Ultimate precision of joint parameter estimation under noisy Gaussian environment

Abstract

The major problem of multiparameter quantum estimation theory is to find an ultimate measurement scheme to go beyond the standard quantum limits that each quasi-classical estimation measurement is limited by. Although, in some specifics quantum protocols without environmental noise, the ultimate sensitivity of a multiparameter quantum estimation can beat the standard quantum limit. However, the presence of noise imposes limitations on the enhancement of precision due to the inevitable existence of environmental fluctuations. Here, we address the motivation behind the usage of Gaussian quantum resources and their advantages in reaching the standard quantum limits under realistic noise. In this context, our work aims to explore the ultimate limits of precision for the simultaneous estimation of a pair of parameters that characterize the displacement channel acting on Gaussian probes and subjected to open dynamics. More precisely, we focus on a general two-mode mixed squeezed displaced thermal state, after reducing it to various Gaussian probes states, like; a two-mode pure squeezed vacuum, two-mode pure displaced vacuum, two-mode mixed displaced thermal, two-mode mixed squeezed thermal. To study the ultimate estimation precision, we evaluate the upper and bottom bound of HCRB in various cases. We find that when the entangled states, two-mode pure squeezed vacuum and two-mode mixed squeezed thermal, are employed as probes states, the upper and bottom bound of HCRB beats the standard quantum limit in the presence of a noisy environment.