Fig. 1

(a) The weak value in Eq. (11) involves five steps: preparation of \(|\psi \rangle\) (red) at the initial time \(t=0\), unitary evolution \(\hat{U}_{{t_R}}\), weak perturbation linked to \(\hat{{O}}\) (green) at time \(t=t_R\), unitary evolution \(\hat{U}_{{t_L}}\) and strong measurement linked to \(\hat{{F}}\) (violet) at the final time \(t=t_R+t_L\). The weak value can be roughly understood as the property of the system during the intermediate green times. (b) The LHD computed when \(t_L\rightarrow 0\) while \(t_R\) is keep constant. (c) The FDLHD is evaluated at the final time and computed from the difference of the weak value \(W_1\) and \(W_2\), which are computed using two different values of \(t_L\) (\(t_L=0\) and \(t_L>0\)). (d) The RHD computed when \(t_R\rightarrow 0\) while \(t_L\) is keep constant. (e) The FDRHD is evaluated at the initial time and computed from the difference of the weak values \(W_1\) and \(W_2\), which are computed using two different values of \(t_R\) (i.e., \(t_R>0\) and \(t_R=0\)).