Complete nondestructive analysis of two-photon six-qubit hyperentangled Bell states assisted by cross-Kerr nonlinearity

Hyperentanglement, the entanglement in several degrees of freedom (DOFs) of a quantum system, has attracted much attention as it can be used to increase both the channel capacity of quantum communication and its security largely. Here, we present the first scheme to completely distinguish the hyperentangled Bell states of two-photon systems in three DOFs with the help of cross-Kerr nonlinearity without destruction, including two longitudinal momentum DOFs and the polarization DOF. We use cross-Kerr nonlinearity to construct quantum nondemolition detectors which can be used to make a parity-check measurement and analyze Bell states of two-photon systems in different DOFs. Our complete scheme for two-photon six-qubit hyperentangled Bell-state analysis may be useful for the practical applications in quantum information, especially in long-distance high-capacity quantum communication.

dot spins inside optical microcavities. Li and Ghose 33 presented a hyper-ECP for multipartite entanglement via linear optics. Some other interesting protocols for hyperentanglement concentration and hyperentanglement purification [34][35][36] were presented in 2015.
In fact, in long-distance high-capacity quantum communication, the complete analysis for all the orthogonal hyperentangled Bell states of photon systems in multiple DOFs is necessary. The 16 orthogonal hyperentangled Bell states of two-photon systems in two DOFs can be distinguished completely if nonlinear optics is introduced. In 2010, Sheng et al. 37 gave the first scheme for the complete hyperentangled-Bell-state analysis (HBSA) for quantum communication with the help of cross-Kerr nonlinearity. In 2012, Ren et al. 38 proposed another complete HBSA scheme for photon systems in both the polarization and the spatial-mode DOFs with the help of giant nonlinear optics in one-sided quantum-dot-cavity systems. Using double-sided quantum-dot-cavity systems, the complete HBSA scheme also can be accomplished 39 . Xia et al. 40 proposed an efficient scheme for hyperentangled Greenberger-Horne-Zeilinger-state analysis with cross-Kerr nonlinearity. Recently, the hyperentangled Bell states for two-photon six-qubit systems were produced in experiments 41,42 , but there are no schemes for the complete analysis on two-photon six-qubit quantum states as they are far more difficult, compared with the Bell states in both one and two DOFs.
In this paper, we give the first scheme to completely distinguish the hyperentangled Bell states of two-photon systems in three DOFs with the help of cross-Kerr nonlinearity without destruction, including a polarization DOF and double longitudinal momentum DOFs. Our HBSA protocol for two-photon six-qubit hyperentangled systems may be useful in the practical applications in quantum information processing, blind quantum computation, distributed quantum computation, and especially long-distance high-capacity quantum communication in the future. With hyperdense coding on two-photon systems entangled in three DOFs simultaneously as an example, we show the principle of the applications of our HBSA protocol in detail.

Results
Complete analysis for the states of a two-photon system in momentum modes. A hyperentangled Bell state of two-photon six-qubit systems in three DOFs can be described as follows: Here the subscripts A and B denote the two photons. H and V represent the horizontal and the vertical polarizations of photons, respectively. The three independent DOFs are polarization and a double longitudinal momentum (r/l and E/I), shown in Fig. 1. The system of the two-photon six-qubit source 42 consists of two type-1 β barium borate (BBO) crystal slabs and a eight-hole screen. When a continuous-wave (cw) vertically polarized Ar + laser beam interacts through spontaneous parametric down-conversation (SPDC) with the two BBO crystal slabs, and the nonlinear interaction between the laser beam and the BBO crystal leads to the production of the degenerate photon pairs, which are entangled in polarization and belong to the surfaces of two emission cones. As shown in Fig. 1(a), the insertion of a eight-hole screen allows us to achieve the double longitudinal momentum entanglement. The labels in Fig. 1(b) are used to identify the selected modes. The internal (I) and the external (E) cones correspond to the first and the second crystals, respectively. Furthermore, l (r) refers to the left (right) side of each cone. The distinction between the internal (I) and the external (E) modes provides us the second longitudinal momentum DOF, while the first longitudinal momentum DOF comes from the distinction between the left (l) and right (r) modes. Therefore, the six-qubit hyperentangled state described in Eq. (1) is given by the product of one polarization entangled state and two longitudinal momentum entangled states of a photon pair. Let us denote the four Bell states in the polarization DOF of two-photon systems as and four Bell states in the first longitudinal momentum DOF as φ ψ = ± = ± ± ± l l r r l r r l while the four Bell states in the second longitudinal momentum DOF can be expressed as Here the subscripts 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. The principle of our scheme for the complete analysis on the quantum states of a two-photon six-qubit system in the first longitudinal momentum DOF is shown in Fig. 2. In detail, one can let the two photons AB pass through the first quantum nondemolition detector (QND 1 ) whose circuit is shown in Fig. 2(a). Based on the principle of cross-Kerr effect (see Methods), the evolution of two-photon six-qubit hyperentangled Bell states and the coherent state can be described as follows: S P S l l r r P S l l e r r e P S P S l r r l P S l r r l P S l r r l Here, P represents the four Bell stats in the polariztion DOF, and S denotes the four Bell states in the second longitudinal momentum DOF. The equation above shows that the Bell states of other two DOFs have not changed. If these two photons are in the same state | 〉 | 〉 r r A B or | 〉 | 〉 l l A B in the first longitudinal momentum DOF, the coherent probe beam will pick up a phase shift + 2θ or − 2θ. If these two photons are in the different states | 〉 | 〉 l r A B or | 〉 | 〉 r l A B , the phase shift of the coherent probe beam will be 0. As the homodyne measurement cannot distinguish + 2θ from − 2θ, there are only two measurement outcomes α and α θ ± e i 2 for the coherent probe beam. Thus, according to the measurement results, one can distinguish the even-parity states φ ± F from the odd-parity states ψ ± F . That is, QND 1 shown in Fig. 2(a) is a quantum nondemolition detector, with which one can distinguish the parity of the two photons A and B in the first longitudinal momentum DOF.
After QND 1 , one can divide the four Bell states in the first longitudinal momentum DOF into two groups, φ ± F and ψ ± F . The next task is to distinguish the different phases in φ ± F and ψ ± F , respectively. By using the 50:50 beam splitters (BSs) shown in Fig. 2(b) on the photons, one can get the following transformations: As the BSs transform the phase difference of the two states from each group into the parity difference, the two Bell states in the same group will belong to different groups after the BSs. Then, if we let photon A and photon B pass through the same quantum circuit as QND 1 shown in Fig. 2(b), the four Bell states can be distinguished com- by BSs in this procedure, respectively, one can use other BSs after the quantum circuit as QND 1 to recover the initial Bell states in the first longitudinal momentum DOF. The relationship between the measurement results of these two QNDs and the corresponding Bell states in the first longitudinal momentum DOF is shown in Table 1. Now, we have finished the distinction of the four Bell states in the first longitudinal momentum, without destroying the hyperentanglement in the other two DOFs. Then we move to the next step to distinguish the four Bell states in the second longitudinal momentum DOF. As the first longitudinal momentum and the second longitudinal momentum are all linear momentum, what we do to realize the next distinction is similar to the analysis protocol of the first longitudinal momentum DOF. The difference is to interchange the path labels r/l to E/I. The principle for distinguishing the four Bell states of the two-photon system in the second longitudinal momentum DOF is shown in Fig. 3. Here, we let the two photons pass through QND 3 and then QND 4 in sequence. With these two QNDs, we can analyze the four Bell states in the second longitudinal momentum DOF completely. The Figure 2. (a) Schematic diagram of QND 1 which is used to distinguish the even-parity states φ ± F from the odd-parity states ψ ± F in the first longitudinal momentum DOF of the two-photon six-qubit system AB. ± θ denotes the cross-Kerr nonlinear media which will make the cohere probe beam α have a phase shift ± θ when there is a signal photon passing through it. X X is the homodyne measurement to discriminate different phase shifts of the coherent probe beam. r and l represent the left and the right sides of each cone from where the photons emit, respectively. The internal (I) and the external (E) cones correspond to the first and the second crystal from which the photons are produced, respectively. on the photon in the first longitudinal momentum DOF. After these two photons pass through the BSs, one can use mirrors to separate the paths of photons. relationship between the measurement results of this scheme and the corresponding Bell states in the second longitudinal momentum DOF is described in Table 2.
Complete six-qubit hyperentangled Bell state analysis scheme for states in polarization. Now, let us move our attention to the last task, which is to distinguish the four Bell states of the two-photon six-qubit system in the polarization DOF. The analysis of the four Bell states in polarization is analogous to that in previous works 37,43 . The schematic diagram for the distinction of the four Bell states in polarization is shown in Figs 4 and 5.
According to QND 5 shown in Fig. 4, the states φ | 〉 | 〉| 〉 ± F S P with the coherent state α evolve as where F represents the four Bell states in the first longitudinal momentum DOF. In these evolutions, the modes . The next step is to distinguish the different relative phases in each of these two groups. This task can be accomplished with the circuit shown in Fig. 5. Here the wave plate R 45 is used to accomplish a Hadamard operation on the polarization of photons. A Hadamard operation on each of the two photons AB will make the following transformations: As R 45 can transform the phase difference into the parity difference, one can then use the same quantum circuit as QND 5 to distinguish the parity difference between the two states in each group. Then we use other R 45 to recover the initial Bell states in polarization DOF. That is, after the photons pass through QND 6 shown in Fig. 5, the two Bell states in the even-parity group φ φ + − { , } P P or the odd-parity one ψ ψ + − { , } P P can be distinguished completely. The relationship between the measurement results of this scheme and the corresponding Bell states in polarization is described in Table 3.
From the analysis above, one can see that the complete nondestructive analysis for two-photon six-qubit hyperentangled Bell states can be accomplished with the sequential connection of the six QNDs. This complete HBSA can be used to complete some other important tasks in high-capacity quantum communication, such as teleportation with photon systems in three DOFs, hyperentanglement swapping, quantum hyperdense coding, and so on.

Discussion
In our six-qubit HBSA scheme, we exploit the cross-Kerr nonlinearity to construct the QNDs to check the parity of the two photons in the three DOFs. Therefore, we should acknowledge that the feasibility of the proposed scheme depends on the nonlinear phase shift of the Kerr media. Although many works have been reported on cross-Kerr nonlinearity 44  on the second longitudinal momentum DOF of photons. of Gea-Banacloche 49 suggested that a large phase shift via a "giant Kerr effect" with single-photon wave packets is impossible at present. Fortunately, our HBSA scheme only requires a small phase shift, as long as it can be distinguished from zero, and much progress has been made on the Kerr nonlinearity and homodyne detection. In 2003, Hofmann et al. 50 demonstrated that a phase shift of π can be achieved with a single two-level atom one-sided cavity system. In 2010, Wittmann et al. 51 investigated the difference between a standard homodyne detector and a displacement-controlled photon number resolving (PNR) detector. They showed that the displacement-based PNR receiver outperforms the standard homodyne detection. Therefore, for a weak cross-Kerr nonlinearity θ  1, if we choose a sufficiently large amplitude of the coherent state, which satisfies the requirement αθ  1 2 , it is possible for us to achieve deterministic distinguishability between the shifted and non-shifted phases in the coherent state. Furthermore, in 2011, He et al. 46 showed that effects due to the transverse degrees of freedom significantly affect the cross-phase modulation process, and made the treatment of single-photon-coherent-state interactions more realistic. In the same year, Feizpour et al. 52 researched the cross-Kerr nonlinearity between continuous-mode coherent states and single photons, and they indicated that a cross-Kerr phase shift is likely to be amplified to observable value with weak-value amplification. Moreover, Zhu and Huang 53 showed that giant Kerr nonlinearity of the probe and the signal pulses may be achieved with nearly vanishing optical absorption. The substantial cross-Kerr nonlinearities 54,55 have already been obtained in the microwave domain using superconducting qubits. In the work by Hoi et al. 54 , the average cross-Kerr phase shift was demonstrated up to 20 degrees per photon with both coherent microwave fields at the single-photon level.

Bell states
Before ending this work, we will briefly discuss the application of our HBSA scheme in quantum hyperdense coding. As quantum hyperdense coding is the generalization of quantum dense coding with photon systems in several DOFs, with our six-qubit HBSA scheme, one can transfer six bits of classical information by sending only one photon. In order to realize quantum hyperdense coding, the sender must choose one of the local 64 operations j I x y z are unitary operations in polarization or one of the two longitudinal momentum DOFs. Here, the unitary operations U x P and U z P can be achieved by a half-wave plate set at 45° and 0°, respectively. The combination of 0° and 45° half-wave plates can be used to perform the unitary operation U y P . One can accomplish the operation U x P , U y P , and U z P by putting appropriate half-wave plates in all the four paths of the photon. . Schematic diagram of QND 5 which is used to distinguish the even-parity states φ ± P from the odd-parity states ψ ± P in polarization DOF of the two-photon six-qubit system AB. PBS represents a polarizing beam splitter which is used to transmit the horizontal (H) polarization photon and reflect the vertical (V) polarization photon, respectively. . Using those operations and our six-qubit HBSA scheme, we can accomplish the six-bit quantum hyperdense coding which will largely improve the capacity of long-distance quantum communication.
In summary, we have proposed an efficient scheme for the complete nondestructive analysis of hyperentanglement of two-photon systems in three DOFs with the help of the cross-Kerr nonlinearity. We use cross-Kerr nonlinearity to construct quantum nondemolition detectors which are used to make a parity-check measurement and analyze Bell states in different DOFs of two-photon systems. We have also presented the applications of our HBSA protocol in quantum hyperdense coding with two-photon systems entangled in three DOFs simultaneously,  which means that our HBSA protocol may be useful for practical applications in quantum information processing, blind quantum computation, distributed quantum computation, and especially long-distance high-capacity quantum communication in future.