A new algorithm for quantum state estimation based on maximum likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix. State reconstruction is a key problem in quantum theory, aiming to determine the maximum amount of information about the system's quantum state. Theoretical predictions and experimental realizations have led to various improvements and new techniques for state reconstruction. Homodyne detection of quadrature operators is used to measure the Wigner function and other quasiprobabilities. Techniques similar to quantum state reconstruction have been used in atomic physics for quantum endoscopy. However, existing reconstruction techniques are improper for quantum theory as they may not guarantee the positive definiteness of the reconstructed density matrix. The proposed algorithm uses maximum likelihood estimation to determine the state that maximizes the likelihood functional. This method is more robust and avoids the issues of positive definiteness. The algorithm is based on the inequality between geometric and arithmetic averages of non-negative numbers. The maximum likelihood estimation is used to determine the desired state, and the uncertainty of such a quantum state estimation is characterized by the likelihood functional. The method is illustrated with examples, showing how it can be applied to reconstruct quantum states from measured data. The proposed method is more accurate and reliable than existing techniques, especially in cases where the data is underdetermined. The method is also applicable to large data sets and realistic measurements. The algorithm is based on the principle of maximum likelihood and is more suitable for quantum state reconstruction than existing techniques. The method is also applicable to the case of multiple detections and can be used to reconstruct the density matrix of a quantum state. The proposed method is more accurate and reliable than existing techniques, especially in cases where the data is underdetermined. The method is also applicable to large data sets and realistic measurements.A new algorithm for quantum state estimation based on maximum likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix. State reconstruction is a key problem in quantum theory, aiming to determine the maximum amount of information about the system's quantum state. Theoretical predictions and experimental realizations have led to various improvements and new techniques for state reconstruction. Homodyne detection of quadrature operators is used to measure the Wigner function and other quasiprobabilities. Techniques similar to quantum state reconstruction have been used in atomic physics for quantum endoscopy. However, existing reconstruction techniques are improper for quantum theory as they may not guarantee the positive definiteness of the reconstructed density matrix. The proposed algorithm uses maximum likelihood estimation to determine the state that maximizes the likelihood functional. This method is more robust and avoids the issues of positive definiteness. The algorithm is based on the inequality between geometric and arithmetic averages of non-negative numbers. The maximum likelihood estimation is used to determine the desired state, and the uncertainty of such a quantum state estimation is characterized by the likelihood functional. The method is illustrated with examples, showing how it can be applied to reconstruct quantum states from measured data. The proposed method is more accurate and reliable than existing techniques, especially in cases where the data is underdetermined. The method is also applicable to large data sets and realistic measurements. The algorithm is based on the principle of maximum likelihood and is more suitable for quantum state reconstruction than existing techniques. The method is also applicable to the case of multiple detections and can be used to reconstruct the density matrix of a quantum state. The proposed method is more accurate and reliable than existing techniques, especially in cases where the data is underdetermined. The method is also applicable to large data sets and realistic measurements.