Here, I denote the initial states of device and quantum system as $|\Phi_\textrm{in}\rangle$ and $|\Psi_\textrm{in}\rangle$.
The measurement interval is $[t_i,t_f]$, after measurement, the device and quantum system will evolve to
$$\exp\left (-\mathrm i\int_{t_i}^{t_f} \hat{H} \mathrm dt\right)|\Psi_\textrm{in}\rangle|\Phi_\textrm{in}\rangle,$$ where $H$ is the coupling Hamiltonian, and then the device gives the results at time $t_m$( $t_m\geq t_f)$.
We make a postselection( a strong measurement) of the state of the quantum system at $t_s\geq t_f$. We mark the particles in a definite state and look at the weak measurement results of these particles.
The state of the device at $t_m$ is $\langle \Psi_f|\exp\left(-\mathrm i\displaystyle\int_{t_i}^{t_f} \hat{H} \mathrm dt\right)|\Psi_\textrm{in}\rangle|\Phi_\textrm{in}\rangle$.
My problem is, according to Copenhagen interpretation, the above expression is valid only when $t_s<t_m$, but it seems it is valid too when $t_s>t_m$. I wonder what replace Copenhagen interpretation here?
What's more, one of the papers I read indicates that both $t_m<t_s$ and $t_m>t_s$ are right. The only difference is that if $t_s<t_m$, we only have to measurement the ensemble corresponding to the specified state; if $t_s>t_m$, we have to measure the whole ensembles and then select the result corresponding to the ensemble in the specified state.
I want to make my question clearer below:
Denote the state at time $t_m$ of the whole ensemble and of the subensemble as $|\Psi\rangle$ and $|\Psi'\rangle$. Assume when $t_f<t<ts$, the state of the whole ensemble is $|\Psi\rangle=1/2 |\Psi_f\rangle +1/2{|\Psi_f\rangle}^{\perp}$, where $|\Psi_f\rangle$ is the state we want to postselect. If $t_s<t_m$, then after strong measurement, the subensemble collapses to a pure state $|\Psi_f\rangle$, the device at later time $t_m$ can be expressed as $\langle \Psi_f|\exp\left(-\mathrm i\displaystyle\int_{t_i}^{t_f} \hat{H} \mathrm dt\right)|\Psi_\textrm{in}\rangle|\Phi_\textrm{in}\rangle$. If $t_m<t_s$, we measure the whole ensemble and then select the result of the subensemble according to the result of postselection. In this case, the state of the whole ensemble and of the subensemble at $t_m$ should not be influenced by the postselection according to Copenhagen Interpretation. Neither the whole ensemble nor the subensemble is in the state $|\Psi_f\rangle$ at earlier time $t_m$, how could we get the same result as the case $t_s<t_m$ by choosing the weak measurement result of the subensemble?
How can the result from postselection influence the result of subensemble?If before the postselection, none of the member of the ensemble is in state $|\Psi_f\rangle$ , then after postselection and selecting the subensemble which collapse to $|\Psi_f\rangle$ , the result after data processing is different with the result of the case the subensemble is in a definite state $|\Psi_f\rangle$ . Extremely, if at earlier time, the ensemble is a pure ensemble in a state $|\Psi\rangle=a |\Psi_f\rangle +b{|\Psi_f\rangle}^{\perp}$ , then the result after data processing is the same with the raw result