Hyperentanglement-assisted hyperdistillation for hyper-encoding photon system

Abstract

In quantum information processing, the quality of photon system is decreased by the inevitable interaction with environment, which will greatly reduce the efficiency and security of quantum information processing. In this paper, we propose hyperentanglement-assisted hyperdistillation schemes to guarantee the quality of hyper-encoding photon system based on the method of quantum hyper-teleportation, which can increase the success probability of hyperdistillation and reduce the resource consumption. First, we propose a hyperentanglement-assisted single-photon hyperdistillation (HASPHD) scheme for polarization and spatial qubits to get rid of the vacuum state component caused by transmission loss, whose success probability can achieve the optimal one by increasing the efficiency of quantum hyper-teleportation. Subsequently, we present two hyperentanglement-assisted hyperentanglement distillation (HAHED) schemes for photon system to protect hyperentanglement from both transmission loss and quantum channel noise, which can recover the less-entangled mixed state to maximally hyperentangled state for known-parameter and unknown-parameter cases with high success probability and low resource consumption. In these hyperdistillation schemes, the influence of imperfect effects of optical elements can be largely decreased by the quantum hyper-teleportation method. These characters make the hyperentanglement-assisted hyperdistillation schemes have potential application prospects in practical quantum information processing.

Graphical abstract

Keywords

hyperdistillation / transmission loss / quantum channel noise / quantum communication / quantum information

1 Introduction

Quantum information process, e.g., quantum communication process or quantum computation process, is an amazing information process by using quantum mechanics theories to improve its security and speed. Photon system is an important information carrier in quantum communication [1], due to its high-speed, manipulability, and high-capacity [2] properties, and it can be used in quantum key distribution [3–6], quantum teleportation [7], quantum dense coding [8, 9], quantum secret sharing [10, 11], quantum secure direct communication [12–18], and some other schemes. Photons have multiple degrees of freedom (DOFs), containing polarization DOF, momentum DOF, time-bin DOF, frequency DOF and orbital angular momentum DOF, and they can be entangled in multiple DOFs simultaneously, which refers to hyperentanglement and has been experimentally demonstrated in many types [19–25]. In the practical long-distance quantum communication processing, the efficiency and distance of communication and the security and integrity of information will be limited and reduced by the transmission loss and quantum channel noise. For instance, the transmission loss caused by the optical fiber attenuation and imperfect photon detectors may lead to vacuum state component, and the quantum channel noise will transform the maximally (hyper)entangled state to partially (hyper)entangled state or mixed (hyper)entangled state.

In order to overcome the transmission loss problem, Ralph and Lund [26] first introduced the concept of heralded amplification based on quantum teleportation model. Since then, a lot of theoretical works [27, 28] and experimental works [29, 30] have been presented for heralded amplification, including heralded amplification for photon qubit. The qubit amplifier can be used to overcome the problem of channel loss in device independent quantum key distribution (DI-QKD) [27, 28, 31] and device-independent quantum secure direct communication (DI-QSDC) [32], which can increase the communication distance of DI-QKD and DI-QSDC. The noiseless linear amplification can also be used to protect single-photon entanglement from photon transmission loss [33–36]. Moreover, the hyperdistillation schemes based on amplification [37–40] have also been proposed for protecting spatial-mode-polarization hyperentangled state from photon loss.

In order to overcome the quantum channel noise problem, (hyper)entanglement purification and (hyper)entanglement concentration are introduced to improve the fidelity of the (hyper)entangled system degraded by the quantum channel noise. Entanglement (hyperentanglement) purification is introduced to distill high-fidelity (hyper)entangled state from mixed (hyper)entangled state [41–52]. Entanglement (hyperentanglement) concentration is introduced to distill maximally (hyper)entangled state from partially (hyper)entangled state [53]. Since the first entanglement concentration protocol was proposed by Bennet et al. using collective measurement [53], many entanglement concentration protocols have been proposed for photon system using linear optical elements [54–58] and nonlinear optical elements [59–63], and some of the entanglement concentration protocols have been demonstrated experimentally using linear optical elements [54, 55]. Hyperentanglement concentration protocols [64–72] have also attracted much attention for recovering the partially hyperentangled state to maximally hyperentangled state with certain success probability.

In this paper, we propose hyperentanglement-assisted hyperdistillation schemes for protecting hyper-encoding photon system from transmission loss and quantum channel noise using quantum hyper-teleportation method, which can increase the success probability of hyperdistillation and reduce the resource consumption. First, we propose a hyperentanglement-assisted single-photon hyperdistillation (HASPHD) scheme for polarization and spatial qubits to get rid of the vacuum state component caused by transmission loss, where the success probability of HASPHD scheme can reach the optimal one by enhancing the efficiency of quantum hyper-teleportation. Then, we show that the quantum circuit of HASPHD scheme can be directly used in hyperentanglement-assisted hyperentanglement distillation (HAHED) scheme for less-entangled mixed state with known parameters to increase the success probability of recovering the less-entangled mixed state to maximally hyperentangled state. Moreover, the HAHED scheme for less-entangled mixed state with unknown parameters can also be implemented easily using quantum hyper-teleportation method with high success probability and low resource consumption. In these hyperentanglement-assisted hyperdistillation schemes, the quantum hyper-teleportation method can depress the imperfect effects of optical elements on the final quantum states of these schemes. All these characters make these schemes have potential application prospects in quantum information process and quantum network.

2 Hyperentanglement-assisted single-photon hyperdistillation for polarization and spatial qubits

In quantum communication process, the photon is initially prepared in the state |φ〉 = (α|H〉+ β|V 〉)_A(γ|a₁〉+ δ|a₂〉), where the four parameters are real for simplicity and satisfy the normalization condition |α|² + |β|² = |γ|² + |δ|² = 1. The states |H〉 and |V 〉 represent the horizontal polarization component and vertical polarization component of a photon, and the states |a₁〉 and |a₂〉 represent the two spatial modes of the photon A. If Alice sends the photon to a remote party Bob by the lossy quantum channel, the state of the photon may decay to ρ₀ = (1−P₀)|vac〉〈vac| + P₀|φ〉〈φ| by the inevitable transmission loss. Here |vac〉 represents the vacuum state caused by the transmission loss, and P₀ represents the fidelity of the state |φ〉 in mixed state ρ₀.

In order to increase the fidelity of the state |φ〉, a HASPHD scheme for polarization and spatial qubits is introduced using quantum hyper-teleportation method. Here we choose a four-photon system BCDE in the hyperentangled Greenberger–Horne–Zeilinger (GHZ) state [|\varphi \begin{array}{l}{\rangle }_{BCDE}=\frac{1}{2}(|HHHH\rangle +|VVVV\rangle {)}_{BCDE}(|b_1{c}_1{d}_1{e}_1\rangle +\hfill \end{array}|b₂c₂d₂e₂〉)] as an auxiliary to implement the HASPHD scheme. (Other type of hyperentangled state as auxiliary is discussed in Section 4.) The setup of HASPHD assisted by hyperentangled GHZ state is shown in Fig. 1. It has ten input ports (marked by a₁, a₂, b₁, b₂, c₁, c₂, d₁, d₂, e₁ and e₂) and two output ports (marked by e₁ and e₂), and it consists of polarization beam splitter (PBS), 50:50 beam splitters (BSs), half-wave plates (HWPs), and photon detectors. PBS can perform the polarization parity-check measurement on the two photons A and B, where the spatial modes of the photons transmitted through PBS are supposed to be invariant and the spatial modes of the photons reflected by PBS are supposed to be changed. BS can perform Hadamard operation on the spatial DOF of the photon. HWP can perform Hadamard operation on the polarization DOF of the photon. Photon detectors can select the successful cases.

The initial state of the five-photon system ABCDE can be described as ρ₀⊗|ϕ〉_BCDE 〈ϕ|. Bob first lets the wave packets in spatial modes |a₁〉, |a₂〉, |b₁〉 and |b₂〉 pass through PBS, and he postselects the cases that the polarization state of the photon system AB is in the even-parity mode (|HH〉 or |V V 〉) and the spatial states of two-photon systems AC and BD are both in even-parity mode (|a₁c₁〉 or |a₂c₂〉, |b₁d₁〉 or |b₂d₂〉), where four photon detectors are triggered (i.e., only one photon can be detected in one of the spatial modes |l_m〉 and |l_n〉, mn = 13, 24, 57, 68). The postselected state of five-photon system ABCDE is

\begin{array}{}\text{(1)}& \begin{array}{ll}|{\text{Φ}}_1\rangle =\hfill & (\alpha |HHHHH\rangle +\beta |VVVVV\rangle )(\gamma |a_1{b}_1{c}_1{d}_1{e}_1\rangle \hfill \\ \hfill & +\delta |a_2{b}_2{c}_2{d}_2{e}_2\rangle )\mathrm{.}\hfill \end{array}\end{array}

Then, Bob lets the wave packets in spatial modes |a₁〉,|a₂〉, |b₁〉, |b₂〉, |c₁〉, |c₂〉, |d₁〉 and |d₂〉 pass through four BSs, respectively, which can make

\begin{array}{}\text{(2)}& \begin{array}{l}|k{\rangle }_{a_1}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_2}-|k{\rangle }_{l_4})\mathrm{,}|k{\rangle }_{a_2}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_1}-|k{\rangle }_{l_3})\mathrm{,}\\ |k{\rangle }_{b_1}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_5}+|k{\rangle }_{l_7})\mathrm{,}|k{\rangle }_{b_2}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_6}+|k{\rangle }_{l_8})\mathrm{,}\\ |k{\rangle }_{c_1}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_1}+|k{\rangle }_{l_3})\mathrm{,}|k{\rangle }_{c_2}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_2}+|k{\rangle }_{l_4})\mathrm{,}\\ |k{\rangle }_{d_1}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_6}-|k{\rangle }_{l_8})\mathrm{,}|k{\rangle }_{d_2}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_5}-|k{\rangle }_{l_7})\mathrm{,}\end{array}\end{array}

where k represents the polarization mode (H or V ) of a photon. After the wave packets in spatial modes pass through four BSs, the postselected state is changed to

\begin{array}{}\text{(3)}& \begin{array}{ll}|{\text{Φ}}_2\rangle =\hfill & \frac{1}{4}\{(\alpha |HHHHH\rangle +\beta |VVVVV\rangle )[\gamma (|l_2\rangle -|l_4\rangle )\hfill \\ \hfill & (|l_5\rangle +|l_7\rangle )(|l_1\rangle +|l_3\rangle )(|l_6\rangle -|l_8\rangle )|e_1\rangle +\delta (|l_1\rangle \hfill \\ \hfill & -|l_3\rangle )(|l_6\rangle +|l_8\rangle )(|l_2\rangle +|l_4\rangle )(|l_5\rangle -|l_7\rangle )|e_2\rangle ]\}\mathrm{.}\hfill \end{array}\end{array}

Finally, Bob lets the photons ABCD pass through eight HWPs [|H\begin{array}{l}\rangle \to \frac{1}{\sqrt{2}}(|H\rangle +|V\rangle )\mathrm{,}|V\rangle \to \frac{1}{\sqrt{2}}(|H\rangle -|V\rangle )]\hfill \end{array}, respectively. Then, the postselected state of five-photon system ABCDE will be changed to

\begin{array}{}\text{(4)}& \begin{array}{ll}|{\text{Φ}}_3\rangle =\hfill & \frac{1}{16}\{[\alpha (|H\rangle +|V\rangle )(|H\rangle +|V\rangle )(|H\rangle +|V\rangle )(|H\rangle \hfill \\ \hfill & +|V\rangle )|H\rangle +\beta (|H\rangle -|V\rangle )(|H\rangle -|V\rangle )(|H\rangle -|V\rangle )\hfill \\ \hfill & (|H\rangle -|V\rangle )|V\rangle ][\gamma (|l_2\rangle -|l_4\rangle )(|l_5\rangle +|l_7\rangle )(|l_1\rangle \hfill \\ \hfill & +|l_3\rangle )(|l_6\rangle -|l_8\rangle )|e_1\rangle +\delta (|l_1\rangle -|l_3\rangle )(|l_6\rangle +|l_8\rangle )\hfill \\ \hfill & (|l_2\rangle +|l_4\rangle )(|l_5\rangle -|l_7\rangle )|e_2\rangle ]\}\mathrm{.}\hfill \end{array}\end{array}

Bob measures the photons ABCD in the eight output modes (|l₁〉, |l₂〉, |l₃〉, |l₄〉, |l₅〉, |l₆〉, |l₇〉 and |l₈〉).

According to the postselection rules, the successful cases of the HASPHD scheme are shown in Table 1. There are four groups in Table 1. In the first group, the polarization states of two-photon systems AC and BD are in the same parity mode, and the spatial states of two-photon systems AC and BD are in the same parity mode. Here, the polarization modes |HH〉 and |V V 〉 (|HV 〉 and |V H〉) are defined as even (odd) parity mode, and the spatial modes |l₁l₂〉, |l₅l₆〉, |l₃l₄〉 and |l₇l₈〉 (|l₁l₄〉, |l₃l₂〉, |l₅l₈〉 and |l₇l₆〉) are defined as even (odd) parity mode. After the photons ABCD are detected, the state of photon E is transformed to |φ_E 〉 = (α|H〉 + β|V 〉)(γ|e₁〉 + δ|e₂〉), which is the initial photon state without vacuum state. In the second group, the polarization states of two-photon systems AC and BD are in different parity modes, and the spatial states of two-photon systems AC and BD are in the same parity mode. After the photons ABCD are detected, the state of photon E is transformed to |{\phi }_E^1\rangle = (α|H〉−β|V 〉)(γ|e₁〉+ δ|e₂〉), and a polarization phase-flip operation {\sigma }_z^P=|H\rangle \langle H|-|V\rangle \langle V| is required to perform on photon E to obtain the state |φ_E 〉. In the third group, the polarization states of two-photon systems AC and BD are in the same parity mode, and the spatial states of two-photon systems AC and BD are in different parity modes. After the photons ABCD are detected, the state of photon E is transformed to \begin{array}{l}|{\phi }_E^2\rangle =(\alpha |H\rangle +\beta |V\rangle )(\gamma |e_1\rangle -\delta |e_2\rangle )\hfill \end{array}, and a spatial phase-flip operation {\sigma }_z^S=|e_1\rangle \langle e_1|-|e_2\rangle \langle e_2| is required to perform on photon E to obtain the state |φ_E 〉. In the fourth group, the polarization states of two-photon systems AC and BD are in different parity modes, and the spatial states of two-photon systems AC and BD are in different parity modes. After the photons ABCD are detected, the state of photon E is transformed to \begin{array}{l}|{\phi }_E^3\rangle =(\alpha |H\rangle -\beta |V\rangle )(\gamma |e_1\rangle -\delta |e_2\rangle )\hfill \end{array}, and polarization phase-flip operation {\sigma }_z^P and spatial phase-flip operation {\sigma }_z^S are required to perform on photon E to obtain the state |φ_E 〉.

If only three photon detectors are triggered or the photons ABCD are detected in the other cases, the HASPHD process is failed, and these cases should be removed. Therefore, the success probability of the HASPHD scheme assisted by hyperentangled GHZ state is \frac{P_0}{4}.

3 Hyperentanglement-assisted polarization-spatial hyperentanglement distillation

In long distance quantum communication, the maximally hyperentangled photon systems (e.g., hyperentangled state |\psi {\rangle }_{A{A}^{\prime }}=\frac{1}{2}(|HH\rangle +|VV\rangle )(|a_1{a}_1^{\prime }\rangle +|a_2{a}_2^{\prime }\rangle )) are prepared locally by Alice. In order to establish the quantum channel, Alice will transmit one photon of hyperentangled photon system (e.g., photon A) to the remote party Bob through the lossy channels with the transmission coefficients of T₁ and T₂ for two spatial modes, respectively (0 < T₁≤1 and 0 < T₂≤1). Therefore, the maximally hyperentangled photon system may be influenced by the decoherence caused by transmission loss and channel noise, and the maximally hyperentangled state |\psi {\rangle }_{A{A}^{\prime }} would deteriorate to less-entangled mixed state ρ₂ = P₁|ψ₁〉〈ψ₁| + (1−P₁)ρ₁, where

\begin{array}{}\text{(5)}& \begin{array}{ll}\hfill & |{\psi }_1\rangle =(\alpha |HH\rangle +\beta |VV\rangle )\otimes (\gamma |a_1{a}_1^{\prime }\rangle +\delta |a_2{a}_2^{\prime }\rangle )\mathrm{,}\hfill \\ \hfill & {\rho }_1=({\alpha }^2|H\rangle \langle H|+{\beta }^2|V\rangle \langle V|)[\frac{1-T_1}{(1-T_1)+(1-T_2)}\hfill \\ \hfill & |a_1^{\prime }\rangle \langle a_1^{\prime }|+\frac{1-T_2}{(1-T_1)+(1-T_2)}|a_2^{\prime }\rangle \langle a_2^{\prime }|]\hfill \\ \hfill & \otimes |vac{\rangle }_A\langle vac|\mathrm{,}\hfill \\ \hfill & P_1=\frac{T_1+T_2}{2}\mathrm{.}\hfill \end{array}\end{array}

Here, the parameters γ and δ are \sqrt{\frac{T_1}{T_1+T_2}} and \sqrt{\frac{T_2}{T_1+T_2}}, respectively. The correlation between the transmission coefficient T_i and the transmission length s_i can be described by T_i={10}^{-{\alpha }_0{s}_i/10} (α₀ =0.2 dB/km, i = 1, 2). If we know the length of each quantum channel, we could calculate the values of T₁ and T₂. In order to depress the effects of transmission loss and channel noise on hyperentangled photon system, we introduce two hyperentanglement-assisted hyperentanglement distillation (HAHED) schemes in this section, which can recover the less-entangled mixed state ρ₂ to maximally hyperentangled state with high success probability and low resource consumption.

3.1 HAHED scheme for less-entangled mixed state with known parameters

We first introduce a HAHED scheme for less-entangled mixed state with parameters α, β, γ and δ known to the remote users Alice and Bob, where the four parameters can be obtained by measuring a large number of sample states [56–58]. In order to obtain the maximally hyperentangled state |ψ〉_AA′ from the mixed hyperentangled state ρ₂, a four-photon system BCDE in the partially hyperentangled GHZ state |ψ′〉_BCDE is chosen as an auxiliary, which can increase the success probability and depress the resource consumption of HAHED scheme in linear optics. Here,

\begin{array}{}\text{(6)}& \begin{array}{ll}|{\psi }^{\prime }{\rangle }_{BCDE}=\hfill & (\beta |HHHH\rangle +\alpha |VVVV\rangle )(\delta |b_1{c}_1{d}_1{e}_1\rangle \hfill \\ \hfill & +\gamma |b_2{c}_2{d}_2{e}_2\rangle )\mathrm{.}\hfill \end{array}\end{array}

The HAHED scheme for less-entangled mixed state with known parameters can also be implemented by using the quantum circuit shown in Fig. 1. The initial state of six-photon system A′ABCDE can be written as ρ₂⊗|ψ′〉_BCDE 〈ψ′|. Bob first lets the spatial modes |a₁〉, |a₂〉, |b₁〉 and |b₂〉 pass through PBS, and he postselects the cases that the polarization state of the photon system AB is in the even-parity mode and the spatial states of two-photon systems AC and BD are both in even-parity mode with four photon detectors triggered (i.e., only one photon can be detected in one of the spatial modes |l_m〉 and |l_n〉, mn = 13, 24, 57, 68). The postselected state of six-photon system A′ABCDE is

\begin{array}{}\text{(7)}& \begin{array}{ll}|{\text{Ψ}}_1\rangle =\hfill & \frac{1}{2}(|HHHHHH\rangle +|VVVVVV\rangle )\hfill \\ \hfill & (|a_1^{\prime }{a}_1{b}_1{c}_1{d}_1{e}_1\rangle +|a_2^{\prime }{a}_2{b}_2{c}_2{d}_2{e}_2\rangle )\mathrm{.}\hfill \end{array}\end{array}

Subsequently, Bob lets the spatial modes |a₁〉, |a₂〉, |b₁〉, |b₂〉, |c₁〉, |c₂〉, |d₁〉, and |d₂〉 pass through BSs and HWPs. The postselected state of six-photon system evolves to

\begin{array}{}\text{(8)}& \begin{array}{ll}|{\text{Ψ}}_2\rangle =\hfill & \frac{1}{32}\{[(|H\rangle +|V\rangle )(|H\rangle +|V\rangle )(|H\rangle +|V\rangle )(|H\rangle \hfill \\ \hfill & +|V\rangle )|HH\rangle +(|H\rangle -|V\rangle )(|H\rangle -|V\rangle )(|H\rangle \hfill \\ \hfill & -|V\rangle )(|H\rangle -|V\rangle )|VV\rangle ]\otimes [(|l_2\rangle -|l_4\rangle )(|l_5\rangle \hfill \\ \hfill & +|l_7\rangle )(|l_1\rangle +|l_3\rangle )(|l_6\rangle -|l_8\rangle )|a_1^{\prime }{e}_1\rangle +(|l_1\rangle \hfill \\ \hfill & -|l_3\rangle )(|l_5\rangle -|l_7\rangle )(|l_2\rangle +|l_4\rangle )(|l_6\rangle +|l_8\rangle )\hfill \\ \hfill & |a_2^{\prime }{e}_2\rangle ]\}\mathrm{.}\hfill \end{array}\end{array}

At last, Bob measures the photons ABCD in the eight output modes (|l₁〉, |l₂〉, |l₃〉, |l₄〉, |l₅〉, |l₆〉, |l₇〉 and |l₈〉). The successful cases of this HAHED scheme is the same as the HASPHD scheme as shown in Table 1. In the first group, the two-photon system A′E can be projected to the maximally hyperentangled state |ψ_A′E〉 = \frac{1}{2}(|HH\rangle +|VV\rangle )(|a_1^{\prime }{e}_1\rangle +|a_2^{\prime }{e}_2\rangle ). In the other groups, polarization phase-flip operation {\sigma }_z^P or spatial phase-flip operation {\sigma }_z^S has to perform on photon E to obtain the maximally hyperentangled state |ψ_A′E〉.

If only three photon detectors are triggered or the photons ABCD are detected in the other cases, the HAHED process is failed, and these cases should be removed. Therefore, the success probability of the HAHED scheme assisted by partially hyperentangled GHZ state |ψ′〉_BCDE is 4P₁|αβγδ|².

3.2 HAHED scheme for less-entangled mixed state with unknown parameters

If the parameters α, β, γ and δ are unknown to the two remote users Alice and Bob, another nonlocal photon pair B′C′ in the less-entangled mixed state {\rho }_2^{\prime }=P_1|{\psi }_1^{\prime }\rangle \langle {\psi }_1^{\prime }|+ (1-P_1){\rho }_1^{\prime } is required to complete the HAHED scheme.Here,

\begin{array}{}\text{(9)}& \begin{array}{ll}|{\psi }_1^{\prime }\rangle =\hfill & (\alpha |HH\rangle +\beta |VV\rangle )\otimes (\gamma |b_1^{\prime }{c}_1^{\prime }\rangle +\delta |b_2^{\prime }{c}_2^{\prime }\rangle )\mathrm{,}\hfill \\ {\rho }_1^{\prime }=\hfill & ({\alpha }^2|H\rangle \langle H|+{\beta }^2|V\rangle \langle V|)\hfill \\ \hfill & \begin{array}{l}\otimes [\frac{1-T_1}{(1-T_1)+(1-T_2)}|b_1^{\prime }\rangle \langle b_1^{\prime }|\\ +\frac{1-T_2}{(1-T_1)+(1-T_2)}|b_2^{\prime }\rangle \langle b_2^{\prime }|]\end{array}\hfill \\ \hfill & \otimes |vac{\rangle }_{C^{\prime }}\langle vac|)\mathrm{.}\hfill \end{array}\end{array}

The two photons B′ and C′ are obtained by the two remote users Alice and Bob respectively. The quantum circuit of HAHED for less-entangled mixed state with unknown parameters is shown in Fig. 2, where the amplification and concentration processes are performed simultaneously to increase the success probability and only one auxiliary hyperentangled GHZ state (|ϕ〉_BCDE = \frac{1}{2}(|HHHH\rangle +|VVVV\rangle {)}_{BCDE}(|b_1{c}_1{d}_1{e}_1\rangle +|b_2{c}_2{d}_2{e}_2\rangle )) is required to depress the resource consumption. The initial state of eight-photon system A′B′C′ABCDE is {\rho }_2\otimes {\rho }_2^{\prime }\otimes |\varphi {\rangle }_{BCDE}\langle \varphi |.

First, Alice and Bob perform the bit-flip operations on the polarization and spatial states of two photons B′ and C′ respectively as shown in Fig. 2. Then, Alice puts the spatial modes |a_1^{\prime }\rangle \mathrm{,}|a_2^{\prime }\rangle \mathrm{,}|b_1^{\prime }\rangle and |b_2^{\prime }\rangle into PBS in Fig. 2(a), and Bob puts the spatial modes |a₁〉 and |c_2^{\prime }\rangle into the PBS of HASPHD in Fig. 2(b), where the principle of HASPHD is introduced in Section 2. Here |c_2^{\prime }\rangle is put into HASPHD by the input port a₂ in Fig. 1. Alice postselects the cases that the polarization state of the photon system A′B′ is in the even-parity mode (one photon can be detected in one of the spatial modes |l_1^{\prime }\rangle and |l_2^{\prime }\rangle), and Bob postselects the successful cases of the HASPHD (the polarization state of the photon system AB is in the even-parity mode and the spatial states of two-photon systems AC and BD are both in even-parity mode with four photon detectors triggered) and the cases that one photon can be detected in one of the spatial modes |l_3^{\prime }\rangle and |l_4^{\prime }\rangle. The postselected state of eight-photon system A′B′C′ABCDE is

\begin{array}{}\text{(10)}& \begin{array}{ll}|{\text{Ψ}}_1^{\prime }\rangle =\hfill & \frac{1}{2}(|HHHHHHHH\rangle +|VVVVVVVV\rangle )\hfill \\ \hfill & \otimes (|a_1^{\prime }{b}_1^{\prime }{c}_1^{\prime }{a}_1{b}_1{c}_1{d}_1{e}_1\rangle +|a_2^{\prime }{b}_2^{\prime }{c}_2^{\prime }{a}_2{b}_2{c}_2{d}_2{e}_2\rangle )\mathrm{.}\hfill \end{array}\end{array}

Subsequently, Alice and Bob put the photons into BSs(BSs in Fig. 2 and HASPHD), which can make

\begin{array}{}\text{(11)}& \begin{array}{l}|k{\rangle }_{a_1}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_2}-|k{\rangle }_{l_4})\mathrm{,}|k{\rangle }_{a_2}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_3^{\prime }}-|k{\rangle }_{l_4^{\prime }})\mathrm{,}\\ |k{\rangle }_{c_1^{\prime }}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_3^{\prime }}+|k{\rangle }_{l_4^{\prime }})\mathrm{,}|k{\rangle }_{c_2^{\prime }}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_1}-|k{\rangle }_{l_3})\mathrm{,}\\ |k{\rangle }_{b_1^{\prime }}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_1^{\prime }}-|k{\rangle }_{l_2^{\prime }})\mathrm{,}|k{\rangle }_{b_2^{\prime }}\to \frac{1}{\sqrt{2}}(|k{\rangle }_{l_1^{\prime }}+|k{\rangle }_{l_2^{\prime }})\mathrm{.}\end{array}\end{array}

The postselected state of eight-photon system A′B′C′ABCDE is changed to

\begin{array}{}\text{(12)}& \begin{array}{ll}|{\text{Ψ}}_2^{\prime }\rangle =\hfill & \frac{1}{16}\{(|HHHHHHHH\rangle +|VVVVVVVV\rangle )\hfill \\ \hfill & [|a_1^{\prime }{e}_1\rangle (|l_2\rangle -|l_4\rangle )(|l_1^{\prime }\rangle -|l_2^{\prime }\rangle )(|l_3^{\prime }\rangle +|l_4^{\prime }\rangle )\hfill \\ \hfill & (|l_5\rangle +|l_7\rangle )(|l_1\rangle +|l_3\rangle )(|l_6\rangle -|l_8\rangle )+|a_2^{\prime }{e}_2\rangle \hfill \\ \hfill & (|l_3^{\prime }\rangle -|l_4^{\prime }\rangle )(|l_1^{\prime }\rangle +|l_2^{\prime }\rangle )(|l_1\rangle -|l_3\rangle )(|l_6\rangle +|l_8\rangle )\hfill \\ \hfill & (|l_2\rangle +|l_4\rangle )(|l_5\rangle -|l_7\rangle )]\}\mathrm{.}\hfill \end{array}\end{array}

Then, Alice and Bob let the photons pass through HWPs (HWPs in Fig. 2 and HASPHD), respectively. The postselected state of eight-photon system A′B′C′ABCDE is changed to

\begin{array}{}\text{(13)}& \begin{array}{ll}|{\text{Ψ}}_3^{\prime }\rangle =\hfill & \frac{1}{128}\{[|HH\rangle (|H\rangle +|V\rangle )(|H\rangle +|V\rangle )(|H\rangle +|V\rangle )\hfill \\ \hfill & (|H\rangle +|V\rangle )(|H\rangle +|V\rangle )(|H\rangle +|V\rangle )+|VV\rangle \hfill \\ \hfill & (|H\rangle -|V\rangle )(|H\rangle -|V\rangle )(|H\rangle -|V\rangle )(|H\rangle -|V\rangle )\hfill \\ \hfill & (|H\rangle -|V\rangle )(|H\rangle -|V\rangle )][|a_1^{\prime }{e}_1\rangle (|l_2\rangle -|l_4\rangle )\hfill \\ \hfill & (|l_1^{\prime }\rangle -|l_2^{\prime }\rangle )(|l_3^{\prime }\rangle +|l_4^{\prime }\rangle )(|l_5\rangle +|l_7\rangle )(|l_1\rangle +|l_3\rangle )\hfill \\ \hfill & (|l_6\rangle -|l_8\rangle )+|a_2^{\prime }{e}_2\rangle (|l_3^{\prime }\rangle -|l_4^{\prime }\rangle )(|l_1^{\prime }\rangle +|l_2^{\prime }\rangle )\hfill \\ \hfill & (|l_1\rangle -|l_3\rangle )(|l_6\rangle +|l_8\rangle )(|l_2\rangle +|l_4\rangle )(|l_5\rangle -|l_7\rangle )]\}\mathrm{.}\hfill \end{array}\end{array}

Finally, Alice and Bob measure the photons in the twelve output modes (|l_1^{\prime }\rangle \mathrm{,}|l_2^{\prime }\rangle \mathrm{,}|l_3^{\prime }\rangle and |l_4^{\prime }\rangle in Fig. 2, and |l₁〉, |l₂〉, |l₃〉, |l₄〉, |l₅〉, |l₆〉, |l₇〉 and |l₈〉 in HASPHD). According to the postselection rules, the successful cases of this HAHED scheme is shown in Table 2. Here, the spatial states |l_1^{\prime }{l}_3^{\prime }\rangle and |l_2^{\prime }{l}_4^{\prime }\rangle ( |l_1^{\prime }{l}_4^{\prime }\rangle and |l_2^{\prime }{l}_3^{\prime }\rangle) are defined as even (odd) parity mode, and De represents the measurement result of the spatial modes |l_1^{\prime }\rangle, |l_2^{\prime }\rangle, |l_3^{\prime }\rangle and |l_4^{\prime }\rangle. There are four groups in Table 2. In the first group, the state of two-photon system A′E is transformed to the maximally hyperentangled state |φ_A′E 〉 = \frac{1}{2}(|HH\rangle +|VV\rangle )(|a_1^{\prime }{e}_1\rangle +|a_2^{\prime }{e}_2\rangle ). In the other groups, polarization phase-flip operation {\sigma }_z^P or spatial phase-flip operation {\sigma }_z^S has to perform on photon E to obtain the maximally hyperentangled state |φ_A′E 〉.

If only three or four photon detectors are triggered in Fig. 2(b) or the photons B′C′ABCD are detected in the other cases, the HAHED process is failed, and these cases should be removed. Therefore, the success probability of the HAHED scheme assisted by hyperentangled GHZ state |ϕ〉_BCDE is P_1^2|\alpha \beta \gamma \delta {|}^2.

4 Discussion and conclusion

In this paper, we present hyperentanglement-assisted hyperdistillation schemes for protecting hyper-encoding photon system form transmission loss and quantum channel noise using quantum hyper-teleportation method. In fact, hyperentangled Bell state is the simplest resource for quantum hyper-teleportation, with which the vacuum state component caused by transmission loss can be eliminated by single-photon-trigger case in the hyperentangled Bell state measurement process of quantum hyper-teleportation. So far, the efficiency of quantum hyper-teleportation using hyperentangled Bell state is not high enough in experiment [73]. Therefore, we choose hyperentangled GHZ state, which has potential to be generated in experiment [74, 75] and can increase the success probability of quantum hyper-teleportation to 25% in linear optics, to implement the HASPHD scheme with the succuss probability of \frac{P_0}{4}. If the efficiency of quantum hyper-teleportation is increased to 100% [76], the succuss probability of HASPHD scheme can achieve P₀, which is the optimal succuss probability to eliminate the vacuum state component caused by transmission loss.

In HASPHD scheme, the transmission coefficients of lossy quantum channel for two spatial modes are supposed to be T₁ = T₂ = T . If the transmission coefficients are T₁ ≠ T₂, the state of the single photon may decay to {\rho }_0^{\prime }=(1-P_0^{\prime })|vac\rangle \langle vac|+P_0^{\prime }|{\phi }^{\prime }\rangle \langle {\phi }^{\prime }| by the inevitable transmission loss, where

\begin{array}{}\text{(14)}& \begin{array}{ll}|{\phi }^{\prime }\rangle =\hfill & (\alpha |H\rangle +\beta |V\rangle {)}_A(\sqrt{\frac{{\gamma }^2{T}_1}{{\gamma }^2{T}_1+{\delta }^2{T}_2}}|a_1\rangle \hfill \\ \hfill & +\sqrt{\frac{{\delta }^2{T}_2}{{\gamma }^2{T}_1+{\delta }^2{T}_2}}|a_2\rangle )\mathrm{,}\hfill \\ P_0^{\prime }=\hfill & {\gamma }^2{T}_1+{\delta }^2{T}_2\mathrm{.}\hfill \end{array}\end{array}

If the transmission coefficients T₁ and T₂ are known, the HASPHD scheme can be completed by changing the parameters of auxiliary hyperentangled GHZ state as shown in Section 3.1. If the transmission coefficients T₁ and T₂ are unknown, the HASPHD scheme can also be completed by the method shown in Section 3.2.

The HASPHD process is important for HAHED schemes. In the previous hyperentanglement distillation scheme with only hyperentanglement concentration process [64], if the transmission loss is happened in quantum channel, the wight of vacuum state will be increased after this hyperentanglement distillation process, so the final state of the scheme will become mixed hyperentangled state instead of maximally hyperentangled state. The fidelity of hyperentangled state after the hyperentanglement distillation with only hyperentanglement concentration process (for the case with unkonwn parameters) is expressed as

\begin{array}{}\text{(15)}& P=\frac{T_2(T_1+T_2)}{T_2(T_1+T_2)+T_2(2-T_1-T_2)+4(1-T_1)(T_1+T_2)}\mathrm{.}\end{array}

The relationships of the fidelities of hyperentangled states after (before) the hyperentanglement distillation processes vs transmission coefficient T (T₁ = T₂ = T ) are shown in Fig. 3. P₁ represents the fidelity of hyperentangled state in the initial state ρ₂ before hyperentanglement distillation. P represents the fidelity of hyperentangled state in final state after hyperentanglement distillation with only hyperentanglement concentration process. P ′ represents the fidelity of hyperentangled state in final state after HAHED process. In Fig. 3, it is obvious that the fidelity of hyperentangled state in final state is significantly decreased after hyperentanglement distillation with only hyperentanglement concentration process. Therefore, it is necessary to use HASPHD to remove the vacuum state, which will improve the fidelity of hyperentangled state. In HAHED scheme, the vacuum state can be completely removed together with the hyperentanglement concentration process, so the fidelity of hyperentangled state in final state is P ′ = 1, which means the less-entangled mixed state is recovered to maximally hyperentangled state.

In the hyperentanglement distillation scheme with only hyperentanglement concentration process [64], the success probabilities are 4|βγ|² (|β| < |α| and |γ| < |δ| with parameter-splitting method) and 4|αβγδ|² for known-parameter and unknown-parameter cases, where the vacuum state component can not been removed. In order to remove the vacuum state component simultaneously, HAHED schemes are introduced with lower success probabilities S₁ = 4P₁|αβγδ|² and S_2=P_1^2|\alpha \beta \gamma \delta {|}^2 for known-parameter and unknown-parameter cases, where the decrease of success probabilities is caused by the process of removing the vacuum state component. The HAHED scheme is not the simple combination of HASPHD and hyperentanglement concentration. For instance, in the previous hyperdistillation protocol for amplification [37], the vacuum state component in hyperentangled mixed state can be completely eliminated with the success probability of \frac{P_1}{4}, but the channel noise effect can not be depressed. In order to depress the channel noise effect and recover the less-entangled state to maximally hyperentangled state, the hyperentanglement concentration process should be performed after the amplification process, and the success probabilities are S₃ = P₁|βγ|² (|β| < |α| and |γ| < |δ| with parameter-splitting method [64]) and S_4=P_1^2|\alpha \beta \gamma \delta {|}^2/4 for known-parameter and unknown-parameter cases, respectively, where the vacuum state component caused by the inefficient detection in parameter-splitting method can not be removed and more resources are required for removing vacuum state components of two copies of less-entangled state in unknown-parameter case. In our HAHED schemes, the HASPHD and hyperentanglement concentration processes are performed simultaneously using quantum hyper-teleportation method, which has largely increased the success probability of recovering the less-entangled state to maximally hyperentangled state and reduced the resource consumption. The relationships of the success probabilities of hyperdistillation schemes vs parameters of less-entangled state are shown in Fig. 4. It is obvious that the success probabilities of recovering the less-entangled state to maximally hyperentangled state can achieve S₅ = 4P₁|βγ|² (|β| < |α| and |γ| < |δ| with parameter-splitting method) and S_6=4P_1^2|\alpha \beta \gamma \delta {|}^2 for known-parameter and unknown-parameter cases with the efficiency of quantum hyper-teleportation increased to 100%. In order to achieve the optimal succuss probabilities and reduce resource consumption, the HASPHD and hyperentanglement concentration processes should be performed simultaneously using quantum hyper-teleportation method as introduced in HAHED schemes instead of amplification-concentration method.

In the HAHED schemes, we suppose the maximally hyperentangled photon system is prepared locally by Alice, where only one photon of hyperentangled photon system will be transmitted to the remote party Bob through the lossy and noisy channel. If the hyperentangled photon source is prepared by Charlie, who is located in the middle point of Alice and Bob, the two photons of hyperentangled photon system will both be transmitted to Alice and Bob by the lossy and noisy channels to establish the quantum channel. Suppose that the transmission coefficients of the channels for spatial modes a₁, a₂,a_1^{\prime }, and a_2^{\prime } are T₁, T₂, T₃, T₄ (0 < T_j < 1, j = 1, 2, 3, 4), respectively. Then the state |ψ_AA′ 〉 will become a mixed hyperentangled state containing the vacuum state component,

\begin{array}{}\text{(16)}& {\rho }_3={\eta }_1|{\text{ϒ}}_1{\rangle }_{A{A}^{\prime }}{\langle {\text{ϒ}}_1|+{\eta }_2{\rho }_4+{\eta }_3{\rho }_5+{\eta }_4|vac\rangle }_{A{A}^{\prime }}\langle vac|\mathrm{,}\end{array}

where

\begin{array}{}\text{(17)}& \begin{array}{ll}\hfill & |{\text{ϒ}}_1{\rangle }_{A{A}^{\prime }}=(\alpha |HH\rangle +\beta |VV\rangle )\otimes (\epsilon |a_1{a}_1^{\prime }\rangle +\theta |a_2{a}_2^{\prime }\rangle )\mathrm{,}\hfill \\ \hfill & {\rho }_4=({\alpha }^2|H\rangle +{\beta }^2|V\rangle )[{\vartheta }^2|a_1\rangle \langle a_1|+{\iota }^2|a_2\rangle \langle a_2|]\hfill \\ \hfill & \otimes |vac{\rangle }_{A^{\prime }}\langle vac|\mathrm{,}\hfill \\ \hfill & {\rho }_5=({\alpha }^2|H\rangle +{\beta }^2|V\rangle )[{\kappa }^2|a_1^{\prime }\rangle \langle a_1^{\prime }|+{\mu }^2|a_2^{\prime }\rangle \langle a_2^{\prime }|]\hfill \\ \hfill & \otimes |vac{\rangle }_A\langle vac|\mathrm{.}\hfill \end{array}\end{array}

The parameters are \epsilon =\sqrt{\frac{T_1{T}_2}{T_1{T}_2+T_3{T}_4}}, \theta =\sqrt{\frac{T_3{T}_4}{T_1{T}_2+T_3{T}_4}}, \vartheta =\sqrt{\frac{T_1(1-T_2)}{T_1(1-T_2)+T_3(1-T_4)}}, \iota =\sqrt{\frac{T_3(1-T_4)}{T_1(1-T_2)+T_3(1-T_4)}}, κ = \sqrt{\frac{(1-T_1)T_2}{(1-T_1)T_2+(1-T_3)T_4}}, \begin{array}{rrr}\hfill \mu & \hfill =& \hfill \sqrt{\frac{(1-T_3)T_4}{(1-T_1)T_2+(1-T_3)T_4}}\end{array}, η₁ = \frac{1}{2}(T_1{T}_3+T_2{T}_4), \begin{array}{l}{\eta }_2=\frac{1}{2}[(1-T_3)T_1+T_2(1-T_4)]\hfill \end{array}, η₃ = \frac{1}{2}{T}_3(1-T_1)+T_4(1-T_2)] and {\eta }_4=\frac{1}{2}[(1-T_1)(1-T_3)+(1−T₂)(1−T₄)], respectively, and the coefficients satisfy the normalization relation as η₁ + η₂ + η₃ + η₄ = 1. In this situation, the HAHED schemes can also be used to recover the less-entangled state ρ₃ to maximally hyperentangled state, where Alice has to perform an additional HASPHD (in Fig. 1) on photon A′ to remove the vacuum state component of photon A′ besides the operations in Fig. 1 and Fig. 2, and the success probabilities will become η₁|αβεθ|² and {\eta }_1^2|\alpha \beta \epsilon \theta {|}^2/4 for known-parameter and unknown-parameter cases. If the efficiency of quantum hyper-teleportation is increased to 100%, the success probabilities of recovering the less-entangled state to maximally hyperentangled state can achieve 4η₁|βγ|² (|β| < |α| and |γ| < |δ| with parameter-splitting method) and 4{\eta }_1^2|\alpha \beta \gamma \delta {|}^2 for known-parameter and unknown-parameter cases.

In hyperdistillation schemes, the efficiencies of linear optical elements (e.g., PBSs, BSs and HWPs) are always assumed to be perfect (approximate to 100%). However, in a practical condition, the linear optical elements do not work in an ideal condition, and the imperfect linear optical elements will affect the final result of hyperdistillation protocol. In our scheme, hyperdistillation protocol is based on quantum hyper-teleportation, and there is no operation performed on the photon E except conditional operations at last. Therefore, the imperfect linear optical elements cause little effects on the final result of our protocol, and the vacuum state can be completely removed.

In summary, we proposed hyperentanglement-assisted hyperdistillation schemes to protect the hyper-encoding photon system from transmission loss and channel noise using quantum hyper-teleportation method, which can increase the success probability of hyperdistillation and reduce the resource consumption. In HASPHD scheme, the vacuum state component casued by transmission loss can be eliminated completely with high success probability by enhancing the efficiency of quantum hyper-teleportation. In the two HAHED schemes, the less-entangled mixed state can be recovered to maximally hyperentangled state with high success probability and low resource consumption. As the quantum hyper-teleportation method is adopted to depress the imperfect effects of optical elements on the final result, these hyperentanglement-assisted hyperdistillation schemes have potential application prospects in the field of practical quantum information processing.

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