Multiparameter quantum estimation under dephasing noise

The simultaneous quantum estimation of multiple parameters has recently become an essential aspect of quantum metrology. Although the ultimate sensitivity of a multiparameter quantum estimation in a noiseless environment can overcome the standard quantum limit that every classical sensor is bounded by, it is unclear whether a quantum sensor has an advantage over a classical sensor under a realistic level of noise. In this study, we present the framework of a simultaneous estimation of multiple parameters using quantum sensors under a specific noisy environment. Three components of an external magnetic field are estimated, and we consider the dephasing noise. We show that there is an optimal sensing time in a time-inhomogeneous noisy environment and that its sensitivity can overcome the standard quantum limit.