Temperature-induced measurement sensitivity enhancement via imaginary weak values

Abstract: We investigate the potential of weak measurement and post-selection to enhance measurement sensitivity when the initial probe state is mixed. In our framework, the mixedness of the probe's density operator is controlled by temperature. We focus on two key quantities: the signal-to-noise ratio and the quantum Fisher information of the final probe state, evaluated after post-selection is applied on the system. Our analysis employs a rigorous, all-order coupling treatment of measurement, demonstrating that the signal-to-noise ratio can be enhanced in certain scenarios by increasing the temperature. However, this enhancement is fundamentally constrained by the validity conditions of the weak measurement regime. Regarding the quantum Fisher information, we find that for a pure probe state, incorporating post-selection does not improve precision beyond the standard (non-post-selected) strategy when the post-selection probability is accounted for. In contrast, when the initial probe state is mixed, the quantum Fisher information for the probe state after post-selection in the system can surpass that of the standard strategy. Notably, we show that the quantum Fisher information might diverge and grow unboundedly with temperature, illustrating a scenario where thermal noise can, counterintuitively, enhance metrological precision.