Self-assisted complete analysis of three-photon hyperentangled Greenberger–Horne–Zeilinger states with nitrogen-vacancy centers in microcavities

Abstract

We propose a self-assisted complete analysis scheme of three-photon hyperentangled Greenberger–Horne–Zeilinger (GHZ) states with nitrogen-vacancy (NV) centers in microcavities (NV center-cavity systems), which is used to distinguish 64 three-photon hyperentangled GHZ states entangled in polarization and spatial-mode degrees of freedom. In our scheme, only three NV center-cavity systems are required for distinguishing the 64 three-photon hyperentangled GHZ states, which is much simpler than the previous schemes. Moreover, the three-photon spatial-mode GHZ states are distinguished with the three NV center-cavity systems without affecting the hyperentangled state of three-photon system, so the three-photon polarization GHZ states can be distinguished with linear optical elements assisted by the spatial-mode entangled state of three-photon system. With our scheme, the difficulty in the experimental realization of complete analysis of hyperentangled GHZ states may be largely decreased, which could potentially improve the performance of high-capacity multi-party quantum communication.

Appendix A

In order to describe the state change of each photon after passing through our hyperentanglement analysis device more clearly, we summarize the detailed steps in the following appendix. Firstly, we put the spatial-modes $a_1$ and $a_2$ of photon A into the quantum circuit shown in Fig. 2. After the spatial mode $a_1$ interacts with NV center-cavity system $\hbox {NV}_1$, the state of the system composed of $\hbox {NV}_1$ and photons A, B, C changes according to:

$$\begin{aligned}&|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_1\rangle |c_1\rangle (|RRR\rangle \mp |LLL\rangle )(|-1\rangle -|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_2\rangle |c_2\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle +|+1\rangle )\},\nonumber \\&|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_1\rangle |c_1\rangle (|LRR\rangle \mp |RLL\rangle ) (|+1\rangle -|-1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_2\rangle |c_2\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle +|+1\rangle )\}. \end{aligned}$$

(A1)

Then, the spatial mode $a_1$ passes through the HWP shown in Fig. 2, which is used to perform a phase-flip operation $Z=|R\rangle \langle R|-|L\rangle \langle L|$ on the polarization DOF of a photon. The state of the system composed of $\hbox {NV}_1$ and photons A, B, and C undergoes the transformation:

$$\begin{aligned}&|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1,\hbox {HWP}} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_1\rangle |c_1\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle -|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_2\rangle |c_2\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle +|+1\rangle )\},\nonumber \\&|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1,\hbox {HWP}} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_1\rangle |c_1\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle -|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_2\rangle |c_2\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle +|+1\rangle )\}. \end{aligned}$$

(A2)

The results of the operations on states $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$, $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$, $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, and $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$ after the spatial mode $a_1$ interacts with NV center-cavity system $\hbox {NV}_1$ and HWP. The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, and $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$ after the spatial mode $a_1$ interacts with NV center-cavity system $\hbox {NV}_1$ and HWP.

Secondly, we put the spatial modes $b_1$ and $b_2$ of photon B into the quantum circuit shown in Fig. 2. After the spatial mode $b_1$ passes through NV center-cavity system $\hbox {NV}_1$ and HWP, the state of the system composed of $\hbox {NV}_1$ and photons A, B, and C changes according to:

$$\begin{aligned}&|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1,\hbox {HWP}} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_1\rangle |c_1\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle +|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_2\rangle |c_2\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle +|+1\rangle )\},\nonumber \\&|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1,\hbox {HWP}} \nonumber \\&\frac{1}{2\sqrt{2}}\{|a_1\rangle |b_2\rangle |c_1\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle -|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_1\rangle |c_2\rangle (|RRR\rangle \pm |LLL\rangle )(|-1\rangle -|+1\rangle )\},\nonumber \\&|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1,\hbox {HWP}} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_1\rangle |c_1\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle +|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_2\rangle |c_2\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle +|+1\rangle )\},\nonumber \\&|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}|\varphi ^{+}\rangle _{1}\xrightarrow {\hbox {NV}_1,\hbox {HWP}} \nonumber \\&\quad \frac{1}{2\sqrt{2}}\{|a_1\rangle |b_2\rangle |c_1\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle -|+1\rangle )\nonumber \\&\quad \pm \,|a_2\rangle |b_1\rangle |c_2\rangle (|LRR\rangle \pm |RLL\rangle )(|-1\rangle -|+1\rangle )\}. \end{aligned}$$

(A3)

The results of the operations on states $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$, $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$, and $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$ after the spatial mode $b_1$ interacts with NV center-cavity system $\hbox {NV}_1$ and HWP. The results of the operations on states $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, and $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$ after the spatial mode $b_1$ interacts with NV center-cavity system $\hbox {NV}_1$ and HWP. The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$, and $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$ after the spatial mode $b_1$ interacts with NV center-cavity system $\hbox {NV}_1$ and HWP. The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, and $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$ after the spatial mode $b_1$ interacts with NV center-cavity system $\hbox {NV}_1$ and HWP. That is, if the spatial-mode DOF of photons A and B is in the odd-parity state ($a_1b_2$ and $a_2b_1$), the electron spin state of $\hbox {NV}_1$ changes to $|\varphi ^{-}\rangle _{1}$.

Thirdly, we put the spatial modes $a_1$ of photon A and $c_1$ of photon C into $\hbox {NV}_2$ and HWP in sequence, as shown in Fig. 2. Then, the state of the system composed of $\hbox {NV}_1$, $\hbox {NV}_2$ and photons A, B, and C changes to:

$$\begin{aligned} |\Phi _{000}^ \pm {\rangle _P}|\Phi _{000}^ \pm {\rangle _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}&\xrightarrow {\hbox {NV}1,\hbox {HWP},\hbox {NV}2,\hbox {HWP}}&|\Phi _{000}^ \pm {\rangle _P}|\Phi _{000}^ \pm {\rangle _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}, \nonumber \\ |\Phi _{000}^ \pm {\rangle _P}|\Phi _{100}^ \pm {\rangle _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}&\xrightarrow {\hbox {NV}1,\hbox {HWP},\hbox {NV}2,\hbox {HWP}}&|\Phi _{000}^ \pm {\rangle _P}|\Phi _{100}^ \pm {\rangle _S}|\varphi ^{-}\rangle _{1}|\varphi ^{-}\rangle _{2}, \nonumber \\ |\Phi _{000}^ \pm {\rangle _P}|\Phi _{010}^ \pm {\rangle _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}&\xrightarrow {\hbox {NV}1,\hbox {HWP},\hbox {NV}2,\hbox {HWP}}&|\Phi _{000}^ \pm {\rangle _P}|\Phi _{010}^ \pm {\rangle _S}|\varphi ^{-}\rangle _{1}|\varphi ^{+}\rangle _{2}, \nonumber \\ |\Phi _{000}^ \pm {\rangle _P}|\Phi _{001}^ \pm {\rangle _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}&\xrightarrow {\hbox {NV}1,\hbox {HWP},\hbox {NV}2,\hbox {HWP}}&|\Phi _{000}^ \pm {\rangle _P}|\Phi _{001}^ \pm {\rangle _S}|\varphi ^{+}\rangle _{1}|\varphi ^{-}\rangle _{2}.\nonumber \\ \end{aligned}$$

(A4)

The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$, and $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{000}\rangle _{S}$ after the spatial modes $a_1$ and $b_1$ interact with NV center-cavity systems $\hbox {NV}_1$, $\hbox {NV}_2$, and HWPs. The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$, and $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{100}\rangle _{S}$ after the spatial modes $a_1$ and $b_1$ interact with NV center-cavity systems $\hbox {NV}_1$, $\hbox {NV}_2$ and HWPs. The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$, and $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{010}\rangle _{S}$ after the spatial modes $a_1$ and $b_1$ interact with NV center-cavity systems $\hbox {NV}_1$, $\hbox {NV}_2$, and HWPs. The results of the operations on states $|\Phi ^{\pm }_{100}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$, $|\Phi ^{\pm }_{010}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$, and $|\Phi ^{\pm }_{001}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$ are the same as the result of the operations on state $|\Phi ^{\pm }_{000}\rangle _{P}|\Phi ^{\pm }_{001}\rangle _{S}$ after the spatial modes $a_1$ and $b_1$ interact with NV center-cavity systems $\hbox {NV}_1$, $\hbox {NV}_2$, and HWPs. According to the response of the electron spin states in $\hbox {NV}_1$ and $\hbox {NV}_2$, we can divide the eight spatial-mode states into four groups, each of which contains two spatial-mode states with different phases.

At last, we let the three photons A, B, and C pass through the first-column beam splitters. Here, the beam splitter (BS) can accomplish a Hadamard operation on the spatial-mode DOF of a photon,

$$\begin{aligned} |x_{1}\rangle&\xrightarrow {\hbox {BS}}\frac{1}{\sqrt{2}}(|x_{1}\rangle +|x_{2}\rangle ),\;\;\;\; |x_{2}\rangle&\xrightarrow {\hbox {BS}}\frac{1}{\sqrt{2}}(|x_{1}\rangle -|x_{2}\rangle ), \end{aligned}$$

(A5)

where x denotes a, b, or c. After the three photons A, B, and C pass through BSs, the eight spatial-mode states will transform as

$$\begin{aligned} |\Phi ^{+}_{000}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{1}\rangle +|a_{2}b_{2}c_{2}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{+}_{000}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{1}c_{1}\rangle +|a_{2}b_{2}c_{1}\rangle \nonumber \\&+\,|a_{1}b_{2}c_{2}\rangle +|a_{2}b_{1}c_{2}\rangle ),\nonumber \\ |\Phi ^{-}_{000}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{1}\rangle -|a_{2}b_{2}c_{2}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{-}_{000}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{2}c_{1}\rangle +|a_{2}b_{1}c_{1}\rangle \nonumber \\&+\,|a_{1}b_{1}c_{2}\rangle +|a_{2}b_{2}c_{2}\rangle ),\nonumber \\ |\Phi ^{+}_{100}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{2}b_{1}c_{1}\rangle +|a_{1}b_{2}c_{2}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{+}_{100}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{1}c_{1}\rangle -|a_{2}b_{2}c_{1}\rangle \nonumber \\&+\,|a_{1}b_{2}c_{2}\rangle -|a_{2}b_{1}c_{2}\rangle ),\nonumber \\ |\Phi ^{-}_{100}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{2}b_{1}c_{1}\rangle -|a_{1}b_{2}c_{2}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{-}_{100}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{2}c_{1}\rangle -|a_{2}b_{1}c_{1}\rangle \nonumber \\&+\,|a_{1}b_{1}c_{2}\rangle -|a_{2}b_{2}c_{2}\rangle ),\nonumber \\ |\Phi ^{+}_{010}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{2}c_{1}\rangle + |a_{2}b_{1}c_{2}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{+}_{010}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{1}c_{1}\rangle -|a_{2}b_{2}c_{1}\rangle -|a_{1}b_{2}c_{2}\rangle \nonumber \\&+\,|a_{2}b_{1}c_{2}\rangle ),\nonumber \\ |\Phi ^{-}_{010}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{2}c_{1}\rangle -|a_{2}b_{1}c_{2}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{-}_{010}\rangle '_{S}=\frac{1}{2}(|a_{2}b_{1}c_{1}\rangle +|a_{1}b_{1}c_{2}\rangle \nonumber \\&-\,|a_{1}b_{2}c_{1}\rangle -|a_{2}b_{2}c_{2}\rangle ),\nonumber \\ |\Phi ^{+}_{001}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{2}\rangle +|a_{2}b_{2}c_{1}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{+}_{001}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{1}c_{1}\rangle +|a_{2}b_{2}c_{1}\rangle \nonumber \\&-\,|a_{1}b_{2}c_{2}\rangle -|a_{2}b_{1}c_{2}\rangle ),\nonumber \\ |\Phi ^{-}_{001}\rangle _{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{2}\rangle -|a_{2}b_{2}c_{1}\rangle )&\xrightarrow {\hbox {BS}}&|\Phi ^{-}_{001}\rangle '_{S}=\frac{1}{2}(|a_{1}b_{2}c_{1}\rangle +|a_{2}b_{1}c_{1}\rangle \nonumber \\&-\,|a_{1}b_{1}c_{2}\rangle -|a_{2}b_{2}c_{2}\rangle ). \end{aligned}$$

(A6)

Then, we let the three photons A, B, and C pass through the quantum circuit shown in Fig. 2 in sequence. That is, the spatial modes $a_2$, $b_2$, and $c_2$ of photons A, B, and C interact with the NV center-cavity system $\hbox {NV}_3$ and HWP, which can distinguish the phase information of the initial spatial-mode GHZ states. Then, the state of the system composed of $\hbox {NV}_1$, $\hbox {NV}_2$, $\hbox {NV}_3$ and three-photon system ABC is transformed into:

$$\begin{aligned}&|\Phi _{000}^ \pm {\rangle _P} |\Phi _{000}^ + {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}}|\Phi _{000}^ \pm {\rangle _P} |\Phi _{000}^ + {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P}|\Phi _{000}^ - {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}}|\Phi _{000}^ \pm {\rangle _P} |\Phi _{000}^ - {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{-}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P}|\Phi _{100}^ +{\rangle '_S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}}|\Phi _{000}^ \pm {\rangle _P} |\Phi _{100}^ + {\rangle ' _S}|\varphi ^{-}\rangle _{1}|\varphi ^{-}\rangle _{2}|\varphi ^{+}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P} |\Phi _{100}^ -{\rangle '_S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}}|\Phi _{000}^ \pm {\rangle _P} |\Phi _{100}^ - {\rangle ' _S}|\varphi ^{-}\rangle _{1}|\varphi ^{-}\rangle _{2}|\varphi ^{-}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P} |\Phi _{010}^ + {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}}|\Phi _{000}^ \pm {\rangle _P} |\Phi _{010}^ + {\rangle ' _S}|\varphi ^{-}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P} |\Phi _{010}^ - {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}}|\Phi _{000}^ \pm {\rangle _P} |\Phi _{010}^ - {\rangle ' _S}|\varphi ^{-}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{-}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P} |\Phi _{001}^ +{\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}} |\Phi _{000}^ \pm {\rangle _P} |\Phi _{001}^ + {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{-}\rangle _{2}|\varphi ^{+}\rangle _{3},\nonumber \\&|\Phi _{000}^ \pm {\rangle _P} |\Phi _{001}^ -{\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{+}\rangle _{2}|\varphi ^{+}\rangle _{3}\nonumber \\&\quad \xrightarrow {\hbox {NV}_1,\hbox {HWP},\hbox {NV}_2,\hbox {HWP},\hbox {BS},\hbox {NV}_3,\hbox {HWP}} |\Phi _{000}^ \pm {\rangle _P} |\Phi _{001}^ - {\rangle ' _S}|\varphi ^{+}\rangle _{1}|\varphi ^{-}\rangle _{2}|\varphi ^{-}\rangle _{3}.\quad \qquad \end{aligned}$$

(A7)

If the polarization DOF of the three-photon system is in one of the states $|\Phi _{100}^ \pm {\rangle _P}$, $|\Phi _{010}^ \pm {\rangle _P}$ and $|\Phi _{001}^ \pm {\rangle _P}$, the phase information of the initial spatial-mode DOF of three-photon system can be distinguished in the same way. Then, we can use the second-column BS’s to restore the spatial-mode state to the initial one. The correspondence between the outcome of the electron spin states of $\hbox {NV}_1$, $\hbox {NV}_2$, $\hbox {NV}_3$ and the initial spatial-mode states of three-photon system is summarized in Table 1.