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. A capture from this paper

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


By "Copenhagen interpretation", I assume that you mean the interpretation with instantaneous "collapse" one usually encounters in an introductory quantum theory course. Such collapse is a useful rule to do calculation but it is only a fiction. What typically happens is that the quantum system is correlated with the macroscopic measurement device and other environmental degrees of freedom that we practically can't and don't keep track of. This process called decoherence, which is not instantaneous but can be very fast, creates the appearance of irreversible collapse while the global time evolution of the system + device + environment is still governed by the Schrödinger equation. Decoherence is a part of quantum theory. No new interpretation is introduced. (If you want to learn more about decoherence, you can learn about density operators first http://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf and then move on to https://arxiv.org/abs/quant-ph/0612118)

What this means for your question is that the process of measurement for all practical purposes can be described by the Schrödinger equation on a larger system and that is exactly what your equation

$$ \exp\left( -i \int_{t_i}^{t_f} dt H \right) |\psi_{in}\rangle |\phi_{in}\rangle $$ describes. The weak measurement doesn't have to "finish" before we are able to do something else with the state. You don't need a new rule to perform a measurement during another measurement.

After the question has been edited and clarified:

In the reference you have added (http://arxiv.org/abs/1109.6315) $t_m$ is when a strong measurement is performed on the device. If both $ t_s, t_m \ge t_f$ (i.e. the weak measurement has already finished), there is no difference whether a measurement is made on the system (at time $ t_s $) or on the device first because the two measurements commute. (Again, no new interpretation is needed.) Now specific to the weak value analysis is that you only care about a subensemble corresponding to a certain post-selected system state $ | \psi_f \rangle $. If $ t_m \ge t_s $, at $ t_m $ you already have the information of which subensemble you are supposed to post-select. If $ t_m < t_s $, at $ t_m $ you have to wait until $ t_s $ to learn the subensemble that you are supposed to post-select. There is no physical difference. The only difference is in how to post-process the data.

What selecting a subensemble means explicitly

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 $|\Psi_f\rangle$...

You seem to think that a subsystem of a pure entangled system is in a definite superposition state and are confused when trying to apply collapse to this superposition state. So let's make this clear.

The situation at hand is that we have two subsystems, the "system" $s$ and the "measurement device" $m$, evolving unitarily (i.e. by a Hamiltonian). The combined system started in a product pure state, so the final state is pure but entangled. This means that the complete statistical description of $s$ or $m$ alone is given by mixed density operator $\rho $; the expectation value of any observable $A$ (strong measurement) can be calculated from $\rho$ by the Born rule $\text{Tr}(\rho A)$. Any measurement statistics for $s$ is determined by the density operator $\rho_s$ and $\rho_s$ alone. It is not affected by $\rho_m$ and measurements on $m$.

We then made measurements $s$ and $m$ separately at time $t_s$ and $t_m$ respectively. (Yes, we are making strong measurement on a "measurement device." The "measurement device" is treated as a quantum system and have its own observables. To read out the desired result we have to measurement an observable of $m$ whose values are correlated with the values of an observable of $s$.)

(If you want more elaboration on the above two paragraphs, you might want to look at the principle of deferred measurement and the principle of implicit measurement in Nielsen & Chuang pp. 186-187.)

Now, here is what selecting a subensemble means. A system described by a mixed density operator can be interpreted as being in an ensemble of different pure states $|\psi_i \rangle $, each occurs with probability $p_i$. Nevertheless $\rho $ is not the same as a classical ensemble because the ensemble is not unique; there is infinitely many ensembles that can be given to a particular $\rho$. Choosing an observable and applying collapse is picking a preferred ensemble interpretation of $\rho $ thus making it a classical ensemble. (Note that the collapse is not applied to a superposition of $|\psi_i \rangle $) Then, only by conditioning on picking a subensemble $|\psi_i \rangle $ for a particular $i$, we are able to say that the combined system is in a product state $|\psi_i \rangle |\phi \rangle $ for some state $|\phi \rangle $ of $m$.

To recapitulate, after $t_f$, no matter if $t_s>t_m$ or $t_s \le t_m$, we can pretend that $s$ is always in a classical ensemble chosen by the observable of the strong measurement on $s$. Then post-selection is just a classical data processing to which no postulate of collapse enters.


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