==== Front Sci Rep Sci Rep Scientific Reports 2045-2322 Nature Publishing Group UK London 78529 10.1038/s41598-020-78529-2 Article Hyperentanglement concentration of nonlocal two-photon six-qubit systems via the cross-Kerr nonlinearity Liu Qian liuhan139@yeah.net 1 Song Guo-Zhu 2 Qiu Tian-Hui 1 Zhang Xiao-Min 1 Ma Hong-Yang 1 Zhang Mei 3 1 grid.412609.80000 0000 8977 2197Research Center for Quantum Optics and Quantum Communication, School of Science, Qingdao University of Technology, Qingdao, 266525 China 2 grid.412735.60000 0001 0193 3951College of Physics and Materials Science, Tianjin Normal University, Tianjin, 300387 China 3 grid.20513.350000 0004 1789 9964Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing, 100875 China 8 12 2020 8 12 2020 2020 10 2144418 8 2020 26 11 2020 © The Author(s) 2020Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.We present an efficient hyperentanglement concentration protocol (hyper-ECP) for two-photon six-qubit systems in nonlocal partially hyperentangled Bell states with unknown parameters. In our scheme, we use two identical partially hyperentangled states which are simultaneously entangled in polarization and two different longitudinal momentum degrees of freedom (DOFs) to distill the maximally hyperentangled Bell state. The quantum nondemolition detectors based on the cross-Kerr nonlinearity are used to realize the parity checks of two-photon systems in three DOFs. The hyper-ECP can extract all the useful entanglement source, and the success probability can reach the theory limit with the help of iteration. All these advantages make our hyper-ECP useful in long-distance quantum communication in the future. Subject terms Quantum informationQuantum mechanicsNational Natural Science Foundation of China11704214119470371160417411605100Program for Innovative Research in University of TianjinTD13-5077http://dx.doi.org/10.13039/501100015642Project of Shandong Province Higher Educational Science and Technology ProgramJ18KZ012issue-copyright-statement© The Author(s) 2020 ==== Body Introduction As one of the striking features of quantum information, entanglement, has been widely used in quantum information processing, such as quantum teleportation1, controlled teleportation2,3, quantum dense coding4,5, quantum key distribution6–9, quantum secret sharing10–12, quantum state sharing13,14, quantum secure direct communication15–20, and so on. Single photons are interesting candidates for quantum communication, and they can carry quantum information in several degrees of freedom (DOFs). The entanglement in which photons are simultaneously entangled in more than one DOF could be called hyperentanglement. Many kinds of hyperentanglement have been discussed, such as polarization-momentum21, polarization-time-bin22, polarization-frequency23, and polarization-orbital-angular-momentum24. The hyperentangled Bell states in both the polarization and two different longitudinal momentum modes DOFs have been introduced in experiments25,26. Hyperentanglement of photon system can increase both the channel capacity of long-distance quantum communication24and its security. It can also be used on complete Bell states analysis21,22,27,28, high-speed quantum computation29,30, superdense coding31, quantum key distribution32, etc. However, the entanglement and the fidelity of the entangled systems inevitably degrade because of the interaction with the environment during the storage and transmission. One of the methods to depress the noise effect on entangled systems is entanglement concentration. It can be used to extract the maximally entangled state from a large number of less entangled pure states. In 1996, Bennett et al.33 proposed the first entanglement concentration protocol (ECP) for two-photon system, which is called as Schmidt projection method. They utilized collective measurement to obtain the coefficient information. As it requires the collective measurement on multiple particles simultaneously, it is difficult to manipulate in experiment. Later, Bose et al.34 designed an efficient ECP with entanglement swapping. In 2000, Shi et al.35 proposed an ECP based on entanglement swapping and collective two-qubit unitary evolution. In 2001, Yamamoto et al.36 and Zhao et al.37 independently presented two ECPs assisted by linear-optical elements and postselection. In 2008, Sheng et al.38 presented an interesting ECP that exploited cross-Kerr nonlinearity, which had a higher efficiency and yield than those with linear optical elements with the help of iteration. Motivated by those innovation works, many interesting ECPs have been presented and discussed for different physical systems and different entangled states39–48. The concentration of hyperentangled states also has been extensively studied in recent years. In 2013, Ren, Du, and Deng49 gave the first hyperentanglement concentration protocol (hyper-ECP) for two-photon four-qubit systems with linear optics. In 2014, Ren and Long50 proposed another hyper-ECP for nonlocal partially hyperentangled Bell states in polarization-spatial mode DOFs assisted by nonlinear interactions. In 2015, Li and Ghose51 presented two hyper-ECPs for time-bin and polarization hyperentangled states with unknown parameters and known parameters, respectively. In 2016, Cao et al.52 presented a hyper-ECP utilizing photonic module system. In 2017, Wang and Ren et al. gave two efficient hyper-ECPs for polarization-spatial-time-bin hyperentangled two-photon six-qubit systems53,54. The hyper-ECP for three-photon partially hyperentangled GHZ states in polarization, spatial-mode and time-bin DOFs with linear optics was also proposed in the next year55. The hyper-ECP for polarization-spatial-time-bin hyperentangled Bell states using cross-Kerr nonlinearity has also been discussed56. Although many hyper-ECPs have been presented, the concentration of two-photon six-qubit hyperentangled Bell states in both the polarization and the double longitudinal momentum modes DOFs has not been researched. In this article, we present an efficient hyper-ECP for partially hyperentangled Bell states of two-photon six-qubit systems. Two photons considered in our hyper-ECP are simultaneously entangled in polarization and two longitudinal momentum DOFs, which is not taken into account in other hyper-ECPs. We use the quantum nondemolition detectors (QNDs) to check the parity of the two-photon in three DOFs to implement our protocol, and therefore the unsuccessful instances in each round can be reused in the next concentration round. The success probability in our scheme can be greatly improved and the maximum of the success probability is nearly 100% with the help of iteration. Moreover, it does not require that the parties know the exact information about the partially hyperentangled Bell states. These good features make our scheme efficient and useful for quantum information processing involving hyperentanglement. Results The hyper-ECP process for two-photon six-qubit systems The hyperentangled Bell state of two-photon six-qubit systems in three DOFs can be described as follows26: 1 |HE6⟩=12(|H⟩A|H⟩B+|V⟩A|V⟩B)⊗12(|l⟩A|r⟩B+|r⟩A|l⟩B)⊗12(|I⟩A|I⟩B+|E⟩A|E⟩B). Here the subscripts A and B denote the two photons. The three independent DOFs are polarization (H/V) and a double longitudinal momentum (r/l and E/I). H and V represent the horizontal and vertical polarization of photons, respectively. l (r) represents the left (right) mode and E (I) represents the external (internal) mode. The four Bell states in the polarization DOF of two-photon systems can be written as 2 |ϕ±⟩ABP=12(|H⟩A|H⟩B±|V⟩A|V⟩B), 3 |ψ±⟩ABP=12(|H⟩A|V⟩B±|V⟩A|H⟩B), and four Bell states in the first longitudinal momentum DOF are 4 |ϕ±⟩ABF=12(|l⟩A|l⟩B±|r⟩A|r⟩B), 5 |ψ±⟩ABF=12(|l⟩A|r⟩B±|r⟩A|l⟩B), while the four Bell states in the second longitudinal momentum DOF can be denoted as 6 |ϕ±⟩ABS=12(|I⟩A|I⟩B±|E⟩A|E⟩B), 7 |ψ±⟩ABS=12(|I⟩A|E⟩B±|E⟩A|I⟩B). Here |ϕ±⟩ABi (i=P,F,S) is the even-parity state of photons in i DOF, while |ψ±⟩ABj (j=P,F,S) is the odd-parity state of photons in j DOF, the superscripts P, F, and S denote the polarization, the first longitudinal momentum, and the second longitudinal momentum DOFs of a two-photon six-qubit system, respectively. In long-distance quantum communication, the maximally hyperentangled Bell state |HE6⟩ may decay to a partially hyperentangled Bell state |ψ⟩AB after passing through the noisy channels. Here, 8 |ψ⟩AB=(α|H⟩A|H⟩B+β|V⟩A|V⟩B)⊗(γ|l⟩A|r⟩B+δ|r⟩A|l⟩B)⊗(ϵ|I⟩A|I⟩B+ε|E⟩A|E⟩B). The parameters α, β, γ, δ, ϵ, ε are unkown and satisfy the normalization condition |α|2+|β|2=|γ|2+|δ|2=|ϵ|2+|ε|2=1. In order to realize the hyperentanglement concentration of the unknown partially hyperentangled Bell state |ψ⟩AB, we use the QNDs shown in Figs. 1, 2, and 3. The P-QND shown in Fig. 1 has been described in Ref.57, and the other two QNDs are improved based on the schemes shown in Ref.57. Based on the principle of the cross-Kerr effect (see the Methods section), if one let the two photons A and B pass through the circuit as P-QND shown in Fig. 1, they can only get two measurement outcomes |α⟩ and |αe±iθ⟩ for the coherent probe beam, with the corresponding polarization states |H⟩A|H⟩B (|V⟩A|V⟩B) and |H⟩A|V⟩B (|V⟩A|H⟩B). It is essentially the polarization parity-check measurement of photons, which can be used to distinguish the even-parity states |ϕ±⟩P from the odd-parity states |ψ±⟩P. F-QND is the first longitudinal momentum parity-check QND, which can be used to distinguish the first longitudinal momentum states |r⟩A|r⟩B (|l⟩A|l⟩B) from |r⟩A|l⟩B (|l⟩A|r⟩B) by different phase shifts 0 and ±θ of the coherent state. The phase shift of the coherent state for the second longitudinal momentum states |I⟩A|I⟩B and |E⟩A|E⟩B is 0, different from phase shift ±θ of the states |I⟩A|E⟩B and |E⟩A|I⟩B, which can be realized in S-QND, the second longitudinal momentum parity-check QND.Figure 1 Schematic diagram of the principle of P-QND constructed by the cross-Kerr nonlinearity, one can distinguish the states |H⟩A|H⟩B and |V⟩A|V⟩B from the states |H⟩A|V⟩B and |V⟩A|H⟩B with different phase shifts ±θ and 0 of the coherent state, respectively. |X⟩⟨X| is the homodyne measurement to distinguish different phase shifts of the coherent probe beam. PBS denotes the polarizing beam splitter which is used to reflect the vertical (V) polarization photon and transmit the horizontal (H) polarization photon, respectively. Figure 2 Schematic diagram of the principle of the F-QND. One can distinguish the states |rr⟩ and |ll⟩ from |lr⟩ and |rl⟩ by the different phase shifts 0 and ±θ of the coherent state. Figure 3 Schematic diagram of S-QND, which is used to distinguish the states |II⟩ and |EE⟩ from |IE⟩ and |EI⟩ by different phase shifts 0 and ±θ of the coherent state. Figure 4 Schematic diagram of our hyper-ECP for a two-photon six-qubit partially hyperentangled Bell state with unknown parameters resorting to cross-Kerr nonlinearity. S1 and S2 are two identical partial hyperentanglement sources. R90 and R45 are half-wave plates, which are used to rotate the polarization of the state by 90o and 45o, respectively. The 50:50 BS is used to accomplish the Hadamard operation for the two longitudinal momentum modes. PBS denotes a polarizing beam splitter which is used to transmit the horizontal polarization component and reflect the vertical polarization component. D represents the single-photon detector. The basic principle of our hyper-ECP for two-photon six-qubit systems in an unknown partially hyperentangled state is shown in Fig. 4. Two photon pairs AB and CD shared by Alice and Bob can be in an identical partially hyperentangled state when they were produced from the same source and passed through the same channel. Thus the state of the photons CD can be denoted as 9 |ψ⟩CD=(α|H⟩C|H⟩D+β|V⟩C|V⟩D)⊗(γ|l⟩C|r⟩D+δ|r⟩C|l⟩D)⊗(ϵ|I⟩C|I⟩D+ε|E⟩C|E⟩D). Before the photons CD are sent to Alice and Bob, the bit-flip operations are performed on the polarization mode of the photons by the half-wave plates R90. The bit-flip operation of the double longitudinal momentum modes of photon C can be achieved by the exchange of the modes |r,E⟩C and |l,I⟩C, |r,I⟩C and |l,E⟩C, respectively. The bit-flip operation of the double longitudinal momentum modes of photon D can be achieved by the same way. Then Bob lets the photons B and D pass through the P-QND, F-QND and S-QND, successively. After the operations above, the whole state of the four photons and three coherent states can evolve into 10 |Ψ⟩ABCD|α⟩1|α⟩2|α⟩3→[(α2|H⟩A|V⟩C|H⟩B|V⟩D+β2|V⟩A|H⟩C|V⟩B|H⟩D)|α⟩1+αβ(|H⟩A|H⟩C|H⟩B|H⟩D|αeiθ⟩1+|V⟩A|V⟩C|V⟩B|V⟩D|αe-iθ⟩1)]⊗[γ2|l⟩A|r⟩C|r⟩B|l⟩D|αeiθ⟩2+δ2|r⟩A|l⟩C|l⟩B|r⟩D|αe-iθ⟩2+γδ(|l⟩A|l⟩C|r⟩B|r⟩D|α⟩2+|r⟩A|r⟩C|l⟩B|l⟩D)|α⟩2]⊗[ϵ2|I⟩A|E⟩C|I⟩B|E⟩D|αe-iθ⟩3+ε2|E⟩A|I⟩C|E⟩B|I⟩D|αeiθ⟩3+ϵε(|I⟩A|I⟩C|I⟩B|I⟩D+|E⟩A|E⟩C|E⟩B|E⟩D)|α⟩3]. The corresponding relation between the states of ABCD, the parity check measurement results of three QNDs and the probability Pi(1) (i=1,2,3,4,5,6,7,8) is shown in Table 1. The superscript “(1)” denotes the first round of concentration. According to the Homodyne measurement results of three QNDs, the eight collapsed states of four-photon system can be divided into four cases.Table 1 The relation between the states of four photons, the parity check measurement results and the probability. State of ABCD P-QND F-QND S-QND Pi(1) |Ψ1⟩ABCD Even Even Even 8|αβγδϵε|2 |Ψ2⟩ABCD Odd Even Even 4|γδϵε|2(|α|4+|β|4) |Ψ3⟩ABCD Even Odd Even 4|αβϵε|2(|γ|4+|δ|4) |Ψ4⟩ABCD Even Even Odd 4|αβγδ|2(|ϵ|4+|ε|4) |Ψ5⟩ABCD Odd Odd Even 2|ϵε|2(|α|4+|β|4)(|γ|4+|δ|4) |Ψ6⟩ABCD odd even odd 2|γδ|2(|α|4+|β|4)(|ϵ|4+|ε|4) |Ψ7⟩ABCD even odd odd 2|αβ|2(|γ|4+|δ|4)(|ϵ|4+|ε|4) |Ψ8⟩ABCD odd odd odd (|α|4+|β|4)(|γ|4+|δ|4)(|ϵ|4+|ε|4) In the first case, all the three parity check measurements give even-parity results. The remaining state of the four-photon system can be described as 11 |Ψ1⟩ABCD=αβP1(1)(|H⟩A|H⟩C|H⟩B|H⟩D+|V⟩A|V⟩C|V⟩B|V⟩D)⊗γδ(|l⟩A|l⟩C|r⟩B|r⟩D+|r⟩A|r⟩C|l⟩B|l⟩D)⊗ϵε(|I⟩A|I⟩C|I⟩B|I⟩D+|E⟩A|E⟩C|E⟩B|E⟩D). The probability that Alice and Bob get the above state is P1(1)=8|αβγδϵε|2. Alice and Bob use R45 to rotate the polarization of the state by 45∘. BSs are used to perform the Hadamard operation on the double longitudinal momentum DOFs of the state. Then the selected term shown in Eq. (11) is transformed into 12 |Ψ1′⟩ABCD=12(|ϕ+⟩ABP|ϕ+⟩CDP+|ϕ-⟩ABP|ψ+⟩CDP)⊗12(|ψ+⟩ABF|ϕ-⟩CDF+|ψ-⟩ABF|ψ-⟩CDF)⊗12(|ϕ+⟩ABS|ϕ+⟩CDS+|ϕ-⟩ABS|ψ+⟩CDS). Table 2 The relation between the states of photon pair CD and the measurement results. States of CD Detectors |ϕ±⟩CDP|ϕ±⟩CDF|ϕ±⟩CDS DC1DD1 DC2DD2 DC3DD3 DC4DD4 DC5DD5 DC6DD6 DC7DD7 DC8DD8 |ϕ±⟩CDP|ϕ±⟩CDF|ψ±⟩CDS DC1DD3 DC3DD1 DC6DD8 DC8DD6 DC2DD4 DC4DD2 DC5DD7 DC7DD5 |ϕ±⟩CDP|ψ±⟩CDF|ϕ±⟩CDS DC2DD7 DC4DD5 DC5DD4 DC7DD2 DC1DD8 DC3DD6 DC6DD3 DC8DD1 |ϕ±⟩CDP|ψ±⟩CDF|ψ±⟩CDS DC2DD5 DC4DD7 DC5DD2 DC7DD4 DC1DD6 DC3DD8 DC6DD1 DC8DD3 |ψ±⟩CDP|ψ±⟩CDF|ϕ±⟩CDS DC1DD2 DC3DD4 DC6DD5 DC8DD7 DC2DD1 DC4DD3 DC5DD6 DC7DD8 |ψ±⟩CDP|ψ±⟩CDF|ψ±⟩CDS DC2DD3 DC4DD1 DC5DD8 DC7DD6 DC1DD4 DC3DD2 DC6DD7 DC8DD5 |ψ±⟩CDP|ψ±⟩CDF|ϕ±⟩CDS DC1DD7 DC4DD6 DC5DD3 DC7DD1 DC2DD8 DC3DD5 DC6DD4 DC8DD2 |ψ±⟩CDP|ψ±⟩CDF|ψ±⟩CDS DC1DD5 DC3DD7 DC6DD2 DC8DD4 DC2DD6 DC4DD8 DC5DD1 DC7DD3 The last step is to distinguish the photons C and D in different polarization and different longitudinal momentum DOFs. The PBSs are used to transmit the horizontal polarization component and reflect the vertical polarization component. The corresponding relation between the measurement results and the states of CD is shown in Table 2. From Table 2, one can see that if both the single-photon detectors DC1 and DD1 click, the photon pair AB is left in the state 13 |ψf⟩AB=|ϕ+⟩ABP|ψ+⟩ABF|ϕ+⟩ABS. That is, the two-photon system AB is projected into the ideal maximally hyperentangled Bell state. Finally, according to the results of the measurement, one can perform corresponding phase-flip operations on the qubits to achieve the ideal state. The phase-flip operations on the qubits can be accomplished by putting conditional half-wave plates in the appropriate paths of photon. The corresponding relation between the final collapsed hyperentangled Bell states of photon pair AB, the half-wave plates and the relevant paths of photon is shown in Table 3. R0 denotes the half-wave plate set at 0o which performs the operation |H⟩→|H⟩, |V⟩→-|V⟩ on the polarization mode of photons. The half-wave plates R90 performs the bit-flip operation on the polarization mode of the photons. If the photon pair AB is in the hyperentangled Bell state |ϕ-⟩ABP|ψ-⟩ABF|ϕ-⟩ABS, Alice can accomplish the phase-flip operations for all the three modes by putting R0 in the paths |l,I⟩ and |r,E⟩, R90, R0 in the paths |r,I⟩ and |l,E⟩ of the photon A, respectively.Table 3 The relation between the states of photon pair AB , the half-wave plates and the relevant paths. State of AB Half-wave plates Paths |ϕ+⟩ABP|ψ+⟩ABF|ϕ+⟩ABS none none |ϕ+⟩ABP|ψ+⟩ABF|ϕ-⟩ABS R0,R90,R0 |r,E⟩, |l,E⟩ |ϕ+⟩ABP|ψ-⟩ABF|ϕ+⟩ABS R0,R90,R0 |r,I⟩, |r,E⟩ |ϕ+⟩ABP|ψ-⟩ABF|ϕ-⟩ABS R0,R90,R0 |r,I⟩, |l,E⟩ |ϕ-⟩ABP|ψ+⟩ABF|ϕ+⟩ABS R0 all the four paths |ϕ-⟩ABP|ψ+⟩ABF|ϕ-⟩ABS R0 |r,I⟩, |l,I⟩ R90,R0 |r,E⟩, |l,E⟩ |ϕ-⟩ABP|ψ-⟩CDF|ϕ+⟩ABS R0 |l,I⟩, |l,E⟩ R90,R0 |r,I⟩, |r,E⟩ |ϕ-⟩ABP|ψ-⟩ABF|ϕ-⟩ABS R0 |l,I⟩, |r,E⟩ R90,R0 |r,I⟩, |l,E⟩ In the second case, one of the parity check measurement results is odd-parity result. The four-photon system is projected into the state |Ψ2⟩ABCD, |Ψ3⟩ABCD or |Ψ4⟩ABCD, respectively, with the corresponding probability Pi(1) (i=2,3,4). We take the state |Ψ2⟩ABCD as an example. Here 14 |Ψ2⟩ABCD=1P2(1)(α2|H⟩A|V⟩C|H⟩B|V⟩D+β2|V⟩A|H⟩C|V⟩B|H⟩D)⊗γδ(|l⟩A|l⟩C|r⟩B|r⟩D+|r⟩A|r⟩C|l⟩B|l⟩D)⊗ϵε(|I⟩A|I⟩C|I⟩B|I⟩D+|E⟩A|E⟩C|E⟩B|E⟩D). Alice and Bob perform Hadamard operations on the three DOFs of photons C and D, respectively, and then the state |Ψ2⟩ABCD can be transformed into the state |Ψ2′⟩ABCD. Here 15 |Ψ2′⟩ABCD=12[(α(2)|H⟩A|H⟩B+β(2)|V⟩A|V⟩B)|ϕ+⟩CDP+(α(2)|H⟩A|H⟩B-β(2)|V⟩A|V⟩B)|ψ+⟩CDP]⊗12(|ψ+⟩ABF|ϕ-⟩CDF+|ψ-⟩ABF|ψ-⟩CDF)⊗12(|ϕ+⟩ABS|ϕ+⟩CDS+|ϕ-⟩ABS|ψ+⟩CDS). Here, α(2)=α2|α|4+|β|4 and β(2)=β2|α|4+|β|4. Then according to the measurement results in Table 2, we can obtain the state |ψ2(2)⟩AB with or without single-photon operations, here 16 |ψ2(2)⟩AB=(α(2)|H⟩A|H⟩B+β(2)|V⟩A|V⟩B)⊗|ψ+⟩ABF|ϕ+⟩ABS. This is a partially hyperentangled Bell-type state with the longitudinal momentum DOFs in a maximally hyperentangled Bell state. For states |Ψ3⟩ABCD and |Ψ4⟩ABCD, one can also obtain a partially hyperentangled Bell-type state with two DOFs in a maximally hyperentangled Bell state. The final state of photon pair AB can be denoted as 17 |ψ3(2)⟩AB=|ϕ+⟩ABP|ϕ+⟩ABS⊗(γ(2)|l⟩A|r⟩B+δ(2)|r⟩A|l⟩B),|ψ4(2)⟩AB=|ϕ+⟩ABP|ψ+⟩ABF⊗(ϵ(2)|I⟩A|I⟩B+ε(2)|E⟩A|E⟩B). Here, γ(2)=γ2|γ|4+|δ|4, δ(2)=δ2|γ|4+|δ|4, ϵ(2)=ϵ2|ϵ|4+|ε|4 and ε(2)=ε2|ϵ|4+|ε|4. In this condition, another round of the hyper-ECP process is required. In the third case, two of the parity check measurement results give odd-parity results. The four-photon system is projected into the state |Ψ5⟩ABCD, |Ψ6⟩ABCD or |Ψ7⟩ABCD. Then after the whole quantum circuit, Alice and Bob can get the two-photon system in the state |ψ5(2)⟩AB, |ψ6(2)⟩AB or |ψ7(2)⟩AB with or without single-photon operations. Here, 18 |ψ5(2)⟩AB=(α(2)|H⟩A|H⟩B+β(2)|V⟩A|V⟩B)⊗(γ(2)|l⟩A|r⟩B+δ(2)|r⟩A|l⟩B)⊗|ϕ+⟩ABS,|ψ6(2)⟩AB=(α(2)|H⟩A|H⟩B+β(2)|V⟩A|V⟩B)⊗(ϵ(2)|I⟩A|I⟩B+ε(2)|E⟩A|E⟩B)|ψ+⟩ABF,|ψ7(2)⟩AB=(γ(2)|l⟩A|r⟩B+δ(2)|r⟩A|l⟩B)⊗(ϵ(2)|I⟩A|I⟩B+ε(2)|E⟩A|E⟩B)⊗|ϕ+⟩ABP. The corresponding probabilities are shown in Table 1. For those partially hyperentangled states, another round of hyper-ECP is needed. In the last case, all the three parity check measurements give odd-parity results. Then after the whole quantum circuit and conditional unitary operations on photon B, the two-photon system can be projected into the partially hyperentangled Bell-type state |ψ8(2)⟩AB with the probability P8(1). Here 19 |ψ8(2)⟩AB=(α(2)|H⟩A|H⟩B+β(2)|V⟩A|V⟩B)⊗(γ(2)|l⟩A|r⟩B+δ(2)|r⟩A|l⟩B)⊗(ϵ(2)|I⟩A|I⟩B+ε(2)|E⟩A|E⟩B). This state can also be used in the next round to obtain the maximally hyperentangled Bell state. For another identical four-photon system A′B′C′D′, the same operations are also performed on photon pairs A′B′ and C′D′ by Alice and Bob. Therefore, we can also obtain eight different collapsed states of photon pair A′B′, and the photon pair A′B′ in less-entangled state can be used as auxiliary photons in the next round. Improving the success probability by iteration The success of the hyperconcentration schemes is based on the three parity checks. When three even-parity outcomes occur, the hyperconcentration schemes succeed with probability P1=P1(1)=8|αβγδϵε|2. Otherwise, these schemes fail. However, the other states can also be used to distill the maximally hyperentangled Bell state. In this subsection, we will use the auxiliary photon pair A′B′ to distill a maximally hyperentangled Bell state from the partially hyperentangled-type state obtained in the first round. That is, we will iterate the hyperentanglement concentration processes to improve the success probability, such method was first proposed in 200838. The principle in the next round is similar to what is shown in Fig. 4. For the second case, the partially hyperentangled Bell-type state is only less entangled in one DOF, while the other two DOFs are in the desired forms. Here, we only discuss the state |ψ2(2)⟩AB in detail, while the other cases can be handled in the similar way. The state of the four-photon system ABA′B′ is |Ψ21⟩ABA′B′, Here 20 |Ψ2(2)⟩ABA′B′=|ψ2(2)⟩AB⊗|ψ2(2)⟩A′B′=(α(2)|H⟩A|H⟩B+β(2)|V⟩A|V⟩B)⊗|ψ+⟩ABF|ϕ+⟩ABS⊗(α(2)|H⟩A′|H⟩B′+β(2)|V⟩A′|V⟩B′)⊗|ψ+⟩A′B′F|ϕ+⟩A′B′S. In this case, we can just perform parity check for the less-entangled mode, hence Bob can let photons B and B′ just pass through the P-QND. If the outcome of the P-QND is even, the two-photon system AB can be projected into the maximally hyperentangled Bell state |ψf⟩, the success probability of this round is 21 P2(2)=P2(1)2|α(2)β(2)|2=P2(1)2|α|4|β|4(|α|4+|β|4)2. For the partially hyperentangled Bell-type states |ψ3⟩AB1 and |ψ4⟩AB1, the success probabilities in the second round are P3(2) and P4(2), respectively. Here 22 P3(2)=P3(1)2|γ(2)δ(2)|2=P3(1)2|γ|4|δ|4(|γ|4+|δ|4)2,P4(2)=P4(1)2|ϵ(2)ε(2)|2=P4(1)2|ϵ|4|ε|4(|ϵ|4+|ε|4)2. Given the analysis of the second round for the second case, we can see that in the mth round, the probabilities of success (failure) of getting the desired state from the (m-1)th round are 23 P2,s(m)=2|α|2m|β|2m(|α|2m+|β|2m)2,P2,f(m)=|α|2m+1+|β|2m+1(|α|2m+|β|2m)2,P3,s(m)=2|γ|2m|δ|2m(|γ|2m+|δ|2m)2,P3,f(m)=|γ|2m+1+|δ|2m+1(|γ|2m+|δ|2m)2,P4,s(m)=2|ϵ|2m|ε|2m(|ϵ|2m+|ε|2m)2,P4,f(m)=|ϵ|2m+1+|ε|2m+1(|ϵ|2m+|ε|2m)2. Here Pi,s(m) and Pi,f(m) (i = 2,3,4) denote the success and failure probabilities for obtaining the maximally hyperentangled state in the mth round, respectively. Then we can compute the success probability Pi (i = 2,3,4) after n (n>2) rounds for the second case, 24 Pi=Pi(1)(Pi,s(2)+Pi,f(2)Pi,s(3)+⋯+Pi,f(2)Pi,f(3)⋯Pi,f(n-1)Pi,s(n)). For the third case, the partially hyperentangled Bell-type state is less entangled in two DOFs, while the third DOF is in maximally entangled state. Thus, in the second round, we will pay attention to two QNDs. Here, we will discuss the state |ψ5(2)⟩AB in detail, the other cases can be handled in the similar way. If the photons B and B′ are in even-parities in polarization and the first longitudinal momentum DOF, we can obtain the maximally hyperentangled Bell state |ψf⟩ with the probability 25 P5(2)=P5(1)P5ee(2)=P5(1)4|α(2)β(2)γ(2)δ(2)|2=P5(1)P2,s(2)P3,s(2). For the state |ψ5(m)⟩AB which corresponds to two odd-parity results in the (m-1)th round, the probabilities of the four parity check results are 26 P5ee(m)=P2,s(m)P3,s(m),P5eo(m)=P2,s(m)P3,f(m),P5oe(m)=P2,f(m)P3,s(m),P5oo(m)=P2,f(m)P3,f(m). The subscripts “ee”, “eo”, “oe” and “oo” indicate the parity check results for two QNDs, with “e” being even and “o” being odd. In this case, the success probability of the mth (m>2) round is 27 P5(m)=P5(1)(P2,s(2)+P2,f(2)P2,s(3)+⋯+P2,f(2)P2,f(3)⋯P2,f(m-2)P2,s(m-1))P3,f(2)P3,f(3)⋯P3,f(m-1)P3,s(m)+P5(1)(P3,s(2)+P3,f(2)P3,s(3)+⋯+P3,f(2)P3,f(3)⋯P3,f(m-2)P3,s(m-1))P2,f(2)P2,f(3)⋯P2,f(m-1)P2,s(m)+P5(1)P2,f(2)P3,f(2)P2,f(3)P3,f(3)⋯P2,f(m-1)P3,f(m-1)P2,s(m)P3,s(m). We can easily achieve the total success probability after n (n>2) rounds 28 P5=∑m=2nP5(m). For the partially hyperentangled Bell-type states |ψ6(2)⟩AB and |ψ7(2)⟩AB, the corresponding success probabilities are 29 P6(2)=P6(1)P6ee(2)=P6(1)4|α(2)β(2)ϵ(2)ε(2)|2=P6(1)P2,s(2)P4,s(2),P7(2)=P7(1)P7ee(2)=P7(1)4|γ(2)δ(2)ϵ(2)ε(2)|2=P7(1)P3,s(2)P4,s(2). For the partially hyperentangled Bell-type states |ψ6(m)⟩AB and |ψ7(m)⟩AB, the probabilities of the four parity check results are 30 P6ee(k)=P2,s(m)P4,s(m),P6eo(k)=P2,s(m)P4,f(m),P6oe(k)=P2,f(m)P4,s(m),P6oo(k)=P2,f(m)P4,f(m),P7ee(k)=P3,s(m)P4,s(m),P7eo(k)=P3,s(m)P4,f(m),P7oe(k)=P3,f(m)P4,s(m),P7oo(k)=P3,f(m)P4,f(m). The corresponding success probabilities of the mth (m>2) round are 31 P6(m)=P6(1)(P2,s(2)+P2,f(2)P2,s(3)+⋯+P2,f(2)P2,f(3)⋯P2,f(m-2)P2,s(m-1))P4,f(2)P4,f(3)⋯P4,f(m-1)P4,s(m)+P6(1)(P4,s(2)+P4,f(2)P4,s(3)+⋯+P4,f(2)P4,f(3)⋯P4,f(m-2)P4,s(m-1))P2,f(2)P2,f(3)⋯P2,f(m-1)P2,s(m)+P6(1)P2,f(2)P4,f(2)P2,f(3)P4,f(3)⋯P2,f(m-1)P4,f(m-1)P2,s(m)P4,s(m),P7(m)=P7(1)(P3,s(2)+P3,f(2)P3,s(3)+⋯+P3,f(2)P3,f(3)⋯P3,f(m-2)P3,s(m-1))P4,f(2)P4,f(3)⋯P4,f(m-1)P4,s(m)+P7(1)(P4,s(2)+P4,f(2)P34,s(3)+⋯+P34,f(2)P4,f(3)⋯P4,f(m-2)P4,s(m-1))P3,f(2)P3,f(3)⋯P3,f(m-1)P3,s(m)+P7(1)P3,f(2)P4,f(2)P3,f(3)P4,f(3)⋯P3,f(m-1)P4,f(m-1)P3,s(m)P4,s(m). By iterating the hyperconcentration process n (n>2)times, the total success probability for each condition is 32 P6=∑m=2nP6(m),P7=∑m=2nP7(m). For the last case, the principle of this step is the same as the first round, except that the photon pair CD is replaced by A′B′. Thus, using the whole quantum circuit shown in Fig. 4, we can achieve the maximally hyperentangled Bell state |ψf⟩ with the probability 33 P8(2)=P8(1)8|α(2)β(2)γ(2)δ(2)ϵ(2)ε(2)|2=P8(1)P2,s(2)P3,s(2)P4,s(2). The success probability of the 3th round is 34 P8(3)=P8(1)(P3,s(2)P4,s(2)P2,f(2)P2,s(3)+P2,s(2)P4,s(2)P3,f(2)P3,s(3)+P2,s(2)P3,s(2)P4,f(2)P4,s(3)+P2,f(2)P3,f(2)P4,s(2)P2,s(3)P3,s(3)+P2,f(2)P4,f(2)P3,s(2)P2,s(3)P4,s(3)+P3,f(2)P4,f(2)P2,s(2)P3,s(3)P4,s(3)+P2,f(2)P3,f(2)P4,f(2)P2,s(3)P3,s(3)P4,s(3)). Then we can obtain the success probability of the mth (m>3) round for the last case 35 P8(m)=P8(1)∑a,b,c[(Pa,s(2)Pb,s(2)+Pa,f(2)Pb,f(2)Pa,s(3)Pb,s(3)+⋯+Pa,f(2)Pb,f(2)⋯Pa,f(m-2)Pb,f(m-2)Pa,s(m-1)Pb,s(m-1))Pc,f(2)⋯Pc,f(m-1)Pc,s(m)+(Pc,s(2)+Pc,f(2)Pc,s(3)+⋯+Pc,f(2)Pc,f(3)⋯Pc,f(m-2)Pc,s(m-1))Pa,f(2)Pb,f(2)⋯Pa,f(m-1)Pb,f(m-1)Pa,s(m)Pb,s(m)]+P8(1)∑i,j,k∑l=2m-2(Pi,s(2)+Pi,f(2)Pi,s(3)+⋯+Pi,f(2)Pi,f(3)⋯Pi,f(l-1)Pi,s(l))Pj,f(2)⋯Pj,f(l)Pj,s(l+1)Pk,f(2)⋯Pk,f(m-1)Pk,s(m)+P8(1)P2,f(2)P3,f(2)P4,f(2)⋯P2,f(m-1)P3,f(m-1)P4,f(m-1)P2,s(m)P3,s(m)P4,s(m). Here, a≠b≠c∈{2,3,4} and satisfy the limitation a