Maximum-likelihood estimation of the density matrix

We present a universal technique for quantum-state estimation based on the maximum-likelihood method. This approach provides a positive-definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multi- mode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementa- tion of the method is based on a particular form of the Gauss decomposition for positive-definite Hermitian matrices.