# Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States
## 1 Introduction  
One of the challenges in developing quantum information technologies is to suppress uncontrollable elements in both classical and quantum devices. This has been an active research subject called state preparation and measurement (SPAM) error [1; 2; 3]. For example, imperfection at the quantum state preparation stage generates unwarranted outlier quantum states at random. Under this circumstance, it is impossible to predict precisely when these outlier quantum states are generated. The resulting quantum state is represented by a convex mixture of the desired quantum state and the outlier quantum states. Thus, the actual model is contaminated by outlier quantum states. No matter how small the occurrence of these outlier quantum states, they affect statistics of measurement outcomes. In this paper, we develop a statistical framework to handle certain types of SPAM errors by applying robust statistics [4; 5; 6; 7; 8]. Data due to undesired quantum states are called _outliers_ in statistics, and they typically lie outside the range of trusted data. The traditional working rule to remove outliers in practice is the so-called 3-\(\sigma\) rule. In short, all data that are 3-\(\sigma\) away from the sample mean are discarded. This is intimately related to the tradition of 3-\(\sigma\) confidence in statistics. However, there is no justification for such heuristic and subjective data processing from the statistical point of view. The current status in the community is in fact that we should not rely on the 3-\(\sigma\) confidence [9; 10; 11]. Robust statistics is a branch of statistics and has been one of the proper tools to remove outliers systematically [4; 5; 6; 7; 8]. Recent success of robust statistics in machine learning is another justification to use it rather than the 3-\(\sigma\) rule. The problem of handling outlier quantum states is a practical and important issue in any quantum communication protocols. Yet, it seems that this problem has not been addressed properly in the framework of robust statistics to the best of our knowledge. The main contribution of this paper is first to present a statistical framework to handle the problem of state estimation in the presence of unknown outlier quantum states. We then demonstrate the usefulness and effectiveness of robust statistics in the quantum case. To this end, we consider a specific problem--phase estimation of coherent states in the presence of outlier quantum states. This problem has many practical applicationsin quantum communication protocols using continuous variables [12; 13; 14]. Measurement data drawn according to this contaminated state contain outliers. We apply two specific M-estimators to make robust estimates from data. We find that a recently proposed robust estimator in [15] based on the generalized divergence performs well. We compare its performance to other methods and evaluate its robustness based on our proposed figure of merit called a \(\varepsilon\)-curve. The system of noisy quantum gaussian states has a long history [16; 17; 18; 19; 20; 21]. Further developments in applications to quantum information processing protocols were studied over the last two decades [12; 13; 14; 20; 21; 22; 23; 24; 25]. We note that our model is described by a different class of noise models studied in the literature. The main difference from the previous studies is that our noise model is not a typical completely positive and trace preserving (CP-TP) map. In particular, the model studied in this paper is not described by a unital CP-TP map, but a mixture of different quantum gaussian states. Another distinction is that we do not need to assume a specific form for outlier quantum states when applying our method to real data. Note that we could apply the conventional parameter estimation method, if we know specific forms of noisy quantum states. For example, we can treat noisy states as nuisance [26; 27]. By contrast, the proposed method of M-estimators can be applied to the occurrence of unknown noisy quantum states.