Optimum Estimation in Quantum Channels by the Generalized Heisenberg Inequality Method

The optimization of indirect quantum measurement at the output of a quantum channel is investigated for a quadratic figure of merit. A lower bound compatible with the Heisenberg uncertainty principle is found for the estimation risk. This lower bound is used in the Gaussian case to establish the linearity of optimum estimation and to give a specific description of an appropriate optimum indirect measurement procedure.