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Due to quantum randomicity, it's impossible to determine the future knowing the present.

But is it possible to determine the past knowing the present?

As far as I understand, it is impossible because of symmetry of quantum laws regarding time. But I am not sure.

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Not sure what you mean by quantum randomicity, but I'll give your question a stab.

It's impossible to determine the future knowing the present in the sense that we can't know for certain the result of all possible measurements in the future (the typical example being that we can't know the momentum and position of a particle simultaneously). But in that sense, we also don't know the present!

All the possible information one can have about a system is in the system's wavefunction. If we know this in the present, we will be able to measure observables and know their values (either with certainty, or with some probability). Further, if we know the Hamiltonian of our system, we can also predict what the wavefunction will be in the future. So if we know the present, we can indeed determine the future.

Something interesting is that to know the present, you have to make a measurement. In making this measurement, you collapsed the wavefunction of the system, and actually erased past information! So interestingly enough, if you make a measurement to determine the present state of a system, you can determine its future state but lost most information about its past state.

As a caveat - what I said is from the standpoint of the Copenhagen interpretation of QM. In this interpretation, there is no symmetry in quantum laws regarding time, because a measurement will break this symmetry. [From another perspective, the observer itself is a quantum system, so it can be argued that there is still time symmetry.]

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  • $\begingroup$ How so? If we erase the past information measuring the current, then archeology would not work. We can determine the past by archeology but can't determine the future this way. It looks like the opposite to what you say! Paradox. Explain. Well, you say we erase a part of the past information. How much remains? $\endgroup$
    – porton
    Commented Jun 22, 2020 at 2:02
  • $\begingroup$ I mean that accodingly to the common sense it is easier to determine the past knowing the present than to determine the future knowing the present. How the physics may say almost the opposite? $\endgroup$
    – porton
    Commented Jun 22, 2020 at 2:04
  • $\begingroup$ If I am understandong well this answer, it is impossible to store any information about a system elsewhere? I would have thought we could store the result of past experiments in a hard disk and then use this information to help to get insights on the past of the system. $\endgroup$
    – untreated_paramediensis_karnik
    Commented Jun 22, 2020 at 5:59
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The future and the past are equally determined by the present, because the laws of QM are time-symmetric. However, neither the future nor the past can be calculated from measurements in the present any more precisely than allowed by Heisenberg's uncertainty principle.

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