# Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States
## 4 Quantum Statistical Model with Outliers
### Quantum Statistical Model with Outlier Quantum States  
We assume that the true quantum state is characterized by a \(d\)-parameter \(\theta=(\theta_{1},\theta_{2},\ldots,\theta_{d})\in\Theta\). The ideal quantum statistical model is given by the family of states:  
\[M=\{\rho_{\theta}|\theta\in\Theta\}. \tag{15}\]  
Let \(\mathcal{G}_{\text{outlier}}\) be a set of possible outlier quantum states. For example, the set \(\mathcal{G}_{\text{outlier}}\) consists of \(L\) elements as:  
\[\mathcal{G}_{\text{outlier}}=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{L}\}. \tag{16}\]  
The case of a continuous set of outlier quantum states can be defined similarly. Following the same philosophy as the classical contaminated model (12), we define a quantum contaminated model by:  
\[\rho_{\theta}^{\varepsilon}=(1-\varepsilon)\rho_{\theta}+\varepsilon\sigma \quad(\sigma\in\mathcal{G}_{\text{outlier}}). \tag{17}\]  
Here, the contamination parameter \(\varepsilon\in[0,1)\) represents the strength of contamination. Importantly, we do not have precise knowledge on the values \(\varepsilon\) or \(\sigma\). This is in contrast to the conventional noise model in the quantum estimation theory, where we assume a parametric family of noise models. Without knowing a specific form of noisy quantum states, we cannot apply the standard methodology of parametric inference to infer the parameters of interest. The simplest case is when there is one particular outlier quantum state \(\sigma\), which may or may not be known to us. In this situation, we have \(\rho_{\theta}^{\varepsilon}=(1-\varepsilon)\rho_{\theta}+\varepsilon\sigma\). When the outlier quantum states exhibit a probabilistic structure, the second term in Equation (17) is expressed as a convex mixture of outlier states. Hence, we can express the above model (17) as:  
\[\rho_{\theta}^{\varepsilon}=(1-\varepsilon)\rho_{\theta}+\varepsilon\int p(s) \sigma(s)ds, \tag{18}\]  
where \(p(s)\) is a probability density function for the occurrence of the parametric family of outlier states \(\{\sigma(s)\}\), and \(s\) is in general a vector-valued quantity. A few remarks about our setting are as follows. First, the question of the optimal estimator. It is in general a difficult task to derive an optimal estimator \(\hat{\theta}\) for our model (17). This is because the MLE, which is asymptotically optimal, is no longer optimal in the presence of unknown outlier quantum states. Second, an additional complication comes in the quantum case due to the measurement degree of freedom. In the quantum estimation theory, we can derive an optimal measurement strategy to extract the maximum information about the parameter \(\theta\) from the ideal model \(M\). However, this optimal measurement is no longer optimal for the quantum contaminated model (17). In fact, it is almost impossible to identify the optimal measurement, when the set \(\mathcal{G}_{\text{outlier}}\) is not fixed but has uncertain elements. The purpose of robust statistics is to estimate the parameter of interest \(\theta\) in the presence of nuisance parameter \(\varepsilon\) and unknown outlier quantum states. With this in mind, we shall not explore an optimal estimation strategy in this paper, but we fix a good measurement for the ideal state. We then apply the method of the M-estimator to make robust estimates on \(\theta\). Third, the model (18) can be understood as a quantum channel (a CP-TP map). This quantum noise model acts on the ideal state \(\rho_{\theta}\). The simplest instance of a single outlier state is expressed as \(\rho_{\theta}^{\varepsilon}=(1-\varepsilon)\rho_{\theta}+\varepsilon\sigma\). Note that this quantum channel is not unital in general. Although this class of non-unital maps is well studied, the more general forms (17) or (18) are not explored in view of robust statistics to the best of our knowledge. Last, measurement outcomes. A measurement is described by a positive operator-valued measure (POVM). Let \(\Pi_{x}\geq 0\) (\(x\in\mathcal{X}\)) be elements of the POVM such that(the identity operator).