Appendix 1
In this appendix, we exhibit the remained outcomes of \(A_{1}\) and \(A_{2}\) and the respective projective measurements of \(B_1\) and \(B_2\) in 2-to-2 JRSP.
Case 2.1 The measurement result of \(A_{1}\) and \(A_{2}\) is \((1,1)\) or \((11,11)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{11}^{0}\rangle , &{} |v_{11}^{1}\rangle , &{} |v_{11}^{2}\rangle , &{} |v_{11}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |1\rangle , &{} |0\rangle , &{} |3\rangle , &{} |2\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(23)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{12}^{0}\rangle , &{} |v_{12}^{1}\rangle , &{} |v_{12}^{2}\rangle , &{} |v_{12}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |1\rangle , &{} |0\rangle , &{} |3\rangle , &{} |2\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(24)
Case 2.2 The measurement result of \(A_{1}\) and \(A_{2}\) is \((2,2)\) or \((8,8)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{21}^{0}\rangle , &{} |v_{21}^{1}\rangle , &{} |v_{21}^{2}\rangle , &{} |v_{21}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |2\rangle , &{} |3\rangle , &{} |0\rangle , &{} |1\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(25)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{22}^{0}\rangle , &{} |v_{22}^{1}\rangle , &{} |v_{22}^{2}\rangle , &{} |v_{22}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |2\rangle , &{} |3\rangle , &{} |0\rangle , &{} |1\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(26)
Case 2.3 The measurement result of \(A_{1}\) and \(A_{2}\) is \((3,3)\) or \((9,9)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{31}^{0}\rangle , &{} |v_{31}^{1}\rangle , &{} |v_{31}^{2}\rangle , &{} |v_{31}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |3\rangle , &{} |2\rangle , &{} |1\rangle , &{} |0\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(27)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{32}^{0}\rangle , &{} |v_{32}^{1}\rangle , &{} |v_{32}^{2}\rangle , &{} |v_{32}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |3\rangle , &{} |2\rangle , &{} |1\rangle , &{} |0\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(28)
Case 2.4 The measurement results of \(A_{1}\) and \(A_{2}\) is \((4,4)\) or \((14,14)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{41}^{0}\rangle , &{} |v_{41}^{1}\rangle , &{} |v_{41}^{2}\rangle , &{} |v_{41}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |1\rangle , &{} |2\rangle , &{} |3\rangle , &{} |0\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(29)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{42}^{0}\rangle , &{} |v_{42}^{1}\rangle , &{} |v_{42}^{2}\rangle , &{} |v_{42}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |1\rangle , &{} |2\rangle , &{} |3\rangle , &{} |0\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(30)
Case 2.5 The measurement results of \(A_{1}\) and \(A_{2}\) is \((5,5)\) or \((15,15)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{51}^{0}\rangle , &{} |v_{51}^{1}\rangle , &{} |v_{51}^{2}\rangle , &{} |v_{51}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |2\rangle , &{} |1\rangle , &{} |0\rangle , &{} |3\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(31)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{52}^{0}\rangle , &{} |v_{52}^{1}\rangle , &{} |v_{52}^{2}\rangle , &{} |v_{52}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |2\rangle , &{} |1\rangle , &{} |0\rangle , &{} |3\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(32)
Case 2.6 The measurement results of \(A_{1}\) and \(A_{2}\) is \((6,6)\) or \((12,12)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{61}^{0}\rangle , &{} |v_{61}^{1}\rangle , &{} |v_{61}^{2}\rangle , &{} |v_{61}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |3\rangle , &{} |0\rangle , &{} |1\rangle , &{} |2\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(33)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{62}^{0}\rangle , &{} |v_{62}^{1}\rangle , &{} |v_{62}^{2}\rangle , &{} |v_{62}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |3\rangle , &{} |0\rangle , &{} |1\rangle , &{} |2\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(34)
Case 2.7 The measurement results of \(A_{1}\) and \(A_{2}\) is \((7,7)\) or \((13,13)\). Then, \(B_{1}\) and \(B_{2}\) will make projective measurements under the following basis
$$\begin{aligned} \left( \begin{array}{cccc} |v_{71}^{0}\rangle , &{} |v_{71}^{1}\rangle , &{} |v_{71}^{2}\rangle , &{} |v_{71}^{3}\rangle \\ \end{array} \right) _{B_{1}}=\left( \begin{array}{cccc} |0\rangle , &{} |3\rangle , &{} |2\rangle , &{} |1\rangle \\ \end{array} \right) _{B_{1}}\mathcal {V}_{1}(\theta )^{T}, \end{aligned}$$
(35)
$$\begin{aligned} \left( \begin{array}{cccc} |v_{72}^{0}\rangle , &{} |v_{72}^{1}\rangle , &{} |v_{72}^{2}\rangle , &{} |v_{72}^{3}\rangle \\ \end{array} \right) _{B_{2}}=\left( \begin{array}{cccc} |0\rangle , &{} |3\rangle , &{} |2\rangle , &{} |1\rangle \\ \end{array} \right) _{B_{2}}\mathcal {V}_{2}(\theta )^{T}. \end{aligned}$$
(36)
Appendix 2
In Eq. (20), \(|\varPhi ^{0}\rangle \), \(\ldots \), \(|\varPhi ^{31}\rangle \) are in the following forms.
$$\begin{aligned} |\varPhi ^{0}\rangle&= a_{1}^{0}a_{2}^{0}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{1}a_{2}^{1}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{3}a_{2}^{3}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{5}a_{2}^{5}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{7}a_{2}^{7}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(37)
$$\begin{aligned} |\varPhi ^{1}\rangle&= a_{1}^{1}a_{2}^{1}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{0}a_{2}^{0}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{2}a_{2}^{2}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{4}a_{2}^{4}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{6}a_{2}^{6}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(38)
$$\begin{aligned} |\varPhi ^{2}\rangle&= a_{1}^{2}a_{2}^{2}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{3}a_{2}^{3}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{1}a_{2}^{1}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{7}a_{2}^{7}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{5}a_{2}^{5}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(39)
$$\begin{aligned} |\varPhi ^{3}\rangle&= a_{1}^{3}a_{2}^{3}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{2}a_{2}^{2}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{0}a_{2}^{0}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{6}a_{2}^{6}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{4}a_{2}^{4}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(40)
$$\begin{aligned} |\varPhi ^{4}\rangle&= a_{1}^{4}a_{2}^{4}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{5}a_{2}^{5}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{7}a_{2}^{7}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{1}a_{2}^{1}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{3}a_{2}^{3}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(41)
$$\begin{aligned} |\varPhi ^{5}\rangle&= a_{1}^{5}a_{2}^{5}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{4}a_{2}^{4}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{6}a_{2}^{6}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{0}a_{2}^{0}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{2}a_{2}^{2}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(42)
$$\begin{aligned} |\varPhi ^{6}\rangle&= a_{1}^{6}a_{2}^{6}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{7}a_{2}^{7}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{5}a_{2}^{5}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{3}a_{2}^{3}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{1}a_{2}^{1}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(43)
$$\begin{aligned} |\varPhi ^{7}\rangle&= a_{1}^{7}a_{2}^{7}(|0,0,0\rangle +|8,8,1\rangle ) +a_{1}^{6}a_{2}^{6}(|1,1,0\rangle +|9,9,1\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|2,2,2\rangle +|10,10,3\rangle ) +a_{1}^{4}a_{2}^{4}(|3,3,2\rangle +|11,11,3\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|4,4,4\rangle +|12,12,5\rangle ) +a_{1}^{2}a_{2}^{2}(|5,5,4\rangle +|13,13,5\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|6,6,6\rangle +|14,14,7\rangle ) +a_{1}^{0}a_{2}^{0}(|7,7,6\rangle +|15,15,7\rangle ), \end{aligned}$$
(44)
$$\begin{aligned} |\varPhi ^{8}\rangle&= a_{1}^{0}a_{2}^{0}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{1}a_{2}^{1}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{3}a_{2}^{3}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{5}a_{2}^{5}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{7}a_{2}^{7}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(45)
$$\begin{aligned} |\varPhi ^{9}\rangle&= a_{1}^{1}a_{2}^{1}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{0}a_{2}^{0}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{2}a_{2}^{2}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{4}a_{2}^{4}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{6}a_{2}^{6}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(46)
$$\begin{aligned} |\varPhi ^{10}\rangle&= a_{1}^{2}a_{2}^{2}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{3}a_{2}^{3}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{1}a_{2}^{1}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{7}a_{2}^{7}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{5}a_{2}^{5}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(47)
$$\begin{aligned} |\varPhi ^{11}\rangle&= a_{1}^{3}a_{2}^{3}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{2}a_{2}^{2}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{0}a_{2}^{0}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{6}a_{2}^{6}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{4}a_{2}^{4}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(48)
$$\begin{aligned} |\varPhi ^{12}\rangle&= a_{1}^{4}a_{2}^{4}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{5}a_{2}^{5}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{7}a_{2}^{7}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{1}a_{2}^{1}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{3}a_{2}^{3}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(49)
$$\begin{aligned} |\varPhi ^{13}\rangle&= a_{1}^{5}a_{2}^{5}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{4}a_{2}^{4}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{6}a_{2}^{6}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{0}a_{2}^{0}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{2}a_{2}^{2}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(50)
$$\begin{aligned} |\varPhi ^{14}\rangle&= a_{1}^{6}a_{2}^{6}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{7}a_{2}^{7}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{5}a_{2}^{5}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{3}a_{2}^{3}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{1}a_{2}^{1}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(51)
$$\begin{aligned} |\varPhi ^{15}\rangle&= a_{1}^{7}a_{2}^{7}(|2,2,0\rangle +|10,10,1\rangle ) +a_{1}^{6}a_{2}^{6}(|3,3,0\rangle +|11,11,1\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|0,0,2\rangle +|8,8,3\rangle ) +a_{1}^{4}a_{2}^{4}(|1,1,2\rangle +|9,9,3\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|6,6,4\rangle +|14,14,5\rangle ) +a_{1}^{2}a_{2}^{2}(|7,7,4\rangle +|15,15,5\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|4,4,6\rangle +|12,12,7\rangle ) +a_{1}^{0}a_{2}^{0}(|5,5,6\rangle +|13,13,7\rangle ), \end{aligned}$$
(52)
$$\begin{aligned} |\varPhi ^{16}\rangle&= a_{1}^{0}a_{2}^{0}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{1}a_{2}^{1}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{3}a_{2}^{3}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{5}a_{2}^{5}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{7}a_{2}^{7}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(53)
$$\begin{aligned} |\varPhi ^{17}\rangle&= a_{1}^{1}a_{2}^{1}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{0}a_{2}^{0}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{2}a_{2}^{2}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{4}a_{2}^{4}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{6}a_{2}^{6}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(54)
$$\begin{aligned} |\varPhi ^{18}\rangle&= a_{1}^{2}a_{2}^{2}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{3}a_{2}^{3}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{1}a_{2}^{1}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{7}a_{2}^{7}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{5}a_{2}^{5}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(55)
$$\begin{aligned} |\varPhi ^{19}\rangle&= a_{1}^{3}a_{2}^{3}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{2}a_{2}^{2}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{0}a_{2}^{0}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{6}a_{2}^{6}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{4}a_{2}^{4}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(56)
$$\begin{aligned} |\varPhi ^{20}\rangle&= a_{1}^{4}a_{2}^{4}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{5}a_{2}^{5}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{7}a_{2}^{7}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{1}a_{2}^{1}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{3}a_{2}^{3}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(57)
$$\begin{aligned} |\varPhi ^{21}\rangle&= a_{1}^{5}a_{2}^{5}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{4}a_{2}^{4}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{6}a_{2}^{6}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{0}a_{2}^{0}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{2}a_{2}^{2}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(58)
$$\begin{aligned} |\varPhi ^{22}\rangle&= a_{1}^{6}a_{2}^{6}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{7}a_{2}^{7}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{5}a_{2}^{5}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{3}a_{2}^{3}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{1}a_{2}^{1}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(59)
$$\begin{aligned} |\varPhi ^{23}\rangle&= a_{1}^{7}a_{2}^{7}(|4,4,0\rangle +|12,12,1\rangle ) +a_{1}^{6}a_{2}^{6}(|5,5,0\rangle +|13,13,1\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|6,6,2\rangle +|14,14,3\rangle ) +a_{1}^{4}a_{2}^{4}(|7,7,2\rangle +|15,15,3\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|0,0,4\rangle +|8,8,5\rangle ) +a_{1}^{2}a_{2}^{2}(|1,1,4\rangle +|9,9,5\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|2,2,6\rangle +|10,10,7\rangle ) +a_{1}^{0}a_{2}^{0}(|3,3,6\rangle +|11,11,7\rangle ), \end{aligned}$$
(60)
$$\begin{aligned} |\varPhi ^{24}\rangle&= a_{1}^{0}a_{2}^{0}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{1}a_{2}^{1}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{3}a_{2}^{3}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{5}a_{2}^{5}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{7}a_{2}^{7}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(61)
$$\begin{aligned} |\varPhi ^{25}\rangle&= a_{1}^{1}a_{2}^{1}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{0}a_{2}^{0}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{2}a_{2}^{2}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{4}a_{2}^{4}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{6}a_{2}^{6}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(62)
$$\begin{aligned} |\varPhi ^{26}\rangle&= a_{1}^{2}a_{2}^{2}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{3}a_{2}^{3}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{1}a_{2}^{1}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{7}a_{2}^{7}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{5}a_{2}^{5}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(63)
$$\begin{aligned} |\varPhi ^{27}\rangle&= a_{1}^{3}a_{2}^{3}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{2}a_{2}^{2}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{0}a_{2}^{0}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{6}a_{2}^{6}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{4}a_{2}^{4}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(64)
$$\begin{aligned} |\varPhi ^{28}\rangle&= a_{1}^{4}a_{2}^{4}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{5}a_{2}^{5}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{6}a_{2}^{6}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{7}a_{2}^{7}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{1}a_{2}^{1}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{3}a_{2}^{3}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(65)
$$\begin{aligned} |\varPhi ^{29}\rangle&= a_{1}^{5}a_{2}^{5}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{4}a_{2}^{4}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{7}a_{2}^{7}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{6}a_{2}^{6}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{0}a_{2}^{0}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{2}a_{2}^{2}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(66)
$$\begin{aligned} |\varPhi ^{30}\rangle&= a_{1}^{6}a_{2}^{6}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{7}a_{2}^{7}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{4}a_{2}^{4}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{5}a_{2}^{5}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{2}a_{2}^{2}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{3}a_{2}^{3}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{0}a_{2}^{0}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{1}a_{2}^{1}(|1,1,6\rangle +|9,9,7\rangle ), \end{aligned}$$
(67)
$$\begin{aligned} |\varPhi ^{31}\rangle&= a_{1}^{7}a_{2}^{7}(|6,6,0\rangle +|14,14,1\rangle ) +a_{1}^{6}a_{2}^{6}(|7,7,0\rangle +|15,15,1\rangle ) \nonumber \\&+\,a_{1}^{5}a_{2}^{5}(|4,4,2\rangle +|12,12,3\rangle ) +a_{1}^{4}a_{2}^{4}(|5,5,2\rangle +|13,13,3\rangle ) \nonumber \\&+\,a_{1}^{3}a_{2}^{3}(|2,2,4\rangle +|10,10,5\rangle ) +a_{1}^{2}a_{2}^{2}(|3,3,4\rangle +|11,11,5\rangle ) \nonumber \\&+\,a_{1}^{1}a_{2}^{1}(|0,0,6\rangle +|8,8,7\rangle ) +a_{1}^{0}a_{2}^{0}(|1,1,6\rangle +|9,9,7\rangle ). \end{aligned}$$
(68)
The remained 7 cases for \(B_{1}\) and \(B_{2}\) to make proper measurements in Two-to-Three JRSP are listed from Case 3.1 to Case 3.7.
Case 3.1 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((1,1)\), \((11,11)\), \((21,21)\) and \((31,31)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{1i}^{0}\rangle \\ |v_{1i}^{1}\rangle \\ \vdots \\ |v_{1i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |1 \rangle \\ |0 \rangle \\ \vdots \\ |14 \rangle \\ \end{array} \right) . \end{aligned}$$
(69)
Case 3.2 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((2,2)\), \((8,8)\), \((22,22)\) and \((28,28)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{2i}^{0}\rangle \\ |v_{2i}^{1}\rangle \\ \vdots \\ |v_{2i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |2 \rangle \\ |3 \rangle \\ \vdots \\ |13 \rangle \\ \end{array} \right) . \end{aligned}$$
(70)
Case 3.3 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((3,3)\), \((9,9)\), \((23,23)\) and \((29,29)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{3i}^{0}\rangle \\ |v_{3i}^{1}\rangle \\ \vdots \\ |v_{3i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |3 \rangle \\ |2 \rangle \\ \vdots \\ |12 \rangle \\ \end{array} \right) . \end{aligned}$$
(71)
Case 3.4 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((4,4)\), \((14,14)\), \((16,16)\) and \((26,26)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{4i}^{0}\rangle \\ |v_{4i}^{1}\rangle \\ \vdots \\ |v_{4i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |4 \rangle \\ |5 \rangle \\ \vdots \\ |11 \rangle \\ \end{array} \right) . \end{aligned}$$
(72)
Case 3.5 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((5,5)\), \((15,15)\), \((17,17)\) and \((27,27)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{5i}^{0}\rangle \\ |v_{5i}^{1}\rangle \\ \vdots \\ |v_{5i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |5 \rangle \\ |4 \rangle \\ \vdots \\ |10 \rangle \\ \end{array} \right) . \end{aligned}$$
(73)
Case 3.6 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((6,6)\), \((12,12)\), \((18,18)\) and \((24,24)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{6i}^{0}\rangle \\ |v_{6i}^{1}\rangle \\ \vdots \\ |v_{6i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |6 \rangle \\ |7 \rangle \\ \vdots \\ |9 \rangle \\ \end{array} \right) . \end{aligned}$$
(74)
Case 3.7 The measurement outcomes of \(A_{1}\) and \(A_{2}\) are \((7,7)\), \((13,13)\), \((19,19)\) and \((25,25)\). The measurement basis performed by \(B_{i}, i=1,2\) is
$$\begin{aligned} \left( \begin{array}{c} |v_{7i}^{0}\rangle \\ |v_{7i}^{1}\rangle \\ \vdots \\ |v_{7i}^{15}\rangle \\ \end{array} \right) =\left( \begin{array}{cc} V_{i1}^{\dagger }(\theta ) &{} V_{i2}^{\dagger }(\theta ) \\ V_{i2}^{\dagger }(\theta ) &{} V_{i1}^{\dagger }(\theta ) \\ \end{array} \right) \left( \begin{array}{c} |7 \rangle \\ |6 \rangle \\ \vdots \\ |8 \rangle \\ \end{array} \right) . \end{aligned}$$
(75)
