The paper by Z. Hradil discusses a new algorithm for quantum state estimation based on maximum likelihood estimation. It critiques existing techniques that rely on the inversion of measured data, which often lead to overestimation and do not guarantee the positive definiteness of the reconstructed density matrix. The author highlights the importance of state reconstruction in contemporary quantum theory and reviews various methods, including those from quantum optics and atomic physics. The key issue with existing techniques is that they can only handle probability distributions that match the exact form given by the probability operator measure (POM), which is not always the case in practical measurements due to sampling and counting errors. The proposed algorithm addresses these issues by using maximum likelihood estimation, which maximizes the likelihood functional based on the observed data. This approach ensures that the reconstructed density matrix is positive definite, even when the data is incomplete or noisy. The method is demonstrated through examples, such as reconstructing the wave function after measuring a Hermitian operator with an orthogonal spectrum and estimating the quantum state after multiple detections of coherent states. The paper emphasizes the significance of proper state description in quantum theory and the potential of the maximum likelihood method for accurate quantum state estimation.The paper by Z. Hradil discusses a new algorithm for quantum state estimation based on maximum likelihood estimation. It critiques existing techniques that rely on the inversion of measured data, which often lead to overestimation and do not guarantee the positive definiteness of the reconstructed density matrix. The author highlights the importance of state reconstruction in contemporary quantum theory and reviews various methods, including those from quantum optics and atomic physics. The key issue with existing techniques is that they can only handle probability distributions that match the exact form given by the probability operator measure (POM), which is not always the case in practical measurements due to sampling and counting errors. The proposed algorithm addresses these issues by using maximum likelihood estimation, which maximizes the likelihood functional based on the observed data. This approach ensures that the reconstructed density matrix is positive definite, even when the data is incomplete or noisy. The method is demonstrated through examples, such as reconstructing the wave function after measuring a Hermitian operator with an orthogonal spectrum and estimating the quantum state after multiple detections of coherent states. The paper emphasizes the significance of proper state description in quantum theory and the potential of the maximum likelihood method for accurate quantum state estimation.