Complete hyperentangled state analysis using high-dimensional entanglement

Abstract

In this paper, we present a novel method for the complete analysis of maximally hyperentangled state of photon system in two degrees of freedom (DOFs), resorting to the auxiliary high-dimensional entanglement in the third DOF. This method not only can be used for complete hyperentangled Bell state analysis of two-photon system, but also can be suitable for complete hyperentangled Greenberger−Horne−Zeilinger (GHZ) state analysis of three-photon system, and can be extended to the complete N-photon hyperentangled GHZ state analysis. In our approach, the parity information of hyperentanglement is determined via the measurement on evolved auxiliary high-dimensional entanglement, and the relative phase information of hyperentanglement is determined via the projective measurement. Moreover, this approach can be accomplished by just using linear optics, and is significant for the investigation of photonic hyperentangled state analysis.

Graphical abstract

Keywords

hyperentangled state analysis / high-dimensional entanglement / GHZ state

1 Introduction

Entanglement is a unique phenomenon in quantum world with no classical counterpart, and it has been widely exploited for entanglement-based quantum information technologies (QITs) [1], such as quantum key distribution [2], quantum dense coding [3], quantum teleportation [4], quantum entanglement swapping [5], quantum imaging [6], and quantum repeater [7]. Photon is regarded as the promising candidate for practical QITs, and several different degrees of freedom (DOFs) of photon have been utilized to encode information, for example, the polarization, the spatial-mode, the time-bin, the frequency and the orbital angular momentum (OAM). As two different types of high-capacity quantum state, photonic hyperentangled state and high-dimensional entangled state have attracted much attentions in the past years. Hyperentanglement is defined as entanglement simultaneously in more than one DOF [8−11], and high-dimensional entanglement represents entanglement in multi-lever quantum system [12]. Recently, these two high-capacity quantum states have found useful applications in the field of quantum imaging and quantum holography [13−20]. In quantum information theory, the distinguishablity of orthogonal entangled basis is an essential and fundamental problem. However, research has shown that the complete Bell state analysis (BSA) for two-photon system is impossible with just linear optics, and the optimal efficiency is only 50% [21]. For the Greenberger−Horne−Zeilinger (GHZ) state analysis (GSA) for multi-photon system, only two GHZ states can be discriminated by just using linear optics, and the maximally efficiency is 25% for three-photon GSA [22]. A useful avenue to achieve the complete BSA and GSA is using quantum hyperentanglement [23−26]. For example, in 1998, Kwiat and Weinfurter [23] showed that a linear-optical polarization BSA is accessible, resorting to polarization-time-bin or polarization-momentum hyperentanglement. In 2014, Zeng et al. [26] proposed the scheme for complete $N$-photon GSA by using polarization-frequency hyperentanglement.

Hyperentangled Bell state analysis (HBSA) is an important procedure in hyperentanglement-based quantum information processing, and a complete HBSA is also impossible with just linear optics [27,28]. In the past decade, the complete HBSA of photon system in two DOFs has been deeply researched [29−42]. Two useful tools have been discovered for the complete analysis of hyperentangled Bell states, i.e. using quantum nonlinear interaction or using auxiliary quantum entanglement. In 2010, Sheng et al. [29] presented the first complete HBSA scheme for photon system in polarization and spatial-mode DOFs via weak cross-Kerr nonlinearity, and given the application of their HBSA scheme in high-capacity quantum communication protocols. Subsequently, researchers have found that quantum-dot spin in optical microcavity and nitrogen−vacancy center in resonator can be efficiently used for complete HBSA [30−32, 35−37]. In the aspect of using auxiliary entanglement, in 2017, Li and Ghose [38] investigated the distinguishablity of a set of mutually orthogonal hyperentangled Bell states, resorting to additional Bell state in other DOF. They showed that given $n$ DOFs, the $4^n$ hyperentangled Bell states can be separated into $x_k=2^{n+k+1}-2^{2k}$ groups via linear optics with the help of $k(k\le n)$ ancillary entangled states [38]. When $k=n$, the complete HBSA scheme in $n$ DOFs can be accomplished [38]. After this significant work, in 2019, Wang et al. [39] presented a simple proposal for complete HBSA of two-photon system in polarization and the first longitudinal momentum DOFs, resorting to an auxiliary hyperentangled Bell state in frequency and the second longitudinal momentum DOFs. Combining these two useful tools for complete hyperentangled state analysis, in 2020, Zeng et al. [41] presented the complete HBSA for polarization and spatial-mode hyperentanglement, resorting to the weak cross-Kerr nonlinearity and auxiliary frequency entanglement. Moreover, the scheme can be generalized to the complete hyperentangled GHZ state analysis (HGSA) of $N$-photon system [41].

However, to our knowledge, all the existing complete HBSA scheme with auxiliary entanglement is using the two-dimensional entanglement, and the role and function of high-dimensional entanglement in the distinguishing of hyperentangled state has not been investigated yet. Compared with traditional two-dimensional entanglement, high-dimensional entanglement has its distinct advantages, from larger information capacity and increased noise resilience, to novel fundamental research possibilities in quantum physics [43]. In this paper, we propose a complete HBSA scheme for two-photon system entangled in polarization and the first spatial-mode DOFs, with the help of auxiliary four-dimensional Bell state in the second spatial-mode DOF. This approach can be extended to the complete HGSA scheme for multi-photon system by using auxiliary four-dimensional GHZ state, and will be useful for the QITs based on high-dimensional hyperentanglement.

2 Complete HBSA using auxiliary four-dimensional Bell state

The hyperentangled Bell state of two-photon system in polarization and two spatial-mode DOFs can be written as

(1)$|\mathrm{\Psi }{⟩}_{AB}=|\mathrm{\Psi }{⟩}_P\otimes |\mathrm{\Psi }{⟩}_F\otimes |\varphi {⟩}_S.$

Here, $A$ and $B$ are the entangled photon pairs. $P$, $F$ and $S$ denote the polarization DOF, the first spatial-mode DOF and the second spatial-mode DOF, respectively. $|\mathrm{\Psi }{⟩}_P$ is arbitrary one of the four Bell states in polarization DOF,

(2)$\begin{array}{rl}& |{\varphi }^{\pm }{⟩}_P=\frac{1}{\sqrt{2}}(|HH⟩\pm |VV⟩{)}_{AB},\\ & |{\psi }^{\pm }{⟩}_P=\frac{1}{\sqrt{2}}(|HV⟩\pm |VH⟩{)}_{AB},\end{array}$

where $|H⟩$ and $|V⟩$ are the horizontal and vertical polarization states of photon, respectively. $|\mathrm{\Psi }{⟩}_F$ is arbitrary one of the four Bell states in the first spatial-mode DOF,

(3)$\begin{array}{rl}& |{\varphi }^{\pm }{⟩}_F=\frac{1}{\sqrt{2}}(|aa⟩\pm |bb⟩{)}_{AB},\\ & |{\psi }^{\pm }{⟩}_F=\frac{1}{\sqrt{2}}(|ab⟩\pm |ba⟩{)}_{AB},\end{array}$

where $|a⟩$ and $|b⟩$ are two different states of photon in the first spatial-mode DOF. Considering both polarization and the first spatial-mode DOFs, there are 16 orthogonal hyperentangled Bell states in two DOFs, which will be completely distinguished in this section. In our complete HBSA scheme for polarization and the first spatial-mode hyperentanglement, an auxiliary high-dimensional entanglement $|\varphi {⟩}_S$ in the second spatial-mode DOF is utilized. $|\varphi {⟩}_S$ is a fixed four-dimensional Bell state, the form of which is

(4)$|\varphi {⟩}_S={\displaystyle \frac{1}{2}}(|00⟩+|11⟩+|22⟩+|33⟩{)}_{AB},$

where $|0⟩$, $|1⟩$, $|2⟩$, and $|3⟩$ are four high-dimensional states of photon in the second spatial-mode DOF. Before describing the process of our complete HBSA scheme in detail, we first demonstrate the construction of high-dimensional quantum gate (HDQG) for single-photon in three different DOFs, which is the core ingredient in our scheme. In our complete HBSA scheme, two different HDQGs are utilized, and they can be built by just using simple linear optical elements.

The setup of HDQG${}_1$ is presented in Fig.1, and its high-dimensional quantum operation can be expressed as

Fig.1 Schematic diagram of the first high-dimensional quantum gate (HDQG${}_1$) for single-photon in three DOFs. The combining of polarization DOF and the first spatial-mode DOF can be viewed as the control mode, and the second spatial-mode DOF in high-dimensional space is the target mode. The half-wave plates (HWPs) implement the bit-flip operation [$|H⟩\leftrightarrow |V⟩$] on the polarization state of photon. The polarization beam splitters (PBSs) transmit the horizontal polarized photon with its path switched, and reflect the vertical one with the path unchanged. With this device, the high-dimensional quantum control of Eq. (5) can be realized.

Full size|PPT slide

(5)$\begin{array}{rl}& |H⟩|a⟩|0⟩\to |H⟩|a⟩|0⟩,|H⟩|a⟩|1⟩\to |H⟩|a⟩|1⟩,\\ & |H⟩|a⟩|2⟩\to |H⟩|a⟩|2⟩,|H⟩|a⟩|3⟩\to |H⟩|a⟩|3⟩,\\ & |V⟩|a⟩|0⟩\to |V⟩|a⟩|2⟩,|V⟩|a⟩|1⟩\to |V⟩|a⟩|3⟩,\\ & |V⟩|a⟩|2⟩\to |V⟩|a⟩|0⟩,|V⟩|a⟩|3⟩\to |V⟩|a⟩|1⟩.\end{array}$

The setup of HDQG${}_2$ is presented in Fig.2, and its high-dimensional quantum operation can be expressed as

Fig.2 Schematic diagram of the second high-dimensional quantum gate (HDQG${}_2$) for single-photon in three DOFs. With this device, the high-dimensional quantum control of Eq. (6) can be realized.

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(6)$\begin{array}{rl}& |H⟩|b⟩|0⟩\to |H⟩|b⟩|1⟩,|H⟩|b⟩|1⟩\to |H⟩|b⟩|0⟩,\\ & |H⟩|b⟩|2⟩\to |H⟩|b⟩|3⟩,|H⟩|b⟩|3⟩\to |H⟩|b⟩|2⟩,\\ & |V⟩|b⟩|0⟩\to |V⟩|b⟩|3⟩,|V⟩|b⟩|1⟩\to |V⟩|b⟩|2⟩,\\ & |V⟩|b⟩|2⟩\to |V⟩|b⟩|1⟩,|V⟩|b⟩|3⟩\to |V⟩|b⟩|0⟩.\end{array}$

These two HDQGs can be viewed as the high-dimensional controlled-flip gate, in which the combination of polarization qubit and the first spatial-mode qubit is control mode and the second spatial-mode is target mode. The arbitrary one of 16 hyperentangled Bell states in polarization and the first spatial-mode DOFs has the unique parity information and relative phase information. In our distinguishing process, the parity information is determined with the assistance of measurement on auxiliary high-dimensional entanglement, and the relative phase information is determined with the assistance of projective measurement.

In order to accomplish the discrimination of 16 hyperentangled Bell states in polarization and the first spatial-mode DOFs, photons $A$ and $B$ are injected into the same setups, one of which the schematic diagram is shown for in Fig.3. After these two photons are manipulated by HDQG${}_1$ and HDQG${}_2$, the state of quantum hyperentanglement in three DOFs evolves as

Fig.3 Schematic diagram of our analyzer for the complete analysis of photonic hyperentangled state in two DOFs. The beam splitters (BSs) implement the Hadamard operation [$|a⟩\to \frac{1}{\sqrt{2}}(|a⟩+|b⟩),|b⟩\to \frac{1}{\sqrt{2}}(|a⟩-|b⟩)$] on the first spatial-mode state of photon. The wave plates (WPs) implement the Hadamard operation [$|H⟩\to \frac{1}{\sqrt{2}}(|H⟩+|V⟩),|V⟩\to \frac{1}{\sqrt{2}}(|H⟩-$$|V⟩)$] on polarization state of photon. Single-photon detectors are utilized for realizing the polarization measurement in $\{|H⟩,|V⟩\}$ basis.

Full size|PPT slide

(7)$\begin{array}{rl}& |{\varphi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F|\varphi {⟩}_S\\ & \to |{\varphi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F\otimes \frac{1}{2}(|00⟩+|11⟩+|22⟩+|33⟩{)}_{AB},\\ & |{\varphi }^{\pm }{⟩}_P|{\psi }^{\pm }{⟩}_F|\varphi {⟩}_S\\ & \to {\varphi }^{\pm }{⟩}_P|{\psi }^{\pm }{⟩}_F\otimes \frac{1}{2}(|01⟩+|10⟩+|23⟩+|32⟩{)}_{AB},\\ & |{\psi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F|\varphi {⟩}_S\\ & \to |{\psi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F\otimes \frac{1}{2}(|02⟩+|13⟩+|20⟩+|31⟩{)}_{AB},\\ & |{\psi }^{\pm }{⟩}_P|{\psi }^{\pm }{⟩}_F|\varphi {⟩}_S\\ & \to |{\psi }^{\pm }{⟩}_P|{\psi }^{\pm }{⟩}_F\otimes \frac{1}{2}(|03⟩+|12⟩+|21⟩+|30⟩{)}_{AB}.\end{array}$

It is easy to find that the hyperentanglement in polarization and the first spatial-mode DOFs are invariant during the above evolution, and the 16 initial states can be classified into four groups through the detection results in auxiliary high-dimensional mode by single-photon detectors. In each group, there are four states that own the same parity information but different relative phase information, which can be discriminated by using the beam splitters (BSs) and wave plates (WPs). Here, we take the state $|{\varphi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F$ as an example to illustrate. After the photon pairs pass through the BSs and WPs, the evolution of hyperentanglement is

(8)$\begin{array}{rl}& |{\varphi }^{+}{⟩}_P|{\varphi }^{+}{⟩}_F\to \frac{1}{2}(|HH⟩+|VV⟩)\otimes (|aa⟩+|bb⟩{)}_{AB},\\ & |{\varphi }^{+}{⟩}_P|{\varphi }^{-}{⟩}_F\to \frac{1}{2}(|HH⟩+|VV⟩)\otimes (|ab⟩+|ba⟩{)}_{AB},\\ & |{\varphi }^{-}{⟩}_P|{\varphi }^{+}{⟩}_F\to \frac{1}{2}(|HV⟩+|VH⟩)\otimes (|aa⟩+|bb⟩{)}_{AB},\\ & |{\varphi }^{-}{⟩}_P|{\varphi }^{-}{⟩}_F\to \frac{1}{2}(|HV⟩+|VH⟩)\otimes (|ab⟩+|ba⟩{)}_{AB}.\end{array}$

Based on the detection results in polarization and the first spatial-mode DOFs, these four states can be completely distinguished with each other. It should be noted that this process is also suitable for the states $|{\varphi }^{\pm }{⟩}_P|{\psi }^{\pm }{⟩}_F$, $|{\psi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F$ and $|{\psi }^{\pm }{⟩}_P|{\psi }^{\pm }{⟩}_F$. In Tab.1, we list the 16 initial hyperentangled Bell states with their corresponding detection results in the three different entangled modes by single-photon detectors.

Tab.1 Corresponding relations between the initial states and possible detection results in polarization mode, the first spatial-mode and the second spatial-mode.

Initial state	$P$ mode	$F$ mode	$S$ mode
$|{\varphi }^{+}{⟩}_P|{\varphi }^{+}{⟩}_F$	$HH,VV$	$aa,bb$	$00,11,22,33$
$|{\varphi }^{+}{⟩}_P|{\varphi }^{-}{⟩}_F$	$HH,VV$	$ab,ba$	$00,11,22,33$
$|{\varphi }^{+}{⟩}_P|{\psi }^{+}{⟩}_F$	$HH,VV$	$aa,bb$	$01,10,23,32$
$|{\varphi }^{+}{⟩}_P|{\psi }^{-}{⟩}_F$	$HH,VV$	$ab,ba$	$01,10,23,32$
$|{\varphi }^{-}{⟩}_P|{\varphi }^{+}{⟩}_F$	$HV,VH$	$aa,bb$	$00,11,22,33$
$|{\varphi }^{-}{⟩}_P|{\varphi }^{-}{⟩}_F$	$HV,VH$	$ab,ba$	$00,11,22,33$
$|{\varphi }^{-}{⟩}_P|{\psi }^{+}{⟩}_F$	$HV,VH$	$aa,bb$	$01,10,23,32$
$|{\varphi }^{-}{⟩}_P|{\psi }^{-}{⟩}_F$	$HV,VH$	$ab,ba$	$01,10,23,32$
$|{\psi }^{+}{⟩}_P|{\varphi }^{+}{⟩}_F$	$HH,VV$	$aa,bb$	$02,13,20,31$
$|{\psi }^{+}{⟩}_P|{\varphi }^{-}{⟩}_F$	$HH,VV$	$ab,ba$	$02,13,20,31$
$|{\psi }^{+}{⟩}_P|{\psi }^{+}{⟩}_F$	$HH,VV$	$aa,bb$	$03,12,21,30$
$|{\psi }^{+}{⟩}_P|{\psi }^{-}{⟩}_F$	$HH,VV$	$ab,ba$	$03,12,21,30$
$|{\psi }^{-}{⟩}_P|{\varphi }^{+}{⟩}_F$	$HV,VH$	$aa,bb$	$02,13,20,31$
$|{\psi }^{-}{⟩}_P|{\varphi }^{-}{⟩}_F$	$HV,VH$	$ab,ba$	$02,13,20,31$
$|{\psi }^{-}{⟩}_P|{\psi }^{+}{⟩}_F$	$HV,VH$	$aa,bb$	$03,12,21,30$
$|{\psi }^{-}{⟩}_P|{\psi }^{-}{⟩}_F$	$HV,VH$	$ab,ba$	$03,12,21,30$

3 Complete HGSA using auxiliary four-dimensional GHZ state

The general form of hyperentangled GHZ state of $N$-photon system in polarization and the first spatial-mode DOFs can be written as

(9)$|\mathrm{\Phi }{⟩}_{AB\cdots Z}=|\mathrm{\Phi }{⟩}_P\otimes |\mathrm{\Phi }{⟩}_F.$

Here, $A,B,\cdots ,Z$ represent the $N$ entangled photons. $|\mathrm{\Phi }{⟩}_P$ ($|\mathrm{\Phi }{⟩}_F$) is arbitrary one of the $2^N$ polarization (the first spatial-mode) GHZ states,

(10)$|{\mathrm{\Phi }}_{ij\cdots k}^{\pm }{⟩}_{P(F)}={\displaystyle \frac{1}{\sqrt{2}}}(|ij\cdots k⟩\pm |\overline{i}\overline{j}\cdots \overline{k}⟩{)}_{AB\cdots Z}.$

Here, $i,j,\cdots ,k\in \{0,1\}$ and $m=1-\overline{m}(m=i,j,\cdots ,k)$. For polarization DOF, $|0⟩\equiv |H⟩$ and $|1⟩\equiv |V⟩$. For the first spatial-mode DOF, $|0⟩\equiv |a⟩$ and $|1⟩\equiv |b⟩$. Considering both polarization and the first spatial-mode DOFs, there are $4^N$ hyperentangled GHZ states for the $N$-photon system, which will be completely distinguished in this section. We first describe the complete HGSA scheme for three-photon system, and then generalize this approach to the complete $N$-photon HGSA scheme directly.

For $N=3$, there are 64 orthogonal hyperentangled GHZ states of three-photon system in two DOFs, one of which can be written as

(11)$\begin{array}{rl}|{\mathrm{\Phi }}_{000}^{+}{⟩}_P\otimes |{\mathrm{\Phi }}_{000}^{+}{⟩}_F=& \frac{1}{\sqrt{2}}(|HHH⟩+|VVV⟩)\\ & \otimes \frac{1}{\sqrt{2}}(|aaa⟩+|bbb⟩{)}_{ABC}.\end{array}$

To accomplish the three-photon complete HGSA, the following auxiliary three-photon four-dimensional GHZ state in the second spatial-mode DOF is utilized,

(12)$|\mathrm{\Phi }{⟩}_S={\displaystyle \frac{1}{2}}(|000⟩+|111⟩+|222⟩+|333⟩{)}_{ABC}.$

Photons $A$, $B$, and $C$ are sent into the same setups, and one of the schematic diagram is shown in Fig.3. After the three photons are manipulated by HDQG${}_1$ and HDQG${}_2$, the evolution of the initial hyperentangled GHZ state $|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{ijk}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S$ is

(13)$\begin{array}{rl}& |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F\otimes \frac{1}{2}(|000⟩+|111⟩+|222⟩+|333⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F\otimes \frac{1}{2}(|001⟩+|110⟩+|223⟩+|332⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F\otimes \frac{1}{2}(|010⟩+|101⟩+|232⟩+|323⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{000}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F\otimes \frac{1}{2}(|011⟩+|100⟩+|233⟩+|322⟩{)}_{ABC}.\end{array}$

The evolution of state $|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{ijk}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S$ is

(14)$\begin{array}{rl}& \\ & |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F\otimes \frac{1}{2}(|002⟩+|113⟩+|220⟩+|331⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F\otimes \frac{1}{2}(|003⟩+|112⟩+|221⟩+|330⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F\otimes \frac{1}{2}(|012⟩+|103⟩+|230⟩+|321⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{001}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F\otimes \frac{1}{2}(|013⟩+|102⟩+|231⟩+|320⟩{)}_{ABC}.\end{array}$

The evolution of state $|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{ijk}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S$ is

(15)$\begin{array}{rl}& \\ & |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F\otimes \frac{1}{2}(|020⟩+|131⟩+|202⟩+|313⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F\otimes \frac{1}{2}(|021⟩+|130⟩+|203⟩+|312⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F\otimes \frac{1}{2}(|030⟩+|121⟩+|212⟩+|303⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{010}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F\otimes \frac{1}{2}(|031⟩+|120⟩+|213⟩+|302⟩{)}_{ABC}.\end{array}$

The evolution of state $|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{ijk}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S$ is

(16)$\begin{array}{rl}& \\ & |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F\otimes \frac{1}{2}(|022⟩+|133⟩+|200⟩+|311⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F\otimes \frac{1}{2}(|023⟩+|132⟩+|201⟩+|310⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F\otimes \frac{1}{2}(|032⟩+|123⟩+|210⟩+|301⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F|\mathrm{\Phi }{⟩}_S\to |{\mathrm{\Phi }}_{100}^{\pm }{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F\otimes \frac{1}{2}(|033⟩+|122⟩+|211⟩+|300⟩{)}_{ABC}.\end{array}$

We can find that the hyperentangled GHZ state in polarization and the first spatial-mode DOFs is invariant during the evolution, and all the 64 initial states can be classified into 16 groups through the detection results in auxiliary high-dimensional mode by single-photon detectors, as shown in Tab.2.

Tab.2 Corresponding relations between the initial GHZ states and possible detection results in the second spatial-mode.

Group	Initial GHZ state	$S$ mode
$1$	$|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$.	$000,111,222,333.$
$2$	$|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$.	$001,110,223,332.$
$3$	$|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$.	$010,101,232,323.$
$4$	$|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$.	$011,100,233,322.$
$5$	$|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$.	$002,113,220,331.$
$6$	$|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$.	$003,112,221,330.$
$7$	$|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$.	$012,103,230,321.$
$8$	$|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$.	$013,102,231,320.$
$9$	$|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$.	$020,131,202,313.$
$10$	$|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$.	$021,130,203,312.$
$11$	$|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$.	$030,121,212,303.$
$12$	$|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$.	$031,120,213,302.$
$13$	$|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{\pm }{⟩}_F$.	$022,133,200,311.$
$14$	$|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{\pm }{⟩}_F$.	$023,132,201,310.$
$15$	$|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{\pm }{⟩}_F$.	$032,123,210,301.$
$16$	$|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$, $|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{\pm }{⟩}_F$.	$033,122,211,300.$

In each group, there are four states that own the same parity information but different relative phase information, which can be discriminated by using beam splitters (BSs) and wave plates (WPs). Here, we take the state $|{\varphi }^{\pm }{⟩}_P|{\varphi }^{\pm }{⟩}_F$ as an example to show the distinguishing principle. After the photons $A$, $B$ and $C$ passing through the BSs and WPs, the evolution of hyperentanglement is

(17)$\begin{array}{rl}& |{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F\to \frac{1}{4}(|HHH⟩+|HVV⟩+|VHV⟩+|VVH⟩)\otimes (|aaa⟩+|abb⟩+|bab⟩+|bba⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F\to \frac{1}{4}(|HHH⟩+|HVV⟩+|VHV⟩+|VVH⟩)\otimes (|aab⟩+|aba⟩+|baa⟩+|bbb⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F\to \frac{1}{4}(|HHV⟩+|HVH⟩+|VHH⟩+|VVV⟩)\otimes (|aaa⟩+|abb⟩+|bab⟩+|bba⟩{)}_{ABC},\\ & |{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F\to \frac{1}{4}(|HHV⟩+|HVH⟩+|VHH⟩+|VVV⟩)\otimes (|aab⟩+|aba⟩+|baa⟩+|bbb⟩{)}_{ABC}.\end{array}$

These four states can be discriminated through the detection results of single-photon detectors in polarization and the first spatial-mode DOFs, with which the 64 initial states can be classified into four groups, as shown in Tab.3. Therefore, with both Tab.2 and Tab.3, we can determine arbitrary one of the 64 three-photon hyperentangled GHZ state in two DOFs by using the fixed auxiliary four-dimensional GHZ state in the third DOF.

Tab.3 Corresponding relations between the initial GHZ states and possible detection results in polarization mode and the first spatial-mode.

Initial GHZ state	$P$ mode	$F$ mode
$|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F,$	$HHH,$	$aaa,$
$|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F,$	$HVV,$	$abb,$
$|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F,$	$VHV,$	$bab,$
$|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F.$	$VVH.$	$bba.$
$|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{000}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F,$	$HHH,$	$aab,$
$|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{001}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F,$	$HVV,$	$aba,$
$|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{010}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F,$	$VHV,$	$baa,$
$|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{100}^{+}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F.$	$VVH.$	$bbb.$
$|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F,$	$HHV,$	$aaa,$
$|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F,$	$HVH,$	$abb,$
$|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F,$	$VHH,$	$bab,$
$|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{+}{⟩}_F,|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{+}{⟩}_F,$ $|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{+}{⟩}_F,|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{+}{⟩}_F.$	$VVV.$	$bba.$
$|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{000}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F,$	$HHV,$	$aab,$
$|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{001}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F,$	$HVH,$	$aba,$
$|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{010}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F,$	$VHH,$	$baa,$
$|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{000}^{-}{⟩}_F,|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{001}^{-}{⟩}_F,$ $|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{010}^{-}{⟩}_F,|{\mathrm{\Phi }}_{100}^{-}{⟩}_P|{\mathrm{\Phi }}_{100}^{-}{⟩}_F.$	$VVV.$	$bbb.$

Our approach can be extended to the complete $N$-photon HGSA scheme, resorting to the following $N$-photon four-dimensional GHZ state in the second spatial-mode DOF,

(18)$\begin{array}{rl}|{\mathrm{\Phi }}^{{}^{\prime }}{⟩}_S=& \frac{1}{2}(|00\cdots 0⟩+|11\cdots 1⟩\\ & +|22\cdots 2⟩+|33\cdots 3⟩{)}_{AB\cdots Z}.\end{array}$

Photons $A$, $B,\cdots ,Z$ are sent into the same setups, and one of the schematic diagram is shown in Fig.3. Assisted by the above auxiliary high-dimensional GHZ state, the parity information of hyperentanglement can be obtained. Specifically speaking, we can determine these four states $|{\mathrm{\Phi }}_{ij\cdots k}^{\pm }{⟩}_P\otimes |{\mathrm{\Phi }}_{ij\cdots k}^{\pm }{⟩}_F$ through the detection results in the second spatial-mode DOF by single-photon detectors, the principle of which is similar to the Eqs. (7), (13)−(16). Subsequently, the four different relative phase information “++”, “+−”, “−+”, and “− −” can be determined by the detection results in polarization and the first spatial-mode DOFs, the principle of which is similar to the Eqs. (8) and (17). With these two independent procedures, the complete analysis of $N$-photon hyperentangled GHZ state in polarization and the first spatial-mode DOFs is accomplished.

4 Discussion and summary

In this paper, we have developed a novel method for the complete analysis of hyperentangled Bell state and GHZ state in two DOFs, resorting to the auxiliary four-dimensional entanglement in the third DOF. The high-dimensional entanglement of quantum system has been well studied and can be efficiently prepared in experiment, for both Bell state and GHZ state [44−48]. In 2017, Wang et al. [44] experimentally generated a complete set of Bell states in four-dimensional Hilbert space, comprising 16 orthogonal high-dimensional Bell states encoded in the OAM DOF of photons. In 2020, Hu et al. [45] showcased entanglement in 32-spatial dimensions with record fidelity to the maximally entangled state. For high-dimensional GHZ state, in 2018, Erhard et al. [47] showed an experimental realization of the OAM GHZ state of three-photon system entangled in three dimensions. In 2022, Cervera-Lierta et al. [48] reported the experimental demonstration and certification of the high-dimensional multipartite GHZ state in a superconducting quantum processor. Moreover, in atomic systems, the high-dimensional quantum states have been theoretically studied with high intrinsic fidelities [49].

The quantum hyperentanglement simultaneously in three different DOFs of photon system is required in our schemes. Therefore, it is necessary to discuss the realization of such hyperentanglement in the real physical situation. Fortunately, in recent years, this type hyperentanglement has been experimentally reported [50−55]. For example, in 2005, Barreiro et al. [50] experimentally demonstrated the first quantum system entangled in polarization, spatial-mode, and time-bin DOFs, in which the polarization state and time-bin state are both two-dimensional state while the spatial-mode state is in high dimensions. In 2009, Vallone et al. [51] presented an optical scheme enabling the simultaneous entanglement of two photons in three independent DOFs, corresponding to polarization and two different spatial-modes derived by the indistinguishable emission of two nonlinear crystals. In 2022, Guo et al. [54] reported the generation of continuous-variable high-dimensional entanglement with three DOFs, including frequency, polarization, and OAM. In 2023, Achatz et al. [55] demonstrated the simultaneous transmission of hyperentanglement in three DOFs through a multicore fiber, in which the polarization and time-bin DOFs are in two dimensions while the spatial-mode DOF is in four dimensions. Based on these important experimental results, we are confident that the high-dimensional hyperentangled state required in our approach can be realized with the development of modern optical quantum technology.

Actually, we have revealed a general fact that high-dimensional entanglement can be helpful for the complete HBSA and HGSA schemes. Although we exploit the hyperentanglement in polarization and two spatial-mode DOFs in this paper, in fact, this approach is also suitable for other type hyperentanglement in other different DOFs, and the key is to construct the high-dimensional quantum gates between two-dimensional control mode and high-dimensional target mode. In summary, we have proposed an efficient approach for the complete analysis of maximally hyperentangled state in two DOFs via linear optics, resorting to the auxiliary high-dimensional entanglement in the third DOF. To our knowledge, this is the first high-dimensional entanglement-based complete hyperentangled state analysis scheme, and is meaningful for the future investigation of complete HBSA and HGSA.

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