Beating the channel capacity limit for superdense coding with entangled ququarts
Xiao-Min Hu Yu Guo Bi-Heng Liu Yun-Feng Huang Chuan-Feng Li Guang-Can Guo









CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, People's Republic of China
These authors contribute equally to this work


CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, People's Republic of China
These authors contribute equally to this work


bhliu@ustc.edu.cn
CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, People's Republic of China


CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, People's Republic of China


cfli@ustc.edu.cn
CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, People's Republic of China


CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, People's Republic of China




Quantum superdense coding protocols enhance channel capacity by using shared quantum entanglement between two users. The channel capacity can be as high as 2 when one uses entangled qubits. However, this limit can be surpassed by using high-dimensional entanglement. Here, we report an experiment that exceeds the limit using high-quality entangled ququarts with fidelities up to 0.98, demonstrating a channel capacity of 2.09±0.01. The measured channel capacity is also higher than that obtained when transmitting only one ququart. We use the setup to transmit a five-color image with a fidelity of 0.952. Our experiment shows the great advantage of high-dimensional entanglement and will stimulate research on high-dimensional quantum information processes.


Beating the channel capacity limit for superdense coding with entangled ququarts
    Guang-Can Guo
    December 30, 2023
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§ INTRODUCTION

Quantum entanglement <cit.> reveals the power of the quantum world. An important application of quantum entanglement is quantum superdense coding (SDC) <cit.>, relying on the shared quantum entanglement between Alice (sender) and Bob (receiver). Alice encodes 2 classical bits of information on her qubit and then sends the qubit to Bob; Bob performs a complete Bell-state measurement on the two qubits and decodes the 2 classical bits. During the process, Alice only sends 1 qubit to Bob, while Bob receives 2 bits of classical information. This protocol has been realized in many systems <cit.>. Considering the photon as the natural flying qubit, the performance of SDC in a linear optical system attracts much attention. However, a complete Bell-state measurement with linear optics is forbidden <cit.>, and the theoretical channel capacity limit of 2 is hard to reach. Previous efforts were made to reach this limit <cit.> even in the presence of noise <cit.>.

Compared with two-qubit entanglement, high-dimensional entanglement is more powerful and increases the channel capacity <cit.>. Recently, high-dimensional entanglement has been demonstrated in optical systems <cit.>. Based on a photon's orbital angular momentum, Zeilinger's group also demonstrated the preparation of a complete four-dimensional Bell-basis <cit.> and quantum gates on single high-dimensional photons <cit.>. These techniques pave the way to high-dimensional quantum communication. However, the fidelity of the high-dimensional Bell-state in the previous works is slightly lower, such as 0.78±0.03 <cit.>, and thus, it is hard to carry out the SDC protocol. On the other hand, we have used beam displacers to build very stable Mach-Zehnder interferometers <cit.> and obtained a high-quality three-dimensional path entanglement with a fidelity of 0.975±0.001 <cit.>. In this article, we report on an experimental demonstration of SDC using path-polarization-encoded entanglement. By encoding the spatial mode and polarization mode of the photon, the system is more compact than the previous one <cit.>, and the fidelity of the four-dimensional entangled state is up to 0.980±0.001. The measured channel capacity is 2.09±0.01, which exceeds the limit of 2 from the qubit system for the first time <cit.>. Moreover, the channel capacity is also higher than the limit of transmitting one ququart <cit.>. We use the setup to transmit a real image and observe a fidelity of 0.952. This makes the efficient SDC using high-dimensional entanglement feasible in the future.



§ RESULTS







To generate high-quality, high-dimensional entangled states, we use a path-polarization hybrid system as shown in Figure 1. We encode horizontally polarized (H) photons in path a1 (a3) as |0⟩, vertically polarized (V) photons in path a1 (a3) as |1⟩, H photons in path a2 (a4) as |2⟩, V photons in path a2 (a4) as |3⟩ (here |0⟩, |1⟩, |2⟩ and |3⟩ represent high-dimensional bases, not Fock states). Thus, the two-photon state is Ψ_11=(|00⟩+|11⟩+|22⟩+|33⟩)/2. Then, Alice uses four computer-controlled liquid crystal variable retarders (LCs, which introduces birefringence between fast axis and slow axis depending on the voltage applied on it) to perform local operations on her own photon and sends it to Bob for Bell-state measurement (Details can be found in Materials and Methods). Although, it is impossible to perform a complete Bell-state measurement on a four-dimensional system with linear optics <cit.>, we can separate the 16 Bell-states into seven classes and select only one state in each class for SDC with the help of photon-number resolving detectors <cit.>. In our system, we choose five states for SDC

    Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩),  
    Ψ_12=1/2(|00⟩-|11⟩+|22⟩-|33⟩),  
    Ψ_13=1/2(|00⟩+|11⟩-|22⟩-|33⟩),  
    Ψ_14=1/2(|00⟩-|11⟩-|22⟩+|33⟩),  
    Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩).


The channel capacity using five-state SDC is log_25≃2.32, which is higher than the limit of 2 set by using entangled qubits and is also higher than the limit of transmitting one qudit. By carefully checking the coincidental events among those single photon detectors, Bob can determine which state Alice prepared, details see the Supplementary Materials. For example, coincidence between D1 and D5 means the state Alice prepared is Ψ_11.









First, we test the state preparation and Bell-state measurement. Alice sends each state to Bob, and Bob performs a measurement and checks the success probabilities. The results are shown in Figure 2. For states Ψ_11, Ψ_12, Ψ_13 and Ψ_14, the average success probability is 0.926±0.002, and for state Ψ_23, the success probability is 0.972±0.002. This is because the first four states suffer from imperfect Hong-Ou-Mandel interference, while the last is just a polarization projection. In our experiment, the measured visibility of Hong-Ou-Mandel interference is 0.962±0.002 when using a 10-mm-long ppKTP crystal. Hence, the channel capacity in our system is measured to be 2.09±0.01, which is slightly lower than the theoretical prediction of 2.32 but still exceeds the limit when using the entangled qubit system. When one transmits one ququart, the channel capacity is log_24=2. Our result also exceeds this limit. This means that our experiment proves that high-dimensional SDC can increase the channel capacity; thus, it is possible to perform quantum communication via efficient SDC even in a high-dimensional system.







Then, we use the encoded information to transmit a real image as shown in Figure 3. Alice encodes a five-color image on her own photon (each color corresponds to a four-dimensional Bell-state, and she can transform the initial state to the others by four-computer-controlled LCs. Alice's encoding process can be found in Materials and Methods) and then transmits the encoded photon to Bob. Bob receives the photon and performs a measurement on the two photons to determine which state Alice encoded and then decodes the information. In our experiment, Alice performs single photon operation on her own photon by four computer-controlled LCs and sends it to Bob, this process will take 0.5 second; then, Bob starts to measure the two-photon state after 0.5 second until he obtains a two-fold coincident event within 1 second. These process is repeated with a repetition rate of 0.5 Hz and the SDC data rate is 0.5 Hz. Theoretically, our setup can distinguish all five of the four-dimensional Bell-states with a single-shot measurement when the detection efficiency is 100%. However, the overall detection efficiency is 0.109±0.002 in our experiment due to the loss of detectors (approximate 50% detection efficiency at 808 nm) and coupling efficiency (approximate 22% including the losses of optical elements), thus Bob needs more time to ensure that he has completed the measurement.




§ DISCUSSION


SDC with high dimensional entanglement is promising in the future. Note that since the photon pairs are generated from ppKTP, the overall detection efficiency can be as high as 78.6% when using careful alignment and superconductor detectors <cit.>. The computer-controlled LC operation rate is approximately 30 Hz; this part can be replaced by a fast electro-optical modulator with an operation rate of 1 GHz. Together with a pulsed laser source with a repetition rate of tens of MHz or even faster, the SDC data rate can be as high as 1 MHz. The path-polarization hyperentanglement can also be distributed to distant users by a multi-core fibre <cit.>, and our SDC with high-dimensional entanglement is feasible in the future for fast, distant quantum communication. Further-more, path-polarization hyperentanglement is possible to connect to integrated photonic chips <cit.>.

In conclusion, we demonstrate the SDC protocol using four-dimensional entanglement for the first time and achieve a channel capacity of 2.09±0.01, which exceeds the limit of SDC using two-dimensional entanglement and the limit from transmitting one ququart. This experiment will stimulate the research of applications based on high-dimensional entanglement.



§ MATERIALS AND METHODS



Alice's encoding process.
The initial state shared by Alice and Bob is Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩), and Alice performs single-photon operations on her own photon and realizes the state transformation from Ψ_11 to the other 4 Bell-states. These operations are realized by four computer-controlled LCs as shown in Figure 4. Then, Alice can easily control the state by applying different voltages on the LCs.
If all the LCs are set to introduce 0 phase, the two-photon state is

    Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩).

If LC2 is set to introduce π phase, while the others remain at 0, then the two-photon state is

    Ψ_12=1/2(|00⟩-|11⟩+|22⟩-|33⟩).

If LC2, LC3 and LC4 are set to introduce π phase, while LC1 remains at 0, then the two-photon state is

    Ψ_13=1/2(|00⟩+|11⟩-|22⟩-|33⟩).

If LC3 and LC4 are set to introduce π phase, while LC1 and LC2 remain at 0, then the two-photon state is

    Ψ_14=1/2(|00⟩-|11⟩-|22⟩+|33⟩).

If all the LCs are set to introduce π phase, then the two-photon state is

    Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩).







Bell-state measurement.
A complete four-dimensional Bell-state measurement is impossible with linear elements <cit.>. However, one can separate the 16 four-dimensional Bell-states into seven classes and select only one state from each class for SDC <cit.>. In our experiment, we only choose five states given in Eq. <ref>. Then Bob can determine which state Alice prepared by the measurement setup shown in Fig. 1. When the two photons inject into PBS1, they are separated into two classes by photon's polarization. For state Ψ_11, Ψ_12, Ψ_13, Ψ_14, one photon is reflected and the other one is transmitted, while for state Ψ_23, both photons are reflected or transmitted. After half wave plate H1 (set at 22.5^∘) and PBS2, state Ψ_11, Ψ_12, Ψ_13, Ψ_14 are separated into two classes by the relative phase between |00⟩ and |11⟩, for example, if the relative phase is 0 (Ψ_11, Ψ_13), then the two photons will both transmit or reflect from PBS2. Here, this separation depends on the two-photon Hong-Ou-Mandel interference <cit.>. Finally, the last part of measurement setup (consists of HWPs, BDs and PBSs) will separate Ψ_11 and Ψ_13 (Ψ_12 and Ψ_14) to two classes by the relative phase between |00⟩ and |22⟩. By carefully checking the coincidental events among those single photon detectors, Bob can determine which state Alice prepared. For example, coincidence between D1 and D5 means the state Alice prepared is Ψ_11. Thus, Bob can determine which state Alice prepared and decode the information that Alice encoded.

Channel capacity.
The channel capacity <cit.> is the maximal mutual information H(x;y). In our experiment, the mutual information can be calculated as

    H(x;y)=∑_y=0^4∑_x=0^4p(x)p(y|x)log(p(y|x)/∑_x=0^4p(x)p(y|x)),

where p(x)={p(Ψ_11),p(Ψ_12),p(Ψ_13),p(Ψ_14),p(Ψ_23)}. The channel capacity in our experiment is 2.09±0.01, which is slightly smaller than the limit of 2.32 owing to the imperfect four-dimensional Bell-state measurement.









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§ ACKNOWLEDGMENTS

Funding: This work was supported by the National Key Research and Development Program of China (No. 2017YFA0304100), NNSFC (Nos. 11374288, 11774335, 61327901, 11474268, 11325419, 11504253), the Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSW-SLH003), the Fundamental Research Funds for the Central Universities, and Anhui Initiative in Quantum Information Technologies (Nos. AHY020100, AHY060300). Author contributions: The experiment was performed by X-M.H., B-H.L., Y.G.. The data analysis was performed by X-M.H., B-H.L. and C-F.L. with input from all authors. The paper was written by X-M.H., B-H.L., and C-F.L. All authors discussed the experimental results. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.








§ APPENDIX





Two-photon Four-dimensional Bell-basis.— For a four-dimensional bipartite system, the complete Bell-basis can be written as <cit.>

    Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩),  
    Ψ_12=1/2(|00⟩-|11⟩+|22⟩-|33⟩),  
    Ψ_13=1/2(|00⟩+|11⟩-|22⟩-|33⟩),  
    Ψ_14=1/2(|00⟩-|11⟩-|22⟩+|33⟩),  
    Ψ_21=1/2(|01⟩+|10⟩+|23⟩+|32⟩),  
    Ψ_22=1/2(|01⟩-|10⟩+|23⟩-|32⟩),  
    Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩),  
    Ψ_24=1/2(|01⟩-|10⟩-|23⟩+|32⟩),  
    Ψ_31=1/2(|02⟩+|13⟩+|20⟩+|31⟩),  
    Ψ_32=1/2(|02⟩-|13⟩+|20⟩-|31⟩),  
    Ψ_33=1/2(|02⟩+|13⟩-|20⟩-|31⟩),  
    Ψ_34=1/2(|02⟩-|13⟩-|20⟩+|31⟩),  
    Ψ_41=1/2(|03⟩+|12⟩+|21⟩+|30⟩),  
    Ψ_42=1/2(|03⟩-|12⟩+|21⟩-|30⟩),  
    Ψ_43=1/2(|03⟩+|12⟩-|21⟩-|30⟩),  
    Ψ_44=1/2(|03⟩-|12⟩-|21⟩+|30⟩).


Bell-state measurement.— In the quantum superdense coding protocol, one need to distinguish all the 16 Bell-basis in a single-shot measurement. However, this is impossible with linear optics <cit.>. A reliable way is to select some of the Bell-basis and determine the basis by a single-shot measurement. Here, we show how to select the special bases and how to distinguish them with linear optics <cit.>.

The experimental setup is shown in Fig. A1.



For each input four-dimensional two-photon state, the measurement result can be described as following.

1. If the input state is Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩), then we can rewrite the state before each line

    Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩)  
       1/2(|H_a1H_a3⟩+|V_a1V_a3⟩+|H_a2H_a4⟩+|V_a2V_a4⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b3+V_b3⟩)+1/2(|H_b1-V_b1⟩ |H_b3-V_b3⟩)) 
       +1/2(|H_b2+V_b2⟩ |H_b4+V_b4⟩)+1/2(|H_b2-V_b2⟩ |H_b4-V_b4⟩)   
       =1/4(|H_b1H_b3⟩+|H_b1V_b3⟩+|V_b1H_b3⟩+|V_b1V_b3⟩+|H_b1H_b3⟩-|H_b1V_b3⟩-|V_b1H_b3⟩+|V_b1V_b3⟩
       +|H_b2H_b4⟩+|H_b2V_b4⟩+|V_b2H_b4⟩+|V_b2V_b4⟩+|H_b2H_b4⟩-|H_b2V_b4⟩-|V_b2H_b4⟩+|V_b2V_b4⟩)  
       =1/2(|H_b1H_b3⟩+|V_b1V_b3⟩+|H_b2H_b4⟩+|V_b2V_b4⟩) 
       1/2(|H_c1H_c5⟩+|V_c2V_c6⟩+|H_c4H_c7⟩+|V_c3H_c8⟩) 
       1/2(|H_d1H_d3⟩+|V_d1V_d3⟩+|H_d2H_d4⟩+|V_d2H_d4⟩) 
        C_D1,D5+C_D2,D6+C_D3,D7+C_D8,D4,

here, H_a1 means a horizontally polarized photon at mode a1, C_D1,D5 means two folds coincidence between detector D1 and D5. One can find that if the input state is Ψ_11, then there have only four possible coincidental events, such as, C_D1,D5, C_D2,D6, C_D3,D7 and C_D8,D4.

2. If the input state is Ψ_12=1/2(|00⟩-|11⟩+|22⟩-|33⟩), then we can rewrite the state before each line

    Ψ_12=1/2(|00⟩-|11⟩+|22⟩-|33⟩)  
       1/2(|H_a1H_a3⟩-|V_a1V_a3⟩+|H_a2H_a4⟩-|V_a2V_a4⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b3+V_b3⟩)-1/2(|H_b1-V_b1⟩ |H_b3-V_b3⟩)) 
       +1/2(|H_b2+V_b2⟩ |H_b4+V_b4⟩)-1/2(|H_b2-V_b2⟩ |H_b4-V_b4⟩)   
       =1/4(|H_b1H_b3⟩+|H_b1V_b3⟩+|V_b1H_b3⟩+|V_b1V_b3⟩-|H_b1H_b3⟩+|H_b1V_b3⟩+|V_b1H_b3⟩-|V_b1V_b3⟩
       +|H_b2H_b4⟩+|H_b2V_b4⟩+|V_b2H_b4⟩+|V_b2V_b4⟩-|H_b2H_b4⟩+|H_b2V_b4⟩+|V_b2H_b4⟩-|V_b2V_b4⟩)  
       =1/2(|H_b1V_b3⟩+|V_b1H_b3⟩+|H_b2V_b4⟩+|V_b2H_b4⟩) 
       1/2(|H_c1H_c7⟩+|V_c2V_c8⟩+|H_c4H_c5⟩+|V_c3V_c6⟩) 
       1/2(|H_d1H_d4⟩+|V_d1V_d4⟩+|H_d2H_d3⟩+|V_d2V_d3⟩) 
        C_D1,D7+C_D2,D8+C_D5,D3+C_D6,D4.


3. If the input state is Ψ_13=1/2(|00⟩+|11⟩-|22⟩-|33⟩), then we can rewrite the state before each line

    Ψ_13=1/2(|00⟩+|11⟩-|22⟩-|33⟩)  
       1/2(|H_a1H_a3⟩+|V_a1V_a3⟩-|H_a2H_a4⟩-|V_a2V_a4⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b3+V_b3⟩)+1/2(|H_b1-V_b1⟩ |H_b3-V_b3⟩)) 
       -1/2(|H_b2+V_b2⟩ |H_b4+V_b4⟩)-1/2(|H_b2-V_b2⟩ |H_b4-V_b4⟩)   
       =1/4(|H_b1H_b3⟩+|H_b1V_b3⟩+|V_b1H_b3⟩+|V_b1V_b3⟩+|H_b1H_b3⟩-|H_b1V_b3⟩-|V_b1H_b3⟩+|V_b1V_b3⟩
       -|H_b2H_b4⟩-|H_b2V_b4⟩-|V_b2H_b4⟩-|V_b2V_b4⟩-|H_b2H_b4⟩+|H_b2V_b4⟩+|V_b2H_b4⟩-|V_b2V_b4⟩)  
       =1/2(|H_b1H_b3⟩+|V_b1V_b3⟩-|H_b2H_b4⟩-|V_b2V_b4⟩) 
       1/2(|H_c1H_c5⟩-|V_c2V_c6⟩+|H_c4H_c7⟩-|V_c3V_c8⟩) 
       1/2(|H_d1H_d3⟩-|V_d1V_d3⟩+|H_d2H_d4⟩-|V_d2V_d4⟩) 
        C_D1,D6+C_D2,D5+C_D8,D4+C_D3,D7.


4. If the input state is Ψ_14=1/2(|00⟩-|11⟩-|22⟩+|33⟩), then we can rewrite the state before each line

    Ψ_14=1/2(|00⟩-|11⟩-|22⟩+|33⟩)  
       1/2(|H_a1H_a3⟩-|V_a1V_a3⟩-|H_a2H_a4⟩+|V_a2V_a4⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b3+V_b3⟩)-1/2(|H_b1-V_b1⟩ |H_b3-V_b3⟩)) 
       -1/2(|H_b2+V_b2⟩ |H_b4+V_b4⟩)+1/2(|H_b2-V_b2⟩ |H_b4-V_b4⟩)   
       =1/4(|H_b1H_b3⟩+|H_b1V_b3⟩+|V_b1H_b3⟩+|V_b1V_b3⟩-|H_b1H_b3⟩+|H_b1V_b3⟩+|V_b1H_b3⟩-|V_b1V_b3⟩
       -|H_b2H_b4⟩-|H_b2V_b4⟩-|V_b2H_b4⟩-|V_b2V_b4⟩+|H_b2H_b4⟩-|H_b2V_b4⟩-|V_b2H_b4⟩+|V_b2V_b4⟩)  
       =1/2(|H_b1V_b3⟩+|V_b1H_b3⟩-|H_b2V_b4⟩-|V_b2H_b4⟩) 
       1/2(|H_c1H_c7⟩+|V_c2V_c8⟩-|H_c4H_c5⟩-|V_c3V_c6⟩) 
       1/2(|H_d1H_D4⟩-|V_d1V_d4⟩+|H_d2H_d3⟩-|V_d2V_d3⟩) 
        C_D1,D8+C_D2,D7+C_D4,D5+C_D3,D6.


5. If the input state is Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩), then we can rewrite the state before each line

    Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩)  
       1/2(|H_a1V_a3⟩+|V_a1H_a3⟩-|H_a2V_a4⟩-|V_a2H_a4⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b1-V_b1⟩)-1/2(|H_b2+V_b2⟩ |H_b2-V_b2⟩)) 
       +1/2(|H_b3+V_b3⟩ |H_b3-V_b3⟩)-1/2(|H_b4+V_b4⟩ |H_b4-V_b4⟩)   
       =1/√(8)(|H_b1H_b1⟩-|V_b1V_b1⟩-|H_b2H_b2⟩+|V_b2V_b2⟩+|H_b3H_b3⟩-|V_b3V_b3⟩-|H_b4H_b4⟩+|V_b4V_b4⟩)  
       1/√(8)(|H_c1H_c1⟩-|V_c2V_c2⟩-|H_c4H_c4⟩+|V_c3V_c3⟩+|H_c5H_c5⟩-|V_c6V_c6⟩-|H_c7H_c7⟩+|V_c8V_c8⟩) 
       1/2(|H_d1V_d1⟩-|H_d2V_d2⟩+|H_d3V_d3⟩-|H_d4V_d4⟩) 
        C_D1,D2+C_D3,D4+C_D5,D6+C_D7,D8.


6. If the input state is Ψ_43=1/2(|03⟩-|21⟩+|12⟩-|30⟩), then we can rewrite the state before each line

    Ψ_43=1/2(|03⟩-|21⟩+|12⟩-|30⟩)  
       1/2(|H_a1V_a4⟩-|H_a2V_a3⟩+|H_a4V_a1⟩-|V_a2H_a3⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b2-V_b2⟩)-1/2(|H_b2+V_b2⟩ |H_b1-V_b1⟩) 
       +1/2(|H_b3-V_b3⟩ |H_b4+V_b4⟩)-1/2(|H_b4-V_b4⟩ |H_b3+V_b3⟩)   
       =1/√(8)(|H_b2V_b1⟩-|H_b1V_b2⟩+|H_b3H_b4⟩-|H_b4V_b3⟩)  
       1/2(|V_c2H_c4⟩-|H_c1V_c3⟩+|H_c5V_c8⟩-|V_c6H_c7⟩) 
       1/2(|V_d1H_D2⟩-|H_d1V_d2⟩+|H_d3V_d4⟩-|H_d4V_d3⟩) 
        C_D2,D3+C_D1,D4+C_D5,D8+C_D6,D7.


7. If the input state is Ψ_21=1/2(|01⟩+|10⟩+|23⟩+|32⟩), then we can rewrite the state before each line

    Ψ_21=1/2(|01⟩+|10⟩+|23⟩+|32⟩)  
       1/2(|H_a1V_a3⟩+|V_a1H_a3⟩+|H_a2V_a4⟩+|V_a2H_a4⟩)  
       1/2(1/2(|H_b1+V_b1⟩ |H_b1-V_b1⟩)+1/2(|H_b2+V_b2⟩ |H_b2-V_b2⟩) 
       +1/2(|H_b3+V_b3⟩ |H_b3-V_b3⟩)+1/2(|H_b4+V_b4⟩ |H_b4-V_b4⟩)   
       =1/√(8)(|H_b1H_b1⟩-|V_b1V_b1⟩+|H_b2H_b2⟩-|V_b2V_b3⟩+|H_b3H_b3⟩-|V_b3V_b3⟩+|H_b4H_b4⟩-|V_b4V_b3⟩
       1/√(8)(|H_c1H_c1⟩+|V_c2V_c2⟩-|H_c4H_c4⟩-|V_c3V_c2⟩+|H_c5H_c5⟩+|V_c6V_c6⟩-|H_c7H_c7⟩-|V_c8V_c8⟩) 
       1/√(8)(|V_d1V_d1⟩+|H_d1H_d1⟩-|V_d2V_d2⟩-|H_d2H_d2⟩+|V_d3V_d3⟩+|H_d3H_d3⟩-|V_d4V_d4⟩-|H_d4H_d4⟩) 
        C_D1,D1+C_D2,D2+C_D3,D3+C_D4,D4+C_D5,D5+C_D6,D6+C_D7,D7+C_D8,D8,

here C_D1,D1 means two photons will arrive at the same detector D1. If one has photon-number-resolving detectors, one can determine the state is Ψ_21 by the measurement result.

Hence, we can divide all the 16 states into 7 classes, such as

1.

    Ψ_11=1/2(|00⟩+|11⟩+|22⟩+|33⟩),  
    Ψ_31=1/2(|02⟩+|13⟩+|20⟩+|31⟩),

the measured evidence is C_D1,D5+C_D2,D6+C_D3,D7+C_D8,D4,

2.

    Ψ_12=1/2(|00⟩-|11⟩+|22⟩-|33⟩),  
    Ψ_32=1/2(|02⟩-|13⟩+|20⟩-|31⟩),

the measured evidence is C_D1,D7+C_D2,D8+C_D5,D3+C_D6,D4,

3.

    Ψ_13=1/2(|00⟩+|11⟩-|22⟩-|33⟩),  
    Ψ_33=1/2(|02⟩+|13⟩-|20⟩-|31⟩),

the measured evidence is C_D1,D6+C_D2,D5+C_D8,D4+C_D3,D7,

4.

    Ψ_14=1/2(|00⟩-|11⟩-|22⟩+|33⟩),  
    Ψ_34=1/2(|02⟩-|13⟩-|20⟩+|31⟩),

the measured evidence is C_D1,D8+C_D2,D7+C_D4,D5+C_D3,D6,


5.

    Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩),  
    Ψ_24=1/2(|01⟩-|10⟩-|23⟩+|32⟩),

the measured evidence is C_D1,D2+C_D3,D4+C_D5,D6+C_D7,D8,

6.

    Ψ_43=1/2(|03⟩+|12⟩-|21⟩-|30⟩),  
    Ψ_44=1/2(|03⟩-|12⟩-|21⟩+|30⟩),

the measured evidence is C_D2,D3+C_D1,D4+C_D5,D8+C_D6,D7,

7.

    Ψ_23=1/2(|01⟩+|10⟩-|23⟩-|32⟩),  
    Ψ_24=1/2(|01⟩-|10⟩-|23⟩+|32⟩),  
    Ψ_41=1/2(|03⟩+|12⟩+|21⟩+|30⟩),  
    Ψ_42=1/2(|03⟩-|12⟩+|21⟩-|30⟩),

the measured evidence is C_D1,D1+C_D2,D2+C_D3,D3+C_D4,D4+C_D5,D5+C_D6,D6+C_D7,D7+C_D8,D8.

In our experiment, we choose Ψ_11, Ψ_12, Ψ_13, Ψ_14, Ψ_23 for quantum superdense coding.



In fact, one can distinguish six classes of state even without number-resolving detectors. To encode the first five states for SDC, one needs four computer-controlled liquid crystal variable retarders (LCs) to realize the state transformation. In order to encode the sixth state, one needs other ten LCs and four beam displacers as shown in Fig. A2. This part can realize a single photon four-dimensional identical operation or U gate <cit.>. The matrix form of U gate can be written as

    U=(
          [ 0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0;   ]).

Together with the first four LCs, Alice can encode six states for SDC. For example, if Alice wants to encode the first five states, she can set the second part to realize an identical operation, if she wants to encode the sixth state Ψ_43, she needs to transform the initial state Ψ_11 to Ψ_13 by the first part, and then realize a U gate by the second part. However, the second part is too complicated and we only use five states in our experiment as a demonstration.
