. The value of the Bell operator (Eq. (2)) is calculated using the probability distribution detected. We can also reconstruct the density of the two-qubit state via quantum state tomography (QST). The value of the Bell operator (Eq. (2)) can be calculated directly using the detected probability distribution of the projectors. Here, we calculated the expected values from the reconstructed state density matrices.
5. Results
The experimental demonstration of hidden nonlocality of states $\rho (q)$ (Eq. (3)) via local filters (Eq. (4)) is realized by rotating the HWPs, $H_{1}$ inside the Sagnac interferometer and $H_{a}$ ($H_{b}$) in the upper path between the two BDs at Alice’s (Bob’s) side. In our experiment, states with $8$ different $q$ for $\epsilon =0.4$ are investigated.
As the state $\rho (q)$ is a mixture of the entangled component $\rho _{ent} = |\psi _{-}\rangle \langle \psi _{-}|$ and the decoherent component $\rho_{deco}=|H\rangle \langle H|\otimes \frac{\textbf{I}_{2}}{2}$, corresponding to two photons $B$ and $A$ output from paths $1$ and $2$, respectively, we first verify the two components by blocking paths $2$ and $1$ and setting $\phi _{a}=\phi _{a}=0$. We obtain the entangled component of the fidelity $95.31\pm (011)\%$ compared with $\rho _{ent}$, and the decoherent component of the fidelity $99.68\pm (008)\%$ compared with $\rho _{deco}$ (see Fig. 4). This is because the photon $A$ in $\rho _{ent}$ would be under some operation owing to the imperfection in the Sagnac interferometer and the beamsplitter, whereas for $\rho _{deco}$, the photon $A$ is filtered into $|H\rangle$ via the plate polarizer with an extinction ratio larger than $10^{3}$, before mixing by the beamsplitter. Here, the fidelity is defined as $F(\rho ,\rho ')\equiv Tr(\sqrt {\sqrt {\rho '}\rho \sqrt {\rho '}})$ [24]. The state parameter $q$ in (Eq. (3)) is tuned by the rotation of $H_{1}$ and is numerically calculated by maximizing $F(\rho ^{exp},\rho ^{theo}(q))$, the fidelity between the measured state $\rho ^{exp}$ and the theoretical one $\rho ^{theo}(p)$, where
and $\rho ^{exp}$, $\rho ^{exp}_{ent}$ and $\rho ^{exp}_{deco}$ are the density matrices measured via QST. Here, all the eight states have fidelities higher than $98.9\%$. (The detailed datas are shown in Appendix A.)Fig. 4. Reconstructed density matrices of the practically prepared maximally entangled state $\rho ^{exp}_{ent}$ and the decoherenced state $\rho _{deco}^{exp}$. (a) Re($\rho _{ent}^{exp}$) and (b) Im($\rho _{ent}^{exp}$) represent the real and imaginary parts of $\rho ^{exp}_{ent}$, respectively. (c) Re($\rho _{deco}^{exp}$) and (d) Im($\rho _{deco}^{exp}$) represent the real and imaginary parts of $\rho _{deco}^{exp}$, respectively.
The local filters are performed by rotating the angles of the HWPs $H_{a}$ and $H_{b}$. Here, by tuning the HWPs $H_{a}$ and $H_{b}$, the parameter of Alice’s filter is fixed at $\epsilon =0.4$, whereas for Bob’s one, $\delta$ is changed with the state $\rho (q)$ according to $\delta =\epsilon /\sqrt {q}$. The Bell inequalities are measured. Furthermore, by setting the rotating angles of both HWPs, $H_{a}$ and $H_{b}$, to $\phi _{a}=\phi _{b}$ ($q=1$), we can perform the standard (unfiltered) Bell test.
The experimental results of the CHSH–Bell tests of projective measurements with and without local filters are shown in Fig. 5. Among the eight states in our experiment, only two can violate the standard CHSH–Bell inequality. Furthermore, with the help of local filters on both sides, five of them show nonlocality. This indicates that the hidden nonlocality of three states is demonstrated successfully. In principle, all the eight states have hidden nonlocality. For example, we numerically simulate the Bell test with the filters $F_{A}(\epsilon )$ and $F_{B}(\delta )$ for $\epsilon =0.1$ (Fig. 2), where all the eight states violate the inequality. However, the transmission through the filters will decrease as the values of the filter parameters $\epsilon$ and $\delta$ become smaller. This would increase the statistical fluctuation of the recorded coincidence counts, and hence, the violation of the Bell inequality would be more difficult to verify. Furthermore, the precise tuning of waveplates $H_{a}$ and $H_{b}$ will be very difficult for the small parameters. The demonstration of the nonlocality will be fragile under a tiny misalignment of elements in practical experiments (See detailed interpretation in Appendix B).
Fig. 5. Experimental results and theoretical predictions of practical mixed-entangled states $\rho ^{exp}.$ The $x$ axis represents the different mixed-entangled states, which are determined by the parameter $q$. The $y$ axis represents the values of the CHSH–Bell inequality. The results of $I_{chsh}$ for $\epsilon = 0.4$ are plotted in Fig. 5. The black solid line represents the theoretical predictions of unfiltered states under our actual experiment, whereas the black rhombuses indicate the corresponding practical measured results of selected states. The red solid line represents the theoretical predictions for $\epsilon = 0.4$, whereas the red squares indicate the experimental results. The error bars are calculated according to Poissonian counting statistics.
6. Conclusion and outlook
In this study, we generate a class of two-photon states $\rho (q)$ with hidden nonlocality, defined in Eq. (3), via the SPDC process and a TDC using linear optical elements. Both the state preparation and local filters operation, we just need to adjust the rotation angle of a single half-wave plate to alter the parameter continuously. Without local filters, all the states with $q\leq 0.7$ admit an LHV model. With the help of local filters, the violation of the CHSH–Bell inequality can be verified for some of these, i.e., hidden nonlocality is demonstrated. Our experimental results confirm the superiority of local filters, and illustrate our understanding of the intriguing phenomena of hidden nonlocality and local states.
It is well known that nonlocality is a momentous resource in quantum mechanics. In a practical implementation, the local filter technique employed here can be applied for device-independent quantum random-number generation and other protocols of quantum communication based on quantum nonlocality.
Appendix A: Detailed Experimental Results
To understand the difference between unfiltered and filtered states, we perform quantum state tomography (QST) to reconstruct the state density matrices. The state parameter is tuned by rotating the HWP $H_{1}$ in the Sagnac interferometer. See Fig. 6 for an example involving the state in Eq. (10) with $q=0.5370$ and the filter parameter $\epsilon =0.4$.
Fig. 6. Reconstructed density matrices of unfiltered and filtered states. (a) and (c) represent the real parts of the unfiltered and filtered states when $\epsilon =0.4$ and $q=0.5370$, respectively. (b) and (d) represent the imaginary parts of the unfiltered and filtered states when $\epsilon =0.4$ and $q=0.5370$, respectively.
The results for all eight states are listed in Table 1, including the state parameter $q$, the fidelity with the theoretical state in Eq. (10), the unfiltered concurrence, and the unfiltered and filtered Bell values $I_{CHSH}$ in Eq. (11). All the calculations were performed using the Monte Carlo method based on the coincident counts.
where $\rho ^{exp}$, $\rho ^{exp}_{ent}$, and $\rho ^{exp}_{deco}$ are the density matrices measured via QST. where the joint expectation value $\langle A_{x}\otimes B_{y} \rangle \equiv \sum _{a,b}(-1)^{a\oplus b}p(a, b|x, y)$.
Table 1. Detailed experimental data on the prepared mixed entangled states
Appendix B: Tolerance Analysis
As shown in Fig. 2, for the state (Eq. (3)) with $q=1$, the Bell violation results are the same with different filters as in the unfiltered case. However, the violations are lower in the filtered than the unfiltered case for the state in our experiment shown in Fig. 5. The reason for this is that the state generated in the experiment is slightly different from the ideal one (Eq. (3)). For the state in Eq. (3), the normalized filtered state has the following form:
Fig. 7. Expected values of the Bell inequality with a filter factor $\epsilon =0.4$. The $x$ axis represents the parameter $q$, and the $y$ axis represents the value of the CHSH–Bell inequality. The black solid line represents the theoretical value of the inequality when using the standard Bell operator, i.e. fixed basis, whereas the red dashed line represents the theoretical value of the inequality when using optimal projective measurements in Eq. (7).
Fig. 8. Expected values of the Bell inequality for the experimental maximally entangled state with and without a filter. The $x$ axis represents the parameter $\delta$, and the $y$ axis represents the value of the CHSH–Bell inequality. The black solid line represents the value of the inequality for $\rho _{exp}$ with the filter, whereas the red solid line represents the value of the inequality for $\rho _{exp}$ without the filter.
Funding
National Key Research and Development Program of China (2017YFA0304100, 2018YFA0306400); National Natural Science Foundation of China (11774180, 61475197, 61590932); Graduate Research and Innovation Projects of Jiangsu Province ((KYCX180915).
Acknowledgment
We thank Drs Chengjie Zhang, Shaoming Fei, and Ming Li for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
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