Quantum metrology in coarsened measurement reference

We investigate the role of coarsened measurement reference, which originates from the coarsened reference time and basis, in quantum metrology. When the measurement is based on one common reference basis, the disadvantage of coarsened measurement can be removed by symmetry. Owing to the coarsened reference basis, the entangled state cannot perform better than the product state for a large number of probe particles in estimating the phase. Given a finite uncertainty of the coarsened reference basis, the optimal number of probe particles is obtained. Finally, we prove that the maximally entangled state always achieves better frequency precision in the case of non-Markovian dephasing than that in the case of Markovian dephasing. The product state is more resistant to the interference of the coarsened reference time than the entangled state.

Figure 1

Precision of frequency $\delta {\omega }^2$ as a function of the uncertainty of reference time $\delta$. Curve 1 represents the precision in the case of Markovian dephasing with the initial product state. Curve 2 represents the case of non-Markovian dephasing with the maximally entangled state. Curve 3 represents ${10}^{-8}$ times the precision in the case of Markovian dephasing with the maximally entangled state, where the scaling factor of ${10}^{-8}$ is used to plot four curves in one diagram because the precision in this case increases considerably faster than in the other cases with the given parameters for the current scope of $\delta$. Curve 4 represents the precision in the case of non-Markovian dephasing with the initial product state. When the uncertainty $\delta$ is larger than a certain value, curve 2 (curve 3) exhibits higher precision than curve 4 (curve 1). The following parameters are used here: $n={10}^4,\gamma \left(0\right)=1,T=1$.