Enhancing quantum coherence and quantum Fisher information by quantum partially collapsing measurements

We consider the enhancement effect of quantum partially collapsing measurements, i.e., weak measurement and quantum measurement reversal, on quantum coherence and quantum Fisher information, both of which are transmitted through a spin-chain channel. For the state parameter lying in the region (π/2,π)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\pi /2,\ \pi )$$\end{document}, weak measurement can enhance quantum coherence and quantum Fisher information. For the state parameter lying in the region (0,π/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\ \pi /2)$$\end{document}, quantum coherence and quantum Fisher information can be enhanced by quantum measurement reversal combined with weak measurement. We assume the probabilistic nature of the method should be responsible for the enhancement.