Quantum metrology in open quantum systems

Estimation of parameters is a pivotal task throughout science and technology. Here, we propose general formulations for open system quantum metrology and particularly obtain two dynamical and a kinematic bounds for the quantum Fisher information. The dynamical bounds are obtained by vectorization method which leads to a dissipative Cramer-Rao bound and by extending the definition of symmetric logarithmic derivative to non-Hermitian domain. The kinematic bound can be obtained for a state with a given convex decomposition, which has two parts: a classical part associated with the Fisher information of the probability distribution of the convex decomposition, and a quantum part given by the average quantum Fisher information of the set of states in this decomposition. When the evolution is given by a quantum channel, using a non-Hermitian symmetric logarithmic derivative in the quantum part leads to the ultimate precision limit for noisy quantum metrology.