# Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States
## 4 Quantum Statistical Model with Outliers
### Quantum Gaussian State with Outliers  
We will now consider a concrete example for a quantum contaminated model for a coherent state. In our model, the ideal state is an unknown coherent state, and outlier quantum states are given by the thermal gaussian states. A motivation for considering this model is that there occur noisy thermal states with some frequency at the state preparation stage. This could be treated as the quantum contaminated model of the form (17). The standard definition of the coherent state, which is characterized by a complex number \(\alpha\in\mathbb{C}\), is:  
\[|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{j=0}^{\infty}\frac{\alpha^{j} }{\sqrt{j!}}|j\rangle, \tag{20}\]  
where \(|j\rangle\) is the Fock state of \(j\) photons. We can also define the coherent state by a unitary shift as follows. Let \(\hat{a}^{\dagger}\) and \(\hat{a}\) be the creation and annihilation operators, respectively, and define the shift operator by:  
\[U(\alpha)=e^{a\alpha^{\dagger}-\alpha^{*}a}\quad(\alpha\in\mathbb{C}). \tag{21}\]  
The coherent state (20) is then expressed as a shifted state by \(\alpha\) from the vacuum state \(|0\rangle\):  
\[|\alpha\rangle=U(\alpha)|0\rangle. \tag{22}\]  
Next, we consider thermal states as outlier quantum states. The thermal state at the inverse temperature \(\beta=\frac{1}{k_{B}T}\) (\(k_{B}\): the Boltzmann constant) is defined by:  
\[\rho_{\beta}^{\text{thermal}}=C_{\beta}\sum_{j=0}^{\infty}e^{-\beta j}|j \rangle\langle j|, \tag{23}\]  
where \(C_{\beta}=1-e^{-\beta}\) and we set the unit energy for the single-mode field without loss of generality, i.e., \(\hbar\omega=1\). Note that the zero temperature limit, \(\beta\rightarrow\infty\), converges to the vacuum state:  
\[\lim_{\beta\rightarrow\infty}\rho_{\beta}^{\text{thermal}}=|0\rangle\langle 0|. \tag{24}\]  
A quantum gaussian shift state at the inverse temperature \(\beta\) is defined by:  
\[\rho_{\beta,\alpha}^{\text{CS}}=U(\alpha)\rho_{\beta}U(\alpha)^{\dagger}, \tag{25}\]  
which is also expressed in the integral form as:  
\[\rho_{\beta,\alpha}^{\text{CS}}=\frac{1}{\pi}\int e^{-\frac{|z-a^{2}}{2\hbar^ {2}}}|z\rangle\langle z|d^{2}z, \tag{26}\]  
with \(2\kappa^{2}=\left(e^{\beta}-1\right)^{-1}\). It is straightforward to see that the zero temperature limit of the quantum gaussian shift state is a coherent state, i.e.,\[\lim_{\beta\to\infty}\rho^{\text{GS}}_{\beta,\alpha}=|\alpha\rangle\langle\alpha|. \tag{27}\]  
The parameter \(\kappa\) (\(\kappa\geq 0\)) represents a dispersion of thermal spreads of coherent states as seen from Equation (26). In the following we mainly use \(\kappa\) and denote the quantum gaussian state state simply by \(\rho_{\alpha,\kappa}\), since there is a one-to-one correspondence between \(\beta\) and \(\kappa\). Let \(\alpha\in\mathbb{C}\) be the complex parameter of interest, and consider quantum outlier states described by the quantum gaussian shift state \(\rho_{z,\kappa}\). With these definitions, our quantum contaminated model is defined by:  
\[\rho^{\varepsilon}_{\alpha}=(1-\varepsilon)|\alpha\rangle\langle\alpha|+ \varepsilon\int p(z,\kappa)\rho_{z,\kappa}d^{2}z\,d\kappa. \tag{28}\]  
In this expression, \(p(z,\kappa)\) describes the probability density function of the outlier quantum state \(\rho_{z,\kappa}\). From this expression, we see that the quantum contaminated model is a mixed quantum gaussian models in our setting. In practice, we take the independent density function as \(p(z,\kappa)=p_{1}(z_{\text{R}})p_{2}(z_{\text{I}})p_{3}(\kappa)\), where \(z_{\text{R}}=\text{Re}\,z\) and \(z_{\text{I}}=\text{Im}\,z\) denote the real and imaginary parts of \(z\), respectively. To implement our numerical simulation of the above model, we consider two different settings as follows.