Classical hypercorrelation and wave-optics analogy of quantum superdense coding

We report the first experimental realization of classical hypercorrelation, correlated simultaneously in every degree of freedom (DOF), from observing a Bell-type inequality violation in each DOF: polarization and orbital angular momentum (OAM). Based on such a classical hypercorrelation, we have realized the analogy of quantum superdense coding in classical optics. Comparing it with quantum superdense coding using pairs of photons simultaneously entangled in polarization and OAM, we find that it exhibits many advantages. It is not only very convenient to realize in classical optics, the attainable channel capacity in the experiment for such a superdense coding can also reach 3 bits, which is higher than that (2.8 bits) of usual quantum one. Our findings can not only give novel insight into quantum physics, they may also open a new field of applications in the classical optical information process.

In order to analyze the hypercorrelation properties between ( , ) E r t a and ( , ) E r t b , they will be divided into two parts. Using one of them to study the OAM correlation as shown in Fig. 1(b), the other part is used to study the polarization correlation as shown in Fig. 1(c). In Fig. 1(b), the beam from ( , ) E r t a is filtered by a H polarizer and passes through a HWP@  45 , which can be expressed as The beam from ( , ) E r t b is filtered by a V polarizer and passes through a HWP@  90 , which can be expressed as In order to analyze the OAM correlation properties between ( , ) E r t a1 and ( , ) E r t b1 , we take OAM basis similar to those in refs 41 and 42. A Bloch sphere is to be introduced to represent the first order transverse Laguerre-Gaussian modes as shown in Fig. 2(a). For a vortex beam in classical optics given order 2, |+ ) 2 and |− ) 2 , correspond to the north and south poles, respectively. In this case, each point on the Bloch sphere stands for a state, which can be described as 2 . In our experiment, the Bloch vector κ | )  is acted as measurement basis, which corresponds to computer phase hologram carried by a spatial light modulator (SLM, Holoeye). With the help of a pin-hole (the function is filtering wave), the computer phase hologram can transform the target spatial mode into the pure Gaussian mode with 15% diffraction efficiency 43 . When the hybrid beams ( ( , ) E r t a1 and ( , ) E r t b1 ) go through the mode splitters as shown in Fig. 1(b), the fields can be expressed in the following forms: where | ) 0 denotes the pure Gaussian mode.
correspond to projection onto a state of the form of Eq. where α β ( , ) P are normalized probabilities of states on the certain measurement basis for the OAM, which can be expressed as In the experiment, the correlation probabilities cannot be directly measured. They can be obtained through measuring the difference of light intensities at two export positions on Mach-Zehnder (MZ) interferometer as shown in Fig. 1 . Here I 1 and I 2 represent the light intensities at two export positions of the MZ interferometer. More information about the measurement method for the first-order field correlation has been given in ref. 27. Then, the CHSH measurement is b correspond to two different measurement states, selected by the holograms on SLM A and SLM B, respectively. Figure 2(b) shows experimental results for the normalized correlation probabilities as a function of θ b at various patterns on SLM B over the full range of possible value (θ π φ = , , ) with fixing the angle of pattern on SLM A. The dots (circle, triangular, square and pentagram) and solid lines represent the experimental measurements and theoretical results, respectively. Here, the theoretical results are normalized by the experimental data. It can be seen that the experimental results are in good agreement with the theoretical calculations. As expected from Eq. (9), we observe sinusoidal fringes in the correlation probabilities. The fringe contrast is about 91.93%, which is much larger than 70.7%, as required for verification of Bell's inequality 45 . The errors can be attributed mostly to misalignment of the interferometer.
The maximum value is obtained at θ π θ = − b a and φ φ = b a , and the minimum value corresponds to the case . From Eq. (10) and experimental results in Fig. 2(a), Bell parameter S can be evaluated by selecting special angles θ a , φ a , θ b and φ b . For example, , which yields the strongest violation of Bell's inequalities in the OAM DOF.
Simultaneously, we can also test the polarization correlation using setup in Fig. 1(c). Here, the analysis of polarization is realized by erasing the distinguishing OAM labels. In Fig. 1(c), the combination of a SPP and a pin-hole, called mode splitter, is utilized to obtain pure Gaussian mode. The beams from ( , ) E r t a and ( , ) E r t b pass through the mode splitter, which can be expressed as   A and θ B ). Subsequently, we perform the demonstration of classical polarization correlation. Similar to the method described in ref. 27, we can demonstrate that the maximum of Bell parameter reaches .
Considering the above two aspects, we are sure that the classical hypercorrelation properties exist in polarization and OAM, which are similar to quantum correlation properties from hyper-entangled photon pairs. This means that the following Bell-like state can be produced in the measurement process of the first-order correlation 27 .
Here the normalizing conditions 〈 have been used. This is highly similar to the production of hyper-entangled photon pairs from spontaneous down-conversion of nonlinear crystal 14 , which means that the classical hyper-correlated state can be constructed. The problem is whether or not some unique phenomena such as superdense coding can be realized by applying such a classical hyper-correlated state, which is similar to the case in the quantum information process.

Experimental realization of classical wave-optics analogy of quantum superdense coding.
In order to study the classical analogy of quantum superdense coding, the experimental layout shown in Fig. 3 is considered. The experiment consists of three distinct parts: the hypercorrelation source generating nonlocal classical optical hypercorrelation states; Bob's station for encoding the messages; finally, Alice's analyzer to identify signals sent by Bob. Such an experimental scheme corresponds to that of the quantum superdense coding described in ref. 13.
Here the simplified form of the correlation source described in Fig. 1 has been used (see Fig. 3(a)), MZIM has been removed, the output fields in such a case are marked by  It is notable that the classical hypercorrelation that violates the Bell-type inequality can also be demonstrated in such a case (see Section II in Supplementary Material). In the following, we use such a hypercorrelation source to study the classical wave-optics analogy of quantum superdense coding. Similar to the scheme of quantum superdense coding by using hyperentanglement 13,14 , in our scheme encoding operations are also globally performed by manipulating separately the two DOFs of the beam ( , ) E r t c , polarization and OAM. In Bob's encoding station, Bob can encode his messages by using two HWPs in two channels where up or down channel can be only chosen for each operation. A DP is inserted in one of the two channels and HWPs are rotated at the correct angle θ HWP (see the Table 1  ( ) These manipulations, which result in eight distinguishable messages corresponding to Eq. (15), can be completed in Bob's encoding station.

E r t E r t H E r t V E r t E r t V E r t H
The output beam from the Bob's encoding station was then combined with the other beam at Alice's analyzer to perform decoding. The process of decoding contains two stages. One is that the polarization-OAM analysis is implemented with an apparatus consisting of a ±2 -OAM splitter and a PBS on each path, as shown in Fig. 3(b). The splitter is composed of a binary forked grating with 30% diffraction efficiency into the first order and a pin-hole, Scientific RepoRts | 5:18574 | DOI: 10.1038/srep18574 which can transform an initial state with + (− ) 2 2 -OAM into a Gaussian mode in the + (− ) 1 1 diffraction order 43 . Subsequently, the diffracting beams filtered by the pin-hole are merged by a PBS. The other is that the first-order correlations are measured. After the polarization-OAM analysis, the beam of port D is combined with the beam from port A and port B though a BS, respectively. Then each beam passes through the optical element group that includes BS, PBS@  0 , and PBS@  45 , as shown in Fig. 3(c). We adjust appropriately the path length difference ∆ of the two beams, which corresponds to the phase δ e i by the relation δ π = ⋅ λ ∆ 2 . The distinguishable messages can be obtained directly by recording intensities I at eight detectors, which has been shown in Table 1 for theoretical results.
In order to demonstrate the above theoretical analysis, we perform the corresponding experiment. The wavelengths of two independent light beams (produced by two independent He-Ne lasers) are still taken as 632.8nm. To characterize the observable intensity at Alice's analyzer, we change the path length difference related to the phase. The results of experimental measurements are shown in Fig. 4. Three colors (blue, gray and green) of histogram represent the case of three-lever intensity output. Here the experimental data are normalized by the average value from all measured dates of detectors under maintaining the stability of interferometer for δ π = . Such   . From the experimental results in Fig. 4, we can determine the message that Bob has sent according to the intensity difference of detectors. Therefore, Alice can distinguish completely the messages sent by Bob, implementing the information transfer of = log 3 2 8 bits. Such high information transfer originates from the classical hypercorrelation as described in the above part.
If we do not use the classical hypercorrelation, consider E c or E d channel separately, the unified operation is used to encode messages in two DOFs, the maximum channel capacity is 2 bits (demonstration has been given in Section III of Supplemental Materials). This means that the present work provides the first demonstration that the superdense coding can be realized in the classical optics, which is analogy of quantum superdense coding using pairs of photons simultaneously entangled in spin and OAM as described in refs 13 and 14.
Comparing the present classical scheme with those for quantum superdense coding described in ref. 13, we find that present scheme posses many advantages. First, the attainable channel capacity in the experiment for such a superdense coding can reach 3 bits, which is higher than that ( ≈ . ) of usual quantum superdense coding using pairs of photons simultaneously entangled in spin and OAM. Second, it is very convenient to realize in optical communication, because the distinguishable messages can be directly identified from the detectors instead of interference and coincidence measurements.
In addition, we would like to point out that the present method is different from the signal demodulation in coherent optical communication although correlation properties of the classical optics have been used in the process [46][47][48] . The traditional coherent demodulation is based on polarization-division multiplexing (PDM) to increase the capacity of optical communication systems. For example, in ref. 46 the information is encoded onto the electrical field and modulated in orthogonal polarizations. In fact, in the present work we have presented a new method to improve the information capacity, which is based on the classical hypercorrelation corresponding to the quantum superdense coding using pairs of photons simultaneously entangled in the polarization and OAM. Such a method does not require the measurement of intensity difference of two outputs from BS 1 (BS 2), because it is enough to distinguish messages only by manipulating and measuring the optical intensity at one arm after the BS 1 (BS 2).

Conclusions
In summary, we have demonstrated experimentally the classical hypercorrelation by implementing the measurement of CHSH inequality in every DOF: polarization and OAM. Such a classical hypercorrelation is similar to the production of hyper-entangled photon pairs from spontaneous down-conversion of nonlinear crystal. They can also be used to increase the information capacity of the system. We have realized experimentally the analogy of quantum superdense coding in the classical optics by using such classical hypercorrelation. Comparing the superdense coding in the quantum case using pairs of photons simultaneously entangled in polarization and OAM, it exhibits many advantages, for example, the attainable channel capacity in the present experiment can reach 3 bits, which is higher than that (2.8 bits) of usual quantum one. It is very convenient to realize in the classical optics, because the distinguishable messages can be directly identified from the detectors instead of interference and coincidence measurements. Thus, our study opens a new way to improve the information capacity in the classical optical communication. It not only provokes deep thought on some basic physical problems such as essence of entanglement and correlation, but also shows potential application in classical optical information processes.