Bell’s measure and implementing quantum Fourier transform with orbital angular momentum of classical light

We perform Bell’s measurement for the non-separable correlation between polarization and orbital angular momentum from the same classical vortex beam. The violation of Bell’s inequality for such a non-separable classical correlation has been demonstrated experimentally. Based on the classical vortex beam and non-quantum entanglement between the polarization and the orbital angular momentum, the Hadamard gates and conditional phase gates have been designed. Furthermore, a quantum Fourier transform has been implemented experimentally.

through a half-wave plate (HWP) and a polarizing beam splitter (PBS) as shown in Fig. 1(a). After the PBS, the light beam is divided into two intensity equaled parts for the two paths with horizontal ( ) h and vertical ( ) v polarizations, respectively. In the path with vertical polarization, a Dove prism is introduced. Then, the two polarized vortex beams in two paths are combined by a beam splitter (BS), and the output of the vortex beam can be expressed as LG 1 p l p l where LG p,l represents the lth Laguerre function and p is the radial mode number. If the horizontal and vertical polarization components of the vortex beam are described by a slightly modified version of the familiar bra-ket notation of quantum mechanics, ) H and ) V 14 , the OAM of LG p,l are expressed by ± ) l , the polarized vortex beam can be described by the ket notation ) = , + ) + , − ) .
( ) E l V l 1 2 [ H ] 2 This representation for the polarized vortex beam is formally equivalent (isomorphic) to a Bell state of two polarized qubits. Hence, the polarization and OAM may be treated as two qubits that are classically entangled. Such an entanglement (local entanglement) can be realized from a single light beam. The problem is whether or not such an entanglement relation can be demonstrated by Bell's measurements.
In order to answer such a problem, an experiment was designed as shown in Fig. 1(b). A polarized vortex beam passes through a HWP and a PBS, and splits into two beams. Then they are manipulated with OAM rotation systems and splitter devices to become four beams. The output intensities of four vortex beams are marked in Fig. 1(b) by I 1 , I 2 , I 3 and I 4 , respectively. In order to perform the Clauser-Horne-Shimony-Holt (CHSH) Bell's measurement, we define the following correlation function 6 : where θ and φ represent polarization and OAM rotated angles in the paths, respectively. The θ φ ( , ) and I = I 1 + I 2 + I 3 + I 4 . After we have obtained C(θ, φ), the CHSH measurement is Figure 2(a,b) show experimental results for the correlation function C(θ, φ) from the same vortex beam with = ± l 1 as a function of θ and φ. In the experiment, the vortex beam is produced by the diffraction from a phase hologram pattern on the SLM as described in ref. 21. Here the OAM rotation has been realized by using Dove prisms and conventional π/2 astigmatic mode converters 31 , and the OAM splitter has been implemented by the SLMs as shown in the experimental setup at the top of Fig. 2. It is well known that the polarization changes when the incident beam passes through the rotated Dove A continuous laser with wavelength 632.8 nm is used. The round dots and solid lines represent the experimental measurements and the theoretical results, respectively. Here, the theoretical results are normalized by the experimental data, such that the amplitude maximum of the theoretical curve are taken to be equal to the maximum of the experimental data. It can be seen that the experimental results are in good agreement with the theoretical calculations in its changing character. If we take θ π = − /8, θ π ″ = /8, φ = 0 and φ π ″ = − /4, = .
± . B 2 407 0 035 is obtained. Although there is some imprecision in the measurements and imperfections in the optical elements, the presented experimental results yield a violation of Bell's inequalities.
The experimental results in Fig. 2 only exhibit the classical correlation between the polarization and the OAM with = ± l 1. In principle, the classical correlation between the polarization and the OAM with any mode number can also be tested through such an experimental design. However, it is very difficult to operate in this scheme. Thus, we take another scheme, for which the detailed design has been given in the Methods section. The experimental results for = ± l 2 are plotted as round dots in Fig. 3. The solid  lines are theoretical results, which are normalized by the experimental data. An agreement between the experimental measurements and theoretical results in changing character is observed again. If we take φ′ = ± .°22 5 , θ′ =°0 and −°45 , = .
± . B 2 101 0 028 is obtained. Comparing the results in Fig. 3 with those for = ± l 1 in Fig. 2, we find that the experimental results still yield a violation of Bell's inequalities, although the loss in the experimental process increases for the present case with = ± l 2. With the increase of l, it becomes more difficult to obtain the measured results with B > 2 because it requires a more accurate measurement of the intensity and precise operation of optical elements. However, the violation of Bell's inequalities can be confirmed.
This means that the presence of local classical entanglement between the polarization and the OAM has been demonstrated. The question is whether or not such classical entanglement can be exploited to perform algorithms for quantum information processing. Because the QFT is the crucial final step in some quantum algorithms such as Shor's algorithm, in the following we explore the possibility to realize QFT by using the local classical entanglement between the polarization and the OAM.
Quantum Fourier transform based on the classical vortex beam. The QFT is a basis transformation in an N-state space that transforms the state k according to where , ∈ k j n { } N and n { } N is a set of complete orthogonal basis vectors with N dimensions, k and j represent an integer ranging from 0 to N − 1. So far, the QFT has been demonstrated experimentally by using a quantum Hadamard gate and conditional phase gates 34,35 . The main advantage of QFT against the classical Fourier transformation is that of higher calculation efficiency, which originates from the quantum correlations.
The above investigations have shown that classically entangled states can exhibit similar correlation properties with the quantum correlations. If we take the classical states ) k and ) j instead of the quantum states k and j , a similar transformation | ) → ∑ ) can be achieved by using the local classical entanglement between the polarization and the OAM. In the following, we take two-qubit as an example to demonstrate such a process. The polarization degree of freedom is marked as the first qubit, that is ) → ) 0 H and ) → ) 1 V, the OAM is marked as the second qubit, ) → + ) 0 1 and ) → − ) 1 1, then the four Bell's states that we use as an example to show our QFT experiment, are expressed as: ,− ) ± , + ) , In order to realize the QFT of the four Bell states, we present an experimental setup as shown in Fig. 4. It consists of two Hadamard gates and one controlled phase gate. The first Hadamard gate is for the OAM, which has been realized by using two Dove prisms and two mode converters. The rotation of OAM by two mode converters (π/2) and a Dove prism with horizontal angle 67.5° can be expressed as a matrix: . In this way, the Hadamard transformation for the OAM is realized. The second Hadamard gate is for the polarization, which can be realized by a HWP. The controlled phase gate consists of one spiral phase plate (SPP) (RPC photonics, VPP-m633) and one SLM. The function of the SPP is to change the order of OAM. Here we use it to realize − ) → ) 1 0 and ) → ) 1 2. Thus, the two light beams with OAM − ) 1 and + ) 1 , which can not be separated in the space, can be transformed to spatially separable modes ) 0 and ) 2 in the space. In such a case, we can do In order to obtain the information from the output states, we measure the intensity by using a PBS and a diaphragm as shown in Fig. 4. That is to say, measure the output intensities for four basis vectors , + ) H 1 , , − ) H 1 , , + ) V 1 and , − ) V 1 . After the controlled phase gate, four basis vectors become , + ) H 2 , , ) H 0 , , + ) V 2 and , ) V 0 , respectively. Then they pass through a PBS and the field intensities with different OAM can be obtained by blocking the light in space. They should correspond to the modular squares of coefficients for the four basis vectors, if the QFT has been realized.
In Fig. 5, we present the comparison between the theoretical computation and the experimental measurements. Figure 5(a-d) show output ratios of different basis vectors for the four kinds of input states, respectively. Here the output ratio represents the ratio of measured intensity for each basis vector and the total output intensity. The blue bars represent theoretical results, and the red bars are experimental results. Comparing them, we find that the agreements between the theoretical results and experimental measurements are in good agreement. This means that the QFT has been realized by our experimental setup. We would like to point out that the above results are only for the classical vortex beam with l = ± 1, and the QFT can be realized by using a classical vortex beam with any mode number if ideal optical elements and methods are achieved.

Conclusions
In summary, Bell's measurement for the non-separable correlation between the polarization and the OAM from the same classical vortex beam has been performed experimentally. The violations of Bell's inequalities for the non-separable classical correlations with various OAM have been demonstrated experimentally. Based on the non-quantum entanglement between the polarization and the OAM in the classical vortex beam, the Hadamard gates and controlled phase gates have been designed, and the QFT has been implemented experimentally. Such an implementation of QFT exhibits many advantages compared with the usual quantum realizations. For example, it is not only easier to implement, the measurement efficiency is also high. Moreover, it sheds light on the new concept of non-quantum entanglement. In the present work, a basic idea has been presented to realize two-qubit QFT via classical beams. In fact, if we can extend such a method to multi-qubit QFT, it could be surely used in a primary case of Shor's algorithm. The expansion of the present work to simulate quantum computing, or perform other quantum information processing task, is a subject for future research.

Methods
In order to perform Bell's measurement for the correlation between the polarization and the OAM with > l 1 more efficient, we consider the experimental setup as shown in Fig. 6, which is similar to that in Ref. 36. In contrast to the scheme in Fig. 2, in the present scheme the measurement basis for the OAM has been generated by controlling the polarization and the projection measurement can be realized by using the interference method. Then, the output intensities ′ I 1 , ′ I 2 and I T as marked in Fig. 6 can be obtained. The calculated processes for ′ I 1 , ′ I 2 and I T are given in the following: The Jones matrix for the optical element group consisting of two HWPs and one PBS can be expressed , where the first line and the second line of the column matrix correspond to ) H and ) V , respectively. From Fig. 6, the ′ I 1 can be calculated in the following process: For the ′ I 2 , we have