# Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States
## 2 Preliminaries
### M-estimator  
An M-estimator is a generalization of the maximum likelihood estimator (MLE) and is defined as follows. Consider a datum \(\mathbf{x}=\{x_{1},\ldots,x_{n}\}\) of the sample size \(n\), which is identically and independently distributed (i.i.d.) according to one-parameter family of probability density functions \(f(x|\theta)\). The MLE to estimate \(\theta\) is defined by:  
\[\hat{\theta}_{\text{MLE}}=\arg\max_{\theta}\sum_{i=1}^{n}\log f\left(x_{i}| \theta\right). \tag{1}\]  
The MLE needs to be a stationary point of the logarithmic likelihood equation:  
\[\sum_{i=1}^{n}\frac{d}{d\theta}\log f(x_{i}|\theta)=0. \tag{2}\]The basic philosophy behind the M-estimator is to generalize this equation by:  
\[\sum_{i=1}^{n}\psi\left(x_{i}|\theta\right)=0, \tag{3}\]  
where \(\psi(x|\theta)\) can be an arbitrary function as long as it satisfies certain conditions [8]. An estimator, which is defined as the above generalized stationary condition (3), is called an _M-estimator_. Equation (3) is called an _M-equation_ in robust statistics. Clearly, the M-estimator depends on a choice of \(\psi\) function. The choice \(\psi(x|\theta)=\frac{d}{d\theta}\log f(x|\theta)\) corresponds to the familiar MLE.  
In the following discussion, we consider estimation of the expectation value \(\mu\) of the model for simplicity. We assume that the true probability density function is a function of \(f(x-\mu)\) and is symmetric at the origin, i.e., \(f(-x)=f(x)\). A typical example of this kind is the normal distribution. Note that it is easy to generalize our setting to an arbitrary location model [8]. Under this assumption, an M-equation to infer the parameter \(\mu\) is a function of \(x-\mu\), and hence we have:  
\[\sum_{i=1}^{n}\psi\left(x_{i}-\mu\right)=0.\]  
The zoology of M-estimators has been studied in robust statistics, see for example studies [5; 7; 8]. For our purpose, we consider two specific M-estimators; bisquare and gamma M-estimators. The \(\psi\) function for the bisquare M-estimator (also known as Tukey's estimator) is given by:  
\[\psi_{\text{bi}}(x-\mu)=\begin{cases}\left(x-\mu\right)\left\{1-\left(\frac{x -\mu}{c}\right)^{2}\right\}^{2}&\quad\left(\left|x-\mu\right|\leq c\right)\\ 0&\quad\left(\left|x-\mu\right|>c\right)\end{cases}, \tag{4}\]  
where \(c\) is a tuning parameter. The basic property of a bisquare M-estimator is to suppress contribution from data that are far away from the true parameter \(\mu\). Clearly, the \(\psi_{\text{bi}}\) function vanishes at \(\left|x-\mu\right|=c\) smoothly.  
In [15], an M-estimator was proposed based on the gamma divergence, which is a generalization of the Kullback-Leibler divergence (also called the relative entropy). Its performance was demonstrated to be more robust than the traditional M-estimators. When the true probability density function obeys the normal distribution, the \(\psi\) function is defined by:  
\[\psi_{\text{gam}}(x-\mu)=\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left\{- \frac{1}{2\sigma^{2}}(x-\mu)^{2}\right\}\right]^{\gamma}(x-\mu), \tag{5}\]  
where \(\gamma>0\) is a tuning parameter appearing in the power of the normal distribution \(\mathcal{N}(\mu,\sigma)\) (\(\mu\) is the expectation value, and \(\sigma\) is the standard deviation). Another tuning parameter, the standard deviation \(\sigma\), needs to be properly chosen as well. This will be discussed later. \(\psi_{\text{gam}}\) approaches 0 as \(\left|x\right|\rightarrow\infty\), and its convergence is exponential.