3 Maximum-Likelihood Methodsin Quantum Mechanics

Abstract

Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its benefits are apparent particularly in the quantum domain. For a given set of measurements, the most likely state is estimated. Though this problem is nonlinear, it can be effectively solved by an iterative algorithm exploiting the convexity of the likelihood functional and the manifold of density matrices. This formulation fully replaces the inverse Radon transformation routinely used for tomographic reconstructions. Moreover, it provides the most efficient estima- tion strategy saturating the Cramer-Rao lower bound asymptotically. In this sense it exploits the acquired data set in the optimal way and minimizes the artifacts associated with the reconstruction procedure. The idea of maximum likelihood reconstruction is further extended to the estimation of quantum processes, measurements, and discrimination between quantum states. This technique is well suited for future applications in quantum information science due to its ability to quantify very subtle and fragile quantum effects. 3.1