Strict hierarchy of optimal strategies for global estimations: Linking global estimations with local ones

A crucial yet challenging issue in quantum metrology is to ascertain the ultimate precision achievable in estimation strategies. While there are two paradigms of estimations, local and global, current research is largely confined to local estimations, which are useful once the parameter of interest is approximately known. In this Letter we target a paradigm shift towards global estimations, which can operate reliably even with a few measurement data and no substantial prior knowledge about the parameter. The key innovation here is to develop a technique, dubbed virtual imaginary-time evolution, which establishes an equality between the information gained in a global estimation and the quantum Fisher information for a virtual local estimation. This offers an intriguing pathway to surmount challenges in the realm of global estimations by leveraging powerful tools tailored for local estimations. We explore our technique to reveal a strict hierarchy of achievable precision for different global estimation strategies and uncover unexpected results contrary to conventional wisdom in local estimations.