Four-dimensional orbital angular momentum Bell-state measurement assisted by the auxiliary polarization and path degrees of freedom

Chaojie Wang; Yuanyuan Chen; Yuanyuan Chen; Lixiang Chen; Lixiang Chen

doi:10.1364/OE.469704

1. Introduction

Bell-state measurement (BSM) plays an critical role in many quantum information processing protocols, such as quantum dense coding [1,2], quantum teleportation [3,4] and entanglement swapping [5,6]. In addition, BSM is also essential in quantum secure direct communication [7], device-independent quantum secure direct communication [8], measurement-device-independent quantum secure dreict communication [9] and quantum key distribution [10,11]. The completion of these quantum communication processes involves quantum entanglement, and a very interesting finding recently is that entanglement can serve as a measure of renormalizability of quantum fields theories [12]. While BSM has been implemented in many photonic degrees of freedom (DOFs), a broad consensus has been that the complete BSM cannot be achieved with linear optics with current technology [13,14]. To tackle this issue, nonlinear optics and auxiliary entanglement have been used as an assistance to enable the deterministic discrimination of all Bell-states in two-dimensional Hilbert space [15–18], which can be applied in the superdense coding that beats the channel capacity limit [19,20].

In the context of quantum communication, photon pairs entangled in high dimensions can carry more quantum information, making them compelling for enhancing quantum channel capacities [21,22], improving noise resilience [23–25], and even speeding up certain tasks in photonic quantum computation [26]. For the implementation of these quantum experiments based on high-dimensional entanglement, entanglement source has been widely investigated in multiple DOFs, including energy-time [27,28], OAM [29,30], and path [31,32]. Thereinto, photons carrying OAM entanglement have attracted great interest in recent years because they are intrinsically suitable for high-dimensional quantum information processing [33–37] and quantum imaging [38,39]. However, deterministic discrimination of the complete Bell-state in high-dimensional OAM domain with linear optics is still a formidable challenge. It is well-known that photons can be entangled in multiple DOFs simultaneously, i.e., hyperentanglement [40], which provides the potential to perform complete BSM for high-dimensional system with merely linear optics. Recently, Zhang et al. designed a clever scheme for realizing three-dimensional BSM by auxiliary entanglement, in which the three-dimensional system state was encoded in the photon path DOFs while auxiliary entanglement in the OAM DOFs were utilized [41]. It is noted that the high-dimensional BSM in the OAM state space remains relatively unexplored.

In this paper, we propose a viable theoretical scheme for four-dimensional OAM BSM using the single-photon hyperentangled state analyzer. Building on the projective measurements of auxiliary polarization and path properties and target OAM property, we are allowed to extract the Bell-state information deterministically and without any requirements for inefficient nonlinear optics. Since the single-photon hyperentangled state analyzer is implemented on paired photons independently, our results may inspire more applications in remote state preparation and secure communication.

2. Basic principle

The schematic representation of four-dimensional BSM by using single photon Bell-state analyzer is shown in Fig. 1. We define that the four-dimensional system states to be measured are encoded in OAM DOFs, and the auxiliary quantum states are encoded in polarization and path DOFs. Such hyperentangled states can be written as:

(1)$$|{\Upsilon}\rangle_{AB}=|{\Pi_{OAM}}\rangle_{AB}\otimes|{\Gamma_{pol}}\rangle_{AB}\otimes|{\Xi_{path}}\rangle_{AB},$$

where $|{\Pi _{OAM}}\rangle _{AB}$ represents 16 four-dimensional Bell states in the OAM DOFs that can be written as:

(2)$$\begin{aligned} & |{\psi^{{\pm}}_{1n}}\rangle_{AB}=\frac{1}{2}(|{-2,-2}\rangle+e^{in\pi}|{-1,-1}\rangle\pm|{1,1}\rangle\pm e^{in\pi}|{2,2}\rangle),\\ & |{\psi^{{\pm}}_{2n}}\rangle_{AB}=\frac{1}{2}(|{-2,-1}\rangle+e^{in\pi}|{-1,-2}\rangle\pm|{1,2}\rangle\pm e^{in\pi}|{2,1}\rangle),\\ & |{\psi^{{\pm}}_{3n}}\rangle_{AB}=\frac{1}{2}(|{-2,1}\rangle+e^{in\pi}|{-1,2}\rangle\pm|{1,-2}\rangle\pm e^{in\pi}|{2,-1}\rangle),\\ & |{\psi^{{\pm}}_{4n}}\rangle_{AB}=\frac{1}{2}(|{-2,2}\rangle+e^{in\pi}|{-1,1}\rangle\pm|{1,-1}\rangle\pm e^{in\pi}|{2,-2}\rangle), \end{aligned}$$

where $n\in \{0,1\}$, and $\ell \in \{-2,-1,1,2\}$ represents the used orthogonal OAM modes in our scheme. $|{\Gamma _{pol}}\rangle _{AB}$ represents four Bell states in the two-dimensional polarization DOFs that can be written as:

(3)$$\begin{aligned} & |{\psi^{{\pm}}_{pol}}\rangle_{AB}=\frac{1}{\sqrt{2}}(|{H}\rangle|{V}\rangle\pm|{V}\rangle|{H}\rangle),\\ & |{\phi^{{\pm}}_{pol}}\rangle_{AB}=\frac{1}{\sqrt{2}}(|{H}\rangle|{H}\rangle\pm|{V}\rangle|{V}\rangle), \end{aligned}$$

where $H$ and $V$ represent the horizontal and vertical single-photon polarization states, respectively. $|{\Xi _{path}}\rangle _{AB}$ represents four Bell states in the two-dimensional path DOFs that can be written as:

(4)$$\begin{aligned} & |{\psi^{{\pm}}_{path}}\rangle_{AB}=\frac{1}{\sqrt{2}}(|{a}\rangle|{b}\rangle\pm|{b}\rangle|{a}\rangle),\\ & |{\phi^{{\pm}}_{path}}\rangle_{AB}=\frac{1}{\sqrt{2}}(|{a}\rangle|{a}\rangle\pm|{b}\rangle|{b}\rangle), \end{aligned}$$

where $a$ and $b$ represent two distinct path modes, respectively. For our BSM scheme, the incident pairs of photons are hyperentangled in the form of $|{\Pi _{OAM}}\rangle _{AB}|{\phi ^{+}_{pol}}\rangle _{AB}|{\phi ^{+}_{path}}\rangle _{AB}$ , where $|{\Pi _{OAM}}\rangle _{AB}$ is one of the 16 four-dimensional OAM Bell states in Eq. (2).

Fig. 1. Schematic of the complete four-dimensional BSM with the assistance of auxiliary DOFs. The incident single photons A and B are hyperentangled in OAM, polarization and path DOFs. CNOT represents the controlled NOT gate, the four-dimensional Hadamard gate is implemented in OAM DOFs, the single photon projective measurement is implemented in four-dimensional OAM, two-dimensional polarization and two-dimensional path DOFs, which results in the corresponding detection of 32 detectors.

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Specifically, the single photon Bell-state analyzer is composed of a CNOT gate controlled by positive and negative OAM modes, a CNOT gate controlled by the parity of OAM modes, a four-dimensional Hadamard gate acted on OAM DOFs, and the complete projective measurement acted on OAM, polarization and path DOFs.

In the implementation of CNOT gate that is controlled by positive and negative OAM modes (see CNOT 1 in Fig. 1), the OAM modes act as the control qubit, and the polarization modes act as the target qubit. Namely, if the OAM mode $\ell >0$, CNOT gate would keep its polarization mode unchanged. On the contrary, if the OAM mode $\ell <0$, CNOT gate would flip its polarization mode as $|{H}\rangle \leftrightarrow |{V}\rangle$. Thus, the operation of CNOT 1 gate can be modeled as

(5)$$\begin{aligned} |{n}\rangle|{H}\rangle & \rightarrow|{n}\rangle|{H}\rangle,\qquad|{-n}\rangle|{H}\rangle\rightarrow|{-n}\rangle|{V}\rangle,\\ |{n}\rangle|{V}\rangle & \rightarrow|{n}\rangle|{V}\rangle,\qquad|{-n}\rangle|{V}\rangle\rightarrow|{-n}\rangle|{H}\rangle,\\ \end{aligned}$$

where $n=1,2$.

In the implementation of CNOT gate that is controlled by the parity of OAM modes (see CNOT 2 in Fig. 1), as similarly to CNOT 1, the OAM modes act as the control qubit, but the distinct path modes act as the target qubit. If the OAM mode is even, CNOT gate would keep its path mode unchanged. On the contrary, if the OAM mode is odd, CNOT gate would flip its path mode as $|{a}\rangle \leftrightarrow |{b}\rangle$. Thus, the operation of CNOT 2 gate can be modeled as

(6)$$\begin{aligned} |{\pm 2}\rangle|{a}\rangle & \rightarrow|{\pm 2}\rangle|{a}\rangle,\qquad|{\pm 2}\rangle|{b}\rangle\rightarrow|{\pm 2}\rangle|{b}\rangle,\\ |{\pm 1}\rangle|{a}\rangle & \rightarrow|{ \pm 1}\rangle|{b}\rangle,\qquad|{\pm 1}\rangle|{b}\rangle\rightarrow|{\pm 1}\rangle|{a}\rangle.\\ \end{aligned}$$

Then the paired photons pass through the four-dimensional Hadamard gate respectively, whose matrix can be expressed as [26]

(7)$$H_4=\frac{1}{2}\left( \begin{array}{cccc} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\\ \end{array} \right),$$

which acts on the OAM modes as the transformation of

(8)$$\begin{aligned} |{-2}\rangle & \rightarrow\frac{1}{2}(|{-2}\rangle+|{-1}\rangle+|{1}\rangle+|{2}\rangle),\\ |{-1}\rangle & \rightarrow\frac{1}{2}(|{-2}\rangle-|{-1}\rangle+|{1}\rangle-|{2}\rangle),\\ |{1}\rangle & \rightarrow\frac{1}{2}(|{-2}\rangle+|{-1}\rangle-|{1}\rangle-|{2}\rangle),\\ |{2}\rangle & \rightarrow\frac{1}{2}(|{-2}\rangle-|{-1}\rangle-|{1}\rangle+|{2}\rangle). \end{aligned}$$

After the operation of CNOT gates and four-dimensional Hadamard gates, the incident 16 hyperentangled Bell states are transformed into

(9)$$\begin{aligned} & |{\psi^{+}_{10}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{10}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{+}_{11}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{20}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{10}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{30}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{11}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{40}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ \\ & |{\psi^{+}_{20}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{11}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{+}_{21}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{21}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{20}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{31}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{21}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{+}_{41}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ \\ & |{\psi^{+}_{30}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{10}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{+}_{31}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{20}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{30}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{30}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{31}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{40}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\psi^{+}_{path}}\rangle_{AB},\\ \\ & |{\psi^{+}_{40}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{11}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{+}_{41}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{21}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{40}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{31}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB},\\ & |{\psi^{-}_{41}}\rangle_{AB}|{\phi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}\rightarrow|{\psi^{-}_{41}}\rangle_{AB}|{\psi^{+}_{pol}}\rangle_{AB}|{\phi^{+}_{path}}\rangle_{AB}.\\ \end{aligned}$$

Now it is obvious that the four-dimensional Bell states in OAM DOFs are distinguishable by using the projective measurement in OAM, polarization and path modes. Therefore, 32 detectors are demonstrated in Fig. 1 to identify the distinct projective measurement results. Building on the coincidence detection between entangled photons, we are allowed to deterministically distinguish the complete four-dimensional Bell states that are encoded in OAM DOFs with the assistance of path and polarization hyperentanglement.

3. Schematic design and discussion

In order to verify the viability of our scheme, we present the potential implementation methods with current available techniques. Spontaneous parametric down conversion in nonlinear crystals representing the present-day gold standard with respect to fiber coupling efficiency, entangled photon pair rates, and entanglement fidelity. As a direct result of the energy and momentum conservation, the OAM and polarization entanglement arises quite naturally between the down-converted photons [42–44]. For example, Chen et al. proposed an efficient scheme to prepare the complete Bell basis in arbitrary dimensional Hilbert space through adaptive pump modulation, and a complete set of four-dimensional OAM Bell states with average fidelity of $0.821\pm 0.0223$ has been implemented in experiment [30]. Additionally, the path-polarization hyperentanglement can be readily implemented by using path-polarization hybrid system, and has been proved in various experiments [45–47]. Thus, we believe that the OAM, path and polarization hyperentanglement as required in our scheme can be readily prepared with current quantum technologies.

In the implementation of the CNOT 1 gate, we first design a modified orbital angular momentum beam splitter (OAM-BS) to sort the photons into opposite spatial modes that are determined by the positive and negative OAM modes. As shown in Fig. 2, the OAM-BS is composed of two modified Mach-Zehnder interferometers (MZI). For an instructive understanding, let us consider an incident photon with the quantum state as $|{\ell }\rangle \otimes |{H}\rangle$, which becomes $|{\ell }\rangle \otimes (|{H}\rangle +|{V}\rangle )/\sqrt {2}$ after passing though a half wave plate (HWP). As this photon enters from input port $A_{in}$, the polarization beam splitter (PBS) would route the photon to the two arms of the MZI. Then two dove prisms (DP) are placed in each arm to introduce a relative phase shift $2\ell \alpha$, where the optical axis of one DP is rotated by an angle of $\alpha$ with respect to another DP [48]. After the combination using PBS2, the state now read

(10)$$|{\ell}\rangle\otimes|{H}\rangle\rightarrow\frac{1}{\sqrt{2}}|{\ell}\rangle\otimes[|{H}\rangle+\exp(i2\ell\alpha)|{V}\rangle].$$

In the case of $\alpha =\pi /(4|\ell |)$, the photons with positive OAM modes become left-handed circular polarized as $|{\ell =\text {positive}}\rangle \otimes |{L}\rangle$, and the photons with negative OAM modes become right-handed circular polarized as $|{\ell =\text {negative}}\rangle \otimes |{R}\rangle$ ($\ell \in \{-2,-1,1,2\}$ and $\alpha \in \{\pi /8, \pi /4\}$ in our scheme). Then a quarter wave plate (QWP) is used to transform the left- and right-handed circular polarization into horizontal polarization and vertical polarization respectively. By the separation of PBS3, the photons with positive OAM modes would exit from output port $A_{out}$, and the photons with negative OAM modes would exit from output port $B_{out}$. Analogously, for the photons that enter from input $B_{in}$, the photons with positive OAM modes would exit from output port $B_{out}$, and the photons with negative OAM modes would exit from output port $A_{out}$. Next, let us consider an incident photons with the quantum state as $|{\ell }\rangle \otimes |{V}\rangle$. By analogous operations, for the case photons are incident on the OAM-BS from input port $A_{in}$, the photons with positive OAM modes would exit from output port $B_{out}$, and the photons with negative OAM modes would exit from output port $A_{out}$. For the case photons are incident on the OAM-BS from input port $B_{in}$, the photons with positive OAM modes would exit from output port $A_{out}$, and the photons with negative OAM modes would exit from output port $B_{out}$. In the meantime, the polarization state may be flipped that is determined by the positive and negative OAM modes as demonstrated in Eq. (5), which fulfills the task of CNOT 1 gate.

Fig. 2. OAM-BS of positive-$\ell$ and negative-$\ell$. (a) Schematic diagram of the experimental setup. This OAM-BS is used to sort the photons into opposite spatial modes that are determined by the positive and negative OAM modes. (b) The incident photons with the quantum state as $|{\ell }\rangle \otimes |{H}\rangle$. In this case, the photons with positive OAM mode are transmitted and remain in some polarization, while the photons with negative OAM mode are reflected and switch the polarization. (c) The incident photons with the quantum state as $|{\ell }\rangle \otimes |{V}\rangle$. In this case, the photons with negative OAM modes are transmitted and switch the polarization, while the photons with positive OAM modes are reflected and remain in some polarization.

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In the implementation of the CNOT 2 gate, a typical OAM-BS is used to sort the photons into opposite spatial modes that are determined by the parity of OAM modes [49,50]. As shown in Fig. 3, a balanced beam splitter (BS) is used to split the incident photons into two arms of a MZI. Then two DP are placed in each arm to introduce a relative phase shift $2\ell \alpha$, which determines the output ports accordingly. More specifically, if $\alpha =\pi /2$, for the photons that are incident from port $A_{in}$, the photons with odd OAM modes would leave the BS from output port $A_{out}$ as a direct result of constructive interference in port $A_{out}$ and destructive interference in port $B_{out}$. On the contrary, the photons with even OAM modes would leave the BS from output port $B_{out}$ as a direct result of constructive interference in port $B_{out}$ and destructive interference in port $A_{out}$. Analogously, for the photons that are incident from port $B_{in}$, the photons with odd OAM modes would leave the BS from output port $B_{out}$ as a direct result of constructive interference in port $B_{out}$, while the photons with even OAM modes would leave the BS from output port $A_{out}$ as a direct result of constructive interference in port $A_{out}$. In the meantime, the path state may be flipped that is determined by the parity of OAM modes as demonstrated in Eq. (6), which fulfills the task of the CNOT 2 gate.

Fig. 3. OAM-BS of even-$\ell$ and odd-$\ell$. (a) Schematic diagram of the experimental setup. This OAM-BS is composed of a MZI with an additional DP in each arm, where the optical axis of one DP is rotated by an angle of $\alpha =\pi /2$ with respect to another DP. (b) Simplified schematic diagram. This interferometer can sort the photons into opposite spatial modes that are determined by the parity of OAM mode, similar to how a PBS sorts individual photons depending on their polarization.

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While the implementation of arbitrary unitary transformations still remains a challenging task, it is possible to perform high dimensional unitary transformations by using multi-plane light conversion [51]. Additionally, higher dimensional quantum Fourier transformation has been widely explored for both theoretical proposals and actual implementation. Thus we believe the four-dimensional Hadamard gate required in our scheme can be realized with relatively little technological effort.

The viable implementation of our scheme is shown in Fig. 4. Paired photons with hyperentangled states $|{\Pi _{OAM}}\rangle _{AB}|{\phi ^{+}_{pol}}\rangle _{AB}|{\phi ^{+}_{path}}\rangle _{AB}$ are routed into two single-photon Bell-state analyzers that are dependent on the distinguishability of path modes. Following the above discussion, the OAM-BS 1 sorts the photons with even and odd OAM modes into distinct spatial modes. By a set of operations acting on the photons, the CNOT 1 gate is implemented. Then the CNOT 2 gate and four-dimensional Hadamard gate acted on OAM DOFs are performed in sequence, which finally transforms the incident Bell states as Eq. (9). Consequently, a complete projective measurement acted on OAM, polarization, and path DOFs is used to distinguish the Bell states. According to the detectors’ clicks, we can extract the full information about the incident Bell states as shown in Table 1.

Fig. 4. Schematic diagram of the complete four-dimensional OAM Bell states measurement. The hyperentangled states source is used to prepare the hyperentangled photons required in our scheme, which can be achieved with current quantum technologies. The single photon Bell-state analyzer is composed of a CNOT gate controlled by positive and negative OAM modes (CNOT 1), a CNOT gate controlled by the parity of OAM modes (CNOT 2), a four-dimensional Hadamard gate acted on OAM DOFs, and the complete projective measurement acted on OAM, polarization and path DOFs.

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Table 1. Detector outcomes for four-dimensional OAM Bell states. $D_{-2H_{a/b}}D_{-2H_{a/b}}$ means coincidence between $D_{-2H_{a}}$ and $D_{-2H_{a}}$ or $D_{-2H_{b}}$ and $D_{-2H_{b}}$.

View Table

4. Summary

In conclusion, we have demonstrated theoretically a method for implementing the deterministic discrimination for the complete four-dimensional OAM Bell states with the assistance of polarization and path hyperentanglement. The complete OAM BSM in our scheme can be implemented with unit probability but without any requirements for inefficient nonlinear optics, which may provide an alternative route toward the practical applications in the near future. Building on the ingenious design of single-photon hyperentangled state analyzer, our scheme has the potential applications in remote state preparation and long-distance quantum communication. Based on the information encoding in four-dimensional OAM entanglement, our scheme provides an alternative platform for enhancing the channel capacity in quantum communication, and increasing the robustness against deleterious noise in practical experiment. Due to the spatial resolution of OAM quantum states, our scheme may inspire more complex quantum imaging applications that providing great advantages in resolution and precision.

Funding

Program for New Century Excellent Talents in University (NCET-13-0495); Natural Science Foundation of Fujian Province of China for Distinguished Young Scientists (2015J06002); Natural Science Foundation of Fujian Province (2020J05004, 2021J02002); Fundamental Research Funds for the Central Universities at Xiamen University (20720190057, 20720200074, 20720210096); National Natural Science Foundation of China (12004318, 12034016, 61975169).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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State	Detector outcomes
$ψ_{10}^{+}$	$D_{- 2 H_{a / b}} D_{- 2 H_{a / b}}$	$D_{- 2 V_{a / b}} D_{- 2 V_{a / b}}$	$D_{- 1 H_{a / b}} D_{- 1 H_{a / b}}$	$D_{- 1 V_{a / b}} D_{- 1 V_{a / b}}$
$ψ_{10}^{+}$	$D_{1 H_{a / b}} D_{1 H_{a / b}}$	$D_{1 V_{a / b}} D_{1 V_{a / b}}$	$D_{2 H_{a / b}} D_{2 H_{a / b}}$	$D_{2 V_{a / b}} D_{2 V_{a / b}}$
$ψ_{11}^{+}$	$D_{- 2 H_{a / b}} D_{- 1 H_{a / b}}$	$D_{- 2 V_{a / b}} D_{- 1 V_{a / b}}$	$D_{- 1 H_{a / b}} D_{- 2 H_{a / b}}$	$D_{- 1 V_{a / b}} D_{- 2 V_{a / b}}$
$ψ_{11}^{+}$	$D_{1 H_{a / b}} D_{2 H_{a / b}}$	$D_{1 V_{a / b}} D_{2 V_{a / b}}$	$D_{2 H_{a / b}} D_{1 H_{a / b}}$	$D_{2 V_{a / b}} D_{1 V_{a / b}}$
$ψ_{10}^{-}$	$D_{- 2 H_{a / b}} D_{1 H_{a / b}}$	$D_{- 2 V_{a / b}} D_{1 V_{a / b}}$	$D_{- 1 H_{a / b}} D_{2 H_{a / b}}$	$D_{- 1 V_{a / b}} D_{2 V_{a / b}}$
$ψ_{10}^{-}$	$D_{2 H_{a / b}} D_{- 2 H_{a / b}}$	$D_{1 V_{a / b}} D_{- 2 V_{a / b}}$	$D_{- 2 H_{a / b}} D_{1 H_{a / b}}$	$D_{- 2 V_{a / b}} D_{1 V_{a / b}}$
$ψ_{11}^{-}$	$D_{- 2 H_{a / b}} D_{2 H_{a / b}}$	$D_{- 2 V_{a / b}} D_{2 V_{a / b}}$	$D_{- 1 H_{a / b}} D_{1 H_{a / b}}$	$D_{- 1 V_{a / b}} D_{1 V_{a / b}}$
$ψ_{11}^{-}$	$D_{1 H_{a / b}} D_{- 1 H_{a / b}}$	$D_{1 V_{a / b}} D_{- 1 V_{a / b}}$	$D_{2 H_{a / b}} D_{- 2 H_{a / b}}$	$D_{2 V_{a / b}} D_{- 2 V_{a / b}}$
$ψ_{20}^{+}$	$D_{- 2 H_{a / b}} D_{- 2 H_{b / a}}$	$D_{- 2 V_{a / b}} D_{- 2 V_{b / a}}$	$D_{- 1 H_{a / b}} D_{- 1 H_{b / a}}$	$D_{- 1 V_{a / b}} D_{- 1 V_{b / a}}$
$ψ_{20}^{+}$	$D_{1 H_{a / b}} D_{1 H_{b / a}}$	$D_{1 V_{a / b}} D_{1 V_{b / a}}$	$D_{2 H_{a / b}} D_{2 H_{b / a}}$	$D_{2 V_{a / b}} D_{2 V_{b / a}}$
$ψ_{21}^{+}$	$D_{- 2 H_{a / b}} D_{- 1 H_{b / a}}$	$D_{- 2 V_{a / b}} D_{- 1 V_{b / a}}$	$D_{- 1 H_{a / b}} D_{- 2 H_{b / a}}$	$D_{- 1 V_{a / b}} D_{- 2 V_{b / a}}$
$ψ_{21}^{+}$	$D_{1 H_{a / b}} D_{2 H_{b / a}}$	$D_{1 V_{a / b}} D_{2 V_{b / a}}$	$D_{2 H_{a / b}} D_{1 H_{b / a}}$	$D_{2 V_{a / b}} D_{1 V_{b / a}}$
$ψ_{20}^{-}$	$D_{- 2 H_{a / b}} D_{1 H_{b / a}}$	$D_{- 2 V_{a / b}} D_{1 V_{b / a}}$	$D_{- 1 H_{a / b}} D_{2 H_{b / a}}$	$D_{- 1 V_{a / b}} D_{2 V_{b / a}}$
$ψ_{20}^{-}$	$D_{2 H_{a / b}} D_{- 2 H_{b / a}}$	$D_{1 V_{a / b}} D_{- 2 V_{b / a}}$	$D_{- 2 H_{a / b}} D_{1 H_{b / a}}$	$D_{- 2 V_{a / b}} D_{1 V_{b / a}}$
$ψ_{21}^{-}$	$D_{- 2 H_{a / b}} D_{2 H_{b / a}}$	$D_{- 2 V_{a / b}} D_{2 V_{b / a}}$	$D_{- 1 H_{a / b}} D_{1 H_{b / a}}$	$D_{- 1 V_{a / b}} D_{1 V_{b / a}}$
$ψ_{21}^{-}$	$D_{1 H_{a / b}} D_{- 1 H_{b / a}}$	$D_{1 V_{a / b}} D_{- 1 V_{b / a}}$	$D_{2 H_{a / b}} D_{- 2 H_{b / a}}$	$D_{2 V_{a / b}} D_{- 2 V_{b / a}}$
$ψ_{30}^{+}$	$D_{- 2 H_{a / b}} D_{- 2 V_{b / a}}$	$D_{- 2 V_{a / b}} D_{- 2 H_{b / a}}$	$D_{- 1 H_{a / b}} D_{- 1 V_{b / a}}$	$D_{- 1 V_{a / b}} D_{- 1 H_{b / a}}$
$ψ_{30}^{+}$	$D_{1 H_{a / b}} D_{1 V_{b / a}}$	$D_{1 V_{a / b}} D_{1 H_{b / a}}$	$D_{2 H_{a / b}} D_{2 V_{b / a}}$	$D_{2 V_{a / b}} D_{2 H_{b / a}}$
$ψ_{31}^{+}$	$D_{- 2 H_{a / b}} D_{- 1 V_{b / a}}$	$D_{- 2 V_{a / b}} D_{- 1 H_{b / a}}$	$D_{- 1 H_{a / b}} D_{- 2 V_{b / a}}$	$D_{- 1 V_{a / b}} D_{- 2 H_{b / a}}$
$ψ_{31}^{+}$	$D_{1 H_{a / b}} D_{2 V_{b / a}}$	$D_{1 V_{a / b}} D_{2 H_{b / a}}$	$D_{2 H_{a / b}} D_{1 V_{b / a}}$	$D_{2 V_{a / b}} D_{1 H_{b / a}}$
$ψ_{30}^{-}$	$D_{- 2 H_{a / b}} D_{1 V_{b / a}}$	$D_{- 2 V_{a / b}} D_{1 H_{b / a}}$	$D_{- 1 H_{a / b}} D_{2 V_{b / a}}$	$D_{- 1 V_{a / b}} D_{2 H_{b / a}}$
$ψ_{30}^{-}$	$D_{2 H_{a / b}} D_{- 2 V_{b / a}}$	$D_{1 V_{a / b}} D_{- 2 H_{b / a}}$	$D_{- 2 H_{a / b}} D_{1 V_{b / a}}$	$D_{- 2 V_{a / b}} D_{1 H_{b / a}}$
$ψ_{31}^{-}$	$D_{- 2 H_{a / b}} D_{2 V_{b / a}}$	$D_{- 2 V_{a / b}} D_{2 H_{b / a}}$	$D_{- 1 H_{a / b}} D_{1 V_{b / a}}$	$D_{- 1 V_{a / b}} D_{1 H_{b / a}}$
$ψ_{31}^{-}$	$D_{1 H_{a / b}} D_{- 1 V_{b / a}}$	$D_{1 V_{a / b}} D_{- 1 H_{b / a}}$	$D_{2 H_{a / b}} D_{- 2 V_{b / a}}$	$D_{2 V_{a / b}} D_{- 2 H_{b / a}}$
$ψ_{40}^{+}$	$D_{- 2 H_{a / b}} D_{- 2 V_{a / b}}$	$D_{- 2 V_{a / b}} D_{- 2 H_{a / b}}$	$D_{- 1 H_{a / b}} D_{- 1 V_{a / b}}$	$D_{- 1 V_{a / b}} D_{- 1 H_{a / b}}$
$ψ_{40}^{+}$	$D_{1 H_{a / b}} D_{1 V_{a / b}}$	$D_{1 V_{a / b}} D_{1 H_{a / b}}$	$D_{2 H_{a / b}} D_{2 V_{a / b}}$	$D_{2 V_{a / b}} D_{2 H_{a / b}}$
$ψ_{41}^{+}$	$D_{- 2 H_{a / b}} D_{- 1 V_{a / b}}$	$D_{- 2 V_{a / b}} D_{- 1 H_{a / b}}$	$D_{- 1 H_{a / b}} D_{- 2 V_{a / b}}$	$D_{- 1 V_{a / b}} D_{- 2 H_{a / b}}$
$ψ_{41}^{+}$	$D_{1 H_{a / b}} D_{2 V_{a / b}}$	$D_{1 V_{a / b}} D_{2 H_{a / b}}$	$D_{2 H_{a / b}} D_{1 V_{a / b}}$	$D_{2 V_{a / b}} D_{1 H_{a / b}}$
$ψ_{40}^{-}$	$D_{- 2 H_{a / b}} D_{1 V_{a / b}}$	$D_{- 2 V_{a / b}} D_{1 H_{a / b}}$	$D_{- 1 H_{a / b}} D_{2 V_{a / b}}$	$D_{- 1 V_{a / b}} D_{2 H_{a / b}}$
$ψ_{40}^{-}$	$D_{2 H_{a / b}} D_{- 2 V_{a / b}}$	$D_{1 V_{a / b}} D_{- 2 H_{a / b}}$	$D_{- 2 H_{a / b}} D_{1 V_{a / b}}$	$D_{- 2 V_{a / b}} D_{1 H_{a / b}}$
$ψ_{41}^{-}$	$D_{- 2 H_{a / b}} D_{2 V_{a / b}}$	$D_{- 2 V_{a / b}} D_{2 H_{a / b}}$	$D_{- 1 H_{a / b}} D_{1 V_{a / b}}$	$D_{- 1 V_{a / b}} D_{1 H_{a / b}}$
$ψ_{41}^{-}$	$D_{1 H_{a / b}} D_{- 1 V_{a / b}}$	$D_{1 V_{a / b}} D_{- 1 H_{a / b}}$	$D_{2 H_{a / b}} D_{- 2 V_{a / b}}$	$D_{2 V_{a / b}} D_{- 2 H_{a / b}}$

Four-dimensional orbital angular momentum Bell-state measurement assisted by the auxiliary polarization and path degrees of freedom

Abstract

1. Introduction

2. Basic principle

3. Schematic design and discussion

4. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Tables (1)

Equations (10)

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