Appendix A: Decomposition of displaced number state in terms of Fock states
The displacement operator \( D\left( \alpha \right) \) is unitary and determined by
$$ D\left( \alpha \right) = \exp \left( {\alpha a^{ + } - \alpha^{*} a} \right), $$
(A1)
with \( \alpha \) the displacement amplitude and \( a (a^{ + } ) \) the bosonic annihilation (creation) operator. Its action on an arbitrary pure state
$$ \left| \psi \right\rangle = \mathop \sum \limits_{n = 0}^{k} a_{n} \left| n \right\rangle , \;{\text{with}}\;\mathop \sum \limits_{n = 0}^{k} \left| {a_{n} } \right|^{2} = 1, $$
(A2)
reads
$$ D\left( \alpha \right)\left| \psi \right\rangle = \mathop \sum \limits_{n = 0}^{k} a_{n} \left| {n,\alpha } \right\rangle , $$
(A3)
where
$$ \left| {n,\alpha } \right\rangle = D\left( \alpha \right)\left| n \right\rangle $$
(A4)
is called a displaced number state, which is characterized by two numbers: a quantum discrete number \( n \) and a classical continuous parameter \( \alpha \) (\( \alpha \) specifies the size of the state [23]). Note that \( \left| {0,\alpha } \right\rangle = D\left( \alpha \right)\left| 0 \right\rangle = \left| \alpha \right\rangle \) is nothing else but the coherent state. For a given \( \alpha \) the states
$$ \left\{ {\left| {n,\alpha } \right\rangle ,n = 0,1,2, \ldots ,\infty } \right\}, $$
(A5)
with the inner products \( \left\langle {n,\alpha |m,\alpha } \right\rangle = \delta_{nm} \) constitute the basic set of the displaced number state. The decomposition of a displaced number state in terms of Fock states is [23]
$$ \left| {l,\alpha } \right\rangle = F\mathop \sum \limits_{n = 0}^{\infty } c_{ln} \left( \alpha \right)\left| n \right\rangle , $$
(A6)
with \( F = \exp \left( { - \left| \alpha \right|^{2} /2} \right) \) and \( c_{ln} \left( \alpha \right) \) are determined in [26] by
$$ c_{ln} \left( \alpha \right) = \frac{{\alpha^{n - l} }}{{\sqrt {l!n!} }}\mathop \sum \limits_{k = 0}^{l} \left( { - 1} \right)^{k} C_{l}^{k} \left| \alpha \right|^{2k} \mathop \prod \limits_{k = 0}^{l - 1} \left( {n - l + k + 1} \right), $$
(A7)
or the same
$$ c_{ln} \left( \alpha \right) = \frac{{\alpha^{n - l} }}{{\sqrt {l!n!} }}\left( \begin{aligned} n\left( {n - 1} \right) \ldots \left( {n - l + 1} \right) - C_{l}^{1} \left| \alpha \right|^{2} n\left( {n - 1} \right) \ldots \left( {n - l + 2} \right) \hfill \\ + C_{l}^{2} \left| \alpha \right|^{4} n\left( {n - 1} \right) \ldots \left( {n - l + 3} \right) + \cdots \hfill \\ + \left( { - 1} \right)^{k} C_{l}^{k} \left| \alpha \right|^{2k} \mathop \prod \limits_{k}^{l - 1} \left( {n - l + k + 1} \right) + \left( { - 1} \right)^{l} \left| \alpha \right|^{2l} \hfill \\ \end{aligned} \right), $$
(A8)
where \( C_{l}^{k} = l!/\left( {k!\left( {l - k} \right)!} \right) \) and
$$ \mathop \prod \limits_{k}^{l - 1} \left( {n - l + k + 1} \right) = n\left( {n - 1} \right) \ldots \left( {n - l + k + 1} \right). $$
(A9)
The matrix elements \( c_{mn} \left( \alpha \right) \) in (A7) satisfy the normalization condition [26]
$$ F^{2} \mathop \sum \limits_{n = 0}^{\infty } \left| {c_{ln} \left( \alpha \right)} \right|^{2} = 1. $$
(A10)
From the explicit expressions (A7) or (A8) of \( c_{ln} \left( \alpha \right) \), it is possible to check the following relation
$$ c_{ln} \left( { - \alpha } \right) = \left( { - 1} \right)^{n - l} c_{ln} \left( \alpha \right). $$
(A11)
Obviously, for even \( l \) \( ({\text{i}} . {\text{e}}., l = 2m \))
$$ c_{2m,n} \left( { - \alpha } \right) = \left( { - 1} \right)^{n} c_{2m,n} \left( \alpha \right), $$
(A12)
and for odd \( l \) \( \left( {{\text{i}} . {\text{e}}., l = 2m + 1} \right) \)
$$ c_{2m + 1,n} \left( { - \alpha } \right) = \left( { - 1} \right)^{n - 1} c_{2m + 1,n} . $$
(A13)
In particular, for \( l = 0 \) and \( l = 1 \) one has [26]
$$ c_{0n} \left( \alpha \right) = \frac{{\alpha^{n} }}{{\sqrt {n!} }}, $$
(A14)
$$ c_{1n} \left( \alpha \right) = \frac{{\alpha^{n - 1} }}{{\sqrt {n!} }}\left( {n - \left| \alpha \right|^{2} } \right), $$
(A15)
satisfying the normalization conditions
$$ F^{2} \mathop \sum \limits_{n = 0}^{\infty } \left| {c_{0n} \left( \alpha \right)} \right|^{2} = 1, $$
(A16)
$$ F^{2} \mathop \sum \limits_{n = 0}^{\infty } \left| {c_{1n} \left( \alpha \right)} \right|^{2} = 1. $$
(A17)
One also has
$$ c_{0n} \left( { - \alpha } \right) = \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right), $$
(A18)
$$ c_{1n} \left( { - \alpha } \right) = \left( { - 1} \right)^{n - 1} c_{1n} \left( \alpha \right). $$
(A19)
Appendix B: DV–CV interaction for generation of hybrid entangled states (9)
The action of HTBS on an even cat (6a) and a dual-rail one-photon state (7),
$$ BS_{13} \left( {\left| {\beta_{ + } } \right\rangle_{1} \left| \varphi \right\rangle_{23} } \right) = N_{ + } \left( {BS_{13} \left( {\left| { - \beta } \right\rangle_{1} \left| \varphi \right\rangle_{23} } \right) + BS_{13} \left( {\left| \beta \right\rangle_{1} \left| \varphi \right\rangle_{23} } \right)} \right), $$
(B1)
can be calculated term by term as follows. For the first term in the RHS of (B1), we have
$$ \begin{aligned} BS_{13} \left( {\left| { - \beta } \right\rangle_{1} \left| \varphi \right\rangle_{23} } \right) & = BS_{13} D_{1} \left( { - \beta } \right)D_{3} \left( { - \alpha } \right)BS_{13}^{ + } BS_{13} \left| 0 \right\rangle_{1} D_{3} \left( \alpha \right)\left| \varphi \right\rangle_{23} \\ & = D_{1} \left( { - \beta /t} \right)D_{3} \left( 0 \right)BS_{13} \left| 0 \right\rangle_{1} \left( {a_{0} \left| 0 \right\rangle_{2} \left| {1,\alpha } \right\rangle_{3} + a_{1} \left| 1 \right\rangle_{2} \left| \alpha \right\rangle_{3} } \right) \\ & = FD_{1} \left( { - \beta /t} \right)\mathop \sum \limits_{n = 0}^{\infty } c_{1n} \left( \alpha \right)\left( {a_{0} \left| 0 \right\rangle_{2} + a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)BS_{13} \left( {\left| {0n} \right\rangle_{13} } \right), \\ \end{aligned} $$
(B2)
while the second term reads
$$ \begin{aligned} BS_{13} \left( {\left| \beta \right\rangle_{1} \left| \varphi \right\rangle_{23} } \right) & = BS_{13} D_{1} \left( \beta \right)D_{3} \left( \alpha \right)BS_{13}^{ + } BS_{13} \left| 0 \right\rangle_{1} D_{3} \left( { - \alpha } \right)\left| \varphi \right\rangle_{23} \\ & = D_{1} \left( {\beta /t} \right)D_{3} \left( 0 \right)BS_{13} \left| 0 \right\rangle_{1} \left( {a_{0} \left| 0 \right\rangle_{2} \left| {1, - \alpha } \right\rangle_{3} + a_{1} \left| 1 \right\rangle_{2} \left| { - \alpha } \right\rangle_{3} } \right) \\ & = FD_{1} \left( { - \beta /t} \right)\mathop \sum \limits_{n = 0}^{\infty } \left( { - 1} \right)^{n - 1} c_{1n} \left( \alpha \right)\left( {a_{0} \left| 0 \right\rangle_{2} - a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)BS_{13} \left( {\left| {0n} \right\rangle_{13} } \right). \\ \end{aligned} $$
(B3)
In (B2) and (B3)
$$ A_{n} \left( \alpha \right) = \frac{{c_{0n} \left( \alpha \right)}}{{c_{1n} \left( \alpha \right)}} = \frac{\alpha }{{n - \left| \alpha \right|^{2} }}. $$
(B4)
Summing up (B2) and (B3) yields
$$ \begin{aligned} & BS_{13} \left( {\left| {\beta_{ + } } \right\rangle_{1} \left| \varphi \right\rangle_{23} } \right) \\ & \quad = FN_{ + } \left( \begin{aligned} \left| {\Delta_{0} \left( { - \beta } \right)} \right\rangle_{123} + \left| {\Delta_{0} \left( \beta \right)} \right\rangle_{123} + r\left( {\left| {\Delta_{1} \left( { - \beta } \right)} \right\rangle_{123} + \left| {\Delta_{1} \left( \beta \right)} \right\rangle_{123} } \right) \hfill \\ + r^{2} \left( {\left| {\Delta_{2} \left( { - \beta } \right)} \right\rangle_{123} + \left| {\Delta_{2} \left( \beta \right)} \right\rangle_{123} } \right) + \cdots \hfill \\ \end{aligned} \right), \\ \end{aligned} $$
(B5)
where
$$ \left| {\Delta_{0} \left( { - \beta } \right)} \right\rangle_{123} = \left| { - \beta /t} \right\rangle_{1} \mathop \sum \limits_{n = 0}^{\infty } c_{1n} \left( \alpha \right)t^{n} \left( {a_{0} \left| 0 \right\rangle_{2} + a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)\left| n \right\rangle_{3} , $$
(B6)
$$ \left| {\Delta_{0} \left( \beta \right)} \right\rangle_{123} = \left| {\beta /t} \right\rangle_{1} \mathop \sum \limits_{n = 0}^{\infty } \left( { - 1} \right)^{n - 1} c_{1n} \left( \alpha \right)t^{n} \left( {a_{0} \left| 0 \right\rangle_{2} - a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)\left| n \right\rangle_{3} , $$
(B7)
$$ \left| {\Delta_{1} \left( { - \beta } \right)} \right\rangle_{123} = \left| {1, - \beta /t} \right\rangle_{1} \mathop \sum \limits_{n = 1}^{\infty } c_{1n} \left( \alpha \right)t^{n - 1} \sqrt n \left( {a_{0} \left| 0 \right\rangle_{2} + a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)\left| {n - 1} \right\rangle_{3} , $$
(B8)
$$ \left| {\Delta_{1} \left( \beta \right)} \right\rangle_{123} = \left| {1,\beta /t} \right\rangle_{1} \mathop \sum \limits_{n = 1}^{\infty } \left( { - 1} \right)^{n - 1} c_{1n} \left( \alpha \right)t^{n - 1} \sqrt n \left( {a_{0} \left| 0 \right\rangle_{2} - a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)\left| {n - 1} \right\rangle_{3} , $$
(B9)
$$ \left| {\Delta_{2} \left( { - \beta } \right)} \right\rangle_{123} = \left| {2, - \beta /t} \right\rangle_{1} \mathop \sum \limits_{n = 2}^{\infty } c_{1n} \left( \alpha \right)t^{n - 2} \sqrt {\frac{{n\left( {n - 1} \right)}}{2!}} \left( {a_{0} \left| 0 \right\rangle_{2} + a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)\left| {n - 2} \right\rangle_{3} , $$
(B10)
$$ \left| {\Delta_{2} \left( \beta \right)} \right\rangle_{123} = \left| {2,\beta /t} \right\rangle_{1} \mathop \sum \limits_{n = 2}^{\infty } \left( { - 1} \right)^{n - 1} c_{1n} \left( \alpha \right)t^{n - 2} \sqrt {\frac{{n\left( {n - 1} \right)}}{2!}} \left( {a_{0} \left| 0 \right\rangle_{2} - a_{1} A_{n} \left| 1 \right\rangle_{2} } \right)\left| {n - 2} \right\rangle_{3} . $$
(B11)
Using the above expressions, we can obtain the formulas (8)–(12) in Sect. 2.1.
The fidelity \( {\text{Fid}}_{n} \) in (23) is derived as follows. If \( n \) photons are registered in mode 3 of Fig. 1(a), then we have, up to the first order of smallness of \( r, \)
$$ \left| {\varPsi_{r}^{\left( n \right)} } \right\rangle_{12} = N_{r}^{\left( n \right)} \left( {\left| {\varPsi_{0}^{\left( n \right)} } \right\rangle_{12} + r\sqrt {n + 1} \frac{{c_{1n + 1} }}{{c_{1n} }}\left| {\varPsi_{1}^{\left( n \right)} } \right\rangle_{12} } \right), $$
(B12)
where
$$ \left| {\varPsi_{0}^{\left( n \right)} } \right\rangle_{12} = N_{n}^{{\left( {\text{tot}} \right)}} \left( {\beta /t} \right)\left( {\left| { - \beta /t} \right\rangle_{1} \left| {\varphi_{n}^{\left( + \right)} } \right\rangle_{2} + \left( { - 1} \right)^{n - 1} \left| {\beta /t} \right\rangle_{1} \left| {\varphi_{n}^{\left( - \right)} } \right\rangle_{2} } \right), $$
(B13)
$$ \left| {\varPsi_{1}^{\left( n \right)} } \right\rangle_{12} = N_{n1}^{{\left( {\text{tot}} \right)}} \left( {\beta /t} \right)\left( {\left| {1, - \beta /t} \right\rangle_{1} \left| {\varphi_{n + 1}^{\left( + \right)} } \right\rangle_{2} + \left( { - 1} \right)^{n} \left| {1,\beta /t} \right\rangle_{1} \left| {\varphi_{n + 1}^{\left( - \right)} } \right\rangle_{2} } \right), $$
(B14)
with \( N_{n}^{{\left( {\text{tot}} \right)}} \) (Eq. (12)) and \( N_{n1}^{{\left( {\text{tot}} \right)}} \) the corresponding normalization factors. As for the normalization factor \( N_{r}^{\left( n \right)} \) in (B12), it is given by
$$ N_{r}^{\left( n \right)} = \left( \begin{aligned} 1 + r^{2} \left( {n + 1} \right)\frac{{\left| {c_{1n + 1} } \right|^{2} }}{{\left| {c_{1n} } \right|^{2} }} \hfill \\ + r\sqrt {n + 1} \left( {\frac{{c_{1n + 1} }}{{c_{1n} }}\left\langle {\varPsi_{0}^{\left( n \right)} |\varPsi_{1}^{\left( n \right)} } \right\rangle + \left( {\frac{{c_{1n + 1} }}{{c_{1n} }}} \right)^{*} \left\langle {\varPsi_{1}^{\left( n \right)} |\varPsi_{0}^{\left( n \right)} } \right\rangle } \right) \hfill \\ \end{aligned} \right)^{ - 1/2} . $$
(B15)
Thus, the analytical expression for the fidelity \( Fid_{n} \) in (23) is
$$ {\text{Fid}}_{n} = \left| {\left\langle {\varPsi_{n} |\varPsi_{r}^{\left( n \right)} } \right\rangle } \right|^{2} = N_{r}^{\left( n \right)2} \left( \begin{aligned} \left| {\left\langle {\varPsi_{n} |\varPsi_{0}^{\left( n \right)} } \right\rangle } \right|^{2} + r\sqrt {n + 1} \hfill \\ \times \left( {\left( {\frac{{c_{1n + 1} }}{{c_{1n} }}} \right)^{*} \left\langle {\varPsi_{n} |\varPsi_{0}^{\left( n \right)} } \right\rangle \left\langle {\varPsi_{n} |\varPsi_{1}^{\left( n \right)} } \right\rangle^{*} + \frac{{c_{1n + 1} }}{{c_{1n} }}\left\langle {\varPsi_{n} |\varPsi_{0}^{\left( n \right)} } \right\rangle^{*} \left\langle {\varPsi_{n} |\varPsi_{1}^{\left( n \right)} } \right\rangle } \right) \hfill \\ \end{aligned} \right), $$
(B16)
up to the first order in \( r \).
Appendix C: DV–CV interaction for generation of hybrid entangled states (16)
Consider the following action of two HTBSs
$$ BS_{15} BS_{26} \left( {\left| {\beta_{ + } } \right\rangle_{1} \left| { - \beta_{1} } \right\rangle_{2} \left| \phi \right\rangle_{3456} } \right) = N_{ + } \left( {BS_{15} BS_{26} \left( {\left| { - \beta } \right\rangle_{1} \left| { - \beta_{1} } \right\rangle_{2} \left| \phi \right\rangle_{3456} } \right) + BS_{15} BS_{26} \left( {\left| \beta \right\rangle_{1} \left| { - \beta_{1} } \right\rangle_{2} \left| \phi \right\rangle_{3456} } \right)} \right). $$
(C1)
Following the steps of calculation as in “Appendix B”, we have
$$ \begin{aligned} & BS_{15} BS_{26} \left( {\left| { - \beta } \right\rangle_{1} \left| { - \beta_{1} } \right\rangle_{2} \left| \phi \right\rangle_{3456} } \right) \\ & \quad = BS_{15} D_{1} \left( { - \beta } \right)D_{5} \left( { - \alpha } \right)BS_{15}^{ + } BS_{26} D_{2} \left( { - \beta_{1} } \right)D_{6} \left( { - \alpha_{1} } \right)BS_{26}^{ + } BS_{15} BS_{26} \left| {00} \right\rangle_{12} D_{5} \left( \alpha \right)D_{6} \left( {\alpha_{1} } \right)\left| \phi \right\rangle_{3456} \\ & \quad = D_{1} \left( { - \beta /t} \right)D_{5} \left( 0 \right)D_{2} \left( { - \beta_{1} /t} \right)D_{6} \left( 0 \right)BS_{15} BS_{26} \left| {00} \right\rangle_{12} \left( {a_{0} \left| {01} \right\rangle_{34} \left| {0,\alpha } \right\rangle_{5} \left| {1,\alpha_{1} } \right\rangle_{6} + a_{1} \left| {10} \right\rangle_{34} \left| {1,\alpha } \right\rangle_{5} \left| {0,\alpha_{1} } \right\rangle_{6} } \right) \\ \end{aligned} $$
(C2)
and
$$ \begin{aligned} & BS_{15} BS_{26} \left( {\left| \beta \right\rangle_{1} \left| { - \beta_{1} } \right\rangle_{2} \left| \phi \right\rangle_{3456} } \right) \\ & \quad = BS_{15} D_{1} \left( \beta \right)D_{5} \left( \alpha \right)BS_{15}^{ + } BS_{26} D_{2} \left( { - \beta_{1} } \right)D_{6} \left( { - \alpha_{1} } \right)BS_{26}^{ + } BS_{15} BS_{26} \left| {00} \right\rangle_{12} D_{5} \left( { - \alpha } \right)D_{6} \left( {\alpha_{1} } \right)\left| \phi \right\rangle_{3456} \\ & \quad = D_{1} \left( {\beta /t} \right)D_{5} \left( 0 \right)D_{2} \left( { - \beta_{1} /t} \right)D_{6} \left( 0 \right)BS_{15} BS_{26} \left| {00} \right\rangle_{12} \left( {a_{0} \left| {01} \right\rangle_{34} \left| {0, - \alpha } \right\rangle_{5} \left| {1,\alpha_{1} } \right\rangle_{6} + a_{1} \left| {10} \right\rangle_{34} \left| {1, - \alpha } \right\rangle_{5} \left| {0,\alpha_{1} } \right\rangle_{6} } \right). \\ \end{aligned} $$
(C3)
Combining the decomposition (A6) with Eqs. (A18) and (A19), we obtain
$$ BS_{15} BS_{26} \left( {\left| {\beta_{ + } } \right\rangle_{1} \left| { - \beta_{1} } \right\rangle_{2} \left| \phi \right\rangle_{3456} } \right) = N_{ + } F^{2} \times \left( \begin{aligned} \left| {\Delta_{0} \left( { - \beta , - \beta_{1} } \right)} \right\rangle_{123456} + \left| {\Delta_{0} \left( {\beta , - \beta_{1} } \right)} \right\rangle_{123456} \hfill \\ + r\left( {\left| {\Delta_{1} \left( { - \beta , - \beta_{1} } \right)} \right\rangle_{123456} + \left| {\Delta_{1} \left( {\beta , - \beta_{1} } \right)} \right\rangle_{123456} } \right) \hfill \\ + r^{2} \left( {\left| {\Delta_{2} \left( { - \beta } \right)} \right\rangle_{123} + \left| {\Delta_{2} \left( \beta \right)} \right\rangle_{123} } \right) + \cdots \hfill \\ \end{aligned} \right), $$
(C4)
where
$$ \left| {\Delta_{0} \left( { - \beta , - \beta_{1} } \right)} \right\rangle_{123456} = \left| { - \beta /t} \right\rangle_{1} \left| { - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 0}^{\infty } \mathop \sum \limits_{m = 0}^{\infty } c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\left( {a_{0} \left| {01} \right\rangle_{34} + a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {nm} \right\rangle_{56} , $$
(C5)
$$ \left| {\Delta_{0} \left( {\beta , - \beta_{1} } \right)} \right\rangle_{123456} = \left| {\beta /t} \right\rangle_{1} \left| { - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 0}^{\infty } \mathop \sum \limits_{m = 0}^{\infty } \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\left( {a_{0} \left| {01} \right\rangle_{34} - a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {nm} \right\rangle_{56} , $$
(C6)
$$ \begin{aligned} &\left| {\Delta_{1} \left( { - \beta , - \beta_{1} } \right)} \right\rangle_{123456} = \left| {1, - \beta /t} \right\rangle_{1} \left| { - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 1}^{\infty } \mathop \sum \limits_{m = 0}^{\infty } c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt n t^{n + m - 1} \left( {a_{0} \left| {01} \right\rangle_{34} + a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {n - 1m} \right\rangle_{56} \\ & \quad + \left| { - \beta /t} \right\rangle_{1} \left| {1, - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 0}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt m t^{n + m - 1} \left( {a_{0} \left| {01} \right\rangle_{34} + a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {nm - 1} \right\rangle_{56} \\ \end{aligned} $$
(C7)
$$ \begin{aligned} &\left| {\Delta_{1} \left( {\beta , - \beta_{1} } \right)} \right\rangle_{123456} = \left| {1,\beta /t} \right\rangle_{1} \left| { - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 1}^{\infty } \mathop \sum \limits_{m = 0}^{\infty } \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt n t^{n + m - 1} \left( {a_{0} \left| {01} \right\rangle_{34} - a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {n - 1m} \right\rangle_{56} \\ & \quad + \left| {\beta /t} \right\rangle_{1} \left| {1, - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 0}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt m t^{n + m - 1} \left( {a_{0} \left| {01} \right\rangle_{34} - a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {nm - 1} \right\rangle_{56} , \\ \end{aligned} $$
(C8)
$$ \begin{aligned} \left| {\Delta_{2} \left( { - \beta , - \beta_{1} } \right)} \right\rangle_{123456} & = \left| {2, - \beta /t} \right\rangle_{1} \left| { - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 2}^{\infty } \mathop \sum \limits_{m = 0}^{\infty } c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt {\frac{{n\left( {n - 1} \right)}}{2!}} t^{n + m - 2} \\ & \quad \quad \left( {a_{0} \left| {01} \right\rangle_{34} + a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {n - 2m} \right\rangle_{56} \\ & \quad + \left| {0, - \beta /t} \right\rangle_{1} \left| {2, - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 0}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt {\frac{{m\left( {m - 1} \right)}}{2!}} t^{n + m - 2} \\ & \quad \quad \left( {a_{0} \left| {01} \right\rangle_{34} + a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {nm - 1} \right\rangle_{56} \\ & \quad + \left| {1, - \beta /t} \right\rangle_{1} \left| {1, - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 1}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt {nm} t^{n + m - 2} \\ & \quad \quad \left( {a_{0} \left| {01} \right\rangle_{34} + a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {n - 1m - 1} \right\rangle_{56} , \\ \end{aligned} $$
(C9)
$$ \begin{aligned} \left| {\Delta_{2} \left( {\beta , - \beta_{1} } \right)} \right\rangle_{123456} & = \left| {2, - \beta /t} \right\rangle_{1} \left| { - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 2}^{\infty } \mathop \sum \limits_{m = 0}^{\infty } \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt {\frac{{n\left( {n - 1} \right)}}{2!}} t^{n + m - 2} \\ & \quad \quad \left( {a_{0} \left| {01} \right\rangle_{34} - a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)|n - 2m_{56} \\ & \quad + \left| {0, - \beta /t} \right\rangle_{{1_{1} }} \left| {2, - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 0}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt {\frac{{m\left( {m - 1} \right)}}{2!}} t^{n + m - 2} \\ & \quad \quad \left( {a_{0} \left| {01} \right\rangle_{34} - a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {nm - 1} \right\rangle_{56} \\ & \quad + \left| {1, - \beta /t} \right\rangle_{1} \left| {1, - \beta_{1} /t} \right\rangle_{2} \mathop \sum \limits_{n = 1}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } \left( { - 1} \right)^{n} c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)\sqrt {nm} t^{n + m - 2} \\ & \quad \quad \left( {a_{0} \left| {01} \right\rangle_{34} - a_{1} A_{nm} \left| {10} \right\rangle_{34} } \right)\left| {n - 1m - 1} \right\rangle_{56} , \\ \end{aligned} $$
(C10)
with
$$ A_{nm} \left( {\alpha ,\alpha_{1} } \right) = \frac{{c_{1n} \left( \alpha \right)c_{0m} \left( {\alpha_{1} } \right)}}{{c_{0n} \left( \alpha \right)c_{1m} \left( {\alpha_{1} } \right)}} = \frac{{\alpha_{1} \left( {n - \left| \alpha \right|^{2} } \right)}}{{\alpha \left( {m - \left| {\alpha_{1} } \right|^{2} } \right)}}. $$
(C11)
In the limit \( t \to 1 \) one obtains formula (15) in Sect. 2.2.
