This is a question about indirect quantum measurement, involving an observable of an object of interest and a probe that is used to measure that observable. In this experiment, an observable of the probe (for example the y component of probe's momentum) becomes entangled with the observable (for example charge) of the object of interest, at time t0. Between t0 and t1, the probe is in transit to the detector. At t1, the probe's momentum is measured, which results in a "state reduction" of the wave function, or density matrix, of the charge of the object of interest. By state reduction, I mean a reduction in size of the continuous range of charge states that have non-zero probabilities.
My question is: At t1, when the probe is measured, does the wave function, or density matrix, of the charge get "reduced" at that time? Or, at t1, when the probe is measured, does the wave function, or density matrix, of the charge get reduced retroactively back in time, at t0, when the entanglement occurred and the probe picked up the info it was carrying about the charge of the object of interest? (I imagine the former, but just checking).
In any case, maybe there have been some experiments to indicate when state reduction occurs for an indirect, probe-based, measurement. If not, I am curious about what you think that your favorite interpretation would say about, once the probe is measured, whether or not the state reduction of the observable of interest that the probe is entangled with takes place 1) retroactively, back in time, at the time of entanglement of the observable of interest with the probe 2) at the time of the measurement of the probe, or 3) at some other time.