1. Introduction

Quantum communication [1–14] is an important branch of quantum information, which introduces a novel way for information processing using the feature of quantum physics. Quantum cryptography [1] provides a reliable and secure way for communication, which combines quantum mechanics with cryptography. The security of quantum cryptography is guaranteed by the uncertainty principle and the no-cloning theorem [15] of quantum mechanics, and the eavesdropper cannot obtain the information of quantum state without disturbing the state. Thus, the authorized users could detect the presence of the eavesdropper by checking for errors to avoid eavesdropping. In 1984, Bennett and Brassard proposed the first quantum key distribution protocol using the single photon source [13]. Subsequently, various quantum key distribution protocols were proposed [16–26], such as "B92" protocol [16] and "Ekert91" protocol [17] based on entanglement. In these protocols, the mutually unbiased basis is one of effective methods to encode information and guarantee the security of communication, which has important applications in quantum key distribution [24–26].

During the communication process, the photons are often served as the carrier of quantum information, and the polarization degree of freedom (DOF) [27] of photon system is commonly used for encoding quantum information. In practical application, the photons are frequently influenced by the environment and channel noises, which will reduce the efficiency and the security of quantum communication protocols. For example, the polarization states need to be defined in a certain frame of reference, but they are easily affected by the rotation noise when transmitted to the parties that are very far apart in the distance. Due to the channel rotation noise or misalignment of reference frame for parties, the polarization states will be distorted when received by parties. This may lead to a significant error rate, and the parties cannot even obtain the key. Many schemes have been proposed to decrease the influence of noise, such as quantum error correction coding (QECC) [28], quantum error rejection [29–34], decoherence-free subspaces (DFSs) [35–47], and entanglement purification [48–52] and concentration [53–56]. In these protocols, it’s usually assumed that the photons are affected by the same noise when the time interval of photons is much shorter than the variation scale of noise fluctuation, which is known as the collective noise model [57]. In 1989, Martinelli proposed an optical scheme to compensate the fluctuation of polarization state in the optical fiber using the Faraday effect [58]. In this scheme, the two-way quantum communication is used, which has the threat of the Trojan horse attacks. The DFS is one of the effective methods against the collective noise [35–47], which can avoid the Trojan horse attacks caused by two-way quantum communication. In 2004, Boileau et al. [36] proposed two quantum key distribution (QKD) protocols against the collective-rotation noise using DFSs constructed by four-photon entangled states and three-photon entangled states respectively. In 2005, Yamamoto et al. [40] proposed the QKD protocol against the collective noise using an auxiliary particle and postselection to pick out the error-free parts. In 2008, Li et al. [41] presented two QKD protocols against the collective noise by using DFSs made up of two polarization Bell states, where the spatial DOF of four-photon system is used to form two nonorthogonal bases. Subsequently, some protocols have shown that the multi-photon entangled states [42,43] can also be used for constructing the DFS.

There are various degrees of freedom (DOFs) in photon system that can be used in quantum communication, such as phase [59], time [35], spatial mode [36], transverse spatial mode [37], and so on. The additional DOFs of photon system are often used for enlarging the channel capacity [60–62] and reducing the sensitivity of photon loss, such as the applications of transverse spatial mode [37] and orbital angular momentum (OAM) DOFs of photons [24–26,46]. In 2007, Aolita and Walborn proposed the single-photon QKD scheme using the polarization and transverse spatial mode DOFs [37]. The DFS against the collective-rotation noise can be constructed with single-photon logical states in multiple DOFs. In 2012, D’Ambrosio et al. [46] exploited the hybrid polarization-OAM states of single photon to construct the DFS against the noise caused by misalignments, and they applied this DFS to QKD protocol. These QKD protocols are based on the Bennett and Brassard (BB84) protocol [13]. The efficiency of BB84 protocol is low since half of the samples cannot be used to generate the key.

In this article, we propose an efficient quantum key distribution protocol against collective-rotation noise using the polarization and transverse spatial mode DOFs of photons. The noiseless subspaces in this scheme are made up of two mutually unbiased bases using two single-photon Bell states and two-photon hyperentangled Bell states, which can reduce the resources required for noiseless subspaces and depress the photonic loss sensitivity compared with the pervious works using multiple photons. As the two single-photon Bell states and two-photon hyperentangled Bell states are symmetrical to the two photons, the relative order of the two photons is not required in our scheme, so the receiver only needs to measure the state of each photon, which makes our protocol easy to execute in experiment than the previous works. Moreover, the authorized users don’t need to switch the measurement bases to get the key, which improves the efficiency of QKD protocol compared with the previously proposed BB84 scheme.

2. Quantum key distribution against collective rotation noise using polarization and transverse spatial mode DOFs

The polarization and transverse spatial mode DOFs of photon system can both be used to encode quantum information. The polarization states and transverse spatial modes are perpendicular to the propagation direction of light, and the character of transverse spatial modes $\rm HG_{10}$ ($\left |h\right \rangle$) and $\rm HG_{01}$ ($\left |v\right \rangle$) is similar to that of horizontal polarization ($\left |H\right \rangle$) and vertical polarization ($\left |V\right \rangle$), as shown in Fig. 1. Here the expression HG$_{nm}$ is the Hermite-Gaussian mode, where $n$ and $m$ are the mode orders related to $x$ and $y$ axes respectively. $\left |h\right \rangle$ and $\left |v\right \rangle$ denote the first-order Hermite-Gaussian modes $\rm HG_{10}$ and $\rm HG_{01}$. Under the influence of collective-rotation noise, the polarization states will be disturbed and evolute as $\left |H\right \rangle \to \cos \theta \left |H\right \rangle +\sin \theta \left |V\right \rangle$ and $\left |V\right \rangle \to \cos \theta \left |V\right \rangle -\sin \theta \left |H\right \rangle$. The transverse spatial modes of photon system will also be affected by the collective-rotation noise, which evolute as $\left |h\right \rangle \to \cos \theta \left |h\right \rangle +\sin \theta \left |v\right \rangle$ and $\left |v\right \rangle \to \cos \theta \left |v\right \rangle -\sin \theta \left |h\right \rangle$.

 

Fig. 1. (a) The polarization states. The first line presents horizontal and vertical polarizations, and the second line presents left and right circular polarizations. (b) The transverse spatial modes. The first line presents $\rm HG_{10}$ mode and $\rm HG_{01}$ mode, and the second line presents $\rm LG_{\pm 1,0}$ modes.

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The Bell states $\left |\phi ^{+}\right \rangle$ ($\left |\phi ^{+}\right \rangle =\frac {1}{\sqrt {2}}\left (\left |0\right \rangle \left |0\right \rangle +\left |1\right \rangle \left |1\right \rangle \right )$) and $\left |\psi ^{-}\right \rangle$ ($\left |\psi ^{-}\right \rangle =\frac {1}{\sqrt {2}}\left (\left |0\right \rangle \left |1\right \rangle -\left |1\right \rangle \left |0\right \rangle \right )$) are invariant under the collective-rotation noise. Thus the single-photon Bell states using polarization and transverse spatial mode DOFs [37] can be used to eliminate the collective-rotation noise, i.e. $|\phi _0\rangle =\frac {1}{\sqrt {2}}(\left |H\right \rangle \left |h\right \rangle +\left |V\right \rangle \left |v\right \rangle )$ and $|\psi _1\rangle =\frac {1}{\sqrt {2}}(\left |H\right \rangle \left |v\right \rangle -\left |V\right \rangle \left |h\right \rangle )$. Similarly, the two-photon Bell states in polarization (transverse spatial mode) DOF can also be used to eliminate the collective-rotation noise, i.e. $|\phi '_0\rangle =\frac {1}{\sqrt {2}}(\left |H\right \rangle \left |H\right \rangle +\left |V\right \rangle \left |V\right \rangle )$ ($|\phi ''_0\rangle =\frac {1}{\sqrt {2}} (\left |h\right \rangle \left |h\right \rangle +\left |v\right \rangle \left |v\right \rangle )$) and $|\psi '_1\rangle =\frac {1}{\sqrt {2}}(\left |H\right \rangle \left |V\right \rangle -\left |V\right \rangle \left |H\right \rangle )$ ($|\psi ''_1\rangle =\frac {1}{\sqrt {2}}(\left |h\right \rangle \left |v\right \rangle -\left |v\right \rangle \left |h\right \rangle )$). In our scheme, the two single-photon Bell states and the two-photon hyperentangled Bell states can be used to construct two mutually unbiased bases, which are used as the logical qubits to against the collective-rotation noise. Here, the hyperentangled Bell state is defined as quantum state entangled in both polarization and transverse spatial mode DOFs.

In QKD protocol against collective-rotation noise, the sender Alice randomly prepares logical qubits using four two-photon states of two mutually unbiased bases (Eqs.(1)–(4)), and she sends them to the receiver Bob over the quantum channel, where the two photons have a time interval $\Delta \textrm {t}$. The four two-photon states of two mutually unbiased bases are expressed as

When the receiver Bob receives the logical qubits, he can randomly choose different measurement bases to detect the logical qubit states. If the measurement basis is chosen in $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$, the logical qubit states can be distinguished by the detection setup shown in Fig. 2. Here the discrimination of logical qubit states can be performed by a Mach-Zehnder interferometer with an additional mirror (MZIM) [67] and polarizing beam splitters (PBSs). The MZIM is composed by two 50:50 beam splitters (BSs) and three high-reflectivity mirrors, and PBS can transmit the horizontally polarized state and reflect the vertically polarized state. When the two photons of logical qubit state pass through the MZIM successively, they will exit from the path $1$ (or path $2$) according to the parity mode of polarization and transverse spatial mode DOFs [67]. If the polarization and transverse spatial mode DOFs of a photon are in the even parity mode ({$\left |H\right \rangle \left |h\right \rangle$ and $\left |V\right \rangle \left |v\right \rangle$}), the photon will exit from the path $1$. If the polarization and transverse spatial mode DOFs of a photon are in the odd parity mode ({$\left |H\right \rangle \left |v\right \rangle$ and $\left |V\right \rangle \left |h\right \rangle$}), the photon will exit from the path $2$. Here the relative phase shift of two arms of MZIM is zero. Combined the MZIM with the PBSs, the four single-photon states $\left |H\right \rangle \left |h\right \rangle$, $\left |V\right \rangle \left |v\right \rangle$, $\left |H\right \rangle \left |v\right \rangle$, and $\left |V\right \rangle \left |h\right \rangle$ can be detected at four output ports $4$, $3$, $6$, and $5$, respectively. At last, Bob can distinguish the states $a$, $b$, $c$, and $d$ according to the detection information of receiving photons and obtain the key information according to the encoding bases information given by Alice as shown in Table 1.

 

Fig. 2. The detection setup for the logical qubit states with the measurement basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$. The detection device includes Mach-Zehnder interferometer with an additional mirror (MZIM) and polarizing beam splitters (PBSs), where the MZIM is composed by two 50:50 beam splitters (BSs) and three high-reflectivity mirrors. The polarizing beam splitter (PBS) is used to transmit the polarization state $\left |H\right \rangle$ and reflect the polarization state $\left |V\right \rangle$.

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Table 1. The decoding results of logical qubit states with measurement basis {$\left |H\right \rangle ,\left |V\right \rangle ,\rm {HG}$}.

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If the measurement basis is chosen in $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$ (shown in Fig. 1), the logical qubit states of receiving photons can be detected by the detection setup shown in Fig. 3. Here $\left |L\right \rangle$ and $\left |R\right \rangle$ represent left and right circularly polarized states respectively, and the LG represents Laguerre-Gaussian transverse spatial modes $\rm {LG}_{\ell {p}}$, where subscripts $\ell$ and $p$ are azimuthal mode index and radial mode index respectively. The relationship between rectilinear polarizations and circular polarizations can be expressed as $\left |L\right \rangle =\frac {1}{\sqrt {2}}\left (\left |H\right \rangle -i\left |V\right \rangle \right )$ and $\left |R\right \rangle =\frac {1}{\sqrt {2}}\left (\left |H\right \rangle +i\left |V\right \rangle \right )$, which can be obtained by performing quarter-wave plate on photons. Similarly, the relationship between the $\rm HG_{10}$, $\rm HG_{01}$ modes and the $\rm LG_{\pm 1,0}$ ($\ell =\pm 1$, $ {p}=0$) modes can be expressed as $\left |l\right \rangle =\frac {1}{\sqrt {2}}\left (\left |h\right \rangle -i\left |v\right \rangle \right )$ and $\left |r\right \rangle =\frac {1}{\sqrt {2}}\left (\left |h\right \rangle +i\left |v\right \rangle \right )$, where the transformation of HG mode and LG mode can be obtained by performing $\frac {\pi }{2}$-mode converter on photons [68–70]. Here $\left |l\right \rangle$ and $\left |r\right \rangle$ refer to first-order left-handed ($\ell =-1$) and right-handed ($\ell =+1$) OAM, respectively. The Eqs.(1)–(4) can be expanded on the basis $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$, which can be rewritten as

 

Fig. 3. The detection setup for the logical qubit states with the measurement basis $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$, including the q-plate (q=1/2), a Mach-Zehnder interferometer with a Dove prism$@45{^\circ }$, and circular-polarization analysis setup. The Mach-Zehnder interferometer with a Dove prism@$45{^\circ }$ is composed by two 50:50 BSs, a Dove prism@45° and two reflective mirrors, and the circular-polarization analysis setup is composed by quarter-wave plate, half-wave plate, and PBS. Dove$@45{^\circ }$ represents the Dove prism aligned at $45{^\circ }$, which can rotate the beam by $\alpha =90{^\circ }$. $\frac {\lambda }{4}$ represents quarter-wave plate aligned at $90{^\circ }$, and $\frac {\lambda }{2}$ represents half-wave plate aligned at $-22.5{^\circ }$.

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The detection setup shown in Fig. 3 includes the q-plate (q=1/2) [71,72], a Mach-Zehnder interferometer with a beam rotator (Dove prism@45${^\circ }$) [73], and circular-polarization analysis setup. Here the Mach-Zehnder interferometer with a Dove prism@$45{^\circ }$ is composed by two 50:50 BSs, a Dove prism$@45{^\circ }$, and two mirrors, and the circular-polarization analysis setup is composed by quarter-wave plates, half-wave plates, and PBSs. The q-plate achieves the transformation of $U_1=\left |R,\ell -2q\right \rangle \left \langle L,\ell \right |+\left |L,\ell +2q\right \rangle \left \langle R,\ell \right |$. After two photons of quantum logical state passing through the q-plate (q=1/2), the photon states $|R,l\rangle$, $|L,r\rangle$, $|R,r\rangle$, and $|L,l\rangle$ will become $|L,\ell =0\rangle$, $|R,\ell =0\rangle$, $|L,\ell =2\rangle$, and $|R,\ell =-2\rangle$ respectively. Then the two photons pass through the Mach-Zehnder interferometer with a Dove prism aligned at 45${^\circ }$. Here the Dove prism rotated at an angle 45${^\circ }$ is used to rotate the state $|\ell \rangle$ (with an OAM of $\ell \hbar$) with an angle 90${^\circ }$, which will introduce a phase factor $\rm {exp}(i\pi \ell /2)$ to the state $|\ell \rangle$. Thus, the photons in $\ell =0$ mode and $\ell =\pm 2$ modes will be led into paths $1$ and $2$ respectively. After passing through circular-polarization analysis setup, the polarization of photons can be distinguished. Here quarter-wave plates aligned at 90${^\circ }$ execute $U_2=|+\rangle \langle R|+|-\rangle \langle L|$ operation, and the half-wave plates aligned at −22.5${^\circ }$ execute $U_3=|V\rangle \langle +|+|H\rangle \langle -|$ operation, where $|+\rangle =(|H\rangle +|V\rangle )/\sqrt {2}$ and $|-\rangle =(|H\rangle -|V\rangle )/\sqrt {2}$ represent diagonally and anti-diagonally polarized states. At last, the initial input states $|R, l\rangle$, $|L, r\rangle$, $|R, r\rangle$, and $|L, l\rangle$ will be detected at $4$, $3$, $6$, and $5$ output ports respectively. Using this detection setup, the receiver Bob can distinguish the states in Eqs.(13)–(15) according to the output ports of the receiving two photons. If the two photons are detected at $3$ and $4$ output ports, the two-photon state is Eq.(13). If the two photons are both detected at $3$ (or $4$) output port, the two-photon state is Eq.(14). If the two photons are detected at $5$ and $6$ output ports, the two-photon state is Eq.(15).

One can see that the logical qubit states $\left |\Phi _{0}\right \rangle$, $\left |\Phi _{1}\right \rangle$, $\left |\Psi _{0}\right \rangle$, and $\left |\Psi _{1}\right \rangle$ can be expanded on the basis {$\left |H\right \rangle ,\left |V\right \rangle ,\rm {HG}$} or $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$. Therefore, the receiver Bob can select different basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$ or $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$ to measure the states of photons for detecting the presence of eavesdropper.

The steps of QKD scheme against collective-rotation noise using two single-photon Bell states and two-photon hyperentangled Bell states based on polarization and transverse spatial mode DOFs are described as follows:

(1) The sender Alice randomly prepares a ($n+2m$) bit string K and a ($n+2m$) bit string B.

(2) The sender Alice encodes the information with the basis $\left \{ \left |\Psi _{0}\right \rangle ,\left |\Psi _{1}\right \rangle \right \}$ (or $\left \{ \left |\Phi _{0}\right \rangle ,\left |\Phi _{1}\right \rangle \right \}$) according to the information of each bit in bit string K, where the selection of encoding basis $\left \{ \left |\Psi _{0}\right \rangle ,\left |\Psi _{1}\right \rangle \right \}$ or $\left \{ \left |\Phi _{0}\right \rangle ,\left |\Phi _{1}\right \rangle \right \}$ depends on the corresponding bit in bit string B. If the corresponding bit in bit string B is $0$, she chooses the basis $\left \{ \left |\Psi _{0}\right \rangle ,\left |\Psi _{1}\right \rangle \right \}$ to encode information; otherwise she chooses the basis $\left \{ \left |\Phi _{0}\right \rangle ,\left |\Phi _{1}\right \rangle \right \}$ to encode information.

(3) The sender Alice sends $n+2m$ two-photon states to the receiver Bob.

(4) The receiver Bob measures the logical qubit states with the detection setups shown in Fig. 2 and Fig. 3 after receiving the two-photon states. He randomly chooses $2m$ two-photon states from the $n+2m$ two-photon states for the eavesdropping check, where $m$ two-photon states are measured with the basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$ using detection setup shown in Fig. 2 and the other $m$ two-photon states are measured with the basis $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$ using detection setup shown in Fig. 3. The remaining $n$ two-photon states are measured with the basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$ using detection setup shown in Fig. 2. Bob records all the results of measurement.

(5) The receiver Bob tells the sender Alice the position of samples for the eavesdropping check through the classical channel. Thus Alice can tell the receiver Bob about the initial states of those samples. Bob compares the results of measurements with the initial states of samples for estimating the error rate. If the error rate is below the threshold value, they continue to the next step. Otherwise, they will abort the protocol and return to the first step.

(6) After the two parties guarantee the security of the communication channel in the quantum communication, the sender Alice will announce the bit string B. Thus Bob can deduce key bit string according to the information of bit string B, and he can take the results as the raw key string. The decoding results are shown in Table 1. At last, they can get the secure secret key after the error correction and privacy amplification protocols.

In the QKD process, the receiver Bob only measures the states of two photons with the linear optical elements. Particularly, in the process of getting keys, the receiver Bob only needs to measure every single photon with the basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$. This QKD scheme can be capable of resisting collective-rotation noise with only two photons, and it doesn’t need to transform the measurement basis in the process of getting the keys, which is more efficient than the previous protocols.

3. Security analysis

In our modified QKD protocol, Alice adopts logical bases $\left |\Psi _0\right \rangle / \left |\Psi _1\right \rangle$ and $\left |\Phi _0\right \rangle / \left |\Phi _1\right \rangle$ to encode the information, and she sends the logical quantum signal to Bob to perform the logical measurement. Different from the procedure using only one DOF in original QKD, the logical measurement firstly sorts the two photons of hybrid entangled state. Since the security check process in our QKD protocol is similar to the BB84 QKD, the security proof of our modified QKD scheme with logical basis is equivalent to the original QKD. The BB84 QKD has been proven unconditionally secure in many protocols [74,75]. The secret key rate is $r=1-H(Q)-I_E(Q)$, where the quantum bit error rate (QBER) is Q=11% for r=0. If the error rate is greater than 11%, the parties cannot obtain secure final key, which will lead to the failure of protocol. Here we use two ways of eavesdropping attack [1] to analyze the security of this scheme. One is the intercept and resend attack, and the other is eavesdropping attack using auxiliary particle.

(1) Security analysis of the QKD protocol for the intercept and resend attack.

The intercept and resend attack contains two different procedures of measurements. One is the product measurement for single photon, and the other is the Bell state measurement. For the product measurement for single photon, as the receiver Bob can obtain the key information by measuring the two photons with the basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$, the eavesdropper can also obtain the information by measuring the two photons with the basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$. When the eavesdropper Eve gets the measurement result of $a$ or $d$, he can directly deduce the key $0$ or $1$. However, if the eavesdropper Eve gets the measurement result of $b$ or $c$, he doesn’t know the information of key unless the sender Alice announces the information of bit string B. To avoid being discovered, Eve fabricates the fake particles and resends them to Bob. He can directly resend two photons in state $a$, $b$, $c$, or $d$ to the receiver Bob according to his measurement results, or he can speculate the initial state according to measurement results and resend it to Bob. If Eve resends two photons in state $a$, $b$, $c$, or $d$ to the receiver Bob, the receiver Bob cannot find the error by measuring the two photons with the basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$, but there is an error rate of 50% by measuring the two photons with the basis $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$. Since Bob randomly chooses the measurement basis for checking errors, the average error rate $\rm {e_{A}}$ is 25%. If Eve speculates the initial state according to his measurement results and resends it to Bob, there is a 50% chance of choosing the correct initial state, which will not be found by Bob. However, under the circumstance of choosing the wrong initial state, there is an error rate of 50% no matter what measurement basis is selected by Bob. Therefore the average error rate $\rm {e_{A}}$ is 25%.

For Bell state measurement (BSM), Eve cannot get the information of the encoding basis selected by Alice, so he can only choose $\left \{ \left |\Psi _{0}\right \rangle , \left |\Psi _{1}\right \rangle \right \}$ or $\left \{ \left |\Phi _{0}\right \rangle , \left |\Phi _{1}\right \rangle \right \}$ randomly. For different encoding basis, the eavesdropper Eve chooses different measurement device. For example, Eve executes single-photon Bell state measurement for $\left \{ \left |\Psi _{0}\right \rangle , \left |\Psi _{1}\right \rangle \right \}$ and only measures one photon due to the symmetry, and he executes two-photon Bell state measurement in polarization (or transverse spatial mode) DOF for $\left \{\left |\Phi _{0}\right \rangle , \left |\Phi _{1}\right \rangle \right \}$. If Eve chooses the correct encoding basis, he can speculate the correct state and resends the correct initial state to Bob with no error. If Eve chooses the wrong encoding basis, he cannot obtain the correct information and has to speculate the initial state to resend it. There is a 50% chance for Eve to realize his mistake and correct the error. But Eve doesn’t know whether encoding key is 0 or 1, and there is still an error rate of 50% when Bob selects the measurement basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$ for the eavesdropping check. If Eve doesn’t realize his mistake and resends the initial state according to measurement results, there is an error rate of 50% no matter what measuring basis is selected by Bob. Therefore the error rate is 25% when Bob selects the measurement basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$, and the error rate is 12.5% when Bob selects the measurement basis $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$. Thus the average error rate $\rm {e_{A}}$ is 18.75%. The error rates for the different ways of eavesdropping attack are shown in the Table 2. Here the e$(\left |H\right \rangle ,\left |V\right \rangle ,\rm HG)$ and e$(\left |R\right \rangle ,\left |L\right \rangle ,\rm LG)$ respectively represent the error rates in the measurement bases $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$ and $\{\left |R\right \rangle ,\left |L\right \rangle ,\rm LG\}$. $\rm {e_{A}}$ represents the average error rate, and MB represents the measurement basis.

Table 2. The error rates for the different ways of eavesdropping attack.

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(2) Security analysis of the QKD protocol for the eavesdropping attack using auxiliary particle.

For the logical qubits encoded in the basis $\left \{ \left |\Psi _{0}\right \rangle ,\left |\Psi _{1}\right \rangle \right \}$, Eve can measure the parity of the polarization and transverse spatial mode DOFs of single photon using auxiliary photon 3 and sends the logical qubits to the receiver Bob after measurement. For example, Eve can perform CNOT gate operations C$_{13}^{p}$ and C$_{13}^{t}$ on the polarization and transverse spatial mode DOFs of photon 1 and polarization DOF of photon 3, where the polarization and transverse spatial mode DOFs of photon 1 are used as the control qubits and the polarization DOF of photon 3 is used as the target qubit. Here the polarization DOF of photon 3 is initially in state $|H\rangle _3$, and it will flip if the polarization DOF of photon 1 is $|V\rangle _1$ or the transverse spatial mode of photon 1 is $|v\rangle _1$. If the polarization state of photon 3 remains unchanged, the polarization and transverse spatial mode DOFs of photon 1 are in even parity mode, which means the value of bit is 0. If the polarization state of photon 3 changes to $|V\rangle _3$, the polarization and transverse spatial mode DOFs of photon 1 are in odd parity mode, which means the value of bit is 1. For the logical qubits encoded in the basis $\left \{ \left |\Phi _{0}\right \rangle ,\left |\Phi _{1}\right \rangle \right \}$, Eve can measure the parity of the polarization DOF of the two photons using auxiliary photon 3 and sends the logical qubits to the receiver Bob after measurement. For example, Eve can perform CNOT gate operations C$_{13}^{p}$ and C$_{23}^{p}$ on the polarization states of photons 1, 2, and 3, where the polarization states of photons 1 and 2 are used as the control qubits and the polarization state of photon 3 is used as the target qubit. If the polarization states of two photons 1 and 2 are in even parity mode ($|HH\rangle _{12}$ or $|VV\rangle _{12}$), the polarization state of the target photon 3 remains unchanged, which means the bit value is 0. If the polarization states of two photons 1 and 2 are in odd parity mode ($|HV\rangle _{12}$ or $|VH\rangle _{12}$), the polarization state of the target photon 3 is changed, which means the bit value is 1. As a result, Eve can measure the state of auxiliary photon 3 to obtain the key information. When Eve chooses the correct encoding basis, he can obtain the information without causing the error. When Eve chooses the wrong encoding basis, the error rate is 50% when Bob selects the measurement basis $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$, and the error rate is 0 when Bob selects the measurement basis $\{\left |H\right \rangle ,\left |V\right \rangle ,\rm HG\}$. Therefore, the average error rate $\rm {e_{A}}$ is 12.5%. The error rates for the different ways of eavesdropping attack are shown in the Table 2.

4. Simulation

Although the logical source encoding process and the logical measurement process are some different from the original QKD, our modified QKD still basically uses BB84 protocol and decoy state method to assure the security. Combined with the decoy state method to overcome the potential photon number splitting (PNS) attack, the yield $Y_1^Z$ and the error rate $e_1^X$ under the practical transmission efficiency can be expressed as,

The simulation curves of key rate $R$ versus the maximal secure transmission distance $L$ are shown in Fig. 4. For simplicity, we ignore mode mismatch in our simulation, and we assume that the system is operated under the normal condition, which means all other parameters are same with the original polarization encoding QKD. The experimental parameters used in the simulations are shown in the Table 3. In our simulation, we adopt a practical assumption for the polarization misalignment. The total error is $e_d=e_{AB}+e_L$. $e_{AB}$ represents the misalignment of Alice-Bob channel transmission, while $e_L$ models the misalignment of the logical state preparation procedure. The probability distribution of $e_k$ is selected as $e_{AB}=0.95e_d$ and $e_L=0.05e_d$ . It is important to notice that the channel transmission misalignment can be removed using the rotation invariant logical qubits. Compared with the key rate of the original QKD scheme, our results show that the modified protocol using hybrid logical basis actually offers a higher performance both in key rate and maximum distance than the polarization encoding scheme owing to the lower quantum bit error rate shown in Fig. 4.

 

Fig. 4. Key rate versus the maximal secure transmission distance L with different protocols. Blue Solid line: QKD protocol using weak coherent source. Red Dash line: modified QKD protocol with logical basis.

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Table 3. The list of experimental parameters used in the simulations

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5. Discussion and summary

In this article, we have proposed an efficient QKD scheme against the collective-rotation noise using the polarization and transverse spatial mode DOFs of photons, where the noiseless subspaces are made up of two mutually unbiased bases, including two single-photon Bell states and two-photon hyperentangled Bell states. The two mutually unbiased bases can be expanded on two different measurement bases. The sender Alice encodes the key bit string using the two mutually unbiased bases, and she sends them to Bob. During the process of transmission, the collective-rotation noise can be removed in principle. When the receiver Bob receives the encoding logical qubit states, he randomly selects the measurement basis {$\left |H\right \rangle ,\left |V\right \rangle ,\rm {HG}$} or $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$ to measure the logical qubit states for security check. In the process of getting keys, the receiver Bob only needs to measure the logical qubit states using the basis {$\left |H\right \rangle ,\left |V\right \rangle ,\rm {HG}$} without changing the measurement basis. Thus, the authorized users don’t switch the measurement bases to get the key, which reduces the complexity of the QKD protocol and increases intrinsic efficiency of QKD protocol compared with BB84-QKD scheme [13,36,39,40].

In this QKD scheme, the polarization and transverse spatial mode DOFs of two photons are used to construct noiseless subspaces instead of multiple photons. The transmission efficiency decreases exponentially with transmission distance when the photons propagate in lossy channel, and the noise influence on transmission efficiency will be greater with the increasing of photon number. Compared with the previous multi-photon QKD schemes [36,41–43], our scheme uses only two photons to construct noiseless subspaces, which reduces the loss sensitivity caused by lossy channel and improves the transmission efficiency.

As the two single-photon Bell states and two-photon hyperentangled Bell states are symmetrical to the two photons, our scheme reduces the requirements for the relative order of the photons, which is easy to execute in the experiment. In the previous multi-photon QKD schemes [36,41–43], the order of photons has important influence on information decoding process. If the receiver measures the photons in the wrong order, he will obtain the wrong information which will lead to the failure of protocol. In our scheme, the information of keys is encoded in the symmetry states of two photons. When the order of photons 1 and 2 is exchanged, the states keep invariant. Therefore the relative order of photons can’t affect the measurement results, which reduces the difficulty in practical application and makes this scheme easy to implement in experiment.

In this scheme, one of the most important devices is MZIM (or the Mach-Zehnder interferometer), which has been demonstrated in many experiments [64,67,77–79]. The visibility of MZIM is around 97$\%$ when it is used to select different components [79], and the visibility can approach unity with advanced experimental technology. Besides, some linear optical elements are used in the scheme, such as q-plate, Dove prism, PBS, and so on. The efficiencies of these optical elements are high enough to execute the scheme. For example, the efficiency of q-plate can reach about 80$\%$ [80], and the tunable liquid crystal q-plate can reach high conversion efficiency (up to 99$\%$) [81]. Therefore, our QKD scheme can work in a high-fidelity way in principle.

In summary, an efficient QKD protocol against the collective-rotation noise is presented using the polarization and transverse spatial mode DOFs of the photons. The sender Alice encodes the information on the noiseless subspaces that are made up of two single-photon Bell states and two-photon hyperentangled Bell states. The receiver Bob can check error (for security check) by randomly choosing the different measuring bases {$\left |H\right \rangle ,\left |V\right \rangle ,\rm {HG}$} and $\{\left |L\right \rangle ,\left |R\right \rangle ,\rm LG\}$, and he only needs to measure the logical qubit states with the basis {$\left |H\right \rangle ,\left |V\right \rangle ,\rm {HG}$} to get keys. This QKD scheme can be achieved by constructing noiseless subspaces with two DOFs of two photons, which reduces the photon loss sensitivity [36,41–43]. Moreover, this scheme has no restriction on the relative order of two photons, and the receiver can obtain the keys by measuring the states of single photons using the linear optical elements, which reduces the difficulty of its implementation in experiment. Therefore, this scheme is a significant step for information security transmission and long-distance quantum communication.