# Error Performance of Amplitude Shift Keying-Type Asymmetric Quantum Communication Systems
## 4 Derivation of Analytical Expression for the Error Probability of the Quasi-Bell State
### Step 1: Representation of the Received Quantum States by an \(8\times 8\) Matrix  
First, if we focus on mode A in the received quantum states \(\Psi^{(0)}_{\rm SA}\) and \(\Psi^{(1)}_{\rm SA}\), we can see that the mode is constructed using the two coherent states \(\{|\pm\alpha\rangle_{\rm A}\}\). In the two-dimensional subspace of a Hilbert space that is spanned by \(\{|\pm\alpha\rangle\}\), the orthonormal basis \(\{|\omega_{0}\rangle,|\omega_{1}\rangle\}\) used in references [24; 25] is the measurement state of the square-root measurement (SRM) [26], which is often applied to the representation of the numerical vector of the quasi-Bell state and to the representation of its density operator. It is known that this approach improves the prospects of the discussion (e.g., [27]). The SRM for \(\{|\pm\alpha\rangle_{\rm A}\}\) is an optimum quantum measurement that minimizes the error probability, and use of these measurement states means that \(\{|\pm\alpha\rangle_{\rm A}\}\) can be expressed as:  
\[|\pm\alpha\rangle_{\rm A}=\frac{1}{\varepsilon_{+}-\varepsilon_{-}}(\sqrt{ \varepsilon_{\pm}}|\omega_{0}\rangle_{\rm A}-\sqrt{\varepsilon_{\mp}}|\omega_ {1}\rangle_{\rm A}). \tag{23}\]  
Therefore, \(\{|\pm\alpha\rangle_{\rm A}\}\) can be represented by a two-dimensional vector, as follows:  
\[|\pm\alpha\rangle_{\rm A}=\frac{1}{\varepsilon_{+}-\varepsilon_{-}}\bigg{[} \begin{array}{c}\sqrt{\varepsilon_{\pm}}\\ -\sqrt{\varepsilon_{\mp}}\end{array} \bigg{]}, \tag{24}\]  
where \(\kappa_{\rm A}=\kappa_{\rm S}=:\kappa\) and \(\varepsilon_{\pm}\) is given as follows:  
\[\varepsilon_{\pm}=\frac{1\pm\sqrt{1-\kappa^{2}}}{2(1-\kappa^{2})}. \tag{25}\]  
Next, if we focus on mode A in the received quantum states \(\Psi^{(0)}_{\rm SA}\) and \(\Psi^{(1)}_{\rm SA}\), we can see that the mode is constructed using the following four coherent states: \(\{|-\sqrt{R_{1}}\alpha\rangle_{\rm S},|-\sqrt{R_{0}}\alpha\rangle_{\rm S},\)\(|\sqrt{R_{0}}\alpha\rangle_{\rm S},\)\(|\sqrt{R_{1}}\alpha\rangle_{\rm S}\}(=:\{|\alpha_{0}\rangle,|\alpha_{1}\rangle,|\alpha_{2} \rangle,|\alpha_{3}\rangle\})\). In the four-dimensional subspace of a Hilbert space that is spanned by these coherent states, we also consider the measurement state of the SRM to be an orthonormal basis. The representation of a four-dimensional vector of these coherent states in the four-dimensional subspace can be obtained immediately from the square root of their Gram matrix. In general, the Gram matrix for \(\{|\alpha_{0}\rangle,|\alpha_{1}\rangle,|\alpha_{2}\rangle,|\alpha_{3}\rangle\}\) is constructed as follows [28]:  
\[\left[ \begin{array}{cccc}\langle\alpha_{0}|\alpha_{0}\rangle&\langle\alpha_ {0}|\alpha_{1}\rangle&\langle\alpha_{0}|\alpha_{2}\rangle&\langle\alpha_{0}| \alpha_{3}\rangle\\ \langle\alpha_{1}|\alpha_{0}\rangle&\langle\alpha_{1}|\alpha_{1}\rangle& \langle\alpha_{1}|\alpha_{2}\rangle&\langle\alpha_{1}|\alpha_{3}\rangle\\ \langle\alpha_{2}|\alpha_{0}\rangle&\langle\alpha_{2}|\alpha_{1}\rangle& \langle\alpha_{2}|\alpha_{2}\rangle&\langle\alpha_{2}|\alpha_{3}\rangle\\ \langle\alpha_{3}|\alpha_{0}\rangle&\langle\alpha_{3}|\alpha_{1}\rangle& \langle\alpha_{3}|\alpha_{2}\rangle&\langle\alpha_{3}|\alpha_{3}\rangle\\ \end{array} \right]=\left[ \begin{array}{cccc}1&b&c&d\\ b&1&a&c\\ c&a&1&b\\ d&c&b&1\end{array} \right], \tag{26}\]  
where  
\[a={\rm e}^{-2R_{0}|a|^{2}},\quad b={\rm e}^{\sqrt{R_{0}R_{1}}|a|^{2}-\frac{1}{2 }|a|^{2}(R_{0}+R_{1})},\]\[c=\mathrm{e}^{-\sqrt{R_{0}R_{1}}|\alpha|^{2}-\frac{1}{2}|\alpha|^{2}(R_{0}+R_{1})}, \quad d=\mathrm{e}^{-2R_{1}|\alpha|^{2}}. \tag{27}\]  
However, because the representation of the square root of the Gram matrix is complicated, the method described in references [29; 30; 31] is applied in this paper.