Appendix A
Proof of Theorem 3
Let \(|\phi \rangle \) be the prepared state of all the qubits except the source qubit. Let \(|\psi \rangle \) be the state of the qubit present at the source node. The unitary operation \(\mathcal {U}\) entangles both qubits into a bipartite quantum state:
$$\begin{aligned} |\varphi \rangle&= \mathcal {U}|\psi \rangle |\phi \rangle ^{\otimes N}. \end{aligned}$$
(124)
The unitary \(\mathcal {U}\) in the traditional MBQC is of the following form:
$$\begin{aligned} \mathcal {U}=\displaystyle \prod _{(p,q)\in \mathcal {E}} \mathcal {U}^{(p,q)}, \end{aligned}$$
(125)
where between every pair of nodes (p, q) connected by an edge, we have
$$\begin{aligned} \mathcal {U}^{(p,q)}=\mathrm {CZ}. \end{aligned}$$
(126)
However, in the generalization, we need not restrict \(\mathcal {U}^{(p,q)}\) to be of the CZ type. That is, we can think of any arbitrary unitary such at \(\mathcal {U}=e^{-iHt}\) where H is the overall Hamiltonian applied for a time duration t. We can write any general unitary acting on the state \(|\varphi \rangle \) in the following form:
$$\begin{aligned} \mathcal {U}&= \displaystyle \sum _{\underline{k}j,\underline{k}'j'} u_{\underline{k}j,\underline{k}'j'} |f_{k_1}\rangle \langle f_{k_1'}| \otimes |f_{k_2}\rangle \langle f_{k_2'}| \otimes \cdots \otimes |f_{k_{N}}\rangle \langle f_{k_{N}'}| \otimes |e_j\rangle \langle e_{j'}|, \end{aligned}$$
where \(\bigg \{\{|f_{k_1}\rangle \},\{|f_{k_2}\rangle \},\ldots ,\{|f_{k_{N}}\rangle \} \bigg \}\) form the basis for the Hilbert space of N qubits. Here each \(k_i \in \{0,1\}\) refers to the two dimensional Hilbert space of each qubit and \(i \in \{1,2,\ldots ,N\}.\) We denote \(\underline{k}=(k_1,k_2,\ldots ,k_{N})\). In terms of density matrix, we can write
$$\begin{aligned} \sigma&= |\varphi \rangle \langle \varphi |,\\&= \mathcal {U} \big ( |\psi \rangle \langle \psi | \otimes |\phi \rangle \langle \phi |^{\otimes N}\big ) \mathcal {U}^{\dagger },\\&= \bigg ( \displaystyle \sum _{\underline{k}j,\underline{k}'j'} u_{\underline{k}j,\underline{k}'j'} |f_{k_1}\rangle \langle f_{k_1'}| \otimes |f_{k_2}\rangle \langle f_{k_2'}| \otimes \cdots \otimes |f_{k_{N}}\rangle \langle f_{k_{N}'}| \otimes |e_j\rangle \langle e_{j'}| \bigg )\\&\quad \times \bigg (|\psi \rangle \langle \psi | \otimes |\phi \rangle \langle \phi |^{\otimes N}\bigg ) \\&\quad \times \bigg ( \displaystyle \sum _{\underline{l}m,\underline{l}'m'} \bar{u}_{\underline{l}m,\underline{l}'m'} |f_{l_1'}\rangle \langle f_{l_1}| \otimes |f_{l_2'}\rangle \langle f_{l_2}| \otimes \cdots \otimes |f_{l_{N}'}\rangle \langle f_{l_{N}}| \otimes |e_m'\rangle \langle e_{m}| \bigg ), \\&= \displaystyle \sum _{\begin{array}{c} \underline{k},\underline{k}',j,j',\\ \underline{l},\underline{l}',m,m' \end{array}} u_{\underline{k}j,\underline{k}'j'} \bar{u}_{\underline{l}m,\underline{l}'m'} |f_{k_1}\rangle \langle f_{k_1'}|\psi \rangle \langle \psi |f_{l_1'}\rangle \langle f_{l_1}| \otimes \\&\quad |f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \langle \phi |f_{l_2'}\rangle \langle f_{l_2}| \otimes \cdots \otimes |f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \langle \phi |f_{l_{N}'}\rangle \langle f_{l_{N}}| \otimes |e_j\rangle \langle e_{j'}|\phi \rangle \langle \phi |e_{m'}\rangle \langle e_m|. \end{aligned}$$
Suppose we perform measurement in the \(\mathcal {B}\) basis. Let the measurement outcome at the ‘i’th node be \(|g_{t_i}\rangle \). Then, the resultant unnormalized joint state becomes:
$$\begin{aligned} \sigma&= \bigg (|g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}| \otimes \cdots \otimes |g_{t_{N}}\rangle \langle g_{t_{N}}| \otimes I\bigg ) \\&\quad \times \bigg (\displaystyle \sum _{\begin{array}{c} \underline{k},\underline{k}',j,j',\\ \underline{l},\underline{l}',m,m' \end{array}} u_{\underline{k}j,\underline{k}'j'} \bar{u}_{\underline{l}m,\underline{l}'m'} |f_{k_1}\rangle \langle f_{k_1'}|\psi \rangle \langle \psi |f_{l_1'}\rangle \langle f_{l_1}| \otimes |f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \langle \phi |f_{l_2'}\rangle \langle f_{l_2}| \otimes \cdots \\&\quad \cdots \otimes |f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \langle \phi |f_{l_{N}'}\rangle \langle f_{l_{N}}| \otimes |e_j\rangle \langle e_{j'}|\phi \rangle \langle \phi |e_{m'}\rangle \langle e_m|\bigg ) \\&\quad \times \bigg (|g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}| \otimes \cdots \otimes |g_{t_{N}}\rangle \langle g_{t_{N}}| \otimes I\bigg ), \\&= \displaystyle \sum _{\begin{array}{c} \underline{k},\underline{k}',j,j',\\ \underline{l},\underline{l}',m,m' \end{array}} u_{\underline{k}j,\underline{k}'j'} \bar{u}_{\underline{l}m,\underline{l}'m'} |g_{t_1}\rangle \langle g_{t_1}|f_{k_1}\rangle \langle f_{k_1'}|\psi \rangle \langle \psi |f_{l_1'}\rangle \\&\quad \langle f_{l_1}|g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}|f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \langle \phi |f_{l_2'}\rangle \langle f_{l_2}|f_{t_2}\rangle \langle g_{t_2}|\\&\quad \otimes \cdots \otimes |g_{t_{N}}\rangle \langle g_{t_{N}}|f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \langle \phi |f_{l_{N}'}\rangle \langle f_{l_{N}}|g_{t_{N}}\rangle \langle g_{t_{N}}|\\&\quad \otimes |e_j\rangle \langle e_{j'}|\phi \rangle \langle \phi |e_{m'}\rangle \langle e_m|,\\&= \bigg (|g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}| \otimes \cdots \otimes |g_{t_n}\rangle \langle g_{t_n}|\bigg ) \otimes \displaystyle \sum _{\begin{array}{c} \underline{k}',j,j',\\ \underline{l}',m,m' \end{array}} u_{\underline{k}j,\underline{k}'j'} \bar{u}_{\underline{t}m,\underline{l}'m'}\\&\quad \bigg (\langle g_{t_1}|f_{k_1}\rangle \langle f_{k_1'}|\psi \rangle \langle \psi |f_{l_1'}\rangle \langle f_{l_1}|g_{t_1}\rangle \\&\quad \langle g_{t_2}|f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \langle \phi |f_{l_2'}\rangle \langle f_{l_2}|g_{t_2}\rangle \cdots \langle g_{t_{N}}|f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \langle \phi |f_{l_{N}'}\rangle \langle f_{l_{N}}|g_{t_{N}}\rangle \bigg )\\&\quad |e_j\rangle \langle e_j'|\phi \rangle \langle \phi |e_m'\rangle \langle e_m| \\&=\bigg ( |g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}| \otimes \cdots \otimes |g_{t_{N}}\rangle \langle g_{t_{N}}| \bigg )\otimes \displaystyle \sum _{\begin{array}{c} \underline{k},\underline{k}',j,j',\\ \underline{l},\underline{l}',m,m' \end{array}} u_{\underline{k}j,\underline{k}'j'} \bar{u}_{\underline{l}m,\underline{l}'m'} |e_j\rangle \langle e_{j'}|\phi \rangle \\&\quad \bigg (\langle g_{t_{N}}|f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \cdots \langle g_{t_2}|f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \bigg )\big (\langle g_{t_1}| f_{k_1}\rangle \langle f_{k_1'}|\psi \rangle \langle \psi |f_{l_1'}\rangle \langle f_{l_1}|g_{t_1}\rangle \big )\\&\quad \bigg (\langle \phi |f_{l_2'}\rangle \langle f_{l_2}|g_{t_2}\rangle \cdots \langle \phi |f_{l_{N}'}\rangle \langle f_{l_{N}}|g_{t_{N}}\rangle \bigg ) \langle \phi |e_{m'}\rangle \langle e_m| \\&= |g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}| \otimes \cdots \otimes |g_{t_{N}}\rangle \langle g_{t_{N}}| \otimes A_{\underline{t}} \rho A^{\dagger }_{\underline{t}} \\&\text {where } A_{\underline{t}} = \displaystyle \sum _{j,j',\underline{k},\underline{k}'} u_{\underline{k}\,j,\underline{k}'j'} |e_j\rangle \langle e_{j'}|\phi \rangle \langle g_{t_{N}}|f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \langle g_{t_{N-1}}|f_{k_{N-1}}\rangle \langle f_{k_{N-1}'}|\phi \rangle \cdots \\&\quad \cdots \langle g_{t_2}|f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \langle g_{t_1}|f_{k_1}\rangle \langle f_{k_1'}|, \\&\quad \text {and}\quad \underline{t}=(t_1,t_2,\ldots ,t_{N}). \end{aligned}$$
Upon normalization, we get
$$\begin{aligned} \sigma '= |g_{t_1}\rangle \langle g_{t_1}| \otimes |g_{t_2}\rangle \langle g_{t_2}| \otimes \cdots \otimes |g_{t_{N}}\rangle \langle g_{t_{N}}|\otimes \frac{A_{\underline{t}} \rho A^{\dagger }_{\underline{t}}}{\mathrm {Tr}(A_{\underline{t}} \rho A^{\dagger }_{\underline{t}})}. \end{aligned}$$
The state of the qubit at node B can be expressed as
$$\begin{aligned} \mathcal {N}(\rho )&= \displaystyle \sum _{\underline{t}} p_{\underline{t}} \frac{A_{\underline{t}} \rho A^{\dagger }_{\underline{t}}}{\mathrm {Tr}(A_{\underline{t}} \rho A^{\dagger }_{\underline{t}})} \quad \text {where}\quad p_{\underline{t}} = \mathrm {Tr}(A_{\underline{t}} \rho A^{\dagger }_{\underline{t}}), \nonumber \\&= \displaystyle \sum _{\underline{t}} A_{\underline{t}} \rho A^{\dagger }_{\underline{t}}. \end{aligned}$$
(127)
The measurement outcome occurs with a certain probability. Averaging over all outcomes we find that between the source and destination nodes, a quantum channel with the following description manifests:
$$\begin{aligned} \mathcal {N}(\rho )=\displaystyle \sum _{\underline{t}} A_{\underline{t}} \rho A_{\underline{t}}^{\dagger }. \end{aligned}$$
(128)
Therefore, we are able to get Kraus like operators, namely \(\{A_{\underline{t}}\}\) that seem to describe a quantum channel. \(\square \)
Appendix B
Proof of Theorem 4
For the above set to be valid Kraus operators, they need to satisfy the completeness relation, namely:
$$\begin{aligned}&\displaystyle \sum _{\underline{t}} A^{\dagger }_{\underline{t}} A_{\underline{t}} = \mathbb {I}_2. \end{aligned}$$
Consider the sum below
$$\begin{aligned}&\displaystyle \sum _{\underline{t}} A^{\dagger }_{\underline{t}} A_{\underline{t}}\\&\quad = \sum _{\underline{t}}\bigg ( \displaystyle \sum _{j,j',\underline{k},\underline{k}'} u_{\underline{k}\,j,\underline{k}'j'} |e_j\rangle \langle e_{j'}|\phi \rangle \langle g_{t_{N}}|f_{k_{N}}\rangle \langle f_{k_{N}'}|\phi \rangle \cdots \langle g_{t_2}|f_{k_2}\rangle \langle f_{k_2'}|\phi \rangle \langle g_{t_1}|f_{k_1}\rangle \langle f_{k_1'}|\bigg )^{\dagger } \\&\qquad \bigg ( \displaystyle \sum _{l,l',\underline{m},\underline{m}'} u_{\underline{m}\,l,\underline{m}'l'} |e_l\rangle \langle e_{l'}|\phi \rangle \langle g_{t_{N}}|f_{m_{N}}\rangle \langle f_{m_{N}'}|\phi \rangle \cdots \langle g_{t_2}|f_{m_2}\rangle \langle f_{m_2'}|\phi \rangle \langle g_{t_1}|f_{m_1}\rangle \langle f_{m_1'}|\bigg ) \\&\quad = \sum _{\underline{t}}\bigg ( \displaystyle \sum _{j,j',\underline{k},\underline{k}'} \bar{u}_{\underline{k}\,j,\underline{k}'j'} |f_{k_1'}\rangle \langle f_{k_1}|g_{t_1}\rangle \langle \phi |f_{k_2'}\rangle \langle f_{k_2}|g_{t_2}\rangle \cdots \langle \phi |f_{k_{N}'}\rangle \langle f_{k_{N}}|g_{t_{N}}\rangle \langle \phi |e_j'\rangle \langle e_j|\bigg ) \\&\qquad \times \bigg ( \displaystyle \sum _{l,l',\underline{m},\underline{m}'} u_{\underline{m}\,l,\underline{m}'l'} |e_l\rangle \langle e_{l'}|\phi \rangle \langle g_{t_{N}}|f_{m_{N}}\rangle \langle f_{m_{N}'}|\phi \rangle \cdots \langle g_{t_2}|f_{m_2}\rangle \langle f_{m_2'}|\phi \rangle \langle g_{t_1}|f_{m_1}\rangle \langle f_{m_1'}|\bigg ) \\&\quad =\displaystyle \sum _{\underline{t}}\sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l,l',\underline{m},\underline{m}' \end{array}} \bar{u}_{\underline{k}j,\underline{k}'j'} u_{\underline{m}l,\underline{m}'l'} \bigg (|f_{k_1'}\rangle \langle f_{k_1}|g_{t_1}\rangle \langle \phi |f_{k_2'}\rangle \langle f_{k_2}|g_{t_2}\rangle \cdots \langle \phi |f_{k_{N}'}\rangle \langle f_{k_{N}}|g_{t_{N}}\rangle \langle \phi |e_j'\rangle \langle e_j| \bigg )\\&\qquad \times \bigg ( |e_l\rangle \langle e_{l'}|\phi \rangle \langle g_{t_{N}}|f_{m_{N}}\rangle \langle f_{m_{N}'}|\phi \rangle \cdots \langle g_{t_2}|f_{m_2}\rangle \langle f_{m_2'}|\phi \rangle \langle g_{t_1}|f_{m_1}\rangle \langle f_{m_1'}| \bigg ) \\&\quad =\displaystyle \sum _{\underline{t}}\sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l,l',\underline{m},\underline{m}' \end{array}} \bar{u}_{\underline{k}j,\underline{k}'j'} u_{\underline{m}l,\underline{m}'l'} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_N}'\rangle \bigg )\langle \phi |e_j'\rangle \bigg (\langle f_{k_1}|g_{t_1}\rangle \langle f_{k_2}|g_{t_2}\rangle \cdots \langle f_{k_{N}}|g_{t_{N}}\rangle \bigg )\\&\qquad \langle e_j|e_l\rangle \bigg (\langle g_{t_{N}}|f_{m_{N}}\rangle \cdots \langle g_{t_2}|f_{m_2}\rangle \langle g_{t_1}|f_{m_1}\rangle \bigg ) \langle e_{l'}|\phi \rangle \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}| \bigg )\\&\quad =\displaystyle \sum _{\underline{t}}\sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l,l',\underline{m},\underline{m}' \end{array}} \bar{u}_{\underline{k}j,\underline{k}'j'} u_{\underline{m}l,\underline{m}'l'} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg ) \langle \phi |e_j'\rangle \bigg (\langle f_{k_1}|g_{t_1}\rangle \langle f_{k_2}|g_{t_2}\rangle \cdots \langle f_{k_{N}}|g_{t_{N}}\rangle \bigg )\delta _{j,l}\\&\qquad \langle g_{t_{N}}|f_{m_{N}}\rangle \cdots \langle g_{t_2}|f_{m_2}\rangle \langle g_{t_1}|f_{m_1}\rangle \langle e_{l'}|\phi \rangle \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}| \bigg ) \end{aligned}$$
$$\begin{aligned}&\quad =\displaystyle \sum _{\underline{t}} \sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l',\underline{m},\underline{m}' \end{array}} \bar{u}_{\underline{k}j,\underline{k}'j'} u_{\underline{m}j,\underline{m}'l'} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg )\langle \phi |e_j'\rangle \bigg (\langle f_{k_1}|g_{t_1}\rangle \langle f_{k_2}|g_{t_2}\rangle \cdots \langle f_{k_{N}}|g_{t_{N}}\rangle \bigg )\\&\qquad \times \bigg (\langle g_{t_{N}}|f_{m_{N}}\rangle \cdots \langle g_{t_2}|f_{m_2}\rangle \langle g_{t_1}|f_{m_1}\rangle \bigg )\langle e_{l'}|\phi \rangle \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}| \bigg ) \\&\quad =\displaystyle \sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l',\underline{m},\underline{m}' \end{array}} \bar{u}_{\underline{k}j,\underline{k}'j'} u_{\underline{m}j,\underline{m}'l'} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg )\langle \phi |e_j'\rangle \bigg (\sum _{t_1}\langle f_{k_1}|g_{t_1}\rangle \\&\qquad \times \bigg (\sum _{t_2}\langle f_{k_2}|g_{t_2}\rangle \cdots \bigg (\sum _{t_{N}}\langle f_{k_{N}}|g_{t_{N}}\rangle \\&\qquad \langle g_{t_{N}}|f_{m_{N}}\rangle \bigg )\cdots \langle g_{t_2}|f_{m_2}\rangle \bigg )\langle g_{t_1}|f_{m_1}\rangle \bigg )\langle e_{l'}|\phi \rangle \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}|\bigg ) \\&\quad =\displaystyle \sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l',\underline{m},\underline{m}' \end{array}} \delta _{\underline{k},\underline{m}} \,\bar{u}_{\underline{k}j,\underline{k}'j'}\, u_{\underline{m}j,\underline{m}'l'} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg ) \langle \phi |e_j'\rangle \langle e_{l'}|\phi \rangle \\&\qquad \times \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}| \bigg ) \\&\quad =\displaystyle \sum _{\begin{array}{c} j,j',\underline{k},\underline{k}',\\ l',\underline{m}' \end{array}} \bar{u}_{\underline{k}j,\underline{k}'j'} u_{\underline{k}j,\underline{m}'l'} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg )\langle \phi |e_j'\rangle \langle e_{l'}|\phi \rangle \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}| \bigg ) \\&\quad =\displaystyle \sum _{\begin{array}{c} j',\underline{k}',\\ l',\underline{m}' \end{array}} \delta _{\underline{k}',\underline{m'}} \delta _{j',l'}\bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg ) \langle \phi |e_j'\rangle \langle e_{l'}|\phi \rangle \bigg (\langle f_{m_{N}'}|\phi \rangle \cdots \langle f_{m_2'}|\phi \rangle \langle f_{m_1'}| \bigg ) \\&\quad =\displaystyle \sum _{\begin{array}{c} \underline{k}' \end{array}} \bigg (|f_{k_1}'\rangle \langle \phi |f_{k_2}'\rangle \cdots \langle \phi |f_{k_{N}}'\rangle \bigg ) \langle \phi | \bigg (\sum _{j'}|e_j'\rangle \langle e_{j'}| \bigg )|\phi \rangle \bigg (\langle f_{k_{N}'}|\phi \rangle \langle f_{k_2'}|\phi \rangle \langle f_{k_1'}| \bigg ) \\&\quad =\displaystyle \sum _{\underline{k}'} \bigg (|f_{k_1'}\rangle \langle \phi |f_{k_2'}\rangle \cdots \langle \phi |f_{k_{N}'}\rangle \langle f_{k_{N}'}|\phi \rangle \cdots \langle f_{k_2'}|\phi \rangle \langle f_{k_1'}|\bigg )\\&\quad =\displaystyle \sum _{k_1'} \Bigg (|f_{k_1'}\rangle \bigg (\sum _{k_2'}\langle \phi |f_{k_2'}\rangle \cdots \bigg (\sum _{k_{N}'} \langle \phi |f_{k_{N}'}\rangle \langle f_{k_{N}'}|\phi \rangle \bigg ) \cdots \langle f_{k_2'}|\phi \rangle \bigg )\langle f_{k_1'}| \Bigg ) \\&\quad =\mathbb {I}_2. \end{aligned}$$
\(\square \)
