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The major problem of multiparameter quantum estimation theory is to find anultimate measurement scheme to go beyond the standard quantum limits that eachquasi-classical estimation measurement is limited by. Although, in somespecifics quantum protocols without environmental noise, the ultimatesensitivity of a multiparameter quantum estimation can beat the standardquantum limit. However, the presence of noise imposes limitations on theenhancement of precision due to the inevitable existence of environmentalfluctuations. Here, we address the motivation behind the usage of Gaussianquantum resources and their advantages in reaching the standard quantum limitsunder realistic noise. In this context, our work aims to explore the ultimatelimits of precision for the simultaneous estimation of a pair of parametersthat characterize the displacement channel acting on Gaussian probes andsubjected to open dynamics. More precisely, we focus on a general two-modemixed squeezed displaced thermal state, after reducing it to various Gaussianprobes states, like; a two-mode pure squeezed vacuum, two-mode pure displacedvacuum, two-mode mixed displaced thermal, two-mode mixed squeezed thermal. Tostudy the ultimate estimation precision, we evaluate the upper and bottom boundof HCRB in various cases. We find that when the entangled states, two-mode puresqueezed vacuum and two-mode mixed squeezed thermal, are employed as probesstates, the upper and bottom bound of HCRB beats the standard quantum limit inthe presence of a noisy environment.

Ultimate precision of joint parameter estimation under noisy Gaussian environment