Non-local classical optical correlation and implementing analogy of quantum teleportation

This study reports an experimental realization of non-local classical optical correlation from the Bell's measurement used in tests of quantum non-locality. Based on such a classical Einstein–Podolsky–Rosen optical correlation, a classical analogy has been implemented to the true meaning of quantum teleportation. In the experimental teleportation protocol, the initial teleported information can be unknown to anyone and the information transfer can happen over arbitrary distances. The obtained results give novel insight into quantum physics and may open a new field of applications in quantum information.

This study reports an experimental realization of non-local classical optical correlation from the Bell's measurement used in tests of quantum non-locality. Based on such a classical Einstein-Podolsky-Rosen optical correlation, a classical analogy has been implemented to the true meaning of quantum teleportation. In the experimental teleportation protocol, the initial teleported information can be unknown to anyone and the information transfer can happen over arbitrary distances. The obtained results give novel insight into quantum physics and may open a new field of applications in quantum information.
Q uantum teleportation provides a means to transport an unknown quantum state from one location to another over arbitrary distances. Since originally being proposed in 1993 1 , the study on quantum teleportation has aroused great interest. Its experimental realizations in various different ways have been reported. For instance, the best known methods have been implemented using photon polarization and optical techniques 2,3 , by means of squeezed states of light 4 , by applying nuclear magnetic resonance (NMR) techniques 5 , and by a hybrid technique 6,7 . Teleportation has also been accomplished between photons and a single atomic ensemble 8,9 , between distant atomic objects [10][11][12][13] , and in a chip-based superconducting circuit architecture 14 . Quantum teleportation is essential for large-scale quantum communication [15][16][17] and distributed quantum networks 18 . It has proven to be a useful tool for realizing universal quantum logic gates in quantum computing and general quantum information manipulation 19 . However, all protocols for accomplishing quantum teleportation require non-local correlations, or Einstein-Podolsky-Rosen (EPR) entanglement, between systems shared by the sender and receiver.

Results and Discussion
Non-local classical optical correlation. Initially two independent light beams, E 1 and E 2 with different wavelengths, are considered and pass through a 50/50 beam splitter (BS) as shown in Fig. 1

MICRO-OPTICS
In order to analyze the correlation properties between E ! ar ,t ð Þ and E ! br ,t ð Þ, a half-wave plate (HWP) and a polarizing beam splitter (PBS) are introduced in each path. They function to rotate angles of the light beam and separateĥ andv polarized light. Here h a and h b represent polarization rotated angles in two paths, respectively. After the HWP and PBS, two polarized beams become four beams. Their fields are described by E ah , E av , E bh and E bv , respectively, which can be modulated by h a and h b . Now we perform the famous Clauser-Horne-Shimony-Holt (CHSH) test in our system as shown in Fig. 1. The correlation function is defined in the following form 20 : where P HH 5 N HH (h a ,h b )/N Total , P VV 5 N VV (h a ,h b )/N Total , P HV 5 N HV (h a ,h b )/N Total , and P VH 5 N VH (h a ,h b )/N Total are normalized correlated probabilities when polarizations of each beam are measured, and N Total 5 . In our experiment, such correlated probabilities can be represented by the first order correlation of electric fields through N IJ (h a ,h b ) (I(J) represents H or V): In the experiment the fields of output lights are not directly measured. However, the first-order field correlation can be obtained through measuring the difference of light intensities at two export positions on Mach-Zehnder interferometer, because Here I 1 and I 2 represent the light intensities at two export positions of the Mach-Zehnder interferometer and DI is the difference between them. More information about the measurement method for the first-order field correlation is provided in the Methods section. Figure 2 shows experimental results for the correlation functions C(h a ,h b ) as a function of polarization rotated angle h a at some certain h b . Here two laser beams (continuous wave mode) with different wavelengths, 532 nm and 632.8 nm, are used. Their intensities are taken as 2 mW. The circle dots and solid lines represent the experimental measurements and theoretical results, respectively. It can be seen that the experimental results are in good agreement with the theoretical calculations. After we have obtained C(h a ,h b ), the CHSH formulation of Bell's measurement is used to quantify such correlation. The CHSH measurement is It is well known that the local realism theories give jBj , 2. However, from the experimental results marked by dashed lines in Fig. 2 Any of the four EPR-Bell states can be constructed by adjusting polarization and phase factor via HWP in the above experimental setup. This is highly similar to the production of polarizationentangled photon pairs from spontaneous down-conversion of nonlinear crystals 37 . For example, the output fields E ! a and E ! b shown in Fig. 1 correspond to the following symmetric Bell-state: Here we use the notation jh) i (jv) i ) to express a classical state 29,31 . That is to say, ifĥ polarization is measured in one beam, the information of v polarization can be ''determined'' in another beam due to the firstorder field correlation, and vice versa. This means that a classical correlation state has been constructed. Because the correlation between two beams is independent on the separation between them, such a correlation is non-local and can be regarded as a classical analogy of EPR entangled state in quantum mechanics. The problem is whether or not some unique phenomena such as quantum teleportation can be realized by applying such a classical EPR correlation state, which is similar to the case in quantum information process.
Classical analogy of quantum teleportation based on non-local classical correlation. In order to study the classical analogy of quantum teleportation, the experimental setup shown in Fig. 3 is considered. The experimental generation of EPR correlation states and Bell-state measurement are two key parts in the teleportation scheme. In this scheme, above classical EPR correlation states are used as the source. A three-photon scheme is referred to for the Bellstate measurement as described by D. Bouwmeester et. al. 2 . This system features the antisymmetric state obtained from four Bellstates. Thus, the antisymmetric Bell-state is used in our ð Þĥ in our classical EPR system, which are shared by Alice and Bob, for example. If Alice want to teleport an initial state jy c ) 5 c 1 jh) 1 c 2 jv) to Bob, she needs perform a joint Bell-state measurement on the initial state and E ! a . In the experiments, the field corresponding to the initial state jy c ) is expressed as: c 1 and c 2 satisfy jc 1 j 2 1 jc 2 j 2 5 1. When c 1 and c 2 are considered real, corresponds to the linear polarization case. Whereas, when c 1 and c 2 are complex, it is in correspondence to the circular polarization case. We first consider linear polarization case, in which c 1 and c 2 can be expressed by the direction angle of polarization h: c 1 5 cos h and c 2 5 sin h. Thus, h will be the teleported information.
In order to perform a joint Bell-state measurement on the initial state and E ! a , an optical element group is constructed that includes a BS and two HWPs as shown in Fig. 3. There are two output ports of the BS, in which one is the sum of two input fields, and the other is the difference of the two. Two HWPs are placed at the output port of the difference. The functions of two HWPs are to realize interchange betweenĥ andv polarizations, and add a p phase for the component ofĥ polarization field. For such an optical element group, if the input ports are in the Bell-states, it is easy to demonstrate by the correlation measurement of the first-order field that only the output of the antisymmetric state is not zero, and the outputs of three symmetric states all are zero (see Methods section). This is similar to the scheme of the Bell measurements in Ref. 2. Referring back to the previous example, and considering synchronicity of Alice's and Bob' measurements, Alice obtains the information E ac as E ! c and E ! a enter the optical element group, and then sends it to Bob through the classical channel. Here E ac represents the amplitude of field without any polarization information (see Supplementary). Owing to lack of technique to measure fields, we had to send the beam to Bob in the experiment. Next, we show that Bob can only use the coherence property of the field rather than the polarization information, and such a process is corresponded to the method used in Ref. 2.
After Bob receives the information that Alice sent, he will perform the correlation measurement of the first-order field by using the information and E ! 0 b in order to obtain the teleported material. This is because there is a strong correlation between E ac and E ! 0 b (see Supplementary). For example, as E ! 0 b passes through a rotated PBS, it will be orthogonally decomposed into E // and E \ as shown in Fig. 3. Our theoretical calculations show that jAEE ac E // aej 2 5 cos 2 (h 2 w) and E ac E \ h i j j 2~s in 2 h{w ð Þ, which are the correlation of field without polarization (see Supplementary). Here w represents the rotated angle of the PBS, and is taken as 0u when the polarization of transmission fields is horizontal. From these relations, it is not difficult to find that h and w are in one-to-one correspondence. If Bob has obtained the maximum of the first-order correlation between E ac and E // for a polarization direction, the corresponding w for such a polarization direction is the teleported information h (see Supplementary).
In the experiment, the first-order correlations between E ac and E // (E \ ) are measured by the difference of light intensities, which is similar to the above method for testing CHSH formulation. Figure 4 displays the experimental results for the first-order correlation degree, DI 5 I 1 2 I 2 , as a function of the angle w. Figure 4(a) and (b) correspond to the case with h 5 0 for E // and E \ , respectively, whereas the corresponding results for h 5 p/4 are plotted in Fig. 4(c) and (d). From Fig. 4(a), the maximum of the first-order correlation   appears at w 5 0, which is in agreement with h 5 0. At the same time, the minimum of the first-order correlation is also found at an orthogonal direction as shown in Fig. 4(b), which validates realization of the teleportation process. Similarly, as h 5 p/4, Bob has measured the maximum of the first-order correlation at w 5 p/4 and the minimum also appears at its orthogonal direction (Fig. 4(c) and (d)). In fact, the above design is suitable for any linear polarization state. The corresponding experiment to teleport a circular polarized initial state was also performed by using the above scheme, similar teleportation process has been realized (see Supplementary). In the teleportation process, it is not necessary for Alice to know where Bob is, the initial polarization state can be unknown to anyone not only Alice. Furthermore, the transfer of information from Alice to Bob can happen over arbitrary distances. These properties for the teleportation in the presented scheme completely agree with those described by D. Bouwmeester et. al., which this work' results focus on 2 . Discovery of other schemes for quantum teleportation by using classical optics warrants further study.

Conclusions
We have demonstrated experimentally the non-local classical optical correlation from the Bell's measurement used in tests of quantum non-locality. Based on such a classical EPR correlation, a classical analogy has been implemented to the true meaning of quantum teleportation. The presented results indicate that some non-local phenomena in quantum machines can be realized in classical optical signal processing. Thus, this study opens a new way to obtain the quantum information process in the classical optical communication network. 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 and quantum information processes.

Methods
Measurement method for the first-order field correlation. In general, the polarization-entangled photon pairs can be produced from spontaneous downconversion of the nonlinear crystal. In such a process, the photon states forĥ andv polarizations are generated with a certain probability at the same time, which the entangled properties can be measured by the coincidence counts. Based on the coherence of light field, we can construct the corresponding non-local classical EPR correlation states as described by Eq.(4), which the correlated properties can be shown by the first-order field correlation. Basically, the measure of the first-order field correlation can be realized by synchronous local measurements and doesn't require direct contact. However, in the experiment it is difficult to obtain the first-order field correlation by directly measuring fields. In addition, the synchronization is very hard to achieve. So, we take the following method, that is, take E ahr ,t ð Þ and E bhr ,t ð Þ for example and do the following Hadamard transformation: then where Re{X} represents the real part of X. From the experimental setup in Fig.1, we can find it is easy to find the following relation: Then That is, the first-order field correlation can be measured by the difference of light intensities and this method is also effective in other measurements of experiments. The above process can be realized by the beam splitter as shown in Figure 5. In the experiment, the same polarization for the light beams is required.
Realization of Bell-state measurement for classical optical entangled states. Bellstate measurement is a key part in the teleportation scheme. In our teleportation scheme, we perform the Bell-state measurement referring to three-photon scheme described in Ref. 2 as shown in Figure 6. The marks p/2 and p/4 in Figure 6 represent the angle between the axis of HWP and the horizontal direction. The functions of two HWPs are to realize interchange betweenĥ andv polarizations, add a p phase for the component ofĥ polarization field. The Jones matrix for the BS is taken in the following form: Two HWPs are put at the output port of the difference. If the input ports are in four Bell-states, the results by the correlation measurement of the first-order field are given in the table 1. Taking the antisymmetric state as an example, in the following we give a demonstration on such a result. The antisymmetric Bell state is jy { Þ~1ffiffi ffi 2 p jhÞ a jvÞ b {jvÞ a jhÞ b À Á : The corresponding field isẼ After passing through the BS, they become Consider the output of the difference, perform interchange betweenĥ andv polarizations and add a p phase for the component ofĥ polarization field by the HWPs, we have