## 3 Deriving Input-Output Mappings for the Two Classes of Systems
### A Connection between the Transfer Functions of the Two Classes of Systems
\begin{bmatrix}\mathbf{X}&\mathbf{Y}\\ \mathbf{Z}&\mathbf{T}\end{bmatrix} ^{-1}= \begin{bmatrix}\mathbf{X}^{-1}+\mathbf{ X}^{-1}\mathbf{Y}\mathbf{S}^{-1}\mathbf{Z}\mathbf{X}^{-1}&-\mathbf{X}^{-1} \mathbf{Y}\mathbf{S}^{-1}\\ -\mathbf{S}^{-1}\mathbf{Z}\mathbf{X}^{-1}&\mathbf{S}^{-1}\end{bmatrix} \text{ where }\mathbf{S}\overset{\text{def}}{=}\mathbf{T}-\mathbf{Z}\mathbf{X}^{-1} \mathbf{Y}\text{.} \tag{32}\]  
Choose now \(\mathbf{X}=s\mathbf{I}_{n}-\mathbf{A}\), \(\mathbf{Y}=-\mathbf{Q}\), \(\mathbf{Z}=\mathbf{0}\) and \(\mathbf{T}=s\mathbf{I}_{n^{2}}-(\mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n} \otimes\mathbf{A})\). Hence, \(\mathbf{S}=\mathbf{T}=s\mathbf{I}_{n^{2}}-(\mathbf{A}\otimes\mathbf{I}_{n}+ \mathbf{I}_{n}\otimes\mathbf{A})\). It follows that:  
\[ \begin{split}\big{(}s\mathbf{I}_{N}-\mathbf{A}^{\otimes}\big{)}^ {-1}=\begin{bmatrix}(s\mathbf{I}_{n}-\mathbf{A})^{-1}&(s\mathbf{I}_{n}- \mathbf{A})^{-1}\mathbf{Q}\big{[}s\mathbf{I}_{n^{2}}-(\mathbf{A}\otimes \mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A})\big{]}^{-1}\\ \mathbf{0}&[s\mathbf{I}_{n^{2}}-(\mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{ n}\otimes\mathbf{A})]^{-1}\end{bmatrix}\end{split}  \tag{33}\]  
We can now compute the first transfer function of the bilinear model as in (4):  
\[\mathbf{H}_{0}(s_{0})=\mathbf{C}^{\otimes}\big{(}s_{0}\mathbf{I}_{N}-\mathbf{A }^{\otimes}\big{)}^{-1}\mathbf{B}^{\otimes}=\mathbf{C}(s_{0}\mathbf{I}_{n}- \mathbf{A})^{-1}\mathbf{B}\text{,} \tag{34}\]  
which is the transfer function of classical LTI systems (here, corresponding to the linear sub-model of the original QBTI system). Additionally, we can write that  
\[ \begin{split}\mathbf{H}_{1}^{\mathbf{N}^{\otimes}}(s_{0},s_{1})& =\mathbf{C}^{\otimes}\big{(}s_{0}\mathbf{I}_{N}-\mathbf{A}^{ \otimes}\big{)}^{-1}\mathbf{N}^{\otimes}\big{(}s_{1}\mathbf{I}_{N}-\mathbf{A}^ {\otimes}\big{)}^{-1}\mathbf{B}^{\otimes}\\ &=\underbrace{\mathbf{C}(s_{0}\mathbf{I}_{n}-\mathbf{A})^{-1} \mathbf{N}(s_{1}\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{B}}_{\mathbf{I}_{1}^{ \mathbf{N}}(s_{0},s_{1})}+\mathbf{K}_{1}(s_{0},s_{1})\text{,}\end{split}  \tag{35}\]  
where \(\mathbf{K}_{1}(s_{0},s_{1})\) is a bivariate function defined as:  
\[ \begin{split}\mathbf{K}_{1}(s_{0},s_{1})&=\mathbf{C} (s_{0}\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{Q}\big{[}s_{0}\mathbf{I}_{n^{2}}-( \mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A})\big{]}^{-1} \\ &(\mathbf{B}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{B} )(s_{1}\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{B}\text{.}\end{split}  \tag{36}\]Now, by using basic properties of the Kronecker product, such as \((\mathbf{v}\otimes\mathbf{I}_{n})\mathbf{w}=\mathbf{v}\otimes\mathbf{w}\) and \((\mathbf{I}_{n}\otimes\mathbf{v})\mathbf{w}=\mathbf{w}\otimes\mathbf{v}\) for any vectors \(\mathbf{v},\mathbf{w}\in\mathbb{C}^{n}\), one can write that:  
\[ \begin{split}\mathbf{K}_{1}(s_{0},s_{1})&=\mathbf{C}( s_{0}\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{Q}[s_{0}\mathbf{I}_{n^{2}}-( \mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes\mathbf{A})]^{-1}\\ &\Big{[}\mathbf{B}\otimes(s_{1}\mathbf{I}_{n}-\mathbf{A})^{-1} \mathbf{B}+(s_{1}\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{B}\otimes\mathbf{B} \Big{]}\\ &=\mathbf{C}(s_{0}\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{Q}[s_{0} \mathbf{I}_{n^{2}}-(\mathbf{A}\otimes\mathbf{I}_{n}+\mathbf{I}_{n}\otimes \mathbf{A})]^{-1}\\ &\Big{[}\mathbf{I}_{n}\otimes(s_{1}\mathbf{I}_{n}-\mathbf{A})^{-1 }+(s_{1}\mathbf{I}_{n}-\mathbf{A})^{-1}\otimes\mathbf{I}_{n}\Big{]}(\mathbf{B }\otimes\mathbf{B}).\end{split}  \tag{37}\]  
Next, by using the identity \((\mathbf{X}\otimes\mathbf{Y})^{-1}=\mathbf{X}^{-1}\otimes\mathbf{Y}^{-1}\) that holds for all invertible matrices \(\mathbf{X},\mathbf{Y}\in\mathbb{R}^{n\times n}\), it follows that  
\[