Experimental simulation of monogamy relation between contextuality and nonlocality in classical light

The Clauser-Horne-Shimony-Holt (CHSH) inequality and the Klyachko-CanBinicioglu-Shumovski (KCBS) inequality present a tradeoff on the no-disturbance (ND) principle. Recently, the fundamental monogamy relation between contextuality and nonlocality in quantum theory has been demonstrated experimentally. Here we show that such a relation and tradeoff can also be simulated in classical optical systems. Using polarization, path and orbital angular momentum of the classical optical beam, in classical optical experiment we have observed the stringent monogamy relation between the two inequalities by implementing the projection measurement. Our results show the application prospect of the concepts developed recently in quantum information science to classical optical system and optical information processing. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (030.0030) Coherence and statistical optics; (260.5430) Polarization. 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In this work, we show that the monogamy relation between the nonlocality and the contextuality, even more stringent relation, can be simulated in the classical optical systems. Using polarization, path and orbital angular momentum (OAM) of the classical optical beam, in classical optical experiments we can observe the tradeoff between the two inequalities by implementing the projection measurement.

Theoretical description on the monogamy relation in classical light
Based on the ND principle, Kurzyński et al proposed a fundamental monogamy relation between the violations of the KCBS inequality and the CHSH inequality [39]. For the KCBS inequality, there are five observables ( 1 A , …, 5 A ), which are pairwise compatible. Each observable has two possible measurement outcomes ± 1. In the noncontextuality case, for the any choice of the outcomes ± 1 for the observables, the KCBS inequality satisfies 3 , where AB β is the average value of CHSH operator. In order to demonstrate the violation of the CHSH inequality, two spatial separated systems (Alice's and Bob's systems) are needed, and the measurements in the two systems can impact mutually. In the Alice's system the observables are The spatial separation and the mutual impact show the nonlocality, and the compatible relations between the observables show the contextuality. According to the two assumptions to measure the four observables 1 B and 2 B , the classical probabilities, which cannot be described by local realistic and noncontextual theories, can be obtained. The CHSH inequality can be violated. The ND principle describes that, for any three observables W, X and Y, such that W and X are compatible, W and Y are compatible, the measurement outcomes of W do not depend on whether W was measured with X or Y. Under the ND principle, the sum of the KCBS inequality and the CHSH inequality is bounded.
Because the Bell inequality is also the noncontextuality inequality, according to Ref [39], we can sum the above two inequalities to produce a new noncontextuality inequality, which is expressed as , the quantum boundary touches the ND boundary, and the inequality becomes an equality. Such a stringent relation has been demonstrated experimentally using quantum states [40]. The KCBS inequality and the CHSH inequality are the bases of the monogamy relation, so we briefly describe the test of the two inequalities in photon experiment.
For the KCBS inequality, there are two measurement methods in the quantum case, joint measurement and sequence measurement. Because our classical light scheme corresponds to the joint measurement method in the quantum case, here we take the joint measurement as an example to describe the implementation process. The experiment process contains the input state preparation and the projection measurement. The correlated photon pairs are generated from the spontaneous parametric down conversion setup, one photon as the trigger, and the other produces the desired qutrit state [40]. In order to measure the compatible observables, their mutual eigenstates are established. When the input state projects onto the mutual eigenstates of the compatible observables, the probabilities of eigenvalues are obtained by the photon coincidence count. Possessing the probabilities of eigenvalues of the compatible observables, each correlation pair can be calculated and the value of KCBS inequality can be gotten further.
For the CHSH scenario, in quantum case it is that the two-qubit input state projects onto the operators corresponding to the four observables, which are two-two spatial separated observables. When the input state projects onto the eigenstates of the observables, these probabilities of eigenvalues are tested by the photon coincidence count. The eigenvalue multiplies by it probability, and we can obtain the observable when the products are summed. Possessing these observables, the result of CHSH operator can also be calculated [40].

3 , A A A A A A A A A A
where A κ ′ is the corresponding form of A κ in Eq. (1), and Eq. (7) is the corresponding expression of the violation of KCBS inequality in the classical light. Its maximum violation is 5 4 5 3.944 − ≈− . In Fig. 1(b), two incompatible observables 1 B′ and 2 B′ were chosen. They are compatible with the i A′ in Fig. 1(a). The compatible rations are represented in Fig. 1 where AB β ′ is the corresponding form of AB β in Eq. (2), and Eq. (8) corresponds to Eq. (2).
Mapping into the classical light system, the compatible relation and the measurement relation are similar, so the ND principle exists similarly. It is similar to the quantum case above-mentioned, and the two inequalities satisfy the monogamy relation 5.
Here Eq. (9) corresponds to Eq. (6). The more stringent monogamy relation between the KCBS inequality and the CHSH inequality also can be represented in classical light. There the observables 1 B′ and 2 B′ in Fig.  1(b) are chosen as two Pauli operators Z ′ and X ′ , respectively. Two overlap regions and the boundaries can be obtained similarly. The detailed descriptions are given in Appendix A. For the boundaries, the two classes of states (un-normalized) in classical light are e e E e e e e ϕ χ ϕ ϕ , the boundary touches the ND boundary. That is, the more stringent monogamy relation can be simulated in the classical optical system, which is attributable to the correlation characteristics of classical optical fields [50]. In the following, we test experimentally the above monogamy relation in the classical light systems.

Experimental demonstration of the monogamy relation in classical light
The experimental monogamy scenario in the classical optical system is designed by analogy of the photon experiment [40]. The experimental scheme is shown in Fig. 3. It consists of three parts: the state preparation, Alice's measurement, and Bob's measurement. Based on such a scheme, the cetrit-cebit system can be established, and the required input state can be obtained. In the Alice's part, we use the joint measurement method to test the KCBS inequality, and then the CHSH inequality is tested using the cetrit-cebit system. The detailed process is described as follow.
In the state preparation stage, two laser beams from He-Ne lasers transmit through Grin lenses (GL) and then the laser beams with horizontal polarization appear. Here the center wavelengths of the laser beams are 633 nm. After they transmit through the vortex phase plate (VPP), the beams carry the OAM with an azimuthal phase structure of exp(ilσ), where σ is the azimuthal angle of polar coordinates, and l is a integer. Adjusting the VPPs, let the first beam carry the OAM with l = 1 and the second OAM beam with l = 2. After two half wave plates (HWPs) and two polarizing beam splitters (PBSs), the beams with different polarizations are combined in a beam-splitter (BS), then two new beams are obtained. The optical field state at the up output port is . After a HWP2 at 90°, the optical field state at the down output port is expressed as h denote the polarized classical states for fields, The superscripts 1, 2 denote the OAM number l, and the meanings of superscripts are same below. If the polarization information for one beam is measured, the polarization for the other beam can be determined, and vice versa. Such a feature can be described by the classical correlation state: . Then one of the correlated beams is transmitted to Alice to produce the cetrit, and the other is transmitted to Bob to produce the classical bit (cebit). After the beam transmitting to Alice passes through a PBS3, then the outputting vertical component passes through a HWP3 and a PBS4, we can obtain three beams of light. The cetrit can be constructed by the fields of the three beams, then the cetritcebit system can be established using this cetrit and the cebit in Bob's part. Thus, the desired input state can be obtained: Fig. 3, the horizontal polarization of path0 and path1, and the vertical polarization of path2 are encoded as 0 | ) ′ ′ is obtained when we sum the products. We establish the eigenstates at the three output port PD1, PD2 and PD3 in Alice's part. The expressions of the optical fields at the output ports and the setting angles of HWPs are summarized in Appendix C. Taking 1 2 A A ′ ′ as example, the output state at the port PD1 is the eigenstate with the eigenvalues 1 1 A′=+ and 2 1 A′ = − ; the output state at the port PD2 corresponds to the eigenstate with the eigenvalues 1 1 A′=− and 2 1 A′ =+ ; the output state at the port PD3 describes the eigenstate with the eigenvalues 1 1 A′= − and 2 1 A′ = − . Here 1 1 A′=+ denotes that the eigenvalue of 1 A′ is + 1. The input state maps to these eigenstates, and the joint probabilities of the correlation 1 2 A A ′ ′ can be obtained. Based on the classical correlation, the OAM components of the eigenstate in the Alice's part correspond to the OAM components in the Bob's part. Thus, by measuring the intensity of optical field of the OAM beam in the Alice's part corresponding to the OAM in the Bob's part, the probability of eigenstate can be tested. There the eigenstates for KCBS operator in the Alice's part are irrelevant with the measurement in the Bob's part. Thereupon we set up the angle of HWP10 at 0°, which corresponds to the measurement for Z ′ (or 1 B′ ) in the Bob's part.
The OAM beams at the output ports D1 and D2 in the Bob's part are the OAM beams of l = 1 and l = 2, respectively. They correspond to the OAM beams in the Alice's part, so the probability is obtained when we test the optical intensity sum of the OAM beams of l = 1 and l = 2 of the eigenstate in the Alice's part. Because the OAM beams of l = 1 and l = 2 are incoherent, the optical intensity sum for the sorting OAM beams equals to the intensity of beam propagating in one beam. Thus we just need to measure the entire optical intensity at one output port rather than to sort the different OAM beams to obtain the probabilities =± . The optical intensities are normalized, and the probabilities of these eigenvalues are obtained finally. Then the correlations For the measurement of CHSH inequality, the case becomes complex. The measurement method is also adopted by analogy of the photon experiment [40]. We need to measure the correlation observables i j A B ′ ′ (i = 1 or 4, j = 1 or 2), thus the observables 1 A′ and 4 A′ in the Alice's part, 1 B′ and 2 B′ in the Bob's part are considered. The observables 1 B′ and 2 B′ in the Bob's part are the Pauli operators Z ′ and X ′ , respectively. Thereupon, the eigenstates and eigenvalues of 1 B′ and 2 B′ can be obtained easily. Similarly, we also need to establish the eigenstates of 1 A′ and 4 A′ in the Alice's part. The input state projects onto the eigenstates of these observables, and the probabilities of eigenvalues can be gotten. Because of the particularity of the observables 1 B′ and 2 B′ , we follow the eigenstates of 1 B′ or 2 B′ in the Bob's part to implement the corresponding measurement in the Alice's part. The field intensities of these eigenstates are test and normalized, and the probabilities of the corresponding eigenvalues can be gotten. Next, the observables i j A B ′ ′ can be calculated by summing the products of eigenvalues and probabilities, and then the result of CHSH operator can be obtained. This process is analogy of those in the photon experiment. We measure concretely these observables as follow. The measurement of observables Z ′ , X ′ can be implemented by a HWP10 and a PBS9 in Bob's part. Similarly, adjusting the angle of the HWP10, the eigenstates corresponding to eigenvalues 1 + and 1 − of the two operators can be obtained at the two output ports D1 and D2. The setting angles of HWP10 are 0° and 22.5° for Z ′ and X ′ , respectively. In order to measure i j A B ′ ′ , the i A′ in the Alice's part is also considered. There the 1 A′ and 4 A′ are needed to test 1 j A B ′ ′ and 4 j A B ′ ′ , and they can be test in the correlation pairs 1 2 A A ′ ′ and 5 4 A A ′ ′, respectively. But the 2 A′ and 5 A′ are not considered and only 1 A′ and 4 A′ are used. For the observable 1 A′ , the eigenstates at the three output ports PD1, PD2 and PD3 correspond to the eigenvalues 1 + , 1 − and 1 − , respectively. For the observable 4 A′ , the eigenstates at the three output ports PD1, PD2 and PD3 correspond to the eigenvalues 1 − , 1 + and 1 − , respectively. The input state maps to these eigenstates in the Alice's part and the Bob's part, the probabilities of eigenvalues can be measured. Because the input state is the classical correlation (entanglement) state that is constituted by the different polarizations and OAMs of the classical beams, the probability is obtained by measuring the intensity of the corresponding optical field. The detailed measurement methods are provided in Appendix D. In fact, if we use field correlation measurement method described in Ref [50]. instead of the above described method, the same results can be obtained.
Under the adjustments of HWPs and the projection measurement, we obtain the experimental results for 11 different input states. The average values of the KCBS and CHSH operators for the different input states are shown in Fig. 2 and Table 1. The tradeoff of the two inequalities for different input states is observed clearly in the Table 1 and Fig. 2. The violation of one inequality forbids the violation of the other and they satisfy the bound imposed by the ND principle. The more stringent boundary between CHSH inequality and KCBS inequality is also observed distinctly in the classical optical experiment. At this point

Conclusion and discussion
Mapping the quantum theory to the classical optical system, the classical eigenstates and correlation states have been constructed by using the polarization, path and OAM of the classical optical beams. The classical input states project onto these eigenstates, and the monogamy relation between the contextuality and nonlocality has been simulated experimentally. Such a relation and tradeoff in the classical optical system are in agreement with the presentation in the quantum experiment. The phenomenon originates from the correspondence between the Maxwell equations describing classical light fields and the Schrodinger equation. Such a correspondence leads to the similarity between coherent processes in quantum mechanics and classical optics. A typical example is the correspondence between the photonic band structure in photonic crystals and the electron band structure in solid state systems [57]. The photonic band structure describes the solutions of the Maxwell equations in periodic dielectric media, and the electron band structure exhibits the solutions of the Schrodinger equation in periodic potentials. Similarly, quantum nonlocality, contextuality and their monogamy relation have been discussed in the quantum scenario. In the present work, we have demonstrated that these phenomena can also be simulated in the classical optical systems. It is generally regarded as an important difference between the classical system and the quantum system. The former admits local realistic and noncontextual description, whereas the latter does not. However, our results have shown that the mathematical machinery developed in quantum information theory is of direct relevance to the discipline of classical optical coherence theory, which can enrich the coherence theory and information optics.

Appendix A: The more stringent monogamy relation in classical light
In this Appendix, we provide the theoretical demonstration about the more stringent monogamy relation in the classical light. In Fig. 1(a) of main body, it includes observables i A′ (i=1,…,5) and a classical trit (cetrit) in the Alice's part, which tries to violate the KCBS inequality. In Fig. 1(b) of main body, it includes the observables Z ′ , X ′ and a classical bit (cebit). The cetrit-cebit system is formed, which tries to violate the CHSH inequality in Alice's part and Bob's part. Corresponding to the quantum theory [39], the KCBS operator is diagonal in the classical case, which can be expressed as 5 So the eigenvalues of the KCBS operator are degenerate However the M is showed in the basis The two N can be showed in basis  ). In this procedure the two parameters reduce to one, namely The boundaries and two regions are showed in the Fig. 2. In fact, the boundaries can be obtained directly by the two classes of normalized states, that is For the term i j A B ′ ′ , i = 1 or 4, j = 1 or 2. Using 1 2 2 3 3 4 4 5

Appendix B: The state preparation and the setting angles of the HWPs for different input states
The classical beams possess correlation property, and they can be used to establish the classical correlation (entanglement) state. Two orthometric (horizontal and vertical) polarization beams transmit through a beam splitter (BS), the classical correlation state can be obtained. There the Jones matrix of BS is 1 . For the rotation angle θ, the Jones matrix of HWP is cos 2 sin 2 sin 2 cos 2 . These formulas are used for the experimental design.
Appropriately setting the angles of HWP1, HWP1 ′ and HWP3, the required input states can be prepared. The angles are list in the follow Table 2. The setting angle θ 1 ′ of HWP1 ′ are same with the angle θ 1 of HWP1, they all are θ 1 . The HWP11 is for the preparation of the state | ) ϕ χ − , and for the state | ) ϕ χ + it is not need (or setting up as 0°).  In order to test the KCBS inequality and the CHSH inequality, we need to measure these observables The input state projects onto the eigenstates of these observables, we measure the probabilities of these eigenvalues, and these observables can be obtained. According to the request described in the main text, the establishments of the eigenstates are an important assignment. Firstly, we describe the eigenstates for the test of the KCBS inequality in Alice's part. Because we adopt the method of joint measurement to test the observable are A 3 1 = + and A 4 1 = − , they are A 3 1 = − and A 4 1 = + at the output port PD2, and A 3 1 = − and A 4 1 = − at the output port PD3. The input state projects onto these eigenstates, and we measure the corresponding optical intensity to obtain the probabilities. After the input state projects on the eigenstates, the optical fields of the three output ports PD1, PD2 and PD3 are  Table 3. The angles of HWP4 and HWP5 are all set as 45° to transform polarization, and the angle of HWP9 is set as 0° for path-length compensation. The HWP4, HWP5 and HWP9 all are same in the entire experiment process.
For the other correlation pairs